Classical Negation and Game-Theoretical Semantics (original) (raw)

Extended Game-Theoretical Semantics

Between Logic and Reality, 2011

A new version of Game-Theoretical Semantics (GTS) is put forward where game rules are extended to the non-logical constants of sentences. The resulting theory, together with a refinement of our criteria of identity for functions, provide the technical basis for a game-based conception of linguistic meaning and interpretation.

The Logic of Conditional Negation

Notre Dame Journal of Formal Logic, 2013

It is argued that the `inner' negation nega\neganega familiar from 3-valued logic can be interpreted as a form of `conditional' negation: negaA\nega AnegaA is read ``$A$ is false \emph{if it has a truth value}''. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language indicative conditional (e.g. ``If Jim went to the party he had a good time''). It is shown that the logic induced by the semantics shares many familiar properties with classical negation, but is orthogonal to both intuitionistic and classical negation: it differs from both in validating the inference from ArightarrownegaBA\rightarrow \nega BArightarrownegaB to nega(ArightarrowB)\nega(A\rightarrow B)nega(ArightarrowB).

An infinite-game semantics for well-founded negation in logic programming

Annals of Pure and Applied Logic, 2008

We present an infinite-game characterization of the well-founded semantics for function-free logic programs with negation. Our game is a simple generalization of the standard game for negation-less logic programs introduced by M.H. van Emden (1986, Journal of Logic Programming, 3(1), 37-53) in which two players, the Believer and the Doubter, compete by trying to prove (respectively disprove) a query. The standard game is equivalent to the minimum Herbrand model semantics of logic programming in the sense that a query succeeds in the minimum model semantics iff the Believer has a winning strategy for the game which begins with the Doubter doubting this query. The game for programs with negation that we propose follows the same rules as the standard one, except that the players swap roles every time the play "passes through" negation. We start our investigation by establishing the determinacy of the new game by using some classical tools from the theory of infinite-games. Our determinacy result immediately provides a novel and purely game-theoretic characterization of the semantics of negation in logic programming. We proceed to establish the connections of the game semantics to the existing semantic approaches for logic programming with negation. For this purpose, we first define a refined version of the game that uses degrees of winning and losing for the two players. We then demonstrate that this refined game corresponds exactly to the infinite-valued minimum model semantics of negation (Rondogiannis & Wadge, 2005, ACM TOCL, 6(2), 441-467). This immediately implies that the unrefined game is equivalent to the well-founded semantics (since the infinite-valued semantics is a refinement of the well-founded semantics).

Some aspects of negation in English

Synthese, 1994

ABSTRACT. I introduce a formal language called the language of informational inde-pendence (IL-language, for short) that extends an ordinary first-order language in a natural way. This language is interpreted in terms of semantical games of imperfect information. In this language, one ...

On the Expressive Power of IF-Logic with Classical Negation

Lecture Notes in Computer Science, 2011

It is well-known that Independence Friendly (IF) logic is equivalent to existential second-order logic (Σ 1 1) and, therefore, is not closed under classical negation. The boolean closure of IF sentences, called Extended IF-logic, on the other hand, corresponds to a proper fragment of ∆ 1 2. In this paper we consider IF-logic extended with Hodges' flattening operator, which allows classical negation to occur also under the scope of IF quantifiers. We show that, nevertheless, the expressive power of this logic does not go beyond ∆ 1 2. As part of the proof, we give a prenex normal form result and introduce a non-trivial syntactic fragment of full second-order logic that we show to be contained in ∆ 1 2 .

A game semantics for linear logic

Annals of Pure and Applied Logic, 1992

Blass, A., A game semantics for linear logic, Annals of Pure and Applied Logic 56 (1992) 183-220. We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard's linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier 'almost' will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective @ is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godel's Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989) fits with game semantics.

A constructive game semantics for the language of linear logic

Annals of Pure and Applied Logic, 1997

I present a semantics for the language of first order additive-multiplicative linear logic, i.e. the language of classical first order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sen-tences=games, sentences=resources or sentences=problems, where "truth" means existence of an effective winning (resource-using, problem-solving) strategy. The paper introduces a decidable first order logic ET in the above language and gives a proof of its soundness and completeness (in the full language) with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic. The semantics presented here is very similar to Blass's game semantics (A.Blass, "A game semantics for linear logic", APAL, 56). Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be adapted to the logic induced by Blass's semantics to show its decidability (via equality to ET), which was a major problem left open in Blass's paper. The reader needs to be familiar with classical (but not necessarily linear) logic and arithmetic.