Book contents
- Frontmatter
- Contents
- Preface
- 1 Overview
- 2 Logical connectives and truth-tables
- 3 Conditional
- 4 Conjunction
- 5 Conditional proof
- 6 Solutions to selected exercises, I
- 7 Negation
- 8 Disjunction
- 9 Biconditional
- 10 Solutions to selected exercises, II
- 11 Derived rules
- 12 Truth-trees
- 13 Logical reflections
- 14 Logic and paradoxes
- Glossary
- Further reading
- References
- Index
14 - Logic and paradoxes
- Frontmatter
- Contents
- Preface
- 1 Overview
- 2 Logical connectives and truth-tables
- 3 Conditional
- 4 Conjunction
- 5 Conditional proof
- 6 Solutions to selected exercises, I
- 7 Negation
- 8 Disjunction
- 9 Biconditional
- 10 Solutions to selected exercises, II
- 11 Derived rules
- 12 Truth-trees
- 13 Logical reflections
- 14 Logic and paradoxes
- Glossary
- Further reading
- References
- Index
Summary
OVERVIEW
The purpose of this final chapter is to look at various paradoxes, some of which call into question our most fundamental logical principles.
A paradox is a piece of reasoning that proceeds from apparently true premises, via seemingly unquestionable logical steps, to an obviously absurd or otherwise unacceptable conclusion. It is natural to hope that any given paradox can be resolved or dissolved, for example by showing that the apparently true premises are actually false or that the apparently valid reasoning is in fact invalid. But there is no guarantee that this hope will always be fulfilled. Nor is it a trivial matter if paradoxes cannot be resolved. Some paradoxes – such as the Liar and the Sorites – have the potential to shake the foundations of our thinking.
Paradoxes are intellectually engaging because there is an obvious urgency to them. How do we get out of the cognitive predicament they put us in? No curious, intelligent individual could come across a paradox and have no interest in its resolution.
THE LIAR PARADOX
The Liar paradox is of ancient heritage (an early version is due to Epimenides the Cretan). Consider the sentence (S):
(S) S is false.
Suppose that (S) is either true or false (in line with the classical principle of bivalence that every sentence [or statement] is true or false).
If (S) is true, then what it says is true. But (S) says that it itself is false. So (S) is false.
If (S) is false, then what it says (i.e. that (S) is false) is false. But, if it's false that (S) is false, then (S) is true. So (S) is true.
In short: if (S) is true, it is false; and if (S) is false, it is true. Assuming that (S) is either true or false, it follows that (S) is both true and false.
We have a classic paradox. From apparently unassailable assumptions by apparently valid reasoning, we have derived an absurd result: that some statement is true and false.
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- Elementary Logic , pp. 155 - 168Publisher: Acumen PublishingPrint publication year: 2012