DISJUNCTION

A simple example of a disjunction or ‘or’-sentence is:

(1) BILL will be at the party or MARY will be at the party

‘BILL will be at the party’ and ‘MARY will be at the party’ are the disjuncts of (1). We represent this logically as:

(1a) B ∨ M

As we know from the truth-table for ‘or’, a disjunction such as (1), or a formula such as (1a), is true if at least one of the disjuncts is true, otherwise false.

Separating out the disjuncts

Sometimes, English sentences may look simple, but in fact are complex. For example:

(2) Bill or Mary will be at the party

may look like a simple sentence, but of course it is not. (The same is true of some ‘and’-sentences, e.g. ‘Bill and Mary went to the party’.) For the purposes of elementary logic, (2) should be explicitly rewritten as (1) then represented as B ∨ M.

However, we do not have ‘or’-sentences that look complex but are actually simple. Th ere are no ‘false disjunctions’ in the way there are ‘false conjunctions’ (e.g. ‘Venus and Serena won the Ladies’ Doubles’, ‘Bill and Ben lift ed the piano’, etc.). ‘Bill and Ben lift ed the piano’ is not a conjunction of two sentences. It would be represented in our formal language by a single letter or atomic formula. In contrast, all occurrences of ‘or’ in English constitute disjunctions of two or more sentences. Hence, all ‘or’-sentences are complex.