We associate the semantic game with chance moves conceived by Blinov with Blamey’s partial logic. We give some equivalent alternatives to the semantic game, some of which are with a third player, borrowing the idea of introducing the pseudo-player called Nature in game theory. We observe that IF propositional logic proposed by Sandu and Pietarinen can be equivalently translated to partial logic, which implies that imperfect information may not be necessary for IF propositional logic. We also indicate that some independent (...) quantifiers can be regarded as dependent quantifiers of indeterminate sequence, using the interjunction connective in partial logic. We conclude our paper by indicating some further research in a more general setting. (shrink)

We define a logic capable of expressing dependence of a variable on designated variables only. Thus has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our avoids some difficulties arising in the original (...) independence friendly logic from coupling the dependence declarations with existential quantifiers. As is the case with independence friendly logic, truth of is definable inside . We give such a definition for in the spirit of [11] and [2] and [1]. (shrink)

In this paper we develop an Ehrenfeucht‐Fraïssé game for. Unlike the standard Ehrenfeucht‐Fraïssé games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of connectives in logic. We prove the adequacy theorem for this game. We also apply the new game to prove complexity results about infinite binary strings.

We introduce a generalization of Independence Friendly logic in which Eloise is restricted to a finite amount of information about Abelard’s moves. This logic is shown to be equivalent to a sublogic of first-order logic, to have the finite model property, and to be decidable. Moreover, it gives an exponential compression relative to logic.

In this paper we study connections between game theoretical concepts and results, and features of IF-predicate logic, extending observations from J. van Benthem (2001) for IF-propositional logic. We highlight how both characteristics of perfect recall can fail in the semantic games for IF-formulas, and we discuss the four Thompson transformations in relation with IF-logic. Many (strong) equivalence schemes for IF-logic correspond to one or more of the transformations. However, we also find one equivalence that does not fit in this picture, (...) by the type of imperfect recall involved. We point out that the connection between the transformations and logical equivalence schemes is less direct in IF-first order logic than in the propositional case. The transformations do not generate a reduced normal form for IF-logic, because the IF-language is not flexible enough. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be (...) so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...) by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

1. Two kinds of logic. To a first approximation there are two main kinds of pursuit in logic. The first is the traditional one going back two millennia, concerned with characterizing the logically valid inferences. The second is the one that emerged most systematically only in the twentieth century, concerned with the semantics of logical operations. In the view of modern, model-theoretical eyes, the first requires the second, but not vice-versa. According to Tarski’s generally accepted account of logical consequence, inference (...) from some statements as hypotheses to a statement as conclusion is logically valid if the truth of the hypotheses ensures the truth of the conclusion, in a way that depends only on the form of the statements involved, not on their content. Interpreted model-theoretically this means that every model of the hypotheses is a model of the conclusion. However, there is an ambiguity in Tarski’s explication, as he himself emphasized, since for the specification of form one needs to determine what are the logical notions. Once those are isolated and their semantical roles are settled, one can see how the truth of a statement is composed from the truth of its basic parts, in whatever way those are specified. The problem of what are the logical notions is an unsettled and controversial one. In the classical truthfunctional perspective, proposals range from those of first-order logic to generalized quantifiers to second and higher-order quantifiers to infinitary languages and beyond. Many of these stronger semantical notions have been treated in the volume Model Theoretic Logics. In a series of singular, thought-provoking publications in recent years, Jaakko Hintikka has vigorously promoted consideration of an extension of first-order logic called IF logic, along with claims that its adoption promises to have revolutionary consequences. (shrink)