Disjunction Glossary
Definition
The disjunction connective 'OR' results in a formula A OR B which is true if and only if at least one of A or B is true. A and B are said to be its disjuncts.
Connectives with similar truth conditions in natural languages are also taken to express disjunction. The word 'or' in English, taken in a truth-functional sense, is the intended analogue of the formal connective.
Comments
The role of disjunction is to make it possible to express partial information: it gives us something to say where we can restrict the truth to a range of possibilities without knowing which one actually holds. For instance, 'The coin will show either heads or tails' tells us what the outcomes of a coin toss might be without requiring us to know which of the two outcomes will actually happen.
Disjunction in English is ambiguous. It sometimes has the inclusive reading as noted above, but is sometimes the exclusive version, true if and only if exactly one of A or B is true. Exclusive disjunction is a perfectly good and useful connective, but it is not the one symbolised by 'OR'.
The truth table is:
| OR | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 1 |
In the natural deduction calculus, the introduction rule for disjunction says that A OR B is an immediate consequence of A, and also an immediate consequence of B. The elimination rule says that whatever follows from A and also from B follows from the assertion that at least one of them holds.
Examples
- 'Canberra is either in Australia or in New Zealand' narrows down the geographical location of Canberra without saying exactly which country it is in. The disjunction is true, because Canberra is actually in Australia, and the truth of one disjunct is enough.
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The following proof illustrates the rules for
disjunction:
SP {q OR r ⊢ (p OR q) OR r} PL {1} {1} {q OR r} {A} PL {2} {2} {q} {A} PL {2} {3} {p OR q} {2 ORI} PL {2} {4} {(p OR q) OR r} {3 ORI} PL {5} {5} {r} {3, 4 ANDI} PL {5} {6} {(p OR q) OR r} {5 ORI} PL {1} {7} {(p OR q) OR r} {1, 4 [2], 6 [5] ORE} EP