Logic is today developing in many, very different directions. But the basic distinction between the semantic approach (truth, "⊨") and the syntactic one (provability, "⊢") is still important and taught in all basic logic courses.

Which of these two approaches came first? Or how did they get disentangled, if they were somehow unified or confused at first? I'd be happy to read your historical summaries and receive literature suggestions.


1 Answer 1


This is something of a chicken and egg problem. In a sense, the circularity is inherent in the very nature of logic. How are we to justify the rules of good reasoning if not by appealing to their meaning? And how are we to make sense of such appeals if not by following the rules of good reasoning?

The crisp semantic/syntactic distinction is recent. It was only explicitly introduced by Tarski, along with the distinction between object language and meta-language, in his paper on truth functional semantics (1933). It was a response to Goedel's incompleteness theorem (1931), which showed that the classical notion of truth can not be captured by a (first order) formal system. One can, perhaps, trace it ideologically to Hilbert, but not much further back. Hilbert combined the "syntactic" view of formal theories with intuitive Kantian "semantics", see Was there a Kantian influence on Hilbert's formalist programme? In his Foundations of Geometry (1899), he gives formal axioms of geometry and provides models for them in a proto-Tarskian style.

The other two forefathers of modern logic, Frege and Russell, opposed such a separation and advocated "universalist" conception of logic, without such distinctions. Frege argued with Hilbert about it at length in personal correspondence. The very notion of "semantic methods", as understood by Tarski, requires a set-theoretic notion of a model, and hence set theory, which Cantor developed not long before Hilbert. Since Hilbert did not formalize semantics, I suppose, one could say that syntax came "first". But this is misleading. At the same time as Tarski developed his semantics, Wittgenstein advocated combining both into a single "grammar" with purely technical/conventional divisions of little philosophical significance. This line of thought continues to this day, see e.g. Rapaport's thesis that semantics is syntax.

And here is another twist that cautions against taking this distinction too seriously. Van Heijenoort and Grattan-Guinness distinguished two traditions in the history of logic, both traceable back to Leibniz, "logic as calculus" vs "logic as language", algebra of logic vs formal logic. Roughly speaking, the "algebraists" took mathematics as more foundational and built logic on it, while the "formalists" went the other way. In the 19th century, the former were represented by Boole and Peirce, and the latter by Frege and Peano. In his review Calculus ratiocinator versus characteristica universalis?, Peckhaus links "semantics" to "algebraists", and "syntax" to "formalists". Then not only does "semantics" comes before "syntax", Boole before Frege, but now Frege finds himself on the opposite side of the supposed divide:

"Frege’s system is closed, nothing can be outside the system. There are no metalogical questions and no separate semantics. ‘The universal formal language supplants the natural language’, he writes... The algebraic propositional calculus uses, on the contrary, a model-theoretic approach. Algebraic systems always need interpretations of operation signs and categories, i.e. classes of concepts. They can clearly be divided into a structural, or syntactic, and a semantical side, the latter providing the interpretations of the figures used."

If we delve further into pre-history, and attempt to impose a modern distinction on distant history, this becomes even more of a chicken and egg problem. Already in Aristotle one can find "syntactic" rules (such as the three laws of thought and syllogism figures) stated right next to "semantic" justifications of them. In Aristotelian medieval logic we also find a mixture of syntactic and semantic notions of logical validity, with little interest in separating them, see What were the historical interpretations of Aristotle's definition of validity/logical consequence?

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