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Index
Title Page Copyright Contents What is Logic? Studying Sentences The Square off Oppositions The Syllogism Connective Logic Leibniz’s Law The Reductio ad Absurdum A “New Organon” Frege’s Quantifiers The Context Principle Propositional Calculus Cantor’s Set Theory The Usefulness of Connectives The Russell Paradox The Fatal Flaw The Problem of Surface Grammar Russell’s System Wittgenstein’s Logical Pictures Carnap and the Vienna Circle The Tolerance Principle Hilbert’s Proof Theory The Arrival of Gödel Gödel’s Incompleteness Theorem The Connections to Proof Theory Wittgenstein’s Table of Logical Connectives Wittgenstein’s Truth Tables Discovering Tautologies The Logic Gates of Digital Electronics A Vending Machine Turing and the “Enigma Code” Euclid’s Axiomatic Method Leibniz’s Proof Method Abuse of Contradiction Rules for Connectives Sensitivity to Grammar Predicate Calculus Model Theoretic Semantics Hilbert’s Recursion Model Finite Rules for Infinite Production Simple Instructions Proof Theory and Formal Language Tarski’s Truth Conditions Formal Semantics in Practice Constructing a Soap Opera Prolog to an AI Soap Opera Turing’s recipe for AI The Problem of Paradoxes Can Paradoxes be Avoided? Theory of Types Tarski’s Solution to the Liar The Unexorcised Paradox Gödel’s Incompleteness Theorem The Consequences of Gödel’s Theorem The “Halting Problem” The Limit of Gödel’s Proof Zeno’s Movement Paradox An Infinite Sum A Convergence on Limits How Much is a “Heap”? The Challenge to Sets Undermining Logic The Fiction of Vague Words What Do Words “Mean”? Fuzzy Logic Fuzzy Heaps Can Logic Escape Paradox? Non-Classical Logics: Intuitionism The Devil’s Argument Intuitionistic Logic Intuitionism versus the Reductio Method The Intuitionistic Fad Addressing Some Old Problems The Value of Possible Truth Values as Numbers The Possible and Non-Contradiction From Classical to Fuzzy Logic Electronic “Possible” States The Fuzzy Logic Search Engine The Fuzzy Logical Machine Logic in the Quantum World The Distributive Law of Quantum Logic Logic by Experiment Logic and Science The Copernican Revolution Galileo’s Revolution Methods of Deduction and Induction Problems with Induction Hume’s Fork Nomological Deduction Induction by Generalization Laws or Empirical Predictions The Raven Paradox A Problem of Cause and Effect Popper’s Answer to Hempel Popper’s Disconfirmation Theory The Probability of Viable Theory Quine’s “Web of Belief” Alterations to the “Web” Insufficient Evidence Quine’s Relativism Feyerabend’s Denial of Scientific Method Davidson’s Reply to Quine The Presentation of Truth Hard-edged Truth versus Relativism Cognitive Science and Logic Chomsky’s Universal Grammar Noun and Verb Categories Recursive Rules of Grammar The X-bar Theory A Logical Theory Problems of Syntax and Semantics Complex Grammatical Structures Problems with “Universal” Grammar The Symbolic Brain Model Training a Neural Net Pattern Recognition The Rational Behaviour Model Practical Reason What is Consciousness? The Place of Logic Wittgenstein’s Change of View Further Reading Index
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