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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2 (view source)

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| <math>\text{Life}\!</math>

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| <math>\text{Sleep}~\!</math>

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A ''subrelation'' of a dyadic relation <math>\underline{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is a dyadic relation <math>\underline{H} = (Y, H) = (H^{(1)}, H^{(2)})</math> that has all of its points and pairs in <math>\underline{G}</math> more precisely, that has all of its points <math>Y \subseteq X</math> and all of its pairs <math>H \subseteq G.</math>

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The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math> In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math> Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>\underline{G}_Y.\!</math>. Using the symbol <math>\bigcap</math> to indicate the intersection of a pair of sets, the construction of <math>\underline{G}_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

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The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math> In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math> Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>\underline{G}_Y.\!</math> Using the symbol <math>\bigcap</math> to indicate the intersection of a pair of sets, the construction of <math>\underline{G}_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

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These definitions for dyadic relations can now be applied in a context where each bit of a sign relation that is being considered satisfies a special set of conditions, namely, if <math>R\!</math> is the relational bit under consideration:

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# Syntactic domain <math>X\!</math> = Sign domain <math>S\!</math> = Interpretant domain <math>I.\!</math>

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# Syntactic domain <math>{X}\!</math> = Sign domain <math>{S}\!</math> = Interpretant domain <math>{I}.\!</math>

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# Connotative component = <math>R_{XX}\!</math> = <math>R_{SI}\!</math> = Equivalence relation <math>E.\!</math>

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# Connotative component = <math>{R_{XX}}\!</math> = <math>{R_{SI}}\!</math> = Equivalence relation <math>{E}.\!</math>

Under these assumptions, and with regard to bits of sign relations that satisfy these conditions, it is useful to consider further selections of a specialized sort, namely, those that keep equivalent signs synonymous.

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For the purposes of this discussion, let it be supposed that each set <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment. In a setting like this it is possible to make a number of useful definitions, to which we now turn.

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The ''negation'' of a sentence <math>s\!</math>, written as <math>^{\backprime\backprime} \, \underline{(} s \underline{)} \, ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \, \operatorname{not}\ s \, ^{\prime\prime}</math>, is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.

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The ''negation'' of a sentence <math>s\!</math>, written as <math>{}^{\backprime\backprime} \, \underline{(} s \underline{)} \, {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \, \operatorname{not}\ s \, {}^{\prime\prime},\!</math> is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.

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The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>^{\backprime\backprime} \, X\!-\!Q \, ^{\prime\prime}</math> and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q\!</math>. When the universe <math>X\!</math> is fixed throughout a given discussion, the complement <math>X\!-\!Q</math> may be denoted either by <math>^{\backprime\backprime} \thicksim \! Q \, ^{\prime\prime}</math> or by <math>^{\backprime\backprime} \, \tilde{Q} \, ^{\prime\prime}</math>. Thus we have the following series of equivalences:

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The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>{}^{\backprime\backprime} \, X\!-\!Q \, {}^{\prime\prime}\!</math> and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q.\!</math> When the universe <math>X\!</math> is fixed throughout a given discussion, the complement <math>X\!-\!Q</math> may be denoted either by <math>{}^{\backprime\backprime} \thicksim \! Q \, {}^{\prime\prime}\!</math> or by <math>{}^{\backprime\backprime} \, \tilde{Q} \, {}^{\prime\prime}.\!</math> Thus we have the following series of equivalences:

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The ''relative complement'' of <math>P\!</math> in <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>^{\backprime\backprime} \, Q\!-\!P \, ^{\prime\prime}</math> and defined as the set of elements in <math>Q\!</math> that do not belong to <math>P,\!</math> that is:

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The ''relative complement'' of <math>P\!</math> in <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, Q\!-\!P \, {}^{\prime\prime}</math> and defined as the set of elements in <math>Q\!</math> that do not belong to <math>P,\!</math> that is:

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The ''intersection'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>^{\backprime\backprime} \, P \cap Q \, ^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to both <math>P\!</math> and <math>Q.\!</math>

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The ''intersection'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, P \cap Q \, {}^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to both <math>P\!</math> and <math>Q.\!</math>

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The ''union'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>^{\backprime\backprime} \, P \cup Q \, ^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to at least one of <math>P\!</math> or <math>Q.\!</math>

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The ''union'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, P \cup Q \, {}^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to at least one of <math>P\!</math> or <math>Q.\!</math>

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The ''symmetric difference'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>^{\backprime\backprime} \, P ~\hat{+}~ Q \, ^{\prime\prime}</math> and is defined as the set of elements in <math>X\!</math> that belong to just one of <math>P\!</math> or <math>Q.\!</math>

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The ''symmetric difference'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, P ~\hat{+}~ Q \, {}^{\prime\prime}\!</math> and is defined as the set of elements in <math>X\!</math> that belong to just one of <math>P\!</math> or <math>Q.\!</math>

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The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a ''sentence'' is, they all rely on the reader's native intuition of what a ''set'' is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance that just about everybody has of the logical connectives ''not'', ''and'', ''or'', as these are expressed in natural language terms.

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The foregoing “definitions” are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a ''sentence'' is, they all rely on the reader's native intuition of what a ''set'' is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance that just about everybody has of the logical connectives ''not'', ''and'', ''or'', as these are expressed in natural language terms.

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As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions. These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, that of seeing informal concepts presented in a formal frame, in a way that demands their increasing clarification. In this style of examination, the frame of the set-builder expression <math>\{ x \in X : \underline{~~~} \}</math> functions like the ''eye of the needle'' through which one is trying to transport a suitably rich import of mathematics.

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As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions. These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, that of seeing informal concepts presented in a formal frame, in a way that demands their increasing clarification. In this style of examination, the frame of the set-builder expression <math>\{ x \in X : \underline{~~~} \}\!</math> functions like the ''eye of the needle'' through which one is trying to transport a suitably rich import of mathematics.

Part the task of the remaining discussion is gradually to formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially formalized conceptions. To this we now turn.

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The ''binary domain'' is the set <math>\mathbb{B} = \{ 0, 1 \}</math> of two algebraic values, whose arithmetic operations obey the rules of <math>\operatorname{GF}(2),</math> the ''galois field'' of exactly two elements, whose addition and multiplication tables are tantamount to addition and multiplication of integers modulo 2.

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The ''binary domain'' is the set <math>{\mathbb{B} = \{ 0, 1 \}}\!</math> of two algebraic values, whose arithmetic operations obey the rules of <math>\operatorname{GF}(2),\!</math> the ''galois field'' of exactly two elements, whose addition and multiplication tables are tantamount to addition and multiplication of integers modulo 2.

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The ''boolean domain'' is the set <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> of two logical values, whose elements are read as ''false'' and ''true'', or as ''falsity'' and ''truth'', respectively.

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The ''boolean domain'' is the set <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}\!</math> of two logical values, whose elements are read as ''false'' and ''true'', or as ''falsity'' and ''truth'', respectively.

At this point, I cannot tell whether the distinction between these two domains is slight or significant, and so this question must evolve its own answer, while I pursue a larger inquiry by means of its hypothesis. The weight of the matter appears to increase as the investigation moves from abstract, algebraic, and formal settings to contexts where logical semantics, natural language syntax, and concrete categories of grammar are compelling considerations. Speaking abstractly and roughly enough, it is often acceptable to identify these two domains, and up until this point there has rarely appeared to be a sufficient reason to keep their concepts separately in mind. The boolean domain <math>\underline\mathbb{B}</math> comes with at least two operations, though often under different names and always included in a number of others, that are analogous to the field operations of the binary domain <math>\mathbb{B},</math> and operations that are isomorphic to the rest of the boolean operations in <math>\underline\mathbb{B}</math> can always be built on the binary basis of <math>\mathbb{B}.</math>

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Of course, as sets of the same cardinality, the domains <math>\mathbb{B}</math> and <math>\underline\mathbb{B}</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction. Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively. The signs <math>^{\backprime\backprime} 0 ^{\prime\prime}</math> and <math>^{\backprime\backprime} 1 ^{\prime\prime},</math> customarily read as nouns but not as sentences, fail to be suitable for this purpose. Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.

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Of course, as sets of the same cardinality, the domains <math>\mathbb{B}\!</math> and <math>\underline\mathbb{B}\!</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction. Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}\!</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively. The signs <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} 1 {}^{\prime\prime},\!</math> customarily read as nouns but not as sentences, fail to be suitable for this purpose. Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.

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The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \lnot x ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math> Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table 8.

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The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>{}^{\backprime\backprime} \underline{(} x \underline{)} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \lnot x {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \operatorname{not}\ x {}^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math> Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table 8.

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It is convenient to transport the product and the sum operations of <math>\mathbb{B}</math> into the logical setting of <math>\underline\mathbb{B},</math> where they can be symbolized by signs of the same character. This yields the following definitions of a ''product'' and a ''sum'' in <math>\underline\mathbb{B}</math> and leads to the following forms of multiplication and addition tables.

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The ''product'' <math>x \cdot y</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given by Table 9. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the product corresponds to the logical operation of ''conjunction'', written <math>^{\backprime\backprime} x \land y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x\!\And\!y ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.</math> In accord with common practice, the multiplication sign is frequently omitted from written expressions of the product.

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The ''product'' <math>x \cdot y</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given by Table 9. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the product corresponds to the logical operation of ''conjunction'', written <math>{}^{\backprime\backprime} x \land y {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} x\!\And\!y {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} x ~\operatorname{and}~ y {}^{\prime\prime}.</math> In accord with common practice, the multiplication sign is frequently omitted from written expressions of the product.

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The ''sum'' <math>x + y\!</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given in Table 10. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the sum corresponds to the logical operation of ''exclusive disjunction'', usually read as <math>^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} ^{\prime\prime}.</math> Depending on the context, other signs and readings that invoke this operation are: <math>^{\backprime\backprime} x \ne y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x \not\Leftrightarrow y ^{\prime\prime},</math> read as <math>^{\backprime\backprime} x ~\text{is not equal to}~ y ^{\prime\prime},</math> <math>^{\backprime\backprime} x ~\text{is not equivalent to}~ y ^{\prime\prime},</math> or <math>^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} ^{\prime\prime}.</math>

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The ''sum'' <math>x + y\!</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given in Table 10. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the sum corresponds to the logical operation of ''exclusive disjunction'', usually read as <math>{}^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} {}^{\prime\prime}.\!</math> Depending on the context, other signs and readings that invoke this operation are: <math>{}^{\backprime\backprime} x \ne y {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} x \not\Leftrightarrow y {}^{\prime\prime},</math> read as <math>{}^{\backprime\backprime} x ~\text{is not equal to}~ y {}^{\prime\prime},</math> <math>{}^{\backprime\backprime} x ~\text{is not equivalent to}~ y {}^{\prime\prime},</math> or <math>{}^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} {}^{\prime\prime}.\!</math>

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<li>

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Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:

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Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}\!</math> that is given by the following formula:

<math>f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y = \underline{1} ~\Leftrightarrow~ x \in Q \}.</math></li>

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The set-builder frame <math>\{ x \in X : \underline{~~~} \}</math> requires a grammatical sentence or a sentential clause to fill in the blank, as with the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> that serves to fill the frame in the initial definition of a logical fiber. And what is a sentence but the expression of a proposition, in other words, the name of an indicator function? As it happens, the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> and the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> represent the very same value to this context, for all <math>x\!</math> in <math>X,\!</math> that is, they will appear equal in their truth or falsity to any reasonable interpreter of signs or sentences in this context, and so either one of them can be tendered for the other, in effect, exchanged for the other, within this context, frame, and reception.

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The set-builder frame <math>\{ x \in X : \underline{~~~} \}\!</math> requires a grammatical sentence or a sentential clause to fill in the blank, as with the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> that serves to fill the frame in the initial definition of a logical fiber. And what is a sentence but the expression of a proposition, in other words, the name of an indicator function? As it happens, the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> and the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> represent the very same value to this context, for all <math>x\!</math> in <math>X,\!</math> that is, they will appear equal in their truth or falsity to any reasonable interpreter of signs or sentences in this context, and so either one of them can be tendered for the other, in effect, exchanged for the other, within this context, frame, and reception.

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The sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> manifestly names the value <math>f(x).\!</math> This is a value that can be seen in many lights. It is, at turns:

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The sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> manifestly names the value <math>f(x).\!</math> This is a value that can be seen in many lights. It is, at turns:

# The value that the proposition <math>f\!</math> has at the point <math>x,\!</math> in other words, the value that <math>f\!</math> bears at the point <math>x\!</math> where <math>f\!</math> is being evaluated, the value that <math>f\!</math> takes on with respect to the argument or the object <math>x\!</math> that the whole proposition is taken to be about.

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# The value that the proposition <math>f\!</math> not only takes up at the point <math>x,\!</math> but that it carries, conveys, transfers, or transports into the setting <math>^{\backprime\backprime} \{ x \in X : \underline{~~~} \} ^{\prime\prime}</math> or into any other context of discourse where <math>f\!</math> is meant to be evaluated.

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# The value that the proposition <math>f\!</math> not only takes up at the point <math>x,\!</math> but that it carries, conveys, transfers, or transports into the setting <math>{}^{\backprime\backprime} \{ x \in X : \underline{~~~} \} {}^{\prime\prime}</math> or into any other context of discourse where <math>f\!</math> is meant to be evaluated.

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# The value that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition <math>f\!</math> and the same object <math>x\!</math> are borne in mind.

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# The value that the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition <math>f\!</math> and the same object <math>x\!</math> are borne in mind.

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# The value that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.

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# The value that the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.

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The sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> indirectly names what the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> more directly names, that is, the value <math>f(x).\!</math> In other words, the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> has the same value to its interpretive context that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> imparts to any comparable context, each by way of its respective evaluation for the same <math>x \in X.</math>

+

The sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> indirectly names what the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> more directly names, that is, the value <math>f(x).\!</math> In other words, the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> has the same value to its interpretive context that the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> imparts to any comparable context, each by way of its respective evaluation for the same <math>x \in X.</math>

What is the relation among connoting, denoting, and ''evaluing'', where the last term is coined to describe all the ways of bearing, conveying, developing, or evolving a value in, to, or into an interpretive context? In other words, when a sign is evaluated to a particular value, one can say that the sign ''evalues'' that value, using the verb in a way that is categorically analogous or grammatically conjugate to the times when one says that a sign ''connotes'' an idea or that a sign ''denotes'' an object. This does little more than provide the discussion with a ''weasel word'', a term that is designed to avoid the main issue, to put off deciding the exact relation between formal signs and formal values, and ultimately to finesse the question about the nature of formal values, the question whether they are more akin to conceptual signs and figurative ideas or to the kinds of literal objects and platonic ideas that are independent of the mind.

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Introducing the realm of ''values'' is a stopgap measure that temporarily permits the discussion to avoid certain singularities in the embedding sign relation, and allowing the process of ''evaluation'' as a compromise mode of signification between connotation and denotation only manages to steer around a topic that eventually has to be mapped in full, but these strategies do allow the discussion to proceed a little further without having to answer questions that are too difficult to be settled fully or even tackled directly at this point. As far as the relations among connoting, denoting, and evaluing are concerned, it is possible that all of these constitute independent dimensions of significance that a sign might be able to enjoy, but since the notion of connotation is already generic enough to contain multitudes of subspecies, I am going to subsume, on a tentative basis, all of the conceivable modes of ''evaluing'' within the broader concept of connotation.

−

With this degree of flexibility in mind, one can say that the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> latently connotes what the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> patently connotes. Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is ''identified by'' the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime},</math> and thus an object that might as well be ''identified with'' the value <math>f(x).\!</math>

+

With this degree of flexibility in mind, one can say that the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> latently connotes what the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> patently connotes. Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is ''identified by'' the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime},</math> and thus an object that might as well be ''identified with'' the value <math>f(x).\!</math>

The upshot of this whole discussion of evaluation is that it allows us to rewrite the definitions of indicator functions and their fibers as follows:

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<li>

−

Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:

+

Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}\!</math> that is given by the following formula:

<math>f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~\Leftrightarrow~ x \in Q \}.</math></li>

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Perhaps this looks like a lot of work for the sake of what seems to be such a trivial form of syntactic transformation, but it is an important step in loosening up the syntactic privileges that are held by the sign of logical equivalence <math>{}^{\backprime\backprime} \Leftrightarrow {}^{\prime\prime},</math> as written between logical sentences, and the sign of equality <math>{}^{\backprime\backprime} = {}^{\prime\prime},</math> as written between their logical values, or else between propositions and their boolean values, respectively. Doing this removes a longstanding but wholly unnecessary conceptual confound between the idea of an ''assertion'' and the notion of an ''equation'', and it allows one to treat logical equality on a par with the other logical operations.

−

As a purely informal aid to interpretation, I frequently use the letters <math>^{\backprime\backprime} p ^{\prime\prime}, ^{\backprime\backprime} q ^{\prime\prime}</math> to denote propositions. This can serve to tip off the reader that a function is intended as the indicator function of a set, and thus it saves us the trouble of declaring the type <math>f : X \to \underline\mathbb{B}</math> each time that a function is introduced as a proposition.

+

As a purely informal aid to interpretation, I frequently use the letters <math>{}^{\backprime\backprime} p {}^{\prime\prime}, {}^{\backprime\backprime} q {}^{\prime\prime}</math> to denote propositions. This can serve to tip off the reader that a function is intended as the indicator function of a set, and thus it saves us the trouble of declaring the type <math>f : X \to \underline\mathbb{B}</math> each time that a function is introduced as a proposition.

Another convention of use in this context is to let underscored letters stand for <math>k\!</math>-tuples, lists, or sequences of objects. Typically, the elements of the <math>k\!</math>-tuple, list, or sequence are all of one type, and the underscored letter is typically the same basic character as the letters that are indexed or subscripted to denote the individual components of the <math>k\!</math>-tuple, list, or sequence. When the dimension of the <math>k\!</math>-tuple, list, or sequence is clear from context, the underscoring may be omitted. For example, the following patterns of construction are very often seen:

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& \text{then} & \underline{f} = (f_1, \ldots, f_k) & : & (X \to Y)^k.

\\

−

\end{array}</math>

+

\end{array}\!</math>

|}

−

There is usually felt to be a slight but significant distinction between a ''membership statement'' of the form <math>^{\backprime\backprime} x \in X \, ^{\prime\prime}</math> and a ''type indication'' of the form <math>^{\backprime\backprime} x : X \, ^{\prime\prime},</math> for instance, as they are used in the examples above. The difference that appears to be perceived in categorical statements, when those of the form <math>^{\backprime\backprime} x \in X \, ^{\prime\prime}</math> and those of the form <math>^{\backprime\backprime} x : X \, ^{\prime\prime}</math> are set in side by side comparisons with each other, is that a multitude of objects can be said to have the same type without having to posit the existence of a set to which they all belong. Without trying to decide whether I share this feeling or even fully understand the distinction in question, I can only try to maintain a style of notation that respects it to some degree. It is conceivable that the question of belonging to a set is rightly regarded as the more serious matter, one that concerns the reality of an object and the substance of a predicate, than the question of falling under a type, that may depend only on the way that a sign is interpreted and the way that information about an object is organized. When it comes to the kinds of hypothetical statements that appear in the present instance, those of the forms <math>^{\backprime\backprime} x \in X ~\Leftrightarrow~ \underline{x} \in \underline{X} \, ^{\prime\prime}</math> and <math>^{\backprime\backprime} x : X ~\Leftrightarrow~ \underline{x} : \underline{X} \, ^{\prime\prime},</math> these are usually read as implying some order of synthetic construction, one whose contingent consequences involve the constitution of a new space to contain the elements being compounded and the recognition of a new type to characterize the elements being moulded, respectively. In these applications, the statement about types is again taken to be less presumptive than the corresponding statement about sets, since the apodosis is intended to do nothing more than abbreviate and summarize what is already stated in the protasis.

+

There is usually felt to be a slight but significant distinction between a ''membership statement'' of the form <math>{}^{\backprime\backprime} x \in X \, {}^{\prime\prime}</math> and a ''type indication'' of the form <math>{}^{\backprime\backprime} x : X \, {}^{\prime\prime},</math> for instance, as they are used in the examples above. The difference that appears to be perceived in categorical statements, when those of the form <math>{}^{\backprime\backprime} x \in X \, {}^{\prime\prime}</math> and those of the form <math>{}^{\backprime\backprime} x : X \, {}^{\prime\prime}</math> are set in side by side comparisons with each other, is that a multitude of objects can be said to have the same type without having to posit the existence of a set to which they all belong. Without trying to decide whether I share this feeling or even fully understand the distinction in question, I can only try to maintain a style of notation that respects it to some degree. It is conceivable that the question of belonging to a set is rightly regarded as the more serious matter, one that concerns the reality of an object and the substance of a predicate, than the question of falling under a type, that may depend only on the way that a sign is interpreted and the way that information about an object is organized. When it comes to the kinds of hypothetical statements that appear in the present instance, those of the forms <math>{}^{\backprime\backprime} x \in X ~\Leftrightarrow~ \underline{x} \in \underline{X} \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} x : X ~\Leftrightarrow~ \underline{x} : \underline{X} \, {}^{\prime\prime},</math> these are usually read as implying some order of synthetic construction, one whose contingent consequences involve the constitution of a new space to contain the elements being compounded and the recognition of a new type to characterize the elements being moulded, respectively. In these applications, the statement about types is again taken to be less presumptive than the corresponding statement about sets, since the apodosis is intended to do nothing more than abbreviate and summarize what is already stated in the protasis.

A ''boolean connection'' of degree <math>k,\!</math> also known as a ''boolean function'' on <math>k\!</math> variables, is a map of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math> In other words, a boolean connection of degree <math>k\!</math> is a proposition about things in the universe <math>X = \underline\mathbb{B}^k.</math>

−

An ''imagination'' of degree <math>k\!</math> on <math>X\!</math> is a <math>k\!</math>-tuple of propositions about things in the universe <math>X.\!</math> By way of displaying the kinds of notation that are used to express this idea, the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> is given as a sequence of indicator functions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = {}_1^k.</math> All of these features of the typical imagination <math>\underline{f}</math> can be summed up in either one of two ways: either in the form of a membership statement, to the effect that <math>\underline{f} \in (X \to \underline\mathbb{B})^k,</math> or in the form of a type statement, to the effect that <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> though perhaps the latter form is slightly more precise than the former.

+

An ''imagination'' of degree <math>k\!</math> on <math>X\!</math> is a <math>k\!</math>-tuple of propositions about things in the universe <math>X.\!</math> By way of displaying the kinds of notation that are used to express this idea, the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> is given as a sequence of indicator functions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = {}_1^k.</math> All of these features of the typical imagination <math>\underline{f}\!</math> can be summed up in either one of two ways: either in the form of a membership statement, to the effect that <math>\underline{f} \in (X \to \underline\mathbb{B})^k,</math> or in the form of a type statement, to the effect that <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> though perhaps the latter form is slightly more precise than the former.

−

The ''play of images'' determined by <math>\underline{f}</math> and <math>x,\!</math> more specifically, the play of the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> that has to do with the element <math>x \in X,</math> is the <math>k\!</math>-tuple <math>\underline{y} = (y_1, \ldots, y_k)</math> of values in <math>\underline\mathbb{B}</math> that satisfies the equations <math>y_j = f_j (x),\!</math> for <math>j = 1 ~\text{to}~ k.</math>

+

The ''play of images'' determined by <math>\underline{f}\!</math> and <math>x,\!</math> more specifically, the play of the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> that has to do with the element <math>x \in X,</math> is the <math>k\!</math>-tuple <math>\underline{y} = (y_1, \ldots, y_k)</math> of values in <math>\underline\mathbb{B}</math> that satisfies the equations <math>y_j = f_j (x),\!</math> for <math>j = 1 ~\text{to}~ k.</math>

−

A ''projection'' of <math>\underline\mathbb{B}^k,</math> written <math>\pi_j\!</math> or <math>\operatorname{pr}_j,\!</math> is one of the maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> that is defined as follows:

+

A ''projection'' of <math>\underline\mathbb{B}^k,\!</math> written <math>\pi_j\!</math> or <math>\operatorname{pr}_j,\!</math> is one of the maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> that is defined as follows:

{| align="center" cellpadding="8" width="90%"

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|}

−

The ''projective imagination'' of <math>\underline\mathbb{B}^k</math> is the imagination <math>(\pi_1, \ldots, \pi_k).</math>

+

The ''projective imagination'' of <math>\underline\mathbb{B}^k</math> is the imagination <math>(\pi_1, \ldots, \pi_k).\!</math>

A ''sentence about things in the universe'', for short, a ''sentence'', is a sign that denotes a proposition. In other words, a sentence is any sign that denotes an indicator function, any sign whose object is a function of the form <math>f : X \to \underline\mathbb{B}.</math>

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Taken in a context of communication, an assertion invites the interpreter to consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or yet again, to invert the indicator function denoted by the sentence with respect to its possible value of truth.

−

A ''denial'' of a sentence <math>s\!</math> is an assertion of its negation <math>^{\backprime\backprime} \, \underline{(} s \underline{)} \, ^{\prime\prime}.</math> The denial acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function denoted by the sentence with respect to its possible value of falsity.

+

A ''denial'' of a sentence <math>s\!</math> is an assertion of its negation <math>{}^{\backprime\backprime} \, \underline{(} s \underline{)} \, {}^{\prime\prime}.</math> The denial acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function denoted by the sentence with respect to its possible value of falsity.

According to this manner of definition, any sign that happens to denote a proposition, any sign that is taken as denoting an indicator function, by that very fact alone successfully qualifies as a sentence. That is, a sentence is any sign that actually succeeds in denoting a proposition, any sign that one way or another brings to mind, as its actual object, a function of the form <math>f : X \to \underline\mathbb{B}.</math>

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Frequently this formula has a ''variable'' in it that ''ranges over'' the universe <math>X.\!</math> A variable is an ambiguous or equivocal sign that can be interpreted as denoting any element of the set that it ranges over.

−

If a sentence denotes a proposition <math>f : X \to \underline\mathbb{B},</math> then the ''value'' of the sentence with regard to <math>x \in X</math> is the value <math>f(x)\!</math> of the proposition at <math>x,\!</math> where <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> is interpreted as ''false'' and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> is interpreted as ''true''.

+

If a sentence denotes a proposition <math>f : X \to \underline\mathbb{B},</math> then the ''value'' of the sentence with regard to <math>x \in X</math> is the value <math>f(x)\!</math> of the proposition at <math>x,\!</math> where <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}</math> is interpreted as ''false'' and <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}</math> is interpreted as ''true''.

Since the value of a sentence or a proposition depends on the universe of discourse to which it is referred, and since it also depends on the element of the universe with regard to which it is evaluated, it is usual to say that a sentence or a proposition ''refers'' to a universe and to its elements, though perhaps in a variety of different senses. Furthermore, a proposition, acting in the role of as an indicator function, ''refers'' to the elements that it ''indicates'', namely, the elements on which it takes a positive value. In order to sort out the possible confusions that are capable of arising here, I need to examine how these various notions of reference are related to the notion of denotation that is used in the pragmatic theory of sign relations.

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One way to resolve the various senses of ''reference'' that arise in this setting is to make the following sorts of distinctions among them. Let the reference of a sentence or a proposition to a universe of discourse, the one that it acquires by way of taking on any interpretation at all, be taken as its ''general reference'', the kind of reference that one can safely ignore as irrelevant, at least, so long as one stays immersed in only one context of discourse or only one moment of discussion. Let the references that an indicator function <math>f\!</math> has to the elements on which it evaluates to <math>\underline{0}</math> be called its ''negative references''. Let the references that an indicator function <math>f\!</math> has to the elements on which it evaluates to <math>\underline{1}</math> be called its ''positive references'' or its ''indications''. Finally, unspecified references to the "references" of a sentence, a proposition, or an indicator function can be taken by default as references to their specific, positive references.

−

The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation. Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables. For instance, even a sentence with no explicit variable, a constant expression like <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{1} ^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math> Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe <math>X.\!</math>

+

The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation. Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables. For instance, even a sentence with no explicit variable, a constant expression like <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math> Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe <math>X.\!</math>

−

Notice that the letters <math>^{\backprime\backprime} p ^{\prime\prime}</math> and <math>^{\backprime\backprime} q ^{\prime\prime},</math> interpreted as signs that denote indicator functions <math>p, q : X \to \underline\mathbb{B},</math> have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions. This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.

+

Notice that the letters <math>{}^{\backprime\backprime} p {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} q {}^{\prime\prime},</math> interpreted as signs that denote indicator functions <math>p, q : X \to \underline\mathbb{B},</math> have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions. This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.

−

To assist the reading of informal examples, I frequently use the letters <math>^{\backprime\backprime} s ^{\prime\prime}</math> and <math>^{\backprime\backprime} t ^{\prime\prime},</math> to denote sentences. Thus, it is conceivable to have a situation where <math>s ~=~ ^{\backprime\backprime} p ^{\prime\prime}</math> and where <math>p : X \to \underline\mathbb{B}.</math> Altogether, this means that the sign <math>^{\backprime\backprime} s ^{\prime\prime}</math> denotes the sentence <math>s,\!</math> that the sentence <math>s\!</math> is the sentence <math>^{\backprime\backprime} p ^{\prime\prime},</math> and that the sentence <math>^{\backprime\backprime} p ^{\prime\prime}</math> denotes the proposition or the indicator function <math>p : X \to \underline\mathbb{B}.</math> In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like <math>{}^{\backprime\backprime} e_1 {}^{\prime\prime}, \, \ldots, \, {}^{\backprime\backprime} e_n {}^{\prime\prime}</math> to refer to the various expressions.

+

To assist the reading of informal examples, I frequently use the letters <math>{}^{\backprime\backprime} s {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} t {}^{\prime\prime},</math> to denote sentences. Thus, it is conceivable to have a situation where <math>s ~=~ {}^{\backprime\backprime} p {}^{\prime\prime}</math> and where <math>p : X \to \underline\mathbb{B}.</math> Altogether, this means that the sign <math>{}^{\backprime\backprime} s {}^{\prime\prime}</math> denotes the sentence <math>s,\!</math> that the sentence <math>s\!</math> is the sentence <math>{}^{\backprime\backprime} p {}^{\prime\prime},</math> and that the sentence <math>{}^{\backprime\backprime} p {}^{\prime\prime}</math> denotes the proposition or the indicator function <math>p : X \to \underline\mathbb{B}.</math> In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like <math>{}^{\backprime\backprime} e_1 {}^{\prime\prime}, \, \ldots, \, {}^{\backprime\backprime} e_n {}^{\prime\prime}</math> to refer to the various expressions.

A ''sentential connective'' is a sign, a coordinated sequence of signs, a significant pattern of arrangement, or any other syntactic device that can be used to connect a number of sentences together in order to form a single sentence. If <math>k\!</math> is the number of sentences that are connected, then the connective is said to be of order <math>k.\!</math> If the sentences acquire a logical relationship by this means, and are not just strung together by this mechanism, then the connective is called a ''logical connective''. If the value of the constructed sentence depends on the values of the component sentences in such a way that the value of the whole is a boolean function of the values of the parts, then the connective is called a ''propositional connective''.

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In particular, a connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math> If these <math>k\!</math> values are values of things in a universe <math>X,\!</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math> Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>

−

To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.</math> Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},</math> a single <math>F\!</math> and many <math>\underline{f},</math> or many <math>F\!</math> and a single <math>\underline{f}</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:

+

To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k\!</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}\!</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.\!</math> Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},\!</math> a single <math>F\!</math> and many <math>\underline{f},\!</math> or many <math>F\!</math> and a single <math>\underline{f}\!</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:

<li>

−

In a general setting, where the connection <math>F\!</math> and the imagination <math>\underline{f}</math> are both permitted to take up a variety of concrete possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> and <math>\underline{f}</math> from <math>X\!</math> to <math>\underline\mathbb{B},</math>'', and write it in the style of a composition as <math>F ~\$~ \underline{f}.</math> This is meant to suggest that the symbol <math>^{\backprime\backprime} $ ^{\prime\prime},</math> here read as ''stretch'', denotes an operator of the form:

+

In a general setting, where the connection <math>F\!</math> and the imagination <math>\underline{f}\!</math> are both permitted to take up a variety of concrete possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> and <math>\underline{f}\!</math> from <math>X\!</math> to <math>\underline\mathbb{B},\!</math>'' and write it in the style of a composition as <math>F ~\$~ \underline{f}.\!</math> This is meant to suggest that the symbol <math>{}^{\backprime\backprime} $ {}^{\prime\prime},\!</math> here read as ''stretch'', denotes an operator of the form:

<math>\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \times (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}).</math></li>

<li>

−

In a setting where the connection <math>F\!</math> is fixed but the imagination <math>\underline{f}</math> is allowed to vary over a wide range of possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> to <math>\underline{f}</math> on <math>X,\!</math>'' and write it in the style <math>F^\$ \underline{f},</math> exactly as if <math>^{\backprime\backprime} F^\$ \, ^{\prime\prime}</math> denotes an operator <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B})</math> that is derived from <math>F\!</math> and applied to <math>\underline{f},</math> ultimately yielding a proposition <math>F^\$ \underline{f} : X \to \underline\mathbb{B}.</math></li>

+

In a setting where the connection <math>F\!</math> is fixed but the imagination <math>\underline{f}\!</math> is allowed to vary over a wide range of possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> to <math>\underline{f}\!</math> on <math>X,\!</math>'' and write it in the style <math>F^\$ \underline{f},\!</math> exactly as if <math>{}^{\backprime\backprime} F^\$ \, {}^{\prime\prime}\!</math> denotes an operator <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B})\!</math> that is derived from <math>F\!</math> and applied to <math>\underline{f},\!</math> ultimately yielding a proposition <math>F^\$ \underline{f} : X \to \underline\mathbb{B}.\!</math></li>

<li>

−

In a setting where the imagination<math>\underline{f}</math> is fixed but the connection <math>F\!</math> is allowed to range over wide variety of possibilities, call <math>p\!</math> the ''stretch of <math>\underline{f}</math> by <math>F\!</math> to <math>\underline\mathbb{B},</math>'' and write it in the style <math>\underline{f}^\$ F,</math> exactly as if <math>^{\backprime\backprime} \underline{f}^\$ \, ^{\prime\prime}</math> denotes an operator <math>\underline{f}^\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \to (X \to \underline\mathbb{B}</math> that is derived from <math>\underline{f}</math> and applied to <math>F,\!</math> ultimately yielding a proposition <math>\underline{f}^\$ F : X \to \underline\mathbb{B}.</math></li>

+

In a setting where the imagination<math>\underline{f}\!</math> is fixed but the connection <math>F\!</math> is allowed to range over wide variety of possibilities, call <math>p\!</math> the ''stretch of <math>\underline{f}\!</math> by <math>F\!</math> to <math>\underline\mathbb{B},\!</math>'' and write it in the style <math>\underline{f}^\$ F,\!</math> exactly as if <math>{}^{\backprime\backprime} \underline{f}^\$ \, {}^{\prime\prime}\!</math> denotes an operator <math>\underline{f}^\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \to (X \to \underline\mathbb{B}\!</math> that is derived from <math>\underline{f}\!</math> and applied to <math>F,\!</math> ultimately yielding a proposition <math>\underline{f}^\$ F : X \to \underline\mathbb{B}.\!</math></li>

</ol>

−

Because this notation is only used in settings where the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> are distinguished by their types, it does not really matter whether one writes <math>^{\backprime\backprime} F ~\$~ \underline{f} \, ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{f} ~\$~ F \, ^{\prime\prime}</math> for the initial composition.

+

Because this notation is only used in settings where the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> are distinguished by their types, it does not really matter whether one writes <math>{}^{\backprime\backprime} F ~\$~ \underline{f} \, {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \underline{f} ~\$~ F \, {}^{\prime\prime}</math> for the initial composition.

Just as a sentence is a sign that denotes a proposition, which thereby serves to indicate a set, a propositional connective is a provision of syntax whose mediate effect is to denote an operation on propositions, which thereby manages to indicate the result of an operation on sets. In order to see how these compound forms of indication can be defined, it is useful to go through the steps that are needed to construct them. In general terms, the ingredients of the construction are as follows:

−

# An imagination of degree <math>k\!</math> on <math>X,\!</math> in other words, a <math>k\!</math>-tuple of propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> or an object of the form <math>\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.</math>

+

# An imagination of degree <math>k\!</math> on <math>X,\!</math> in other words, a <math>k\!</math>-tuple of propositions <math>f_j : X \to \underline\mathbb{B},\!</math> for <math>j = 1 ~\text{to}~ k,\!</math> or an object of the form <math>\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.\!</math>

−

# A connection of degree <math>k,\!</math> in other words, a proposition about things in <math>\underline\mathbb{B}^k,</math> or a boolean function of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math>

+

# A connection of degree <math>k,\!</math> in other words, a proposition about things in <math>\underline\mathbb{B}^k,\!</math> or a boolean function of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.\!</math>

−

From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),</math> for all <math>x \in X.</math> The desired construction is determined as follows:

+

From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}\!</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),\!</math> for all <math>x \in X.\!</math> The desired construction is determined as follows:

−

The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))</math> on the functional image <math>f!(w)\!</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))</math> under <math>\underline{f},</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>

+

The cartesian power <math>\underline\mathbb{B}^k,\!</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)\!</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,\!</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)\!</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,\!</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))\!</math> on the functional image <math>{f!(w)}\!</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))\!</math> under <math>\underline{f},\!</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>

−

Just to be on the safe side, I state this again in more standard terms. The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of <math>k\!</math> projection maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> along with the property that any <math>k\!</math> maps <math>f_j : W \to \underline\mathbb{B},</math> from an arbitrary set <math>W\!</math> to <math>\underline\mathbb{B},</math> determine a unique map <math>f! : W \to \underline\mathbb{B}^k</math> such that <math>\pi_j (f!(w)) = f_j (w),\!</math> for all <math>j = 1 ~\text{to}~ k,</math> and for all <math>w \in W.</math>

+

Just to be on the safe side, I state this again in more standard terms. The cartesian power <math>\underline\mathbb{B}^k,\!</math> as a cartesian product, is characterized by the possession of <math>k\!</math> projection maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},\!</math> for <math>j = 1 ~\text{to}~ k,\!</math> along with the property that any <math>k\!</math> maps <math>f_j : W \to \underline\mathbb{B},\!</math> from an arbitrary set <math>W\!</math> to <math>\underline\mathbb{B},\!</math> determine a unique map <math>f! : W \to \underline\mathbb{B}^k\!</math> such that <math>\pi_j (f!(w)) = f_j (w),\!</math> for all <math>j = 1 ~\text{to}~ k,\!</math> and for all <math>w \in W.\!</math>

−

Now suppose that the arbitrary set <math>W\!</math> in this construction is just the relevant universe <math>X.\!</math> Given that the function <math>f! : X \to \underline\mathbb{B}^k</math> is uniquely determined by the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> that is, by the <math>k\!</math>-tuple of propositions <math>\underline{f} = (f_1, \ldots, f_k),</math> it is safe to identify <math>f!\!</math> and <math>\underline{f}</math> as being a single function, and this makes it convenient on many occasions to refer to the identified function by means of its explicitly descriptive name <math>^{\backprime\backprime} (f_1, \ldots, f_k) \, ^{\prime\prime}.</math> This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

+

Now suppose that the arbitrary set <math>W\!</math> in this construction is just the relevant universe <math>X.\!</math> Given that the function <math>f! : X \to \underline\mathbb{B}^k\!</math> is uniquely determined by the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k,\!</math> that is, by the <math>k\!</math>-tuple of propositions <math>\underline{f} = (f_1, \ldots, f_k),\!</math> it is safe to identify <math>f!\!</math> and <math>\underline{f}\!</math> as being a single function, and this makes it convenient on many occasions to refer to the identified function by means of its explicitly descriptive name <math>{}^{\backprime\backprime} (f_1, \ldots, f_k) \, {}^{\prime\prime}.\!</math> This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

====2.2.7. Stretching Operations====

Line 1,422: Line 1,422:

Thus, <math>F^\$</math> is what a propositional connective denotes, a particular way of connecting the propositions that are denoted by a number of sentences into a proposition that is denoted by a single sentence.

−

Now <math>^{\backprime\backprime} f_Q \, ^{\prime\prime}</math> is sign that denotes the proposition <math>f_Q,\!</math> and it certainly seems like a sufficient sign for it. Why is there is a need to recognize any other signs of it?

+

Now <math>{}^{\backprime\backprime} f_Q \, {}^{\prime\prime}</math> is sign that denotes the proposition <math>f_Q,\!</math> and it certainly seems like a sufficient sign for it. Why is there is a need to recognize any other signs of it?

If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a higher order sign relation.

Line 1,443: Line 1,443:

| <math>f^{-1} (y)\!</math>

| <math>f\!</math>

−

| <math>^{\backprime\backprime} f \, ^{\prime\prime}</math>

+

| <math>{}^{\backprime\backprime} f \, {}^{\prime\prime}</math>

|-

| <math>Q\!</math>

| <math>\underline{1}</math>

−

| <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math>

+

| <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}</math>

|-

| <math>{}^{_\sim} Q</math>

| <math>\underline{0}</math>

−

| <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math>

+

| <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}</math>

|}

Line 1,467: Line 1,467:

In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over. For this reason, I express their usage a bit more carefully as follows:

−

# Let the ''down hooks'' <math>\downharpoonleft \cdots \downharpoonright</math> be placed around the name of a sentence <math>s,\!</math> as in the expression <math>^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime},</math> or else around a token appearance of the sentence itself, to serve as a name for the proposition that <math>s\!</math> denotes.

+

# Let the ''down hooks'' <math>\downharpoonleft \cdots \downharpoonright</math> be placed around the name of a sentence <math>s,\!</math> as in the expression <math>{}^{\backprime\backprime} \downharpoonleft s \downharpoonright \, {}^{\prime\prime},</math> or else around a token appearance of the sentence itself, to serve as a name for the proposition that <math>s\!</math> denotes.

−

# Let the ''up hooks'' <math>\upharpoonleft \cdots \upharpoonright</math> be placed around a name of a set <math>Q,\!</math> as in the expression <math>^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime},</math> to serve as a name for the indicator function <math>f_Q.\!</math>

+

# Let the ''up hooks'' <math>\upharpoonleft \cdots \upharpoonright</math> be placed around a name of a set <math>Q,\!</math> as in the expression <math>{}^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, {}^{\prime\prime},</math> to serve as a name for the indicator function <math>f_Q.\!</math>

Table 12 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.

Line 1,495: Line 1,495:

| <math>[| q |]\!</math>

| <math>q\!</math>

−

| <math>^{\backprime\backprime} q \, ^{\prime\prime}</math>

+

| <math>{}^{\backprime\backprime} q \, {}^{\prime\prime}</math>

|-

| <math>[| f_Q |]\!</math>

| <math>f_Q\!</math>

−

| <math>^{\backprime\backprime} f_Q \, ^{\prime\prime}</math>

+

| <math>{}^{\backprime\backprime} f_Q \, {}^{\prime\prime}</math>

|-

| <math>Q\!</math>

| <math>\upharpoonleft Q \upharpoonright</math>

−

| <math>^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime}</math>

+

| <math>{}^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, {}^{\prime\prime}</math>

|}

Line 1,515: Line 1,515:

|-

|   ||   || where

−

| <math>q : X \to \underline\mathbb{B}.</math>

+

| <math>q : X \to \underline\mathbb{B}.\!</math>

|-

|  

Line 1,521: Line 1,521:

|-

|   ||  

−

| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q.</math>

+

| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q.\!</math>

|-

| valign="top" | 2.

Line 1,527: Line 1,527:

|-

|   ||   || where

−

| <math>q : X \to \underline\mathbb{B}</math>

+

| <math>q : X \to \underline\mathbb{B}\!</math>

|-

|   ||   || and

−

| <math>[| q |] ~=~ q^{-1} (\underline{1}) ~=~ Q \subseteq X.</math>

+

| <math>[| q |] ~=~ q^{-1} (\underline{1}) ~=~ Q \subseteq X.\!</math>

|-

|  

Line 1,536: Line 1,536:

|-

|   ||  

−

| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q ~=~ f_Q ~=~ \upharpoonleft Q \upharpoonright.</math>

+

| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q ~=~ f_Q ~=~ \upharpoonleft Q \upharpoonright.\!</math>

|-

| valign="top" | 3.

Line 1,546: Line 1,546:

|  

| align="center" | <math>=\!</math>

−

| <math>[| \upharpoonleft X \upharpoonright |] ~=~ \upharpoonleft X \upharpoonright^{-1} (\underline{1})</math>

+

| <math>[| \upharpoonleft X \upharpoonright |] ~=~ \upharpoonleft X \upharpoonright^{-1} (\underline{1})\!</math>

|-

|  

| align="center" | <math>=\!</math>

−

| <math>[| f_Q |] ~=~ f_Q^{-1} (\underline{1}).</math>

+

| <math>[| f_Q |] ~=~ f_Q^{-1} (\underline{1}).\!</math>

|-

| valign="top" | 4.

−

| align="center" | <math>\upharpoonleft Q \upharpoonright</math>

+

| align="center" | <math>\upharpoonleft Q \upharpoonright\!</math>

| align="center" | <math>=\!</math>

−

| <math>\upharpoonleft \{ x \in X : x \in Q \} \upharpoonright</math>

+

| <math>\upharpoonleft \{ x \in X : x \in Q \} \upharpoonright\!</math>

|-

|  

| align="center" | <math>=\!</math>

−

| <math>\downharpoonleft x \in Q \downharpoonright</math>

+

| <math>\downharpoonleft x \in Q \downharpoonright\!</math>

|-

|  

Line 1,569: Line 1,569:

|}

−

Now if a sentence <math>s\!</math> really denotes a proposition <math>q,\!</math> and if the notation <math>^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime}</math> is merely meant to supply another name for the proposition that <math>s\!</math> already denotes, then why is there any need for the additional notation? It is because the interpretive mind habitually races from the sentence <math>s,\!</math> through the proposition <math>q\!</math> that it denotes, and on to the set <math>Q = q^{-1} (\underline{1})</math> that the proposition <math>q\!</math> indicates, often jumping to the conclusion that the set <math>Q\!</math> is the only thing that the sentence <math>s\!</math> is intended to denote. This higher order sign situation and the mind's inclination when placed in its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from <math>s\!</math> to <math>q\!</math> to <math>Q.\!</math>

+

Now if a sentence <math>s\!</math> really denotes a proposition <math>q,\!</math> and if the notation <math>{}^{\backprime\backprime} \downharpoonleft s \downharpoonright \, {}^{\prime\prime}\!</math> is merely meant to supply another name for the proposition that <math>s\!</math> already denotes, then why is there any need for the additional notation? It is because the interpretive mind habitually races from the sentence <math>s,\!</math> through the proposition <math>q\!</math> that it denotes, and on to the set <math>Q = q^{-1} (\underline{1})\!</math> that the proposition <math>q\!</math> indicates, often jumping to the conclusion that the set <math>Q\!</math> is the only thing that the sentence <math>s\!</math> is intended to denote. This higher order sign situation and the mind's inclination when placed in its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from <math>s\!</math> to <math>q\!</math> to <math>Q.\!</math>

===2.3. The Cactus Patch===

Line 1,639: Line 1,639:

1. &

\operatorname{If} &

−

^{\backprime\backprime}\operatorname{A}^{\prime\prime} &

+

{}^{\backprime\backprime}\operatorname{A}{}^{\prime\prime} &

\rightarrow &

\operatorname{Ann}, \\

&

\operatorname{that~is}, &

−

^{\backprime\backprime}\operatorname{A}^{\prime\prime} &

+

{}^{\backprime\backprime}\operatorname{A}{}^{\prime\prime} &

\operatorname{denotes} &

\operatorname{Ann}, \\

Line 1,659: Line 1,659:

&

\operatorname{Thus} &

−

^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} &

+

{}^{\backprime\backprime}\operatorname{Ann}{}^{\prime\prime} &

\rightarrow &

\operatorname{A}, \\

&

\operatorname{that~is}, &

−

^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} &

+

{}^{\backprime\backprime}\operatorname{Ann}{}^{\prime\prime} &

\operatorname{denotes} &

\operatorname{A}. \\

−

\end{array}</math>

+

\end{array}\!</math>

|}

Line 1,677: Line 1,677:

\operatorname{Bob} &

\leftarrow &

−

^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\

+

{}^{\backprime\backprime}\operatorname{B}{}^{\prime\prime}, \\

&

\operatorname{that~is}, &

\operatorname{Bob} &

\operatorname{is~denoted~by} &

−

^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\

+

{}^{\backprime\backprime}\operatorname{B}{}^{\prime\prime}, \\

&

\operatorname{then} &

Line 1,697: Line 1,697:

\operatorname{B} &

\leftarrow &

−

^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}, \\

+

{}^{\backprime\backprime}\operatorname{Bob}{}^{\prime\prime}, \\

&

\operatorname{that~is}, &

\operatorname{B} &

\operatorname{is~denoted~by} &

−

^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}. \\

+

{}^{\backprime\backprime}\operatorname{Bob}{}^{\prime\prime}. \\

\end{array}</math>

|}

Line 1,711: Line 1,711:

|

<math>\begin{array}{lll}

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime} &

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} &

\leftarrow &

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \\

\\

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} &

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} &

\rightarrow &

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime} \\

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} \\

\end{array}</math>

|}

Line 1,726: Line 1,726:

|

<math>\begin{array}{lllll}

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime}

& \leftarrow &

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime}

& = &

−

^{\backprime\backprime}\operatorname{b}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{b}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{l}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{l}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{a}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{a}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{n}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{n}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{k}^{\prime\prime} \\

+

{}^{\backprime\backprime}\operatorname{k}{}^{\prime\prime} \\

\end{array}</math>

|}

Line 1,743: Line 1,743:

|

<math>\begin{array}{lll}

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \\

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} & = & \operatorname{blank} \\

\end{array}</math>

|}

Line 1,752: Line 1,752:

|

<math>\begin{array}{lclcl}

−

^{\backprime\backprime}\operatorname{~~}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{~~}{}^{\prime\prime}

& = &

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime}

& = &

\operatorname{blank} \, \cdot \, \operatorname{blank} \\

\\

−

^{\backprime\backprime}\operatorname{~blank}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{~blank}{}^{\prime\prime}

& = &

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime}

& = &

\operatorname{blank} \, \cdot \,

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \\

\\

−

^{\backprime\backprime}\operatorname{blank~}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{blank~}{}^{\prime\prime}

& = &

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \, \cdot \,

−

^{\backprime\backprime}\operatorname{~}^{\prime\prime}

+

{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime}

& = &

−

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \,

+

{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \, \cdot \,

\operatorname{blank}

\end{array}</math>

Line 1,807: Line 1,807:

{| align="center" cellpadding="4" style="text-align:center" width="90%"

|-

−

| <math>\varepsilon</math>

+

| <math>\varepsilon\!</math>

| =

−

| <math>^{\backprime\backprime\prime\prime}</math>

+

| <math>{}^{\backprime\backprime\prime\prime}\!</math>

| =

| align="left" | the empty string.

|-

−

| <math>\underline\varepsilon</math>

+

| <math>\underline\varepsilon\!</math>

| =

−

| <math>\{ \varepsilon \}</math>

+

| <math>\{ \varepsilon \}\!</math>

| =

| align="left" | the language consisting of a single empty string.

Line 1,857: Line 1,857:

& = &

\{ &

−

^{\backprime\backprime} \, \operatorname{~} \, ^{\prime\prime} & , &

+

{}^{\backprime\backprime} \, \operatorname{~} \, {}^{\prime\prime} & , &

−

^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & , &

+

{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} & , &

−

^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} & , &

+

{}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} & , &

−

^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} &

+

{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} &

\} \\

& = &

Line 1,878: Line 1,878:

</ol>

−

The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})</math> is to indicate the general sorts of operations that suffice to construct the greater share of its sentences from the specified few of its sentences that require a special election. In accord with this manner of proceeding, I introduce a family of operations on strings of <math>\mathfrak{A}^*</math> that are called ''syntactic connectives''. If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),</math> then these operations are tantamount to ''sentential connectives'', and if the syntactic sentences, considered as abstract strings of meaningless signs, are given a semantics in which they denote propositions, considered as indicator functions over some universe, then these operations amount to ''propositional connectives''.

+

The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})\!</math> is to indicate the general sorts of operations that suffice to construct the greater share of its sentences from the specified few of its sentences that require a special election. In accord with this manner of proceeding, I introduce a family of operations on strings of <math>\mathfrak{A}^*\!</math> that are called ''syntactic connectives''. If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),\!</math> then these operations are tantamount to ''sentential connectives'', and if the syntactic sentences, considered as abstract strings of meaningless signs, are given a semantics in which they denote propositions, considered as indicator functions over some universe, then these operations amount to ''propositional connectives''.

Rather than presenting the most concise description of these languages right from the beginning, it serves comprehension to develop a picture of their forms in gradual stages, starting from the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.

−

The first step is to define two sets of basic operations on strings of <math>\mathfrak{A}^*.</math>

+

The first step is to define two sets of basic operations on strings of <math>\mathfrak{A}^*.\!</math>

Line 1,889: Line 1,889:

The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math>

−

The ''concatenation'' of two strings <math>s_1, s_2\!</math> is the string <math>s_1 \cdot s_2.\!</math>

+

The ''concatenation'' of two strings <math>{s_1, s_2}\!</math> is the string <math>{s_1 \cdot s_2}.\!</math>

−

The ''concatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>s_1 \cdot \ldots \cdot s_k.\!</math></li>

+

The ''concatenation'' of the <math>k\!</math> strings <math>{(s_j)_{j = 1}^k}\!</math> is the string of the form <math>{s_1 \cdot \ldots \cdot s_k}.\!</math></li>

<li>

−

The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

+

The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.\!</math>

−

The ''surcatenation'' of two strings <math>s_1, s_2\!</math> is <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

+

The ''surcatenation'' of two strings <math>{s_1, s_2}\!</math> is <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_2 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.\!</math>

−

The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

+

The ''surcatenation'' of the <math>k\!</math> strings <math>{(s_j)_{j = 1}^k}\!</math> is the string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, \ldots \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_k \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.\!</math></li>

</ol>

Line 1,923: Line 1,923:

−

<li><math>\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

+

<li><math>\operatorname{Surc}_{j=1}^1 s_j \ = \ {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></li>

<li>

For <math>\ell > 1,\!</math>

−

<math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

+

<math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></li>

</ol></ol>

Line 1,968: Line 1,968:

The conception of the <math>k\!</math>-place surcatenation operation can be extended to include its natural "prequel":

−

<math>\operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math>

+

<math>\operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}.</math>

Finally, the construction of the <math>k\!</math>-place surcatenation can be broken into stages by means of the following conceptions:

Line 1,975: Line 1,975:

<li>

−

A ''subclause'' in <math>\mathfrak{A}^*</math> is a string that ends with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

+

A ''subclause'' in <math>\mathfrak{A}^*</math> is a string that ends with a <math>{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></li>

<li>

The ''subcatenation'' <math>\operatorname{Subc} (s_1, s_2)</math> of a subclause <math>s_1\!</math> by a string <math>s_2\!</math> is the string that is defined as follows:

−

<math>\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

+

<math>\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_2 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>

<li>

−

The ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated subcatenation over the sequence of <math>k+1\!</math> strings that starts with the string <math>s_0 \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</li>

+

The ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated subcatenation over the sequence of <math>k+1\!</math> strings that starts with the string <math>s_0 \ = \ \operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</li>

<li>

−

<math>\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></li>

+

<math>\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}.</math></li>

<li>

Line 2,001: Line 2,001:

Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language <math>\mathfrak{L} = \mathfrak{A}^*,</math> it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

−

If <math>\mathfrak{L}</math> is an arbitrary formal language over an alphabet of the sort that

+

If <math>\mathfrak{L}\!</math> is an arbitrary formal language over an alphabet of the sort that

−

we are talking about, that is, an alphabet of the form <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},</math> then there are a number of basic structural relations that can be defined on the strings of <math>\mathfrak{L}.</math>

+

we are talking about, that is, an alphabet of the form <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},\!</math> then there are a number of basic structural relations that can be defined on the strings of <math>\mathfrak{L}.\!</math>

{| align="center" cellpadding="4" width="90%"

Line 2,029: Line 2,029:

| 4. || <math>s\!</math> is a ''subclause'' of <math>\mathfrak{L}</math> if and only if

|-

−

|   || <math>s\!</math> is a sentence of <math>\mathfrak{L}</math> and <math>s\!</math> ends with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

+

|   || <math>s\!</math> is a sentence of <math>\mathfrak{L}</math> and <math>s\!</math> ends with a <math>{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>

|-

| 5. || <math>s\!</math> is the ''subcatenation'' of <math>s_1\!</math> by <math>s_2\!</math> if and only if

Line 2,035: Line 2,035:

|   || <math>s_1\!</math> is a subclause of <math>\mathfrak{L},</math> <math>s_2\!</math> is a sentence of <math>\mathfrak{L},</math> and

|-

−

|   || <math>s = s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

+

|   || <math>s = s_1 \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_2 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>

|-

| 6. || <math>s\!</math> is the ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>

|-

−

|   || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>j = 1 \ldots k,\!</math> and

+

|   || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>{j = 1 \ldots k},\!</math> and

|-

−

|   || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

+

|   || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, \ldots \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_k \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>

|}

Line 2,065: Line 2,065:

|}

−

As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math> An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> or a ''cactus expression'', for short. Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>

+

As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.\!</math> An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),\!</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> or a ''cactus expression'', for short. Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>

A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \varnothing.</math> A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).</math> This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math> A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.

Line 2,095: Line 2,095:

# To specify the intension or to signify the intention that every string that fits the conditions of the abstract type <math>T\!</math> must also fall under the grammatical heading of a sentence, as indicated by the type <math>S,\!</math> all within the target language <math>\mathfrak{L}.</math>

−

In these types of situation the letter <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> that signifies the type of a sentence in the language of interest, is called the ''initial symbol'' or the ''sentence symbol'' of a candidate formal grammar for the language, while any number of letters like <math>^{\backprime\backprime} T \, ^{\prime\prime}</math> signifying other types of strings that are necessary to a reasonable account or a rational reconstruction of the sentences that belong to the language, are collectively referred to as ''intermediate symbols''.

+

In these types of situation the letter <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> that signifies the type of a sentence in the language of interest, is called the ''initial symbol'' or the ''sentence symbol'' of a candidate formal grammar for the language, while any number of letters like <math>{}^{\backprime\backprime} T \, {}^{\prime\prime}</math> signifying other types of strings that are necessary to a reasonable account or a rational reconstruction of the sentences that belong to the language, are collectively referred to as ''intermediate symbols''.

−

Combining the singleton set <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \}</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}</math> that assembles together all of the intermediate symbols results in the set <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}</math> of ''non-terminal symbols''. Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''. In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}.</math> Finally, it is convenient to refer to all of the symbols in <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> as the ''augmented strings'', in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars. In certain settings it becomes desirable to separate the augmented strings that contain the symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> from all other sorts of augmented strings. In these situations the strings in the disjoint union <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar.

+

Combining the singleton set <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \}\!</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}\!</math> that assembles together all of the intermediate symbols results in the set <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q}\!</math> of ''non-terminal symbols''. Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''. In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q}.</math> Finally, it is convenient to refer to all of the symbols in <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> as the ''augmented strings'', in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars. In certain settings it becomes desirable to separate the augmented strings that contain the symbol <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> from all other sorts of augmented strings. In these situations the strings in the disjoint union <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar.

In forming a grammar for a language statements of the form <math>W :> W',\!</math> where <math>W\!</math> and <math>W'\!</math> are augmented strings or sentential forms of specified types that depend on the style of the grammar that is being sought, are variously known as ''characterizations'', ''covering rules'', ''productions'', ''rewrite rules'', ''subsumptions'', ''transformations'', or ''typing rules''. These are collected together into a set <math>\mathfrak{K}</math> that serves to complete the definition of the formal grammar in question.

Line 2,116: Line 2,116:

=====2.3.1.1. Grammar 1=====

−

Grammar 1 is something of a misnomer. It is nowhere near exemplifying any kind of a standard form and it is only intended as a starting point for the initiation of more respectable grammars. Such as it is, it uses the terminal alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> it specifies <math>\mathfrak{Q} = \varnothing,</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}</math> as listed in the following display.

+

Grammar 1 is something of a misnomer. It is nowhere near exemplifying any kind of a standard form and it is only intended as a starting point for the initiation of more respectable grammars. Such as it is, it uses the terminal alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}\!</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> it specifies <math>\mathfrak{Q} = \varnothing,\!</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}\!</math> as listed in the following display.

Line 2,131: Line 2,131:

& S

& :>

−

& m_1 \ = \ ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}

+

& m_1 \ = \ {}^{\backprime\backprime} \operatorname{~} {}^{\prime\prime}

\\

2.

Line 2,146: Line 2,146:

& S

& :>

−

& \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}

+

& \operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}

\\

5.

Line 2,156: Line 2,156:

& S

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

\end{array}</math>

Line 2,169: Line 2,169:

<li value="5"> The concept of a sentence in <math>\mathfrak{L}</math> covers any concatenation of sentences in <math>\mathfrak{L},</math> in effect, any number of freely chosen sentences that are available to be concatenated one after another.</li>

−

<li value="6"> The concept of a sentence in <math>\mathfrak{L}</math> covers any surcatenation of sentences in <math>\mathfrak{L},</math> in effect, any string that opens with a <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime},</math> continues with a sentence, possibly empty, follows with a finite number of phrases of the form <math>^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> and closes with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

+

<li value="6"> The concept of a sentence in <math>\mathfrak{L}</math> covers any surcatenation of sentences in <math>\mathfrak{L},</math> in effect, any string that opens with a <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime},</math> continues with a sentence, possibly empty, follows with a finite number of phrases of the form <math>{}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S,</math> and closes with a <math>{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></li>

</ol>

Line 2,207: Line 2,207:

|}

−

There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math> In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math> As a result, this style of characterization has to be classified as ''true but not very informative''. If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity. In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.

+

There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math> In particular, since it implies that <math>S :> \underline\varepsilon,\!</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},\!</math> for any formal language <math>\mathfrak{L},\!</math> the empty string <math>\varepsilon\!</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math> As a result, this style of characterization has to be classified as ''true but not very informative''. If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity. In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.

Suppose that one tries to deal with this problem by eliminating each use of the kleene star operation, by reducing it to a purely finitary set of steps, or by finding an alternative way to cover the sublanguage that it is used to generate. This amounts, in effect, to ''recognizing a type'', a complex process that involves the following steps:

Line 2,217: Line 2,217:

In sum, one introduces a non-terminal symbol for each type of sentence and each ''part of speech'' or sentential component that is generated by means of iteration or recursion under the ruling constraints of the grammar. In order to do this one needs to analyze the iteration of each grammatical operation in a way that is analogous to a mathematically inductive definition, but further in a way that is not forced explicitly to recognize a distinct and separate type of expression merely to account for and to recount every increment in the parameter of iteration.

−

Returning to the case of the cactus language, the process of recognizing an iterative type or a recursive type can be illustrated in the following way. The operative phrases in the simplest sort of recursive definition are its ''initial part'' and its ''generic part''. For the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:

+

Returning to the case of the cactus language, the process of recognizing an iterative type or a recursive type can be illustrated in the following way. The operative phrases in the simplest sort of recursive definition are its ''initial part'' and its ''generic part''. For the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:

{| align="center" cellpadding="8" width="90%"

Line 2,234: Line 2,234:

& \operatorname{Surc}_{j=1}^0

& =

−

& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}

\\ \\

& \operatorname{Surc}_{j=1}^k S_j

Line 2,261: Line 2,261:

& \operatorname{Surc}

& :>

−

& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}

\\ \\

& \operatorname{Surc}

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, S \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\ \\

& \operatorname{Surc}

& :>

−

& \operatorname{Surc} \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& \operatorname{Surc} \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\end{array}</math>

|}

Line 2,282: Line 2,282:

One way to analyze the surcatenation of any number of sentences is to introduce an auxiliary type of string, not in general a sentence, but a proper component of any sentence that is formed by surcatenation. Doing this brings one to the following definition:

−

A ''tract'' is a concatenation of a finite sequence of sentences, with a literal comma <math>^{\backprime\backprime} \operatorname{,} ^{\prime\prime}</math> interpolated between each pair of adjacent sentences. Thus, a typical tract <math>T\!</math> takes the form:

+

A ''tract'' is a concatenation of a finite sequence of sentences, with a literal comma <math>{}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}</math> interpolated between each pair of adjacent sentences. Thus, a typical tract <math>T\!</math> takes the form:

{| align="center" cellpadding="8" width="90%"

Line 2,291: Line 2,291:

& S_1

& \cdot

−

& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}

+

& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}

& \cdot

& \ldots

& \cdot

−

& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}

+

& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}

& \cdot

& S_k

Line 2,312: Line 2,312:

<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 2}\!</math>

| align="right" style="border-right:1px solid black;" width="50%" |

−

<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>

+

<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math>

|-

| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |

Line 2,339: Line 2,339:

& S

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

6.

Line 2,349: Line 2,349:

& T

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S

+

& T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S

\\

\end{array}</math>

Line 2,356: Line 2,356:

−

In this rendition, a string of type <math>T\!</math> is not in general a sentence itself but a proper ''part of speech'', that is, a strictly ''lesser'' component of a sentence in any suitable ordering of sentences and their components. In order to see how the grammatical category <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.</math> In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> with the substitutions <math>T = \varepsilon</math> and <math>S = \varepsilon</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.</math>

+

In this rendition, a string of type <math>T\!</math> is not in general a sentence itself but a proper ''part of speech'', that is, a strictly ''lesser'' component of a sentence in any suitable ordering of sentences and their components. In order to see how the grammatical category <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.\!</math> In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S,\!</math> with the substitutions <math>T = \varepsilon\!</math> and <math>S = \varepsilon\!</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}.~\!</math>

−

Grammar 2 achieves a portion of its success through a higher degree of intermediate organization. Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math> but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly. Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further.

+

Grammar 2 achieves a portion of its success through a higher degree of intermediate organization. Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math> but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly. Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further.

=====2.3.1.3. Grammar 3=====

Line 2,370: Line 2,370:

|}

−

When there is no possibility of confusion, the letter <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes. The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math> In effect, <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math> In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>

+

When there is no possibility of confusion, the letter <math>{}^{\backprime\backprime} R \, {}^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes. The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>{}^{\backprime\backprime} R \, {}^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math> In effect, <math>{}^{\backprime\backprime} R \, {}^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math> In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>

−

A ''foil'' is a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract. Thus, a typical foil <math>F\!</math> has the form:

+

A ''foil'' is a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime},</math> where <math>T\!</math> is a tract. Thus, a typical foil <math>F\!</math> has the form:

{| align="center" cellpadding="8" width="90%"

Line 2,379: Line 2,379:

F

& =

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime}

& \cdot

& S_1

& \cdot

−

& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}

+

& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}

& \cdot

& \ldots

& \cdot

−

& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}

+

& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}

& \cdot

& S_k

& \cdot

−

& ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

\end{array}</math>

|}

−

This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math> Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math> Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language

+

This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math> Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>{}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}.</math> Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>{}^{\backprime\backprime} F \, {}^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language

−

<math>\mathfrak{C} (\mathfrak{P}).</math> All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter <math>^{\backprime\backprime} F \, ^{\prime\prime}.</math>

+

<math>\mathfrak{C} (\mathfrak{P}).</math> All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter <math>{}^{\backprime\backprime} F \, {}^{\prime\prime}.</math>

Line 2,405: Line 2,405:

<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!</math>

| align="right" style="border-right:1px solid black;" width="50%" |

−

<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>

+

<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} F \, {}^{\prime\prime}, \, {}^{\backprime\backprime} R \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math>

|-

| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |

Line 2,447: Line 2,447:

& F

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

9.

Line 2,457: Line 2,457:

& T

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S

+

& T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S

\\

\end{array}</math>

Line 2,464: Line 2,464:

−

In Grammar 3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types. Rules 4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types. Rule 8 characterizes a foil <math>F\!</math> as a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract. The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.

+

In Grammar 3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types. Rules 4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types. Rule 8 characterizes a foil <math>F\!</math> as a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime},</math> where <math>T\!</math> is a tract. The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.

−

At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.

+

At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),\!</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.

Any degree of reflection on this difficulty raises the general question: What is a practical strategy for accounting for the empty string in the organization of any formal language that counts it among its sentences? One answer that presents itself is this: If the empty string belongs to a formal language, it suffices to count it once at the beginning of the formal account that enumerates its sentences and then to move on to more interesting materials.

−

Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> in other words, the formal language <math>\operatorname{PARCE}</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:

+

Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> in other words, the formal language <math>\operatorname{PARCE}\!</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:

Line 2,499: Line 2,499:

For brevity in the present case, and to serve as a generic device in any similar array of situations, let <math>S\!</math> be the type of an arbitrary sentence, possibly empty, and let <math>S'\!</math> be the type of a specifically non-empty sentence. In addition, let <math>\underline\varepsilon</math> be the type of the empty sentence, in effect, the language

−

<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign <math>^{\backprime\backprime} + ^{\prime\prime}</math> signify a disjoint union of types. In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:

+

<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> signify a disjoint union of types. In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:

{| align="center" cellpadding="8" width="90%"

Line 2,509: Line 2,509:

=====2.3.1.4. Grammar 4=====

−

If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> and <math>^{\backprime\backprime} T \, ^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>^{\backprime\backprime} S \, ^{\prime\prime}, \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.

+

If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} T \, {}^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}, \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T' \, {}^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T' \, {}^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.

Line 2,517: Line 2,517:

<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!</math>

| align="right" style="border-right:1px solid black;" width="50%" |

−

<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \}</math>

+

<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T' \, {}^{\prime\prime} \, \}</math>

|-

| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |

Line 2,544: Line 2,544:

& S'

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

6.

Line 2,564: Line 2,564:

& T'

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S

+

& T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S

\\

\end{array}</math>

Line 2,575: Line 2,575:

There does not appear to be anything radically wrong with trying this approach to types. It is reasonable and consistent in its underlying principle, and it provides a rational and a homogeneous strategy toward all parts of speech, but it does require an extra amount of conceptual overhead, in that every non-trivial type has to be split into two parts and comprehended in two stages. Consequently, in view of the largely practical difficulties of making the requisite distinctions for every intermediate symbol, it is a common convention, whenever possible, to restrict intermediate types to covering exclusively non-empty strings.

−

For the sake of future reference, it is convenient to refer to this restriction on intermediate symbols as the ''intermediate significance'' constraint. It can be stated in a compact form as a condition on the relations between non-terminal symbols <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}</math> and sentential forms <math>W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.</math>

+

For the sake of future reference, it is convenient to refer to this restriction on intermediate symbols as the ''intermediate significance'' constraint. It can be stated in a compact form as a condition on the relations between non-terminal symbols <math>q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q}</math> and sentential forms <math>W \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.</math>

Line 2,598: Line 2,598:

& q

& =

−

& ^{\backprime\backprime} S \, ^{\prime\prime}

+

& {}^{\backprime\backprime} S \, {}^{\prime\prime}

\\

\end{array}</math>

Line 2,605: Line 2,605:

−

If this is beginning to sound like a monotone condition, then it is not absurd to sharpen the resemblance and render the likeness more acute. This is done by declaring a couple of ordering relations, denoting them under variant interpretations by the same sign, <math>^{\backprime\backprime}\!< \, ^{\prime\prime}.</math>

+

If this is beginning to sound like a monotone condition, then it is not absurd to sharpen the resemblance and render the likeness more acute. This is done by declaring a couple of ordering relations, denoting them under variant interpretations by the same sign, <math>{}^{\backprime\backprime}\!< \, {}^{\prime\prime}.</math>

−

# The ordering <math>^{\backprime\backprime}\!< \, ^{\prime\prime}</math> on the set of non-terminal symbols, <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},</math> ordains the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> to be strictly prior to every intermediate symbol. This is tantamount to the axiom that <math>^{\backprime\backprime} S \, ^{\prime\prime} < q,</math> for all <math>q \in \mathfrak{Q}.</math>

+

# The ordering <math>{}^{\backprime\backprime}\!< \, {}^{\prime\prime}</math> on the set of non-terminal symbols, <math>q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q},</math> ordains the initial symbol <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> to be strictly prior to every intermediate symbol. This is tantamount to the axiom that <math>{}^{\backprime\backprime} S \, {}^{\prime\prime} < q,</math> for all <math>q \in \mathfrak{Q}.</math>

−

# The ordering <math>^{\backprime\backprime}\!< \, ^{\prime\prime}</math> on the collection of sentential forms, <math>W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,</math> ordains the empty string to be strictly minor to every other sentential form. This is stipulated in the axiom that <math>\varepsilon < W,</math> for every non-empty sentential form <math>W.\!</math>

+

# The ordering <math>{}^{\backprime\backprime}\!< \, {}^{\prime\prime}</math> on the collection of sentential forms, <math>W \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,</math> ordains the empty string to be strictly minor to every other sentential form. This is stipulated in the axiom that <math>\varepsilon < W,</math> for every non-empty sentential form <math>W.\!</math>

Given these two orderings, the constraint in question on intermediate significance can be stated as follows:

Line 2,628: Line 2,628:

& q

& >

−

& ^{\backprime\backprime} S \, ^{\prime\prime}

+

& {}^{\backprime\backprime} S \, {}^{\prime\prime}

\\

\text{then}

Line 2,657: Line 2,657:

& :>

& W,

−

& \text{with} \ q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+

+

& \text{with} \ q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+

\\

\end{array}</math>

Line 2,664: Line 2,664:

A grammar that fits into this mold is called a ''context-free grammar''. The first type of rewrite rule is referred to as a ''special production'', while the second type of rewrite rule is called an ''ordinary production''. An ''ordinary derivation'' is one that employs only ordinary productions. In ordinary productions, those that have the form <math>q :> W,\!</math> the replacement string <math>W\!</math> is never the empty string, and so the lengths of the augmented strings or the sentential forms that follow one another in an ordinary derivation, on account of using the ordinary types of rewrite rules, never decrease at any stage of the process, up to and including the terminal string that is finally generated by the grammar. This type of feature is known as the ''non-contracting property'' of productions, derivations, and grammars. A grammar is said to have the property if all of its covering productions, with the possible exception of <math>S :> \varepsilon,</math> are non-contracting. In particular, context-free grammars are special cases of non-contracting grammars. The presence of the non-contracting property within a formal grammar makes the length of the augmented string available as a parameter that can figure into mathematical inductions and motivate recursive proofs, and this handle on the generative process makes it possible to establish the kinds of results about the generated language that are not easy to achieve in more general cases, nor by any other means even in these brands of special cases.

−

Grammar 5 is a context-free grammar for the painted cactus language that uses <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with <math>\mathfrak{K}</math> as listed in the next display.

+

Grammar 5 is a context-free grammar for the painted cactus language that uses <math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \},</math> with <math>\mathfrak{K}</math> as listed in the next display.

Line 2,672: Line 2,672:

<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 5}\!</math>

| align="right" style="border-right:1px solid black;" width="50%" |

−

<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>

+

<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math>

|-

| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |

Line 2,704: Line 2,704:

& S'

& :>

−

& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}

\\

7.

& S'

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

8.

& T

& :>

−

& ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}

\\

9.

Line 2,724: Line 2,724:

& T

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}

+

& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}

\\

11.

& T

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S'

+

& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, S'

\\

\end{array}</math>

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=====2.3.1.6. Grammar 6=====

−

Grammar 6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.

+

Grammar 6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} F \, {}^{\prime\prime}, \, {}^{\backprime\backprime} R \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.

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{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"

| align="left" style="border-left:1px solid black;" width="50%" |

−

<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}\!</math>

+

<math>{\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}}\!</math>

| align="right" style="border-right:1px solid black;" width="50%" |

−

<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>

+

<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} F \, {}^{\prime\prime}, \, {}^{\backprime\backprime} R \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}\!</math>

|-

| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |

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& F

& :>

−

& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}

\\

10.

& F

& :>

−

& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}

\\

11.

& T

& :>

−

& ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}

+

& {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}

\\

12.

Line 2,815: Line 2,815:

& T

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}

+

& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}

\\

14.

& T

& :>

−

& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S'

+

& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, S'

\\

\end{array}</math>

Line 2,849: Line 2,849:

|-

|  

−

| <math>\operatorname{Surc}_{j=1}^k S_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}</math>

+

| <math>\operatorname{Surc}_{j=1}^k S_j \ = \ {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, S_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, \ldots \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, S_k \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}</math>

|-

|  

Line 2,859: Line 2,859:

It is fitting to wrap up the foregoing developments by summarizing the notion of a formal grammar that appeared to evolve in the present case. For the sake of future reference and the chance of a wider application, it is also useful to try to extract the scheme of a formalization that potentially holds for any formal language. The following presentation of the notion of a formal grammar is adapted, with minor modifications, from the treatment in (DDQ, 60–61).

−

A ''formal grammar'' <math>\mathfrak{G}</math> is given by a four-tuple <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> that takes the following form of description:

+

A ''formal grammar'' <math>\mathfrak{G}</math> is given by a four-tuple <math>\mathfrak{G} = ( \, {}^{\backprime\backprime} S \, {}^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> that takes the following form of description:

−

<li><math>^{\backprime\backprime} S \, ^{\prime\prime}</math> is the ''initial'', ''special'', ''start'', or ''sentence'' symbol. Since the letter <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> serves this function only in a special setting, its employment in this role need not create any confusion with its other typical uses as a string variable or as a sentence variable.</li>

+

<li><math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> is the ''initial'', ''special'', ''start'', or ''sentence'' symbol. Since the letter <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> serves this function only in a special setting, its employment in this role need not create any confusion with its other typical uses as a string variable or as a sentence variable.</li>

−

<li><math>\mathfrak{Q} = \{ q_1, \ldots, q_m \}</math> is a finite set of ''intermediate symbols'', all distinct from <math>^{\backprime\backprime} S \, ^{\prime\prime}.</math></li>

+

<li><math>\mathfrak{Q} = \{ q_1, \ldots, q_m \}</math> is a finite set of ''intermediate symbols'', all distinct from <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}.</math></li>

−

<li><math>\mathfrak{A} = \{ a_1, \dots, a_n \}</math> is a finite set of ''terminal symbols'', also known as the ''alphabet'' of <math>\mathfrak{G},</math> all distinct from <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> and disjoint from <math>\mathfrak{Q}.</math> Depending on the particular conception of the language <math>\mathfrak{L}</math> that is ''covered'', ''generated'', ''governed'', or ''ruled'' by the grammar <math>\mathfrak{G},</math> that is, whether <math>\mathfrak{L}</math> is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe <math>\mathfrak{A}</math> as the ''alphabet'', ''lexicon'', ''vocabulary'', ''liturgy'', or ''phrase book'' of both the grammar <math>\mathfrak{G}</math> and the language <math>\mathfrak{L}</math> that it regulates.</li>

+

<li><math>\mathfrak{A} = \{ a_1, \dots, a_n \}</math> is a finite set of ''terminal symbols'', also known as the ''alphabet'' of <math>\mathfrak{G},</math> all distinct from <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> and disjoint from <math>\mathfrak{Q}.</math> Depending on the particular conception of the language <math>\mathfrak{L}</math> that is ''covered'', ''generated'', ''governed'', or ''ruled'' by the grammar <math>\mathfrak{G},</math> that is, whether <math>\mathfrak{L}</math> is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe <math>\mathfrak{A}</math> as the ''alphabet'', ''lexicon'', ''vocabulary'', ''liturgy'', or ''phrase book'' of both the grammar <math>\mathfrak{G}</math> and the language <math>\mathfrak{L}</math> that it regulates.</li>

<li><math>\mathfrak{K}</math> is a finite set of ''characterizations''. Depending on how they come into play, these are variously described as ''covering rules'', ''formations'', ''productions'', ''rewrite rules'', ''subsumptions'', ''transformations'', or ''typing rules''.</li>

Line 2,877: Line 2,877:

−

<li>The symbols in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> form the ''augmented alphabet'' of <math>\mathfrak{G}.</math></li>

+

<li>The symbols in <math>\{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> form the ''augmented alphabet'' of <math>\mathfrak{G}.</math></li>

−

<li>The symbols in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}</math> are the ''non-terminal symbols'' of <math>\mathfrak{G}.</math></li>

+

<li>The symbols in <math>\{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q}</math> are the ''non-terminal symbols'' of <math>\mathfrak{G}.</math></li>

<li>The symbols in <math>\mathfrak{Q} \cup \mathfrak{A}</math> are the ''non-initial symbols'' of <math>\mathfrak{G}.</math></li>

−

<li>The strings in <math>( \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> are the ''augmented strings'' for <math>\mathfrak{G}.</math></li>

+

<li>The strings in <math>( \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> are the ''augmented strings'' for <math>\mathfrak{G}.</math></li>

−

<li>The strings in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*</math> are the ''sentential forms'' for <math>\mathfrak{G}.</math></li>

+

<li>The strings in <math>\{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*</math> are the ''sentential forms'' for <math>\mathfrak{G}.</math></li>

</ol>

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In this scheme, <math>S_1\!</math> and <math>S_2\!</math> are members of the augmented strings for <math>\mathfrak{G},</math> more precisely, <math>S_1\!</math> is a non-empty string and a sentential form over <math>\mathfrak{G},</math> while <math>S_2\!</math> is a possibly empty string and also a sentential form over <math>\mathfrak{G}.</math>

−

Here also, <math>q\!</math> is a non-terminal symbol, that is, <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},</math> while <math>Q_1, Q_2,\!</math> and <math>W\!</math> are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, <math>Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.</math>

+

Here also, <math>q\!</math> is a non-terminal symbol, that is, <math>q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q},</math> while <math>Q_1, Q_2,\!</math> and <math>W\!</math> are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, <math>Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.</math>

In practice, the couplets in <math>\mathfrak{K}</math> are used to ''derive'', to ''generate'', or to ''produce'' sentences of the corresponding language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}).</math> The language <math>\mathfrak{L}</math> is then said to be ''governed'', ''licensed'', or ''regulated'' by the grammar <math>\mathfrak{G},</math> a circumstance that is expressed in the form <math>\mathfrak{L} = \langle \mathfrak{G} \rangle.</math> In order to facilitate this active employment of the grammar, it is conventional to write the abstract characterization <math>(S_1, S_2)\!</math> and the specific characterization <math>(Q_1 \cdot q \cdot Q_2, \ Q_1 \cdot W \cdot Q_2)</math> in the following forms, respectively:

Line 2,919: Line 2,919:

In this usage, the characterization <math>S_1 :> S_2\!</math> is tantamount to a grammatical license to transform a string of the form <math>Q_1 \cdot q \cdot Q_2</math> into a string of the form <math>Q1 \cdot W \cdot Q2,</math> in effect, replacing the non-terminal symbol <math>q\!</math> with the non-initial string <math>W\!</math> in any selected, preserved, and closely adjoining context of the form <math>Q1 \cdot \underline{~~~} \cdot Q2.</math> In this application the notation <math>S_1 :> S_2\!</math> can be read to say that <math>S_1\!</math> ''produces'' <math>S_2\!</math> or that <math>S_1\!</math> ''transforms into'' <math>S_2.\!</math>

−

An ''immediate derivation'' in <math>\mathfrak{G}</math> is an ordered pair <math>(W, W')\!</math> of sentential forms in <math>\mathfrak{G}</math> such that:

+

An ''immediate derivation'' in <math>\mathfrak{G}\!</math> is an ordered pair <math>(W, W^\prime)\!</math> of sentential forms in <math>\mathfrak{G}\!</math> such that:

{| align="center" cellpadding="8" width="90%"

Line 2,952: Line 2,952:

|}

−

The language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle</math> that is ''generated'' by the formal grammar <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> is the set of strings over the terminal alphabet <math>\mathfrak{A}</math> that are derivable from the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> by way of the intermediate symbols in <math>\mathfrak{Q}</math> according to the characterizations in <math>\mathfrak{K}.</math> In sum:

+

The language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle</math> that is ''generated'' by the formal grammar <math>\mathfrak{G} = ( \, {}^{\backprime\backprime} S \, {}^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> is the set of strings over the terminal alphabet <math>\mathfrak{A}</math> that are derivable from the initial symbol <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> by way of the intermediate symbols in <math>\mathfrak{Q}</math> according to the characterizations in <math>\mathfrak{K}.</math> In sum:

{| align="center" cellpadding="8" width="90%"

−

| <math>\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, ^{\backprime\backprime} S \, ^{\prime\prime} \, :\!*\!:> \, W \, \}.</math>

+

| <math>\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, :\!*\!:> \, W \, \}.</math>

|}

Line 2,998: Line 2,998:

Any style of declarative programming, also called ''logic programming'', depends on a capacity, as embodied in a programming language or other formal system, to describe the relation between problems and solutions in logical terms. A recurring problem in building this capacity is in bridging the gap between ostensibly non-logical orders and the logical orders that are used to describe and to represent them. For instance, to mention just a couple of the most pressing cases, and the ones that are currently proving to be the most resistant to a complete analysis, one has the orders of dynamic evolution and rhetorical transition that manifest themselves in the process of inquiry and in the communication of its results.

−

This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).</math> It is the intention of this work to interpret this language for propositional logic, and thus to use it as a sentential calculus, an order of reasoning that forms an active ingredient and a significant component of all logical reasoning. To describe this language, the standard devices of formal grammars and formal language theory are more than adequate, but this only raises the next question: What sorts of devices are exactly adequate, and fit the task to a "T"? The ultimate desire is to turn the tables on the order of description, and so begins a process of eversion that evolves to the point of asking: To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation? In other words: How well can the language be described by using the language itself to do so?

+

This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).\!</math> It is the intention of this work to interpret this language for propositional logic, and thus to use it as a sentential calculus, an order of reasoning that forms an active ingredient and a significant component of all logical reasoning. To describe this language, the standard devices of formal grammars and formal language theory are more than adequate, but this only raises the next question: What sorts of devices are exactly adequate, and fit the task to a "T"? The ultimate desire is to turn the tables on the order of description, and so begins a process of eversion that evolves to the point of asking: To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation? In other words: How well can the language be described by using the language itself to do so?

In order to speak to these questions, I have to express what a grammar says about a language in terms of what a language can say on its own. In effect, it is necessary to analyze the kinds of meaningful statements that grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements that the syntactic ''sentences'' of the cactus language might be interpreted as making about the very same topics. So far in the present discussion, the sentences of the cactus language do not make any meaningful statements at all, much less any meaningful statements about languages and their constitutions. As of yet, these sentences subsist in the form of purely abstract, formal, and uninterpreted combinatorial constructions.

Line 3,028: Line 3,028:

Having broached the distinction between propositions and sentences, one can see its similarity to the distinction between numbers and numerals. What are the implications of the foregoing considerations for reasoning about propositions and for the realm of reckonings in sentential logic? If the purpose of a sentence is just to denote a proposition, then the proposition is just the object of whatever sign is taken for a sentence. This means that the computational manifestation of a piece of reasoning about propositions amounts to a process that takes place entirely within a language of sentences, a procedure that can rationalize its account by referring to the denominations of these sentences among propositions.

−

The application of these considerations in the immediate setting is this: Do not worry too much about what roles the empty string <math>\varepsilon \, = \, ^{\backprime\backprime\prime\prime}</math> and the blank symbol <math>m_1 \, = \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}</math> are supposed to play in a given species of formal languages. As it happens, it is far less important to wonder whether these types of formal tokens actually constitute genuine sentences than it is to decide what equivalence classes it makes sense to form over all of the sentences in the resulting language, and only then to bother about what equivalence classes these limiting cases of sentences are most conveniently taken to represent.

+

The application of these considerations in the immediate setting is this: Do not worry too much about what roles the empty string <math>\varepsilon \, = \, ^{\backprime\backprime\prime\prime}</math> and the blank symbol <math>m_1 \, = \, {}^{\backprime\backprime} \operatorname{~} {}^{\prime\prime}</math> are supposed to play in a given species of formal languages. As it happens, it is far less important to wonder whether these types of formal tokens actually constitute genuine sentences than it is to decide what equivalence classes it makes sense to form over all of the sentences in the resulting language, and only then to bother about what equivalence classes these limiting cases of sentences are most conveniently taken to represent.

These concerns about boundary conditions betray a more general issue. Already by this point in discussion the limits of the purely syntactic approach to a language are beginning to be visible. It is not that one cannot go a whole lot further by this road in the analysis of a particular language and in the study of languages in general, but when it comes to the questions of understanding the purpose of a language, of extending its usage in a chosen direction, or of designing a language for a particular set of uses, what matters above all else are the ''pragmatic equivalence classes'' of signs that are demanded by the application and intended by the designer, and not so much the peculiar characters of the signs that represent these classes of practical meaning.

Line 3,063: Line 3,063:

|}

−

It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differ. The production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math> Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' the qualifiers ''possible'' and ''possibly'' are the critical elements in these statements, and they are crucial to the meaning of what is actually being implied. In effect, these qualifications reverse the direction of implication, yielding <math>^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}</math> as the best analogue for the sense of the production.

+

It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differ. The production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math> Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' the qualifiers ''possible'' and ''possibly'' are the critical elements in these statements, and they are crucial to the meaning of what is actually being implied. In effect, these qualifications reverse the direction of implication, yielding <math>{}^{\backprime\backprime} \, q \Leftarrow W \, {}^{\prime\prime}</math> as the best analogue for the sense of the production.

−

One way to sum this up is to say that non-terminal symbols have the significance of hypotheses. The terminal strings form the empirical matter of a language, while the non-terminal symbols mark the patterns or the types of substrings that can be noticed in the profusion of data. If one observes a portion of a terminal string that falls into the pattern of the sentential form <math>W,\!</math> then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type <math>q\!</math> but even comes to be generated under the auspices of the non-terminal symbol <math>^{\backprime\backprime} q ^{\prime\prime}.</math>

+

One way to sum this up is to say that non-terminal symbols have the significance of hypotheses. The terminal strings form the empirical matter of a language, while the non-terminal symbols mark the patterns or the types of substrings that can be noticed in the profusion of data. If one observes a portion of a terminal string that falls into the pattern of the sentential form <math>W,\!</math> then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type <math>q\!</math> but even comes to be generated under the auspices of the non-terminal symbol <math>{}^{\backprime\backprime} q {}^{\prime\prime}.</math>

A moment's reflection on the issue of style, giving due consideration to the received array of stylistic choices, ought to inspire at least the question: "Are these the only choices there are?" In the present setting, there are abundant indications that other options, more differentiated varieties of description and more integrated ways of approaching individual languages, are likely to be conceivable, feasible, and even more ultimately viable. If a suitably generic style, one that incorporates the full scope of logical combinations and operations, is broadly available, then it would no longer be necessary, or even apt, to argue in universal terms about which style is best, but more useful to investigate how we might adapt the local styles to the local requirements. The medium of a generic style would yield a viable compromise between additive and multiplicative canons, and render the choice between parallel and serial a false alternative, at least, when expressed in the globally exclusive terms that are currently most commonly adopted to pose it.

Line 3,103: Line 3,103:

|}

−

The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2</math> of the formal languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2</math> without the extra <math>\times</math>'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear. One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.

+

The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2\!</math> of the formal languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2\!</math> without the extra <math>\times\!</math>’s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear. One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.

A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified. It is assumed that one knows enough to tell if two strictures are equivalent as pieces of information, but any more determinate indications, like names for the places that are mentioned in the stricture, or bounds on the number of places that are involved, are regarded as being extraneous impositions, outside the proper concern of the definition, no matter how convenient they are found to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape:

:{| cellpadding="8"

−

| <math>^{\backprime\backprime}</math>

+

| <math>{}^{\backprime\backprime}</math>

| <math>\ldots \times X \times Q \times X \times \ldots</math>

−

| <math>^{\prime\prime}</math>

+

| <math>{}^{\prime\prime}</math>

|}

Line 3,126: Line 3,126:

|

<math>\begin{array}{ccccccc}

−

^{\backprime\backprime} Q ^{\prime\prime}

+

{}^{\backprime\backprime} Q {}^{\prime\prime}

& , &

−

^{\backprime\backprime} X \times Q \times X ^{\prime\prime}

+

{}^{\backprime\backprime} X \times Q \times X {}^{\prime\prime}

& , &

−

^{\backprime\backprime} X \times X \times Q \times X \times X ^{\prime\prime}

+

{}^{\backprime\backprime} X \times X \times Q \times X \times X {}^{\prime\prime}

& , &

\ldots

Line 3,162: Line 3,162:

|

<math>\begin{matrix}

−

^{\backprime\backprime} X ^{\prime\prime} &

+

{}^{\backprime\backprime} X {}^{\prime\prime} &

−

^{\backprime\backprime} P ^{\prime\prime} &

+

{}^{\backprime\backprime} P {}^{\prime\prime} &

−

^{\backprime\backprime} Q ^{\prime\prime} \\

+

{}^{\backprime\backprime} Q {}^{\prime\prime} \\

\\

−

^{\backprime\backprime} X \times X ^{\prime\prime} &

+

{}^{\backprime\backprime} X \times X {}^{\prime\prime} &

−

^{\backprime\backprime} X \times P ^{\prime\prime} &

+

{}^{\backprime\backprime} X \times P {}^{\prime\prime} &

−

^{\backprime\backprime} X \times Q ^{\prime\prime} \\

+

{}^{\backprime\backprime} X \times Q {}^{\prime\prime} \\

\\

−

^{\backprime\backprime} P \times X ^{\prime\prime} &

+

{}^{\backprime\backprime} P \times X {}^{\prime\prime} &

−

^{\backprime\backprime} P \times P ^{\prime\prime} &

+

{}^{\backprime\backprime} P \times P {}^{\prime\prime} &

−

^{\backprime\backprime} P \times Q ^{\prime\prime} \\

+

{}^{\backprime\backprime} P \times Q {}^{\prime\prime} \\

\\

−

^{\backprime\backprime} Q \times X ^{\prime\prime} &

+

{}^{\backprime\backprime} Q \times X {}^{\prime\prime} &

−

^{\backprime\backprime} Q \times P ^{\prime\prime} &

+

{}^{\backprime\backprime} Q \times P {}^{\prime\prime} &

−

^{\backprime\backprime} Q \times Q ^{\prime\prime} \\

+

{}^{\backprime\backprime} Q \times Q {}^{\prime\prime} \\

\end{matrix}</math>

|}

Line 3,186: Line 3,186:

<math>\begin{array}{lcccc}

\text{High:}

−

& ^{\backprime\backprime} P \times P ^{\prime\prime}

+

& {}^{\backprime\backprime} P \times P {}^{\prime\prime}

−

& ^{\backprime\backprime} P \times Q ^{\prime\prime}

+

& {}^{\backprime\backprime} P \times Q {}^{\prime\prime}

−

& ^{\backprime\backprime} Q \times P ^{\prime\prime}

+

& {}^{\backprime\backprime} Q \times P {}^{\prime\prime}

−

& ^{\backprime\backprime} Q \times Q ^{\prime\prime}

+

& {}^{\backprime\backprime} Q \times Q {}^{\prime\prime}

\\

\text{Med:}

−

& ^{\backprime\backprime} P ^{\prime\prime}

+

& {}^{\backprime\backprime} P {}^{\prime\prime}

−

& ^{\backprime\backprime} X \times P ^{\prime\prime}

+

& {}^{\backprime\backprime} X \times P {}^{\prime\prime}

−

& ^{\backprime\backprime} P \times X ^{\prime\prime}

+

& {}^{\backprime\backprime} P \times X {}^{\prime\prime}

\\

\text{Med:}

−

& ^{\backprime\backprime} Q ^{\prime\prime}

+

& {}^{\backprime\backprime} Q {}^{\prime\prime}

−

& ^{\backprime\backprime} X \times Q ^{\prime\prime}

+

& {}^{\backprime\backprime} X \times Q {}^{\prime\prime}

−

& ^{\backprime\backprime} Q \times X ^{\prime\prime}

+

& {}^{\backprime\backprime} Q \times X {}^{\prime\prime}

\\

\text{Low:}

−

& ^{\backprime\backprime} X ^{\prime\prime}

+

& {}^{\backprime\backprime} X {}^{\prime\prime}

−

& ^{\backprime\backprime} X \times X ^{\prime\prime}

+

& {}^{\backprime\backprime} X \times X {}^{\prime\prime}

\\

\end{array}</math>

Line 3,233: Line 3,233:

# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.

−

Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively. In effect, the subscripting by the indices <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.

+

Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}\!</math> and <math>(\mathfrak{L}_2)_{[2]},\!</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively. In effect, the subscripting by the indices <math>{}^{\backprime\backprime} [1] {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} [2] {}^{\prime\prime}\!</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.

In order to systematize the relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:

−

The <math>j^\text{th}\!</math> ''excerpt'' of a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime},</math> regarded within a frame of discussion where the number of places is limited to <math>k,\!</math> is the stricture of the form <math>^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, ^{\prime\prime}.</math> In the proper context, this can be written more succinctly as the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.

+

The <math>j^\text{th}\!</math> ''excerpt'' of a stricture of the form <math>{}^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, {}^{\prime\prime},</math> regarded within a frame of discussion where the number of places is limited to <math>k,\!</math> is the stricture of the form <math>{}^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, {}^{\prime\prime}.</math> In the proper context, this can be written more succinctly as the stricture <math>{}^{\backprime\backprime} \, (S_j)_{[j]} \, {}^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.

−

The <math>j^\text{th}\!</math> ''extract'' of a strait of the form <math>S_1 \times \ldots \times S_k,\!</math> constrained to a frame of discussion where the number of places is restricted to <math>k,\!</math> is the strait of the form <math>X \times \ldots \times S_j \times \ldots \times X.</math> In the appropriate context, this can be denoted more succinctly by the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.

+

The <math>j^\text{th}\!</math> ''extract'' of a strait of the form <math>S_1 \times \ldots \times S_k,\!</math> constrained to a frame of discussion where the number of places is restricted to <math>k,\!</math> is the strait of the form <math>X \times \ldots \times S_j \times \ldots \times X.</math> In the appropriate context, this can be denoted more succinctly by the stricture <math>{}^{\backprime\backprime} \, (S_j)_{[j]} \, {}^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.

−

In these terms, a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}</math> can be expressed in terms of simpler strictures, to wit, as a conjunction of its <math>k\!</math> excerpts:

+

In these terms, a stricture of the form <math>{}^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, {}^{\prime\prime}</math> can be expressed in terms of simpler strictures, to wit, as a conjunction of its <math>k\!</math> excerpts:

{| align="center" cellpadding="8" width="90%"

|

<math>\begin{array}{lll}

−

^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}

+

{}^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, {}^{\prime\prime}

& = &

−

^{\backprime\backprime} \, (S_1)_{[1]} \, ^{\prime\prime}

+

{}^{\backprime\backprime} \, (S_1)_{[1]} \, {}^{\prime\prime}

\, \land \, \ldots \, \land \,

−

^{\backprime\backprime} \, (S_k)_{[k]} \, ^{\prime\prime}.

+

{}^{\backprime\backprime} \, (S_k)_{[k]} \, {}^{\prime\prime}.

\end{array}</math>

|}

Line 3,276: Line 3,276:

In this Subsection, I discuss the ''mechanics'' of parsing the cactus language into the corresponding class of computational data structures. This provides each sentence of the language with a translation into a computational form that articulates its syntactic structure and prepares it for automated modes of processing and evaluation. For this purpose, it is necessary to describe the target data structures at a fairly high level of abstraction only, ignoring the details of address pointers and record structures and leaving the more operational aspects of implementation to the imagination of prospective programmers. In this way, I can put off to another stage of elaboration and refinement the description of the program that constructs these pointers and operates on these graph-theoretic data structures.

−

The structure of a ''painted cactus'', insofar as it presents itself to the visual imagination, can be described as follows. The overall structure, as given by its underlying graph, falls within the species of graph that is commonly known as a ''rooted cactus'', and the only novel feature that it adds to this is that each of its nodes can be ''painted'' with a finite sequence of ''paints'', chosen from a ''palette'' that is given by the parametric set <math>\{ \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.</math>

+

The structure of a ''painted cactus'', insofar as it presents itself to the visual imagination, can be described as follows. The overall structure, as given by its underlying graph, falls within the species of graph that is commonly known as a ''rooted cactus'', and the only novel feature that it adds to this is that each of its nodes can be ''painted'' with a finite sequence of ''paints'', chosen from a ''palette'' that is given by the parametric set <math>\{ \, {}^{\backprime\backprime} \operatorname{~} {}^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.</math>

It is conceivable, from a purely graph-theoretical point of view, to have a class of cacti that are painted but not rooted, and so it is frequently necessary, for the sake of precision, to more exactly pinpoint the target species of graphical structure as a ''painted and rooted cactus'' (PARC).

Line 3,339: Line 3,339:

|}

−

Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).</math> One way to do this proceeds as follows:

+

Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),\!</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}\!</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})\!</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).\!</math> One way to do this proceeds as follows:

Line 3,410: Line 3,410:

A ''substructure'' of a PARC is defined recursively as follows. Starting at the root node of the cactus <math>C,\!</math> any attachment is a substructure of <math>C.\!</math> If a substructure is a blank or a paint, then it constitutes a minimal substructure, meaning that no further substructures of <math>C\!</math> arise from it. If a substructure is a lobe, then each one of its accoutrements is also a substructure of <math>C,\!</math> and has to be examined for further substructures.

−

The concept of substructure can be used to define varieties of deletion and erasure operations that respect the structure of the abstract graph. For the purposes of this depiction, a blank symbol <math>^{\backprime\backprime} ~ ^{\prime\prime}</math> is treated as a ''primer'', in other words, as a ''clear paint'' or a ''neutral tint''. In effect, one is letting <math>m_1 = p_0.\!</math> In this frame of discussion, it is useful to make the following distinction:

+

The concept of substructure can be used to define varieties of deletion and erasure operations that respect the structure of the abstract graph. For the purposes of this depiction, a blank symbol <math>{}^{\backprime\backprime} ~ {}^{\prime\prime}</math> is treated as a ''primer'', in other words, as a ''clear paint'' or a ''neutral tint''. In effect, one is letting <math>m_1 = p_0.\!</math> In this frame of discussion, it is useful to make the following distinction:

# To ''delete'' a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.

Line 3,652: Line 3,652:

|+ '''Table 16. Boolean Functions on Zero Variables'''

|- style="background:whitesmoke"

−

| width="14%" | <math>F\!</math>

+

| width="14%" | <math>F~\!</math>

−

| width="14%" | <math>F\!</math>

+

| width="14%" | <math>F~\!</math>

−

| width="48%" | <math>F()\!</math>

+

| width="48%" | <math>F()~\!</math>

−

| width="24%" | <math>F\!</math>

+

| width="24%" | <math>F~\!</math>

|-

| <math>\underline{0}</math>

| <math>F_0^{(0)}\!</math>

| <math>\underline{0}</math>

−

| <math>(~)</math>

+

| <math>\texttt{(~)}\!</math>

|-

| <math>\underline{1}</math>

| <math>F_1^{(0)}\!</math>

| <math>\underline{1}</math>

−

| <math>((~))</math>

+

| <math>\texttt{((~))}\!</math>

|}

Line 3,676: Line 3,676:

Column 3 lists the functional values for each boolean function, or possibly a boolean element appearing in the guise of a function, for each combination of its domain values.

−

Column 4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats. Here I illustrate also the convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

+

Column 4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats. Here I illustrate also the convention of using the expression <math>{}^{\backprime\backprime} ((~)) {}^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

Table 17 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four.

Line 3,696: Line 3,696:

| width="24%" |  

|-

−

| <math>F_0^{(1)}\!</math>

+

| <math>{F_0^{(1)}}\!</math>

−

| <math>F_{00}^{(1)}\!</math>

+

| <math>{F_{00}^{(1)}}\!</math>

| <math>\underline{0}</math>

−

| <math>(~)</math>

+

| <math>\texttt{(~)}\!</math>

|-

−

| <math>F_1^{(1)}\!</math>

+

| <math>{F_1^{(1)}}\!</math>

−

| <math>F_{01}^{(1)}\!</math>

+

| <math>{F_{01}^{(1)}}\!</math>

| <math>\underline{0}</math>

| <math>\underline{1}</math>

−

| <math>(x)\!</math>

+

| <math>\texttt{(} x \texttt{)}\!</math>

|-

−

| <math>F_2^{(1)}\!</math>

+

| <math>{F_2^{(1)}}\!</math>

−

| <math>F_{10}^{(1)}\!</math>

+

| <math>{F_{10}^{(1)}}\!</math>

| <math>\underline{1}</math>

| <math>\underline{0}</math>

| <math>x\!</math>

|-

−

| <math>F_3^{(1)}\!</math>

+

| <math>{F_3^{(1)}}\!</math>

−

| <math>F_{11}^{(1)}\!</math>

+

| <math>{F_{11}^{(1)}}\!</math>

| <math>\underline{1}</math>

−

| <math>((~))</math>

+

| <math>\texttt{((~))}\!</math>

|}

Line 3,766: Line 3,766:

|-

| <math>F_{0}^{(2)}\!</math>

−

| <math>F_{0000}^{(2)}\!</math>

+

| <math>F_{0000}^{(2)}~\!</math>

| <math>\underline{0}</math>

Line 3,774: Line 3,774:

|-

| <math>F_{1}^{(2)}\!</math>

−

| <math>F_{0001}^{(2)}\!</math>

+

| <math>F_{0001}^{(2)}~\!</math>

| <math>\underline{0}</math>

Line 3,782: Line 3,782:

|-

| <math>F_{2}^{(2)}\!</math>

−

| <math>F_{0010}^{(2)}\!</math>

+

| <math>F_{0010}^{(2)}~\!</math>

| <math>\underline{0}</math>

Line 3,790: Line 3,790:

|-

| <math>F_{3}^{(2)}\!</math>

−

| <math>F_{0011}^{(2)}\!</math>

+

| <math>F_{0011}^{(2)}~\!</math>

| <math>\underline{0}</math>

Line 3,798: Line 3,798:

|-

| <math>F_{4}^{(2)}\!</math>

−

| <math>F_{0100}^{(2)}\!</math>

+

| <math>F_{0100}^{(2)}~\!</math>

| <math>\underline{0}</math>

| <math>\underline{1}</math>

Line 3,806: Line 3,806:

|-

| <math>F_{5}^{(2)}\!</math>

−

| <math>F_{0101}^{(2)}\!</math>

+

| <math>F_{0101}^{(2)}~\!</math>

| <math>\underline{0}</math>

| <math>\underline{1}</math>

Line 3,814: Line 3,814:

|-

| <math>F_{6}^{(2)}\!</math>

−

| <math>F_{0110}^{(2)}\!</math>

+

| <math>F_{0110}^{(2)}~\!</math>

| <math>\underline{0}</math>

| <math>\underline{1}</math>

Line 3,822: Line 3,822:

|-

| <math>F_{7}^{(2)}\!</math>

−

| <math>F_{0111}^{(2)}\!</math>

+

| <math>F_{0111}^{(2)}~\!</math>

| <math>\underline{0}</math>

| <math>\underline{1}</math>

Line 3,830: Line 3,830:

|-

| <math>F_{8}^{(2)}\!</math>

−

| <math>F_{1000}^{(2)}\!</math>

+

| <math>F_{1000}^{(2)}~\!</math>

| <math>\underline{1}</math>

| <math>\underline{0}</math>

Line 3,838: Line 3,838:

|-

| <math>F_{9}^{(2)}\!</math>

−

| <math>F_{1001}^{(2)}\!</math>

+

| <math>F_{1001}^{(2)}~\!</math>

| <math>\underline{1}</math>

| <math>\underline{0}</math>

Line 3,846: Line 3,846:

|-

| <math>F_{10}^{(2)}\!</math>

−

| <math>F_{1010}^{(2)}\!</math>

+

| <math>F_{1010}^{(2)}~\!</math>

| <math>\underline{1}</math>

| <math>\underline{0}</math>

Line 3,854: Line 3,854:

|-

| <math>F_{11}^{(2)}\!</math>

−

| <math>F_{1011}^{(2)}\!</math>

+

| <math>F_{1011}^{(2)}~\!</math>

| <math>\underline{1}</math>

| <math>\underline{0}</math>

Line 3,862: Line 3,862:

|-

| <math>F_{12}^{(2)}\!</math>

−

| <math>F_{1100}^{(2)}\!</math>

+

| <math>F_{1100}^{(2)}~\!</math>

| <math>\underline{1}</math>

Line 3,870: Line 3,870:

|-

| <math>F_{13}^{(2)}\!</math>

−

| <math>F_{1101}^{(2)}\!</math>

+

| <math>F_{1101}^{(2)}~\!</math>

| <math>\underline{1}</math>

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|-

| <math>F_{14}^{(2)}\!</math>

−

| <math>F_{1110}^{(2)}\!</math>

+

| <math>F_{1110}^{(2)}~\!</math>

| <math>\underline{1}</math>

Line 3,886: Line 3,886:

|-

| <math>F_{15}^{(2)}\!</math>

−

| <math>F_{1111}^{(2)}\!</math>

+

| <math>F_{1111}^{(2)}~\!</math>

| <math>\underline{1}</math>

Line 3,932: Line 3,932:

For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table 18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>

−

Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.

+

Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,\!</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}\!</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}\!</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),\!</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.

When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>

Line 3,938: Line 3,938:

For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:

−

: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}</math>

+

: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}\!</math>

−

The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.</math> There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)</math> that is otherwise known as <math>x + y\!.</math>

+

The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}\!</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},\!</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.\!</math> There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},\!</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)\!</math> that is otherwise known as <math>x + y.\!</math>

−

The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:

+

The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>{}^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, {}^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:

{| align="center" cellpadding="8" width="90%"

Line 3,976: Line 3,976:

|}

−

Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math> This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}</math> into the context <math>^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.

+

Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math> This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>{}^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, {}^{\prime\prime}</math> into the context <math>{}^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, {}^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.

If the appropriate safeguards can be kept in mind, avoiding all danger of confusing propositions with sentences and sentences with assertions, then the marks of these distinctions need not be forced to clutter the account of the more substantive indications, that is, the ones that really matter. If this level of understanding can be achieved, then it may be possible to relax these restrictions, along with the absolute dichotomy between algebraic and logical values, which tends to inhibit the flexibility of interpretation.

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|- style="height:40px"

|  

−

| <math>\text{such that:}\!</math>

+

| <math>\text{such that:}~\!</math>

|  

|}

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| <math>\operatorname{R3c.}</math>

| <math>\upharpoonleft Q \upharpoonright (x) ~=~ \underline{1}</math>

−

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R3c~:~R2b}</math>

+

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R3c~:~R2b}</math>

|}

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Besides linking rules together into extended sequences of equivalents, there is one other way that is commonly used to get new rules from old. Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule and then stating their equality in the appropriate form of equation.

−

For example, extracting the expressions <math>\text{R3a}\!</math> and <math>\text{R3c}\!</math> that are given as equivalents in Rule 3 and explicitly stating their equivalence produces the equation recorded in Corollary 1.

+

For example, extracting the expressions <math>\text{R3a}~\!</math> and <math>\text{R3c}~\!</math> that are given as equivalents in Rule 3 and explicitly stating their equivalence produces the equation recorded in Corollary 1.

Line 4,443: Line 4,443:

−

The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule 3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math> At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For example, the expression <math>^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, ^{\prime\prime}</math> ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition but can at most be a sign of one.

+

The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule 3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math> At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For example, the expression <math>{}^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, {}^{\prime\prime}</math> ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition but can at most be a sign of one.

The use of the basic logical connectives can be expressed in the form of an STR as follows:

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|- style="height:48px"

|  

−

| <math>\text{such that:}\!</math>

+

| <math>\text{such that:}~\!</math>

|  

|- style="height:48px"

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|- style="height:40px; text-align:center"

| width="80%" |  

−

| width="20%" | <math>\operatorname{Definition~7}</math>

+

| width="20%" | <math>\operatorname{Definition~7}\!</math>

|}

|-

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| width="2%" style="border-top:1px solid black" |  

| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>

−

| width="80%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in the universe}~ X</math>

+

| width="80%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in the universe}~ X\!</math>

|- style="height:40px"

|  

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|- style="height:40px; text-align:center"

| width="80%" |  

−

| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~7}</math>

+

| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~7}\!</math>

|}

|-

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| <math>\operatorname{R8f.}</math>

| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright (x) ~,~ \downharpoonleft t \downharpoonright (x) ~\underline{))}</math>

−

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8f~:~R7e}</math>

+

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8f~:~R7e}\!</math>

|- style="height:20px"

|  

Line 5,305: Line 5,305:

|  

| <math>\operatorname{R10c.}</math>

−

| <math>\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright</math>

+

| <math>\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright\!</math>

−

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R8b}</math>

+

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R8b}\!</math>

|- style="height:20px"

|  

Line 5,458: Line 5,458:

|- style="height:40px"

|  

−

| <math>\operatorname{R11f.}</math>

+

| <math>\operatorname{R11f.}\!</math>

−

| <math>\upharpoonleft Q \upharpoonright ~=~ \downharpoonleft s \downharpoonright</math>

+

| <math>\upharpoonleft Q \upharpoonright ~=~ \downharpoonleft s \downharpoonright\!</math>

−

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R11f~:~\_\_?\_\_}</math>

+

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R11f~:~\_\_?\_\_}\!</math>

|}

Line 5,509: Line 5,509:

| style="width:2%; border-top:1px solid black" |  

| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>

−

| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math>

+

| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)\!</math>

| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |

−

<math>\operatorname{F1a~:~R9a}</math>

+

<math>\operatorname{F1a~:~R9a}\!</math>

|- style="height:20px"

|  

Line 5,675: Line 5,675:

|  

| <math>\operatorname{D8e.}</math>

−

| <math>\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}</math>

+

| <math>\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}\!</math>

|}

Line 5,805: Line 5,805:

−

The dyadic relation <math>L_{SO}\!</math> that is the converse of the denotative relation <math>L_{OS}\!</math> can be defined directly in the following fashion:

+

The dyadic relation <math>{L_{SO}}\!</math> that is the converse of the denotative relation <math>L_{OS}\!</math> can be defined directly in the following fashion:

{| align="center" cellpadding="8" width="90%"

Line 5,840: Line 5,840:

| width="2%" style="border-top:1px solid black" |  

| width="18%" style="border-top:1px solid black" | <math>\operatorname{D11a.}</math>

−

| width="80%" style="border-top:1px solid black" | <math>L_{SO}\!</math>

+

| width="80%" style="border-top:1px solid black" | <math>{L_{SO}}\!</math>

|- style="height:50px"

|  

| <math>\operatorname{D11b.}</math>

−

| <math>\overset{\smile}{L_{OS}}</math>

+

| <math>\overset{\smile}{L_{OS}}\!</math>

|- style="height:50px"

|  

| <math>\operatorname{D11c.}</math>

−

| <math>\overset{\smile}{\operatorname{Den}^L}</math>

+

| <math>\overset{\smile}{\operatorname{Den}^L}\!</math>

|- style="height:50px"

|  

| <math>\operatorname{D11d.}</math>

−

| <math>\overset{\smile}{\operatorname{Den}(L)}</math>

+

| <math>\overset{\smile}{\operatorname{Den}(L)}\!</math>

|- style="height:50px"

|  

| <math>\operatorname{D11e.}</math>

−

| <math>\operatorname{proj}_{SO}(L)</math>

+

| <math>\operatorname{proj}_{SO}(L)\!</math>

|- style="height:50px"

|  

| <math>\operatorname{D11f.}</math>

−

| <math>\operatorname{Conv}(\operatorname{Den}(L))</math>

+

| <math>\operatorname{Conv}(\operatorname{Den}(L))\!</math>

|- style="height:50px"

|  

| <math>\operatorname{D11g.}</math>

−

| <math>\{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}</math>

+

| <math>\{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}\!</math>

|}

Line 5,915: Line 5,915:

| width="2%" style="border-top:1px solid black" |  

| width="18%" style="border-top:1px solid black" | <math>\operatorname{D12a.}</math>

−

| width="80%" style="border-top:1px solid black" | <math>L_{OS} \cdot x</math>

+

| width="80%" style="border-top:1px solid black" | <math>L_{OS} \cdot x\!</math>

|- style="height:40px"

|  

Line 5,951: Line 5,951:

Signs are ''equiferent'' if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be ''denotatively equivalent'' or ''referentially equivalent'', but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.

−

To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>x ~\overset{L}{=}~ y,\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.

+

To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>{x ~\overset{L}{=}~ y},\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).\!</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.

For each sign relation <math>L,\!</math> this yields a binary relation <math>\operatorname{Der}(L) \subseteq S \times I</math> that is defined as follows:

Line 6,031: Line 6,031:

'''Transitive property.'''

−

Does <math>x ~\overset{L}{=}~ y</math> and <math>y ~\overset{L}{=}~ z</math> imply <math>x ~\overset{L}{=}~ z</math> for all <math>x, y, z \in S</math>?

+

Does <math>{x ~\overset{L}{=}~ y}\!</math> and <math>y ~\overset{L}{=}~ z</math> imply <math>{x ~\overset{L}{=}~ z}\!</math> for all <math>x, y, z \in S\!</math>?

−

To belabor the point, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)</math> and <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, z)</math> imply <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, z)</math> for all <math>x, y, z \in S</math>?

+

To belabor the point, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)\!</math> and <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, z)\!</math> imply <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, z)\!</math> for all <math>x, y, z \in S\!</math>?

Yes, once again, under the stated conditions.</li>

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|- style="height:60px"

|  

−

| valign="top" | <math>\operatorname{F2.1f.}</math>

+

| valign="top" | <math>\operatorname{F2.1f.\!}</math>

| valign="top" |

<math>\begin{array}{ll}

Line 6,308: Line 6,308:

\end{array}</math>

| style="border-left:1px solid black; text-align:center" |

−

<math>\operatorname{F2.2c~:~R11c}</math>

+

<math>\operatorname{F2.2c~:~R11c}</math>

|- style="height:20px"

| colspan="3" |  

Line 6,456: Line 6,456:

& & & \\

\end{array}</math>

−

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3c~:~Log}</math>

+

| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3c~:~Log}</math>

|- style="height:20px"

| colspan="3" |  

Jon Awbrey

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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2 (view source)

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