Substitutivity in Semantic Logic

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Substitutivity

The problem of substitutivity has always been a thorn in the side of the study of semantic logic. Why does it sometimes appear that terms that refer to identical objects cannot be replaced with each other in propositions without altering the truth value or meaning of said proposition? Leibniz's Law would seem to ensure that we could perform such an action without anything significant having changed, but this is clearly not so. I intend to look at the history, not only of this problem, but of the theories that have created an atmosphere in which these questions can be contemplated. Finally, I will offer some of my own insights and perceived problems.

Gottlob Frege (1) asserts that, in addition to the denotation of a sign (2), there is also a meaning attached. (3) The denotation is the specific thing that a sign refers to. The meaning, however, is the actual intention behind using the sign. It follows from this distinction that signs may be identical, but have different functions in a proposition. When it comes to meanings, it is not necessarily the case that the respective denotations of the same sign are equivalent. In fact, Frege argues, there may well be meaning without any denotation whatsoever. As an example, he uses 'the celestial body most distant from the Earth'. While this expression most likely has meaning, it is doubtful that it denotes anything. Frege concludes, "In grasping a (meaning), one is not certainly assured of (denoting) anything". (4) Yet, Frege admits, we must presuppose that there is, indeed a denotation, in order to grasp the meaning. He admits that while we may be mistaken in our assumption, the importance lies in the intention of the speaker, not in the existence of the denotation.

Frege next turns his attention to full sentences. He claims that the thought conveyed by a sentence cannot be its denotation, for the same denotation may be expressed by very different thoughts. The thought, therefore, must be the meaning. Yet Frege admits that we are justified in desiring a denotation for our sentences, and he claims that we do so in order to determine truth. His conclusion is that the denotation of a sentence is its truth value. He claims that, "Every assertoric sentence concerned with what its words (denote) is therefore to be regarded as a proper name, and its (denotation), if it has one, is either the True or the False".

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