The proof-theoretic square

0573743 - FLÚ 2024 RIV DE eng J - Journal Article
Piccolomini d’Aragona, Antonio
The proof-theoretic square.
Synthese. Roč. 201, č. 6 (2023), č. článku 219. ISSN 0039-7857. E-ISSN 1573-0964
Grant - others:AV ČR(CZ) LQ300092101
Program: Prémie Lumina quaeruntur
Institutional support: RVO:67985955
Keywords : Prawitz * proof-theoretic semantics * justification * validity * inference
OECD category: Philosophy, History and Philosophy of science and technology
Impact factor: 1.5, year: 2022
Method of publishing: Limited access
https://doi.org/10.1007/s11229-023-04203-5

In Prawitz’s semantics, the validity of an argument may be defined, either relatively to an atomic base which determines the meaning of the non-logical terminology, or relatively to the whole class of atomic bases, namely as logical validity. In the first case, which may be qualified as local, one has to choose whether validity of arguments is or not monotonic over expansions of bases, while in the second case, which may be qualified as global, one has to choose whether the reduction functions which justify non-primitive inferences are or not base-depending. I claim that these oppositions enjoy some conceptual symmetries, and that these symmetries may be understood as putting restrictions on one’s choice both at the local and at the global level. This produces two different proof-theoretic semantics in line with Prawitz’s tenets. However, I also argue that the symmetries stem from a deeper interaction within Prawitz’s semantics, i.e. non-logical meanings vs interpretation of rules. This interaction is in turn based on some primitive ingredients which, when suitably combined, produce other two Prawitz-compatible readings. The four readings form a diagram where some order relations hold. I finally claim that this diagram is complete, namely that, if we combine the ingredients in ways other than those giving rise to the four readings above, we obtain nothing new, or nothing compatible with general semantic requirements. Thus, symmetries between the local and the global level, and interaction between non-logical meaning and interpretation of rules, produce a complete classification of potential Prawitzian semantics.
Permanent Link: https://hdl.handle.net/11104/0344328