Sundholm's semantics: logical atavism and the nature of proofs

Philosophy of Science Colloquium
The Institute Vienna Circle holds a Philosophy of Science Colloquium with talks by our present fellows.

Date: 20/06/2024

Time: 16h45

Venue: New Institute Building (NIG), Universitätsstraße 7, 1010 Wien, HS 3F

Abstract:

I aim at providing an overview of some of Sundholm's observations on the philosophy and history of logic, which I overall refer to as Sundholm's semantics. In the reconstruction I propose, the latter is based on two main tenets, namely, that we had better jettison the object-language/meta-language distinction, and that the careful meaning-explanation which formalisms must come with render the axioms and rules of inference evident. The first tenet goes hand in hand with the detection of what Sundholm call the Bolzano reductions, namely, two methodological tenets according to which, first, correctness of assertions is reduced to propositional truth and, second, inferential validity is reduced to logical consequence. In Sundholm's opinion, these Bolzano reductions hide crucial epistemic and pragmatic aspects of logic, when compared to what is the case in "atavistic" approaches like Frege's, where formalisms are indeed understood as meaningful languages. Thus, Sundholm's semantics does not consist in a meta-linguistic attribution of meaning to uninterpreted sets of strings, but in an issuing of the intended meaning of contentual formalism, including semantic values for derivations. The latter are dealt with by Sundholm through a distinction between proof-objects and proof-acts, which Sundholm himself had put stressed in the context of his early discussion of BHK-semantics. When cast as a constructive reading of the so-called truth-maker principle, via a Martin-Löfian rendering of assertions as existence of proof(-object)s for given propositions, the proof-object/proof-act distinction leads to a fresh semantic account of Gentzen's 1932 and 1936 versions of Natural Deduction, as well as to doubts regarding their often asserted meta-theoretical equivalence.