Language, Proof and Logic

By Kemeny, John G.

Book Id: WPLBN0000660740
Format Type: PDF eBook
File Size: 3.72 MB
Reproduction Date: 2005

Title:	Language, Proof and Logic
Author:	Kemeny, John G.
Volume:
Language:	English
Subject:	Science., Mathematics, Logic
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Citation APA MLA Chicago Kemeny, J. G. (n.d.). Language, Proof and Logic. Retrieved from http://gutenberg.cc/

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: The special role of logic in rational inquiry. What do the fields of astronomy, economics, finance, law, mathematics, medicine, physics, and sociology have in common? Not much in the way of subject matter, that's for sure. And not all that much in the way of methodology. What they do have in common, with each other and with many other fields, is their dependence on a certain standard of rationality. In each of these fields, it is assumed that the participants can differentiate between rational argumentation based on assumed principles or evidence, and wild speculation or non sequiturs, claims that in no way follow from the assumptions. In other words, these fields all presuppose an underlying acceptance of basic principles of logic.

Table of Contents
Contents Acknowledgements iii Introduction 1 The special role of logic in rational inquiry . . . . . . . . . . . . . . 1 Why learn an arti?cial language? . . . . . . . . . . . . . . . . . . . . 2 Consequence and proof . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Instructions about homework exercises (essential! ) . . . . . . . . . . 5 To the instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Web address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 I Propositional Logic 17 1 Atomic Sentences 19 1.1 Individual constants . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 Predicate symbols . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Atomic sentences . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4 General ?rst-order languages . . . . . . . . . . . . . . . . . . . 28 1.5 Function symbols (optional ) . . . . . . . . . . . . . . . . . . . . 31 1.6 The ?rst-order language of set theory (optional ) . . . . . . . . 37 1.7 The ?rst-order language of arithmetic (optional) . . . . . . . . 38 1.8 Alternative notation (optional ) . . . . . . . . . . . . . . . . . . 40 2 The Logic of Atomic Sentences 41 2.1 Valid and sound arguments . . . . . . . . . . . . . . . . . . . . 41 2.2 Methods of proof . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Formal proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.4 Constructing proofs in Fitch . . . . . . . . . . . . . . . . . . . . 58 2.5 Demonstrating nonconsequence . . . . . . . . . . . . . . . . . . 63 2.6 Alternative notation (optional ) . . . . . . . . . . . . . . . . . . 66 3 The Boolean Connectives 67 3.1 Negation symbol: : . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 Conjunction symbol: ^ . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Disjunction symbol: _ . . . . . . . . . . . . . . . . . . . . . . . 74 3.4 Remarks about the game . . . . . . . . . . . . . . . . . . . . . 77