Game-theoretic semantics for non-distributive logics

We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for the logics of lattices with weak negation and modal operators (Modal Lattice Logic).

Semantics, game-theoretic

Game-theoretic semantics (GTS) uses concepts from game theory to study how the truth and falsity of the sentences of a language depend upon the truth and falsity of the language’s atomic sentences (or upon its sub-sentential expressions). Unlike the Tarskian method (which uses recursion clauses to determine satisfaction conditions for nonatomic sentences in terms of the satisfaction conditions of their component sentences, then defines truth in terms of satisfaction), GTS associates with each sentence its own semantic game played on sentences of the language. This game defines truth in terms of the existence of a winning strategy for one of the players involved. The structure of the game is determined by the sentence’s structure, and thus the semantic properties of the sentence in question can be studied by attending to the properties of its game.

Semantics for Hyperclassical Logic and the Problem of Negation in the Formal Language

The application of game-theoretic semantics for first-order logic is based on a certain kind of semantic assumptions, directly related to the asymmetry of the definition of truth and lies as the winning strategies of the Verifier (Abelard) and the Counterfeiter (Eloise). This asymmetry becomes apparent when applying GTS to IFL. The legitimacy of applying GTS when it is transferred to IFL is based on the adequacy of GTS for FOL. But this circumstance is not a reason to believe that one can hope for the same adequacy in the case of IFL. Then the question arises if GTS is a natural semantics for IFL. Apparently, the intuitive understanding of negation in natural language can be explicated in formal languages in various ways, and the result of an incomplete grasp of the concept in these languages can be considered a certain kind of anomalies, in view of the apparent simplicity of the explicated concept. Comparison of the theoretical-model and game theoretic semantics in application to two kinds of language – the first-order language and friendly-independent logic – allows to discover the causes of the anomaly and outline ways to overcome it.

Conjunction is Parallel Computation

This paper proposes a new, game theoretical, analysis of conjunction which provides a single logical translation of AND in its sentential, predicate, and NP uses, including both Boolean and non-Boolean cases. In essence it analyzes conjunction as parallel composition, based on game-theoretic semantics and logical syntax by Abramsky (2007).

Conjunction is Parallel Computation

This paper proposes a new, game theoretical, analysis of conjunction which provides a single logical translation of AND in its sentential, predicate, and NP uses, including both Boolean and non-Boolean cases. In essence it analyzes conjunction as parallel composition, based on game-theoretic semantics and logical syntax by Abramsky (2007).

Conjunction is Parallel Computation

This paper proposes a new, game theoretical, analysis of conjunction which provides a single logical translation of and in its sentential, predicate, and NP uses, including both Boolean and non-Boolean cases. In essence it analyzes conjunction as parallel composition, based on game-theoretic semantics and logical syntax by Abramsky (2007).

LOGICS AND ALGEBRAS FOR MULTIPLE PLAYERS

We study a generalization of the standard syntax and game-theoretic semantics of logic, which is based on a duality between two players, to a multiplayer setting. We define propositional and modal languages of multiplayer formulas, and provide them with a semantics involving a multiplayer game. Our focus is on the notion of equivalence between two formulas, which is defined by saying that two formulas are equivalent if under each valuation, the set of players with a winning strategy is the same in the two respective associated games. We provide a derivation system which enumerates the pairs of equivalent formulas, both in the propositional case and in the modal case. Our approach is algebraic: We introduce multiplayer algebras as the analogue of Boolean algebras, and show, as the corresponding analog to Stone’s theorem, that these abstract multiplayer algebras can be represented as concrete ones which capture the game-theoretic semantics. For the modal case we prove a similar result. We also address the computational complexity of the problem whether two given multiplayer formulas are equivalent. In the propositional case, we show that this problem is co-NP-complete, whereas in the modal case, it is PSPACE-hard.