The Psychology of Proof

Preface I Preliminaries 1 Psychological Approaches to Deduction Some Examples of Deductive Reasoning Example 1: Sentential Reasoning Example 2: Reasoning with Instances Some Preliminary Observations The Centrality of Deductive Reasoning Theoretical Evidence for the Importance of Deduction Empirical Evidence Main Trends in Reasoning Research The Psychometric Approach The Experimental Approach Conclusions and Prospects 2 Reasoning and Logic Formal Proof The Axiomatic Method The Natural-Deduction Method The CPL (Classical Predicate Logic) System Rules for Connectives Rules for Quantifiers The Place of Logic in a Theory of Reasoning 3 Reasoning and Computation Problems of Search in Deduction Proof Heuristics: LT and GPS Improved Algorithms for Proofs Non-Resolution Methods and Revisions to the Natural-Deduction Rules Sentential Rules Quantifiers Solving Problems by Proving Theorems Sample Uses of Deduction Methods in Deductive Problem Solving II A Psychological Theory Of Deduction 4 Mental Proofs and Their Formal Properties Overview of the Core Theory Core Assumptions Assumptions about Memory Assumptions about Inference Routines Assumptions about Control Formal Properties PSYCOP's Incompleteness with Respect to Classical Logic A Halting Theorem for PSYCOP and PSYCOP + Canonical Proofs Completeness and Decidability of PSYCOP + Summary Appendix: Proofs of the Major Propositions HP-1: PSYCOP+'s rules create only finitely many domains for a given argument. HP-II: The backward rules produce only afinite number of subgoals within a domain. HP-III: The forward rules produce only a finite number of assertions within a domain. CP-I: An algorithm for translating tree proofs to natural-deduction structures. CP-II: Canonical proofs are deducible via the rules of PSYCOP+. 5 Mental Proofs and Their Empirical Consequences Some Tests of the Proof Model Evidence from Evaluation of Arguments Evidence from Proof Comprehension Evidence from Memory for Proofs Summary Consistency with Earlier Findings Inferences with Negatives Inferences with Conditionals Summary 6 Variables in Reasoning Extending the Core System to Variables A Sample Proof with Variables and Temporary Names Matching Modifications to Rules Formal Properties Soundness of Matching Soundness of Deduction Rules Incompleteness Appendix: Proofs of the Major Propositions 7 Reasoning with Variables Empirical Predictions for Sentences with a Single Variable Presuppositions and Implicatures of Categorical Sentences Categorical Syllogisms Producing Conclusions to Syllogistic Premises Predictions for Multi-Variable Contexts An Experiment on Matching Time Complex Multi-Variable Arguments Summary Appendix: Syllogism Results III Implications and Extensions 8 The Role of Deduction in Thought Deduction as a Cognitive Architecture Problem Solving by Deduction Deduction in Classification Frames, Schemas, Theories, and Deduction Natural Deduction, Nonmonotonic Logic, and Truth Maintenance Nonmonotonic Logic Truth Maintenance Open Questions 9 Alternative Psychological Theories: Rule-Based Systems Alternative Theories Based on Natural Deduction GPS Revisited Osherson's Models of Deduction Natural Logic According to Braine, Reiser, and Rumain Summary Pragmatic Schemas Schemas According to Cheng and Holyoak The Status of Pragmatic Schemas Morals Concerning Standard Deduction Rules Social Contracts Costs-and-Benefits' Costs and Benefits Morals Concerning Standard Deduction Rules Summary 10 Alternative Psychological Theories: Ruleless Systems Heuristic-Based Theories and ''Content'' Effects Deduction and Availability ''Content'' in Deduction Can Availability Explain Deduction? Reasoning by Diagrams and Models Diagrammatic Approaches Models and Mental Models The Mental-Models Hypothesis How Plausible Are Mental Diagrams and Models as Explanations of Reasoning? Models, Inference Rules, and Content Effects Models, Inference Rules, and Truth Mental Models, Inference Rules, and Acquisition 11 Perspectives on Reasoning Ability Aren't You Making People Out to Be More Logical or Rational Than They Really Are? Aren't You Asserting the Discredited Doctrine of Psychologism (or that the Laws of Logic Are the La... Aren't You Assuming That People Can't Possess ''Incorrect'' Deduction Rules? Doesn't the Account of Performance Beg the Question about the Source of Subjects' Errors on Deducti... Aren't You Assuming a ''Logicist'' Approach That Has Already Proved Unwieldy in AI? (Aren't You Assum... Aren't You Making People Out to Be More Illogical or Irrational Than They Really Are? Aren't You Assuming That People Are Programmed to Be Irrational? Aren't You Assuming That People Should Interpret Connectives in a Truth-Functional Manner and Then ... Summary Notes References Index