Abstract not found

We describe a logic which is the same as first-order logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gives a compositional meaning to formulas of the `information-friendly' language of Hintikka and Sandu. For first-order formulas the semantics reduces to Tarski's semantics for first-order logic. We prove that two formulas have the same interpretation in all structures if and only if replacing an occurrence of one by an occurrence of the other in a sentence never alters the truth-value of the sentence in any structure.

We discuss the differences between first-order set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former. 1

Abstract. In this paper it is argued that Hintikka’s game theoretical semantics for Independence Friendly logic does not formalize the intuitions about independent choices; it rather is a formalization of imperfect information. Furthermore it is shown that the logic has several strange properties (e.g. renaming of bound variables is not allowed). An alternative semantics is proposed which formalizes intuitions about independence.

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this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary first-order logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the first-order level. This eliminates once and for all the need of set theory for the purposes of a metatheory of logic.

Semantic games are an important evaluation method for a wide range of logical languages, and are frequently resorted to when traditional methods do not easily apply. A case in point is a family of independence-friendly (IF) logics which allow regulation over information flow in formulas, and thus perfect information fails in the games associated with such formulas. This mechanism of imperfect information is studied in this paper. It is noted that imperfect information of players often gives rise to the game-theoretic phenomenon of imperfect recall. Furthermore, independence-friendliness in epistemic logic is investigated. We also discuss a couple of misunderstandings that have occurred in the literature concerning IF first-order logics and gametheoretical semantics, related to such issues as intuitionism, constructivism, truth-definitions, mathematical prose, and the status of set theory. By straighten out these misunderstandings, we hope to show the importance of the role semantics ga...

We study partiality in propositional logics containing formulas with either unde ned or over-de ned truth-values. Unde ned values are created by adding a four-place connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a game-theoretic perspective, emerging from a two-stage extensive form semantic game of imperfect information between two players. This game-theoretic approach yields an interpretation where partiality is generated as a property of non-determinacy of games. Over-de ned values are produced by adding a weak, contradictory negation or, alternatively, by relaxing the assumption that games are strictly competitive. In general, particular forms of extensive imperfect information games give rise to a generalised propositional logic where various forms of informational dependencies and independencies of connectives can be studied.

Abstract. We give a Dialectica-style interpretation of first-order classical affine logic. By moving to a contraction-free logic, the translation (a.k.a. D-translation) of a firstorder formula into a higher-type ∃∀-formula can be made symmetric with respect to duality, including exponentials. It turned out that the propositional part of our Dtranslation uses the same construction as de Paiva’s dialectica category GC and we show how our D-translation extends GC to the first-order setting in terms of an indexed category. Furthermore the combination of Girard’s?!-translation and our D-translation results in the essentially equivalent ∃∀-formulas as the double-negation translation and Gödel’s original D-translation.

Hintikka describes the semantical games for Independence Friendly logic (IF-logic) in terms of the game rules. In this paper we elaborate in detail how the standard extensive game model serves as a mathematical model for these games. We intend the game model to be a framework in which we can reason with mathematical rigor about strategies, hence about truth and falsity in GTS. We discuss negation normal forms, and compare the notion of Skolem function with the game theoretical notion of strategy.