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
12 - Truth-trees
- 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
In this chapter we compare and contrast our natural deduction proof method with another way of proving sequents: the truth-tree method. The method of truthtrees (or semantic tableaux as it is also called) was developed by the Dutch logician E. W. Beth (1908–64). We also show how truth-tables can be used to test for validity and invalidity. Both methods – truth-trees and truth-tables – have similar advantages and disadvantages.
SOME COMMENTS ON TRUTH-TREES
The tree method of proof has two interesting features. First, it is entirely mechanical. No creativity or ingenuity is required in order to complete a tree. Second, the tree method, implemented correctly, is guaranteed to determine whether a given sequent of elementary logic is valid or invalid.
Our natural deduction system has neither of these features. It can take insight to see how to prove a valid sequent using our inference rules. And our natural deduction system can only show a valid sequent to be valid. It cannot prove an invalid sequent to be invalid. But the tree method can conclusively show an invalid sequent to be invalid.
These noteworthy features of the tree method come at a cost. First, using the tree method we can easily determine whether a given sequent is valid or invalid, but we get no sense of why it is valid or invalid. In contrast, natural deduction proofs allow us to see, step by step, that the conclusion of a valid sequent follows from its premises, and why. Second, tree proofs can become very unwieldy, especially in the case of more complex sequents. Natural deduction proofs, in contrast, are relatively compact and easy to follow.
- Type
- Chapter
- Information
- Elementary Logic , pp. 128 - 138Publisher: Acumen PublishingPrint publication year: 2012