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Part 3. the Logic of Arguments Logic for Liberal Arts Students Part 3. The Logic of Arguments nto the assembly of the gods came Dialectic, a woman whose weapons are Icomplex and knotty utterances....In her left hand she held a snake twined in immense coils; in her right hand a set of formulas, carefully inscribed on wax tablets, which were adorned with the beauty of contrasting colors, was held on the inside by a hidden hook; but since her left hand kept the crafty device of the snake hidden under her cloak, her right was offered to one and all. Then if anyone took one of those formulas, he was soon caught on the hook and dragged toward the poisonous coils of the hidden snake, which presently emerged and after first biting the man relentlessly with the venomous points of its sharp teeth then gripped him in its many coils and compelled him to the intended position. If no one wanted to take any of the formulas, Dialectic confronted them with some other questions; or secretly stirred the snake to creep up on them until its tight embrace strangled those who were caught and compelled them to accept the will of their interrogator. Martianus Capella, The Seven Liberal Arts, 327-329 (310-339 A.D.) Dialectica with Serpent resting on Aristotle with the Organon. Chartres Cathedral West Façade, Right Portal, ca. 1145-1155. Version2/10/2008 Table of Contents Part 3. The Logic of Arguments .........................................................................................1 Introduction......................................................................................................................1 Lecture 12. Validity, Consistency, and Logical Truth......................................................5 Semantic Entailment....................................................................................................5 The Semantic Definition of Validity...........................................................................5 Showing Arguments are Valid ..................................................................................7 Schema for Proofs of Validity .................................................................................10 Showing Arguments are Invalid..............................................................................10 Consistency ...............................................................................................................13 The Semantic Definition of Consistency.................................................................13 Interdefinablity of Consistency and Validity ............................................................14 Logical Truth ..............................................................................................................15 The Semantic Definition of Logical Truth................................................................15 Interdefinablity of Validity, Consistency and Logical Truth .....................................15 Lecture 13. Categorical Logic: Validity .........................................................................17 Categorical Logic .......................................................................................................17 Immediate Inference ..................................................................................................20 The Square of Opposition ......................................................................................23 ∗The Logic of Empty Terms and Negations ...............................................................26 Obversion...............................................................................................................27 Terms with Empty Extensions ................................................................................28 Formal Syntax and Semantics ...............................................................................28 Square of Opposition for Predicate Negations .......................................................31 The Syllogistic............................................................................................................32 Basic Concepts ......................................................................................................32 Proving Syllogisms are Valid..................................................................................35 Proofs of the Validity of Selected Syllogisms .........................................................36 The Names for the Valid Moods.............................................................................39 ∗Lecture 14. Categorical Logic: Invalidity .....................................................................42 Traditional Term Rules for Invalid Syllogisms ........................................................43 Definition of Distribution .........................................................................................44 Leibniz’s Term Rules..............................................................................................45 Examples of Applications of the Rules.......................................................................46 Rule 1. Undistributed Middle ..................................................................................46 Rule 2. Distributed Term in the Conclusion ...........................................................48 Rule 3. Affirmative premise ...................................................................................49 Rule 4. Negative Conclusion .................................................................................50 Rule 5. Particular Premise.....................................................................................51 Rule 6. Negative Premise......................................................................................52 Rule 7. Universal Premise.....................................................................................54 ∗Syllogism with Empty Subject Terms .......................................................................55 Summary ...................................................................................................................58 Part 3, Page ii Version2/10/2008 Lecture 15. Propositional and First-Order Logic: Validity ............................................60 Propositional Logic ....................................................................................................60 The Truth-Table Test for Validity............................................................................60 Examples ...............................................................................................................61 Proving Invalidity by Truth-Tables ..........................................................................62 Showing Consistency and Inconsistency ...............................................................64 First-Order Logic........................................................................................................65 Validity and Logical Entailment ..............................................................................65 Examples of First-Order Validity Metatheorems.....................................................69 Proofs of the Metatheorems ...................................................................................70 Lecture 16. Proof Theory..............................................................................................75 The Axiomatic Method ...............................................................................................75 Euclid’s Postulates for Plane Geometry .................................................................76 Newton's Three Laws of Motion .............................................................................78 Spinoza’s Philosophical Axioms.............................................................................79 Non-Euclidean Geometry...........................................................................................80 Playfair’s Version of Euclid’s Fifth Postulate ..........................................................81 Postulate 5 in Lobachevskian Geometry...............................................................81 Postulate 5 in Riemanninan Geometry...................................................................82 Axiom Systems and Proofs........................................................................................84 Axiom System ........................................................................................................88 Derivation ...............................................................................................................89 Certainty.................................................................................................................90 Soundness and Completeness ..............................................................................91 Definitions ..............................................................................................................92 Summary ...................................................................................................................94 Lecture 17. Reductions to the Perfect Syllogisms .......................................................96 Syllogistic Reduction..................................................................................................96 Transposition..........................................................................................................97 Simple Conversion .................................................................................................97
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