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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

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|+ <math>\text{Table 43.}~~\text{Composite and Compiled Order Relations}</math>

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|+ <math>\text{Table 42.}~~\text{Composite and Compiled Order Relations}</math>

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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (44)

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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (43)

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| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}</math>

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| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}</math>

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| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}</math>

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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (47)

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| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}</math>

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|+ style="height:30px" | <math>\text{Table 48.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>

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|+ style="height:30px" | <math>\text{Table 47.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>

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| style="border-bottom:1px solid black" | <math>p\!</math>

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|+ style="height:30px" | <math>\text{Table 49.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 48.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>

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| style="border-bottom:1px solid black" | <math>p\!</math>

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|+ style="height:30px" | <math>\text{Table 50.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 49.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>

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| style="border-bottom:1px solid black" | <math>p\!</math>

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|+ style="height:30px" | <math>\text{Table 51.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 50.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>

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| style="border-bottom:1px solid black" | <math>p\!</math>

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These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of it. In particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.

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For ease of reference in the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}</math> such that <math>f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism map'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to its fiber of truth <math>[| \operatorname{syll} |] = \operatorname{syll}^{-1}(1)</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 52 shows <math>\operatorname{Syll}</math> as a relational dataset.

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For ease of reference in the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}</math> such that <math>f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism map'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to its fiber of truth <math>[| \operatorname{syll} |] = \operatorname{syll}^{-1}(1)</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 51 shows <math>\operatorname{Syll}</math> as a relational dataset.

<br>

{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"

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|+ style="height:30px" | <math>\text{Table 52.} ~~ \text{Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Syllogism Relation}</math>

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| style="border-bottom:1px solid black" | <math>p\!</math>

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One of the first questions that we might ask about a 3-adic relation, in this case <math>\operatorname{Syll},</math> is whether it is ''determined by'' its 2-adic projections. I will illustrate what this means in the present case.

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Table 53 repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns. For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{Syll}</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time. Likewise, <math>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{Syll}_{qr}</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}</math> that indicate the 2-adic relations that result from these projections.

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Table 52 repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns. For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{Syll}</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time. Likewise, <math>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{Syll}_{qr}</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}</math> that indicate the 2-adic relations that result from these projections.

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

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|+ style="height:30px" | <math>\text{Table 53.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 52.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

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| <math>\operatorname{Syll}</math>

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Figure 28-a. Boolean 3-Cube B^3

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Table ~~28-b~~ shows the 3-adic relation <math>\operatorname{Syll} \subseteq \mathbb{B}^3</math> again, and Figure ~~28-c~~ shows it plotted on a 3-cube template.

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Table 54 shows the 3-adic relation <math>\operatorname{Syll} \subseteq \mathbb{B}^3</math> again, and Figure 55 shows it plotted on a 3-cube template.

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Table ~~28-b~~. Syll c B^3

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Table 54. Syll c B^3

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Figure ~~28-c~~. Triadic Relation Syll c B^3

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Figure 55. Triadic Relation Syll c B^3

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Table ~~29-a~~. Syll c B^3

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Table 56. Syll c B^3

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Table ~~29-b~~. Dyadic Projections of Syll

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Table 57. Dyadic Projections of Syll

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In showing the 2-adic projections of a 3-adic relation <math>L \subseteq \mathbb{B}^3,</math> I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.

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Figure ~~29-c~~ shows <math>\operatorname{Syll}</math> and its three 2-adic projections:

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Figure 58 shows <math>\operatorname{Syll}</math> and its three 2-adic projections:

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Figure ~~29-c~~. Syll c B^3 and its Dyadic Projections

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Figure 58. Syll c B^3 and its Dyadic Projections

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Table ~~30-a~~. Syll c B^3

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Table 59. Syll c B^3

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Table ~~30-b~~. Dyadic Projections of Syll

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Table 60. Dyadic Projections of Syll

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Table ~~30-c~~. Tacit Extensions of Projections of Syll

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Table 61. Tacit Extensions of Projections of Syll

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Figure ~~30-d~~. Tacit Extension te_12_3 (Syll_12)

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Figure 62. Tacit Extension te_12_3 (Syll_12)

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Figure ~~30-e~~. Tacit Extension te_13_2 (Syll_13)

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Figure 63. Tacit Extension te_13_2 (Syll_13)

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Figure ~~30-f~~. Tacit Extension te_23_1 (Syll_23)

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Figure 64. Tacit Extension te_23_1 (Syll_23)

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Figure 31. Syll = te(Syll_12) |^| te(Syll_23)

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Figure 65. Syll = te(Syll_12) |^| te(Syll_23)

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In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.

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At first or maybe second sight, the relationships seem easy enough to write out. Figure 32 shows how the various logical expressions are related to each other: The expressions "(p (q))" and "(q (r))" are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression "(p (q)) (q (r))" that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference "(p (r))" by the most conventional implication.

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At first or maybe second sight, the relationships seem easy enough to write out. Figure 66 shows how the various logical expressions are related to each other: The expressions "(p (q))" and "(q (r))" are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression "(p (q)) (q (r))" that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference "(p (r))" by the most conventional implication.

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o-------------------o

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Figure 32. Expressive Aspects of Transitive Inference

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Figure 66. Expressive Aspects of Transitive Inference

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Most of the customary names for this type of process have turned out to have misleading connotations, and so I will experiment with calling it the ''expressive'' aspect of the various rules for transitive inference, simply to emphasize the fact that rules can be given for it that operate solely on signs and expressions, without necessarily needing to look at the objects that are denoted by these signs and expressions.

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In the way of many experiments, the word ''expressive'' does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective. Aside from coining a word like ''ennotative'', some other options would be ''connotative'', ''hermeneutic'', ''semiotic'', ''syntactic'' — each of which works in some range of interpretation but fails in others~~. Trial 2~~. Let's try ''formulaic''.

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In the way of many experiments, the word ''expressive'' does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective. Aside from coining a word like ''ennotative'', some other options would be ''connotative'', ''hermeneutic'', ''semiotic'', ''syntactic'' — each of which works in some range of interpretation but fails in others. Let's try ''formulaic''.

Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith. Thus we engage the ''denotative semantics'' or the ''model theory'' of these extremely simple programs that we call ''propositions''.

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Table 33 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference. A couple of alternative notations are introduced in this Table:

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Table 67 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference. A couple of alternative notations are introduced in this Table:

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

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Figure 33. Denotative Aspects of Transitive Inference

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Figure 67. Denotative Aspects of Transitive Inference

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An abstract reference to a point of <math>X\!</math> is a triple in <math>\mathbb{B}^3.</math> A concrete reference to a point of <math>X\!</math> is a conjunction of signs from the dimensions <math>P^\ddagger, Q^\ddagger, R^\ddagger,</math> picking exactly one sign from each dimension.

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To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 35 translates the contents of Figure 33 into the new language.

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To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 68 translates the contents of Figure 33 into the new language.

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o---------o---------o o---------o---------o

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Figure 35. Denotative Aspects of Transitive Inference

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Figure 68. Denotative Aspects of Transitive Inference

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Jon Awbrey

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