Since the first chapter of this book presents an intro duction to the present state of game-theoretical semantics (GTS), there is no point in giving a briefer survey here. Instead, it may be helpful to indicate what this volume attempts to do. The first chapter gives a short intro duction to GTS and a survey of what is has accomplished. Chapter 2 puts the enterprise of GTS into new philo sophical perspective by relating its basic ideas to Kant's phi losophy (...) of mathematics, space, and time. Chapters 3-6 are samples of GTS's accomplishments in understanding different kinds of semantical phenomena, mostly in natural languages. Beyond presenting results, some of these chapters also have other aims. Chapter 3 relates GTS to an interesting line of logical and foundational studies - the so-called functional interpretations - while chapter 4 leads to certain important methodological theses. Chapter 7 marks an application of GTS in a more philo sophical direction by criticizing the Frege-Russell thesis that words like "is" are multiply ambiguous. This leads in turn to a criticism of recent logical languages (logical notation), which since Frege have been based on the ambi guity thesis, and also to certain methodological sug gestions. In chapter 8, GTS is shown to have important implications for our understanding of Aristotle's doctrine of categories, while chapter 9 continues my earlier criticism of Chomsky's generative approach to linguistic theorizing. (shrink)

Attempts to indentify the fundamental concepts of language, argues that the study of language reveals hidden facts about the mind, and looks at the impact of propaganda.

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous (...) impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities. (shrink)

We discuss the differences between first-order set theory and second-order 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. (shrink)

In game-theoretical semantics, perfectlyclassical rules yield a strong negation thatviolates tertium non datur when informationalindependence is allowed. Contradictorynegation can be introduced only by a metalogicalstipulation, not by game rules. Accordingly, it mayoccur (without further stipulations) onlysentence-initially. The resulting logic (extendedindependence-friendly logic) explains several regularitiesin natural languages, e.g., why contradictory negation is abarrier to anaphase. In natural language, contradictory negationsometimes occurs nevertheless witin the scope of aquantifier. Such sentences require a secondary interpretationresembling the so-called substitutionalinterpretation of quantifiers.This interpretation is sometimes impossible,and (...) it means a step beyond thenormal first-order semantics, not an alternative to it. (shrink)

Let us assume that you are entrusted by UNESCO with an important task. You are asked to devise a universal logical language, a Begriffsschrift in Frege's sense, which is to serve the purposes of science, business and everyday life. What requirements should such a “conceptual notation” satisfy? There are undoubtedly many relevant desiderata, but here I am focusing on one unmistakable one. In order to be a viable lingua universalis, your language must in any case be capable of representing any (...) possible configuration of dependence and independence between different variables. For if such a configuration is possible in principle, there is no guarantee that it might not one day show up among the natural, human or social phenomena we have to study.But how are dependencies and independencies between variables expressed in our familiar logical notation? Every logician worth his or her truth-table knows the answer. Dependencies between two variables are expressed by dependencies between the quantifiers to which they are bound. For instance, inthe variable y depends on x, while inz depends on x but not on y, while u depends on both x and y.But how is the dependence of a quantifier on another one expressed in familiar logical languages? Obviously by occurring in its scope, indicated by the pair of parentheses following it. But the nesting of scopes is a transitive and antisymmetrical relation which allows branching only in one direction. Hence other kinds of structures of dependence and independence between variables are not representable in the received logical notation. Such previously inexpressible structures form the subject matter of what has been referred to as independence-friendly logic. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical terms, (...) since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)

The notion of scope as it relates to a model of logical form is discussed. The inability of the accepted definition of scope to account for the contrast between priority scope - the logical priority of different quantifiers & other logical notions via rule ordering - & binding scope - the identification of the connection between variables of quantification & a particular quantifier - is demonstrated. The semantic ambiguity of this dichotomy of scope is explored via examination of donkey sentences. (...) A formal representation of this dichotomy is suggested. The government-binding approach & principle of compositionality that handle scope relations via the logical form is rejected due to the complexity of the mechanism by which syntactic form results in logical forms. A discourse representational approach is also rejected. It is concluded that the interaction of syntax & semantics is instead better represented by a game-theoretical treatment of scope & anaphora. 26 References. T. Rosenberg. (shrink)

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous (...) impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities. (shrink)