Isomorphic formulae in classical propositional logic

Abstract

Key words isomorphic formulae, classical propositional logic, classical linear propositional logic, categories, equality of deductions, identity of proofs, categorial coherence. MSC (2010) 03F03, 03F07, 03F52, 03G30Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

Grounding rules and (hyper-)isomorphic formulas

“…We now bring together the proof-theoretic tradition represented by the work of Došen and Petríc (2011) with the ground-theoretic tradition exemplified by the grounding rules of Poggiolesi. As already explained, the aim is to study the relation between grounding rules and the meaning of the constants they provide the grounds for.…”

“…Proof-theoretic semantics, a flourishing and thriving domain of research (see Francez (2015), Schroeder-Heister (2018)), is built on the (Wittgenstein) thesis that use determines meaning, and that therefore the meaning of logical connectives is determined by their (logical) use in inference rules. In particular, there exists a branch of proof-theoretic semantics, mainly developed by Došen (2019); Došen and Petríc (2011) and recently taken up by Restall (2019), which aims at identifying in a precise mathematical manner those formulas of a certain logic L that have the same meaning according to this conception: that is, those formulas that behave identically in the inference rules of L. Such formulas are called isomorphic formulas of L.…”

Logics of Synonymy

“…Lemma 1 For every sentence ϕ, L(ϕ) is the set of literals that occur in ϕ max . 17 Thus, the definition of L(ϕ) via valence or via the literals in ϕ max are equivalent. The former might be conceptually more useful (so we'll use it in the main text), while the later is more useful in proofs (so we'll use it in the Appendix).…”

“…Sixth, what's beyond AC? There is, for example, isomorphism in the category of classical proofs which coincides with equivalence in multiplicative linear logic [17,52], factual equivalence [15], equivalence in exact entailment [25], equivalence in some impossible world semantics [6], or syntactic identity. However, for reasons of space, we won't further investigate those.…”

“…23 Now, following [52], "essentially" can be spelled out in different ways: roughly, it could be exact identity or it could allow treating equally occurrences of the same atom (either all or only those of the same polarity). If we choose identity, the resulting notion of synonymy is that of isomorphism in the category of classical proofs as mentioned above [17,52]. If we choose same-polarity occurrences, we get a notion of synonymy equivalent to AC [52].…”