This paper is at the same time a first step towards an "implementation" of the inferentialist view of meaning and a first proposal for a logical structure which describes an argumentation. According to inferentialism the meaning of a statement lies in its argumentative use, its justifications, its refutations and more generally its deductive relation to other statements. In this first step we design a simple notion of argumentative dialogue. Such dialogues can be either carried in purely logical terms or in natural language. Indeed, a sentence can be mapped to logical formulas representing the possible meanings of the sentence, as implemented with some categorial parsers. We then present our version of dialogical logic, which we recently proved complete for first order classical logic. Next we explain, through examples, how argumentative dialogues can be modeled within our version of dialogical logic.Finally, we discuss how this framework can be extended to real argumentative dialogues, in particular with a proper treatment of axioms.
In this paper we provide the first game semantics for the constructive modal logic CK. We first define arenas encoding modal formulas, we then define winning innocent strategies for games on these arenas, and finally we characterize the winning strategies corresponding to proofs in the logic CK. To prove the full-completeness of our semantics, we provide a sequentialization procedure of winning strategies. We conclude the paper by proving their compositionality and showing how our results can be extend to the constructive modal logic CD.
We consider an extended version of sabotage games played over Attack Graphs. Such games are two-player zero-sum reachability games between an Attacker and a Defender. This latter player can erase particular subsets of edges of the Attack Graph. To reason about such games we introduce a variant of Sabotage Modal Logic (that we call Subset Sabotage Modal Logic) in which one modality quantifies over non-empty subset of edges. We show that we can characterize the existence of winning Attacker strategies by formulas of Subset Sabotage Modal Logic.
In this chapter, we introduce a new dialogical system for first order classical logic which is close to natural language argumentation, and we prove its completeness with respect to usual classical validity. We combine our dialogical system with the Grail syntactic and semantic parser developed by the second author in order to address automated textual entailment, that is, we use it for deciding whether or not a sentence is a consequence of a short text. This work -which connects natural language semantics and argumentation with dialogical logic -can be viewed as a step towards an inferentialist view of natural language semantics.
In this paper we provide two new semantics for proofs in the constructive modal logics CK and CD.The first semantics is given by extending the syntax of combinatorial proofs for propositional intuitionistic logic, in which proofs are factorised in a linear fragment (arena net) and a parallel weakening-contraction fragment (skew fibration). In particular we provide an encoding of modal formulas by means of directed graphs (modal arenas), and an encoding of linear proofs as modal arenas equipped with vertex partitions satisfying topological criteria.The second semantics is given by means of winning innocent strategies of a two-player game over modal arenas. This is given by extending the Heijltjes-Hughes-Straßburger correspondence between intuitionistic combinatorial proofs and winning innocent strategies in a Hyland-Ong arena. Using our first result, we provide a characterisation of winning strategies for games on a modal arena corresponding to proofs with modalities.
The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the field of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the Journal in 1936 until the end of 1999. The Association moved the reviews to this Bulletin, beginning in 2000. The Reviews Section is edited by Clinton Conley (Managing Editor)
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