Same Same But Different: An Alphabetically Innocent Compositional Predicate Logic

Abstract

Compositionality is at the heart of model theoretical semantics, and the most common way of doing truth value semantics for natural language (NL) is to translate a fragment of NL into some extensions of classical predicate logic. Yet, somewhat ironically and strangely enough, predicate logic itself is not compositional, because the truth conditions for quantification as usually stated are not a function of the denotations of its parts but depend on value assignments for variables. That this kind of dependence on value assignments leads to non-compositionality is well-known and will be demonstrated explicitly in section 1.3. One could, as is also well-known, remedy this awkwardness by considering not truth values as denotations of formulas but sets of value assignments for variables. As we will show in 1.4, such a semantics is compositional, but now an additional problem emerges, namely the lack of alphabetical “innocence” (or “invariance”) in that the denotation of P(x) is different from that of P(y) although the formulas are mere alphabetic variants of each other.1 This new problem is related to what Fine (2007), p. 7, calls the “antinomy of the variable”: