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Independence-Friendly Logic

The present work is a critical analysis of IF ("Independence-Friendly") logic, a logical system which was introduced by the Finnish philosopher Jaakko Hintikka towards the end of the '80s. It is a syntactical variant of Leon Henkin's logic of the branching quantifiers; Henkin's and Hintikka's logics are both characterized by the great freedom they allow in expressing functional dependences and independences among individual variables. Relationships of this kind were already implicit in classical logic; yet, by means of the classical first-order syntax only a small and rather arbitrary subset of all dependence and independence relationships can be expressed.
Hintikka claims that his logic plays a fundamental role in the foundations of mathematics; he even suggested that it may be the case that IF logic should replace first-order classical logic. The purpose of the present work is a formal clarification of the definition and peculiarities of Independence-Friendly logic, aimed at a full understanding of Hintikka's assertions. This seems to be a necessary step towards the confirmation or refutation of Hintikka's claims.
Chapter One introduces Hintikka's syntax, and the most traditional semantics for IF logic; then comes an overview of the main syntactical variants which occur in literature. Following Hintikka, we describe another semantics, which is based on Game Theory; we prove its equivalence with the previous semantics. We prove that IF logic is also equivalent with a significant fragment of second-order logic. Then, we present a generalized IF logic, and its semantics as formalized by W.Hodges; we explain the relationship which bounds together IF and generalized IF logic; through Hodges' semantics we obtain an answer to the problem of "compositionality" of IF logic.
Chapter Two is devoted to an analysis of the differences between IF and first-order classical logic. The focus is mainly on the difficulties which arise in the treatment of concepts such as negation, equivalence and implication. Towards the end of the chapter, we shift our attention to some relevant remarks of Theo Janssen.
A few of his remarks reveal bizarre features of IF logic, and cast doubts on its meanings. Others seem to be simply wrong; yet, a careful examination of these argumentations shows that IF logic is extremely sensitive to slight changes in syntax.
In Chapter Three we make use of the formalization which was developed so far in order to re-examine some of the main claims of Hintikka's "The Principles of Mathematics Revisited".

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0.1 Introduction A series of recent papers of the Finnish logicians Jaakko Hintikka and Gabriel Sandu - and, in particular, Hintikka’s 1996 treatise The Principles of Mathe- matics Revisited - revived the interest in a minor generalization of first-order logic, the logic of branching quantifiers. Introduced in the early sixties by Leon Henkin ([6]), the logic of branching quantifiers was shown not to be recursively axiomatizable; it turned out to be equivalent, from a semantical viewpoint, to a fragment of second-order logic known as Σ11 logic; and it was even employed to prove an old conjecture on first-order logic ([20]). The intuition behind the branching quantifiers is that quantifiers establish dependences among variables; whenever an existential quantifier occurs within the scope of a universal quan- tifier, the existentially quantified variable depends on the universally quantified variable; the branching quantifiers permit any possible pattern of dependence, while first-order syntax allows only patterns with a peculiar structure. Hintikka and Sandu introduced a new, linear notation for branching quantifi- cation, featuring tokens like ∃y/∀x1, . . . ,∀xn (y is specified to be independent of x1, . . . , xn); they called the logic that bears this syntax Independence-Friendly (IF) logic. They also provided an alternative semantics based on game theory; and they investigated the role of negation. Allowing negations in front of IF sentences leads out of Σ11 logic; the new logic thus obtained was called extended IF logic. But, above all, Jaakko Hintikka tried to to persuade mathematicians and philosophers of mathematics that IF logic had a central role in the foundations of mathematics; that it was, in a sense, the true first-order logic, freed from unnatural restrictions. It was hard to judge of the correctness of his claims, because of two reasons at least. The first is the style adopted in The Princi- ples of Mathematics Revisited : lots of mathematical results are quoted without demonstrations, and the argumentations seem to be chosen and ordered so as to convince rather than clarify. Other difficulties spring from IF logic itself. The law of the excluded middle turned out to fail in IF logic; that is, there are IF sentences that are neither true nor false. As a consequence, “false” and “non-true”, “true” and “non- false” become distinct concepts; the crucial semantic notions of equivalence and logical implication have no more a univocal, “canonical” definition. On such unusual grounds, it is an hard matter to decide whether IF logic really resembles first-order logic or not, and whether results concerning IF logic have the same philosophical import than they would have if they were proved for more regular logics. The purpose of the present work is to provide the reader with some of the instruments that are needed to understand the debates around IF logic. The presentation is more formal than Hintikka’s; proofs are often provided, except for when the results are already well-known from more traditional areas of logic. Hintikka’s arguments in favor of IF logic are presented and criticized. But they come in the end, in the third and last chapter, when the technical and problematic aspects of IF logic have already been discussed. It is proved that 3

Laurea liv.II (specialistica)

Facoltà: Scienze Matematiche, Fisiche e Naturali

Autore: Fausto Barbero Contatta »

Composta da 70 pagine.


Questa tesi ha raggiunto 261 click dal 27/02/2009.

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