The controversy between syntax and semantics in Post, Lewis and Wittgenstein
DOI:
https://doi.org/10.22370/sst.2020.8.4921Keywords:
history of logic, truth tables, protosemantics, completeness, PrincipiaAbstract
In logic, the idea that the syntax and the calculi provide an accuracy and a rigour which is lacking in semantics is relatively common. The purpose of the present paper is to analyze the historical roots of this idea. In that connection, our starting point will be Post’s doctoral dissertation, where soundness and completeness are proved for propositional logic. Truth tables were developed with a cautious attitude towards meanings, which are considered “extraneous” to the formal system. This attitude comes from Lewis, who argued that his “heterodox conception” of logic should be confronted with Principia’s philosophy of mathematics. For this concepction, a formal system is a string of signs which can be manipulated by means of purely syntactic rules. We claim that this point of view is shared not only by Post and Lewis, but also by Wittgenstein. However, the achievement of major metalogical results (like the completeness of first-order logic) will requiere new semantic tools which were mastered in the Hilbert school.
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