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A ''propositional language'' is a syntactic system that mediates the reasonings of a ''propositional logic''.  The objects of the language and the logic, that is, 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 <math>\mathbb{B}^n \to \mathbb{B}\!</math> and remaining subject only to suitable choices of the parameter <math>n.\!</math>

A ''propositional language'' is a syntactic system that mediates the reasonings of a ''propositional logic''.  The objects of the language and the logic, that is, 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 <math>\mathbb{B}^n \to \mathbb{B}\!</math> and remaining subject only to suitable choices of the parameter <math>n.\!</math>

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.

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".

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.

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".

# 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''.

 

# 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''.

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".

# 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.

 

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.

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.