Grounding axioms for (relevant) implication

Most of the logics of grounding that have so far been proposed contain grounding axioms, or grounding rules, for the connectives of conjunction, disjunction and negation, but little attention has been dedicated to the implication connective. The present paper aims at repairing this situation by proposing adequate grounding axioms for relevant implication. Because of the interaction between negation and implication, new grounding axioms concerning negation will also arise

Display calculi and other modal calculi: a comparison

International audienceIn this paper we introduce and compare four different syntactic methods for generating sequent calculi for the main systems of modal logic: the multiple sequents method, the higher-arity sequents method, the tree-hypersequents method and the display method. More precisely we show how the first three methods can all be translated in the fourth one. This result sheds new light on these generalisations of the sequent calculus and raises issues that will be examined in the last section

Grounding rules and (hyper-)isomorphic formulas

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

The Method of Tree-hypersequents for Modal Propositional Logic

Paper from the Studia Logica conference Trends in Logic IVIn this paper we present a method, that we call the tree-hypersequent method, for generating contraction-free and cut-free sequent calculi for modal propositional logics. We show how this method works for the systems K, KD, K4 and KD4, by giving a sequent calculus for these systems which are normally presented in the Hilbert style, and by proving all the main results in a purely syntactical way

Modal Truths from an Analytic-Synthetic Kantian Distinction

In the article of 1965 Are logical truths analytic? Hintikka deals with the problem of establishing whether the logical truths of first-order logic areanalytic or synthetic. In order to provide an answer to this issue, Hintikka firstly distinguishes two different notions of analyticity, and then he showsthat the sentences of first-order logic are analytic in one sense, but synthetic in another. This interesting result clearly illustrates the non-triviality of thequestion. In this paper we aim at answering the question Are modal truths analytic? In order to elaborate a satisfactory answer to this question, wewill follow the strategy of Hintikka and we will exploit some recent results on the proof theory for modal logic. Finally, our conclusions will shed newlights on the links between first-order logic and modal logic

Reflecting the Semantic Features of S5 at the Syntactic Level

In this paper we present two different sequent calculi for modal logic S5, each of which reflects, at the syntactic level, one of the two ways of describing S5 semantically. We will analyze both these sequent calculi in detail and we will briefly sketch the proofs of: (i) adequacy of the calculi, (ii) admissibility of the structural rules, cut-rule included. All results are proved in a purely syntactic way. Contraction-free, Cut-free, Modal logic, Sequent Calculus, Tree-hypersequents

A purely syntactic and cut-free sequent calculus for the modal logic of provability

International audienceIn this paper we present a sequent calculus for the modal propositional logic GL (the logic of provability) obtained by means of the tree-hypersequent method, a method in which the metalinguistic strength of hypersequents is improved, so that we can simulate trees shapes. We prove that this sequent calculus is sound and complete with respect to the Hilbert-style system GL, that it is contraction free and cut free and that its logical and modal rules are invertible. No explicit semantic element is used in the sequent calculus and all the results are proved in a purely syntactic way

Grounding rules and (hyper-)isomorphic formulas

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

Three solutions to the knower paradox

National audienceIn this paper I shall present three solutions to the Knower Paradox which, despite important points in common, differ in several respects. The first solution, proposed by C. A. Anderson [1] is a hierarchical solution, developed in the framework of first-order arithmetic. However I will try to show that this solution is based on an incorrect argument. The second solution, inspired by a book of R.M. Smullyan [14], is developed in the framework of modal logic and it is based on the idea of interpreting one of the basic systems of the modal logic of provability in an epistemic way. I shall give arguments in support of this solution. The third solution, proposed by P. Egrèé [8] attempts to connect the first and the second solutions. I will show that this attempt fails for philosophical and formal reasons

From a single agent to multi-agent via hypersequents

International audienceIn this paper we present a sequent calculus for the multi-agent system S5 m . First, we introduce a particularly simple alternative Kripke semantics for the system S5 m . Then, we construct a hypersequent calculus for S5 m that reflects at the syntactic level this alternative interpretation. We prove that this hypersequent calculus is theoremwise equivalent to the Hilbert-style system S5 m , that it is contraction-free and cut-free, and finally that it is decidable. All results are proved in a purely syntactic way and the cut-elimination procedure yields an upper bound of ip 2 (n, 0) where ip 2 is an hyperexponential function of base 2