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===6.15. Propositional Calculus===

===6.15. Propositional Calculus===

The order of reasoning called ''propositional logic'', as it is pursued from various perspectives, concerns itself with three domains of objects, with all three domains having analogous structures in the relationships of their objects to each other.  There is a domain of logical objects called ''properties'' or ''propositions'', a domain of functional objects called ''binary-'', ''boolean-'', or ''truth-valued'' functions, and a domain of geometric objects called ''regions'' or ''subsets'' of the relevant universe of discourse.  Each domain of objects needs a domain of signs to refer to its elements, but if one's interest lies mainly in referring to the common aspects of structure exhibited by these domains, then it serves to maintain a single notation, variously interpreted for all three domains.

The order of reasoning called "propositional logic", as it is pursued from various perspectives, concerns itself with three domains of objects, with all three domains having analogous structures in the relationships of their objects to each other.  There is a domain of logical objects called "properties" or "propositions", a domain of functional objects called "binary", "boolean", or "truth valued" functions, and a domain of geometric objects called "regions" or "subsets" of the relevant universe of discourse.  Each domain of objects needs a domain of signs to refer to its elements, but if one's interest lies mainly in referring to the common aspects of structure exhibited by these domains, then it serves to maintain a single notation, variously interpreted for all three domains.

The first order of business is to comment on the logical significance of the rhetorical distinctions that appear to prevail among these objects.  My reason for introducing these distinctions is not to multiply the number of entities beyond necessity but merely to summarize the variety of entities that have been used historically, to figure out a series of conversions between them, and to integrate suitable analogues of them within a unified system.

The first order of business is to comment on the logical significance of the rhetorical distinctions that appear to prevail among these objects.  My reason for introducing these distinctions is not to multiply the number of entities beyond necessity but merely to summarize the variety of entities that have been used historically, to figure out a series of conversions between them, and to integrate suitable analogues of them within a unified system.

For many purposes the distinction between a property and a proposition does not affect the structural aspects of the domains being considered.  Both properties and propositions are tantamount to fictional objects, made up to supply general signs with singular denotations, and serving as indirect ways to explain the "plural indefinite references" (PIRs) of general signs to the multitudes of their ultimately denoted objects.  A property is signfied by a sign called a "term" that achieves by a form of indirection a PIR to all the elements in a class of "things".  A proposition is signified by a sign called a "sentence" that achieves by a form of indirection a PIR to all the elements in a class of "situations".  But "things" are any objects of discussion and thought, in other words, a perfectly general category, and "situations" are just special cases of these "things".

For many purposes the distinction between a property and a proposition does not affect the structural aspects of the domains being considered.  Both properties and propositions are tantamount to fictional objects, made up to supply general signs with singular denotations, and serving as indirect ways to explain the ''plural indefinite references'' (PIRs) of general signs to the multitudes of their ultimately denoted objects.  A property is signified by a sign called a ''term'' that achieves by a form of indirection a PIR to all the elements in a class of ''things''.  A proposition is signified by a sign called a ''sentence'' that achieves by a form of indirection a PIR to all the elements in a class of ''situations''.  But ''things'' are any objects of discussion and thought, in other words, a perfectly general category, and ''situations" are just special cases of these ''things''.

There is still something left to the logical distinction between properties and propositions, but it is largely immaterial to the order of reasoning that is found reflected in propositional logic.  When it is useful to emphasize their commonalities, properties and propositions can both be referred to as "Props".  As a handle on the aspects of structure that are shared between these two domains and as a mechanism for ignoring irrelevant distinctions, it also helps to have a single term for a "domain of properties" (DOP) and a "domain of propositions" (DOP).

There is still something left to the logical distinction between properties and propositions, but it is largely immaterial to the order of reasoning that is found reflected in propositional logic.  When it is useful to emphasize their commonalities, properties and propositions can both be referred to as ''Props''.  As a handle on the aspects of structure that are shared between these two domains and as a mechanism for ignoring irrelevant distinctions, it also helps to have a single term for a ''domain of properties'' (DOP) and a ''domain of propositions'' (DOP).

Because a Prop is introduced as an intermediate object of reference for a general sign, it factors a PIR of a general sign across two stages, the first appearing as a reference of a general sign to a singular Prop, and the second appearing as an application of a Prop to its proper objects.  This affords a point of articulation that serves to unify and explain the manifold of references involved in a PIR, but it requires a distinction to be fashioned between the intermediate objects, whether real or invented, and the original, further, or ultimate objects of a general sign.

Because a Prop is introduced as an intermediate object of reference for a general sign, it factors a PIR of a general sign across two stages, the first appearing as a reference of a general sign to a singular Prop, and the second appearing as an application of a Prop to its proper objects.  This affords a point of articulation that serves to unify and explain the manifold of references involved in a PIR, but it requires a distinction to be fashioned between the intermediate objects, whether real or invented, and the original, further, or ultimate objects of a general sign.

Next, it is necessary to consider the stylistic differences among the logical, functional, and geometric conceptions of propositional logic.  Logically, a domain of properties or propositions is known by the axioms it is subject to.  Concretely, one thinks of a particular property or proposition as applying to the things or situations it is true of.  With the synthesis just indicated, this can be expressed in a unified form:  In abstract logical terms, a DOP is known by the axioms it is subject to.  In concrete functional or geometric terms, a particular element of a DOP is known by the things it is true of.

Next, it is necessary to consider the stylistic differences among the logical, functional, and geometric conceptions of propositional logic.  Logically, a domain of properties or propositions is known by the axioms it is subject to.  Concretely, one thinks of a particular property or proposition as applying to the things or situations it is true of.  With the synthesis just indicated, this can be expressed in a unified form:  In abstract logical terms, a DOP is known by the axioms to which it is subject.  In concrete functional or geometric terms, a particular element of a DOP is known by the things of which it is true.

With the appropriate correspondences between these three domains in mind, the general term "proposition" can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects.  Thus, a locution like "the proposition F" can be interpreted in three ways, literally, to denote a logical proposition, functionally, to denote a mapping from a space X of propertied or proposed objects to the domain B = {0, 1} of truth values, and geometrically, to denote the so called "fiber of truth" F 1(1) as a region or a subset of X.  For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition F can be recast as a boolean function F : X >B, and understood to indicate the region of the space X that is ruled by F.

With the appropriate correspondences between these three domains in mind, the general term ''proposition'' can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects.  Thus, a locution like <math>{}^{\backprime\backprime} \text{the proposition}~ F {}^{\prime\prime}\!</math> can be interpreted in three ways:  (1) literally, to denote a logical proposition, (2) functionally, to denote a mapping from a space <math>X\!</math> of propertied or proposed objects to the domain <math>\mathbb{B} = \{ 0, 1 \}\!</math> of truth values, and (3) geometrically, to denote the so-called ''fiber of truth'' <math>F^{-1}(1)\!</math> as a region or a subset of <math>X.\!</math> For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition <math>F\!</math> can be recast as a boolean function <math>F : X \to \mathbb{B},\!</math> and understood to indicate the region of the space <math>X\!</math> that is ruled by <math>F.\!</math>

Generally speaking, it does not seem possible to disentangle these three domains from each other or to determine which one is more fundamental.  In practice, due to its concern with the computational implementations of every concept it uses, the present work is biased toward the functional interpretation of propositions.  From this point of view, the abstract intention of a logical proposition F is regarded as being realized only when a program is found that computes the function F : X >B.

Generally speaking, it does not seem possible to disentangle these three domains from each other or to determine which one is more fundamental.  In practice, due to its concern with the computational implementations of every concept it uses, the present work is biased toward the functional interpretation of propositions.  From this point of view, the abstract intention of a logical proposition F is regarded as being realized only when a program is found that computes the function F : X >B.