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
3 - Conditional
- 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
We now begin the project of constructing natural deduction proofs. Each of our five connectives is associated with two inference rules: an introduction rule and an elimination rule. An introduction rule allows us to introduce the relevant connective from formulae not containing the connective. An elimination rule allows us to derive formulae not containing the connective from formulae that do. We begin gently with the very simple elimination rule associated with → (also known as the rule of modus ponens). In Chapter 5 we examine the more complicated introduction rule governing →: the rule of conditional proof.
CONDITIONAL SENTENCES
A simple conditional sentence
A paradigmatic English conditional sentence is:
(1) If some Canberrans vote for ABBOTT then some CANBERRANS are Liberals.
A conditional has an antecedent and a consequent. In (1) the antecedent is ‘some Canberrans vote for ABBOTT’ and the consequent is ‘some CANBERRANS are Liberals’.
In general, in any conditional of the form ‘if P then Q’, P is the antecedent and Q the consequent.
Sentence (1) would be represented in our symbolic language as:
(1a) A → C
In this formula, A is the antecedent and C the consequent.
More complicated conditional sentences
Not all English conditional sentences come neatly packaged with the antecedent between the ‘if’ and the ‘then’ and the consequent following the ‘then’. Indeed, some conditional sentences do not contain the words ‘if’ and ‘then’.
For example, consider the following sentences:
(2) MARY inherits if BILL dies.
(3) BILL dies only if MARY inherits.
(4) Only if MARY inherits, BILL dies.
(5) MARY inherits provided that BILL dies.
(6) Provided that BILL dies, MARY inherits.
(7) BILL'S dying is sufficient for MARY'S inheriting.
(8) MARY's inheriting is necessary for BILL's dying.
These sentences are, in fact, simply different ways of saying:
(9) If BILL dies, then MARY inherits.
Sentences (2)–(9) should all be represented as:
(9a) B → M
- Type
- Chapter
- Information
- Elementary Logic , pp. 22 - 34Publisher: Acumen PublishingPrint publication year: 2012