Book contents
- Frontmatter
- Contents
- Preface
- PART I First steps in logical reasoning
- 1 Starting points
- 2 Rules of proof
- 3 Natural deduction
- 4 Proof search
- 5 Classical natural deduction
- 6 Proof search in classical logic
- 7 The semantics of propositional logic
- Part II Logical reasoning with the quantifiers
- Part III Beyond pure logic
- Part IV Complementary topics
- Suggestions for the use of this book
- Further reading
- Bibliography
- Index of names
- Index of subjects
2 - Rules of proof
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface
- PART I First steps in logical reasoning
- 1 Starting points
- 2 Rules of proof
- 3 Natural deduction
- 4 Proof search
- 5 Classical natural deduction
- 6 Proof search in classical logic
- 7 The semantics of propositional logic
- Part II Logical reasoning with the quantifiers
- Part III Beyond pure logic
- Part IV Complementary topics
- Suggestions for the use of this book
- Further reading
- Bibliography
- Index of names
- Index of subjects
Summary
Logical reasoning proceeds from given assumptions to some sought conclusion. The essence of assumptions is that they are hypothetical so that it is not determined if they hold, and the point with the steps of reasoning is that they produce correct conclusions whenever the assumptions are correct. These steps are two-fold: In one direction, we analyse the assumptions into their simpler parts, in another direction, we look at the conditions from which the sought for conclusion can be synthesized. The aim is to make these ends meet. Some examples lead us to a small collection of basic steps and it turns out that all logical arguments based on the connectives can be reproduced as combinations of the basic steps.
Steps in proofs
Consider our bather in Cap Breton. The argument was: We have assumptions of the forms A ⊃ B and ¬ B. Now a is added to these assumptions, and a contradiction follows. The argument can be presented as a succession of steps each one of which is in itself hard to doubt. We write the steps one after another together with a justification at right:
Example argument 2.1. Proof of a contradiction from A ⊃ B, ¬ B, and A.
A ⊃ B by assumption
¬B by assumption
A assumed with the aim of proving a contradiction
B from 1 and 3
B & ¬ B from 4 and 2
- Type
- Chapter
- Information
- Elements of Logical Reasoning , pp. 15 - 30Publisher: Cambridge University PressPrint publication year: 2014