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
9 - Biconditional
- 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 introduce our final connective – the biconditional – and the two inference rules that govern it. The biconditional – symbolized as ↔ – will be our last logical constant. Unlike our other connectives, the biconditional is explicitly defined in terms of two other connectives (& and →). It is thus a derivative connective.
BICONDITIONAL
In English, this connective is expressed by phrases such as ‘if and only if’, ‘just if’, ‘precisely if’ and ‘exactly if’. We shall take ‘if and only if’ as our canonical expression of the biconditional.
As noted, this connective is explicitly defined in terms of two other connectives: & and →.
‘P if and only if Q’ or ‘P iff Q’
is equivalent to:
‘if P then Q’ and ‘if Q then P’.
Thus P ↔ Q is equivalent to (P →Q) & (Q → P).
BICONDITIONAL
An English biconditional, such as ‘MARY will go to the party if and only if RICHARD goes’, is rendered in our symbolic language as: M ↔ R. It is logically equivalent to the conjunction (M → R) & (R → M). According to the truth-table for ↔, ‘M ↔ R’ is true if M and R have the same truth-value, otherwise false. Relevance logicians who think that → does not capture the meaning of ‘if … then …’ in English will also think that ↔ does not capture the meaning of ‘if and only if’ in English. They will not think that an English biconditional is automatically true if both component sentences are true (or if both are false).
- Type
- Chapter
- Information
- Elementary Logic , pp. 95 - 99Publisher: Acumen PublishingPrint publication year: 2012