Having broached the distinction between objective propositions and syntactic sentences, its analogy to the distinction between numbers and numerals becomes clear.  What are the implications of that distinction for the realm of reasoning about propositions and its representation in sentential logic?

If the purpose of a sentence is precisely to denote a proposition then the proposition is simply the object of whatever sign is taken for the sentence.  The computational manifestation of a piece of reasoning about propositions thus amounts to a process taking place entirely within a language of sentences, being a procedure which can rationalize its account by referring to the denominations of sentences among propositions.

As far as it bears on our current context of problems, the upshot is this:  Do not worry too much about what roles the empty string and the blank symbol are supposed to play in a given species of formal language.  As it happens, it is far less important to wonder whether those types of formal tokens actually constitute genuine sentences than it is to decide what equivalence classes it makes sense to form over all the sentences in the resulting language, and only then to bother about what equivalence classes those limiting cases of sentences are most conveniently taken to represent.

Resources

• Cactus Language • Pragmatics
• Survey of Animated Logical Graphs
• Survey of Theme One Program

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