Grounding rules and (hyper-)isomorphic formulas

Abstract


 
 
 An oft-defended claim of a close relationship between Gentzen inference rules and the meaning of the connectives they introduce and eliminate has given rise to a whole domain called proof-theoretic semantics, see Schroeder- Heister (1991); Prawitz (2006). A branch of proof-theoretic semantics, mainly developed by Dosen (2019); Dosen and Petric (2011), isolates in a precise mathematical manner formulas (of a logic L) that have the same meaning. These isomorphic formulas are defined to be those that behave identically in inferences. The aim of this paper is to investigate another type of recently discussed rules in the literature, namely grounding rules, and their link to the meaning of the connectives they provide the grounds for. In particular, by using grounding rules, we will refine the notion of isomorphic formulas through the notion of hyper-isomorphic formulas. We will argue that it is actually the notion of hyper-isomorphic formulas that identify those formulas that have the same meaning.
 
 

Logical Grounding: The Case of “if‐then‐else”

“…However, it is mostly agreed that commutativity and associativity of conjunction and disjunction give rise to t-equivalent formulas. A comprehensive discussion can be found in Poggiolesi (2020b). For a view that φ and ¬¬φ are t-equivalent, contrasting a general opinion, see Francez (2020a); I adopt here this view.…”

“…However, it is mostly agreed that commutativity and associativity of conjunction and disjunction give rise to ‐equivalent formulas. A comprehensive discussion can be found in Poggiolesi (2020b). For a view that and are ‐equivalent, contrasting a general opinion, see Francez (2020a); I adopt here this view.The exact definition of same truths for classical logic is also orthogonal to my current concerns, so I will deal with a minimal fragment where the issue does not arise in its full generality.…”

“…However, it is mostly agreed that commutativity and associativity of conjunction and disjunction give rise to ‐equivalent formulas. A comprehensive discussion can be found in Poggiolesi (2020b).…”