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===6.15. Propositional Calculus===
===6.15. Propositional Calculus===
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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.
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" (PIR's) 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".
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.
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.
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.
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.
The functional interpretation of propositional calculus goes hand in hand with an approach to logical reasoning that incorporates "semantic" or "model theoretic" methods, as distinguished from the purely "syntactic" or "proof theoretic" option. Indeed, the functional conception of a proposition is model theoretic in a double sense, not only because its notations denote functions as their semantic objects, but also because the domains of these functions are spaces of logical interpretations for the propositions, with the points of the domain that lie in the inverse image of truth under the function being the "models" of the proposition.
One of the reasons for pursuing a pragmatic hybrid of semantic and syntactic approaches, rather than keeping to the purely syntactic ways of manipulating meaningless tokens according to abstract rules of proof, is that the model theoretic strategy preserves the form of connection that exists between an agent's concrete particular experiences and the abstract propositions and general properties that it uses to describe its experience. This makes it more likely that a hybrid approach will serve in the realistic pursuits of inquiry, since these efforts involve the integration of deductive, inductive, and abductive sources of knowledge.
In this approach to propositional logic, with a view toward computational realization, one begins with a space X, called a "universe of discourse", whose points can be reasonably well described by means of a finite set of logical features. Since the points of the space X are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections xi : X >B, for i = 1 to n, for some n, that can serve to describe the points of X. This means that there is a computable coordinate representation for X, in other words, a computable map T : X >Bn that describes the points of X insofar as they are known. Thus, each proposition F : X >B can be factored through the coordinate representation T : X >Bn to yield a related proposition f : Bn >B, one that speaks directly about coordinate n tuples but indirectly about points of X. Composing maps on the right, the mapping f is defined by the equation F = T o f. For all practical purposes served by the representation T, the proposition f can be taken as a proxy for the proposition F, saying things about the points of X by means of X's encoding to Bn.
Working under the functional perspective, the formal system known as "propositional calculus" is introduced as a general system of notations for referring to boolean functions. Typically, one takes a space X and a coordinate representation T : X >Bn as parameters of a particular system and speaks of the propositional calculus on a finite set of variables {xi}. In objective terms, this constitutes the "domain of propositions" on the basis {xi}, notated as "DOP{xi}". Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone. In practice, this means that the formalism and its computational implementation should allow for the automatic enmbedding of DOP(X) into DOP(Y) whenever X c Y.
The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.
This section reviews the elements of a calculus for propositional logic that I initially presented in two earlier papers (Awbrey, 1989 & 1994). This calculus belongs to a family of formal systems that hark back to C.S. Peirce's "existential graphs" (PEG) and it draws on ideas from Spencer Brown's "Laws of Form" (LOF). A feature that distinguishes the use of these formalisms can be summed up by saying that they treat logical expressions primarily as elements of a "language" and only secondarily as elements of an "algebra". In other words, the most important thing about a logical expression is the logical object it denotes. To the extent that the object can be represented in syntax, this attitude puts the focus on the "logical equivalence class" (LEC) to which the expression belongs, relegating to the background the whole variety of ways that the expression can be generated from algebraically conceived operations.
One of the benefits of this notation is that it facilitates the development of a "differential extension" (DEX) for propositional logic that can be used to reason about changing universes of discourse.
A "propositional language" (PL) is a syntactic system that mediates the reasonings of a "propositional logic" (PL). The objects of a PL, the logical entities denoted by the language and invoked by the operations of the logic, can be conceived to rest at various levels of abstraction, residing in spaces of functions that are basically of the types Bn >B and remaining subject only to suitable choices of the parameter n.
Persistently reflective engagement in logical reasoning about any domain of objects leads to the identification of generic patterns of inference that appear to be universally valid, never disappointing the trust that is placed in them. After a time, a formal system naturally arises that commemorates one's continuing commitment to these patterns of logical conduct, and acknowledges one's conviction that further inquiry into their utility can be safely put beyond the reach of everyday concerns. At this juncture each descriptive pattern becomes a normative template, regulating all future ventures in reasoning until such time as a clearly overwhelming mass of doubtful outcomes cause one to question it anew.
Propositions about a coherent domain of objects tend to gather together and express themselves collectively in organized bodies of statements known as "theories". As theories grow in size and complexity, one is faced with massive collections of propositional constraints and complex chains of logical inferences, and it becomes useful to support reasoning with the implementation of a "propositional calculator".
At this point, variations in common and technical usage of the term "proposition" require a few comments on terminology. The heart of the issue is how to maintain a proper distinction between the logical form and the rhetorical style of a proposition, that is, how best to mark the difference between its invariant contents and its variant expressions. There are many ways to draw the required form of distinction between the objective situation and the significant expression in this relation. Here, I outline a compromise strategy that incorporates the advantages of several options and makes them available to intelligent choice as best fits the occasion.
1. According to a prevailing technical usage, a "proposition" is a categorical object of abstract thought, something that is tantamount to an objective situation, a statistical event, or a state of affairs of a specified type. In distinction to the abstract proposition, a statement that a situation of the proposed type is actually in force is expressed in the form of a syntactic formula called a "sentence".
2. Another option enjoys a set of incidental advantages that makes it worth mentioning here and also worth exploring in a future discussion. Under this alternative, one refers to the signifying expressions as "propositions", deliberately conflating propositions and sentences, but then introduces the needed distinction at another point of articulation, referring to the signified objects as "positions".
3. Attempting to strike a compromise with common usage, I often allow the word "proposition" to exploit the full range of its senses, denoting either object or sign according to context, and resorting to the phrase "propositional expression" whenever it is necessary to emphasize the involvement of the sign.
The operative distinction in every case, propositional or otherwise, is the difference in roles between objects and signs, not the names they are called by. To reconcile a logical account with the pragmatic theory of signs, one entity is construed as the "propositional object" (PO) and the other entity is recognized as the "propositional sign" (PS) at each moment of interpretation in a propositional sign relation. Once these roles are assigned, all the technology of sign relations applies to the logic of propositions as a special case. In the context of propositional sign relations, a "semantic equivalence class" (SEC) is referred to as a "logical equivalence class" (LEC). Each propositional object can then be associated, or even identified for all informative and practical puposes, with the LEC of its propositional signs. Accordingly, the proposition is reconstituted from its sentences in the appropriate way, as an abstract object existing in a semantic relation to its signs.
Taking this topic, "the representation of sign relations", and seeking a computational formulation of its theory, leads to certain considerations about the best approach to the subject. Computational formulations are those with no recourse but to finitary resources. In setting up a computational formulation of any theory, one has to specify the finite set of axioms that are constantly available to subsequent reasoning. This makes it advisable to approach the topic of representations at a level of generality that will give the resulting theory as much power as possible, the kind of power to which inductive hypotheses can have easy and constant recourse. In order to furnish these resources with an ample supply of theoretical power ...
In doing this, it is expeditious, if not absolutely necessary, to broaden the focus on sign relations in two ways: (1) to expand its extension from a special class of triadic relations to the wider sphere of n place relations, and (2) to diffuse its intension from fully specified and concretely presented relations to incompletly specified and abstractly described relations.
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===6.16. Recursive Aspects===
===6.16. Recursive Aspects===
