The Principles of Mathematics Revisited

Abstract

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.

Compositional Natural Language Semantics using Independence Friendly Logic or Dependence Logic

“…[13] or [15]). In this extension of predicate logic it can be indicated that a quantifier is independent of some of the quantifiers with larger scope.…”

“…The first example (in 1974) concerned natural language semantics: the branching quantifier sentences [12]. Later the de dicto -de re ambiguity and other scope ambiguities were mentioned, and as examples in mathematics the definitions of continuity and uniform continuity [13]. Many more examples of applications are given in [13].…”

On the Expressive Power of IF-Logic with Classical Negation

“…Independence Friendly logic [4] (IF, for short) is an extension of first-order logic (FO) where each disjunction and each existential quantifier may be decorated with denotations of universally-quantified variables, as in:…”

“…Since the expressive power of IF corresponds to that of existential secondorder logic (Σ 1 1 ) [4,5] and Σ 1 1 is not closed by (classical) negation, it is clear that classical negation cannot be defined in IF.…”

“…In [4], he claims that "virtually all off classical mathematics can in principle be done in extended IF first-order logic" (in a way that is ultimately "reducible" to plain IF logic). What he calls "(truth-functionally) extended IF logic" is the closure of the set of IF-sentences with operators ¬, ∧ and ∨.…”