The Principles of Mathematics RevisitedCambridge University Press, 28 ապր, 1998 թ. - 288 էջ 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. |
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The functions of logic and the problem of truth | 1 |
The game of logic | 23 |
Independencefriendly logic | 46 |
Some uses of IF logic | 72 |
The complexities of completeness | 88 |
Whos afraid of Alfred Tarski? Truth definitions | 105 |
Negation in IF logic | 131 |
Fraenkelsteins monster? | 163 |
IF logic as a framework for mathematical theorizing | 183 |
10 | 195 |
Constructivism reconstructed | 211 |
The epistemology of mathematical objects | 235 |
Appendix by Gabriel Sandu | 254 |
271 | |
281 | |
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Common terms and phrases
assumptions axiom of choice axiom system axiomatic set theory Chapter combinatorial compositionality concept of truth constructivistic contradictory negation deductively complete defined descriptively complete diagonal lemma elementary arithmetic epistemic interpretation epistemic logic excluded middle existence existential quantifier expressed extended IF first-order fact falsifier finite first-order language first-order sentence foundations of mathematics function of logic Gödel number Hence higher-order logic Hilbert's Hintikka idea incompleteness inference initial verifier instance instantiation intended models intuitionists kind language games law of excluded liar paradox logic and mathematics logical truth logicians mathematical theories mathematicians means model theory natural languages natural numbers negation normal form nonlogical notation notion of truth ordinary first-order logic paradox philosophers player principle problems propositions purely logical question recursive result S₁ S₂ scope second-order logic second-order translation semantical games sense set-theoretical Skolem functions structures T-schema Tarski-type truth definitions Tarski's theorem tion true truth definition truth predicate truth-conditions variable winning strategy