What Is a Disjunction? The Logical Or and Its Two Meanings

The inclusive reading holds that the combined statement is true if either part is true or if both are, and it is false only when both are false. The exclusive reading holds that it is true when exactly one part is true and false when both are. English uses both without marking the difference, which is why an offer of tea or coffee generally means one of them while a requirement to hold a passport or a driving licence is satisfied by holding both. Formal logic adopted the inclusive reading as standard, partly because it composes more simply with the other connectives and partly because the exclusive version can be built from the inclusive one while the reverse is also possible but less convenient. Legal and technical writing handles the ambiguity by stating which is meant, which is where the awkward construction joining both with a slash comes from.

The connective follows rules that make it tractable and occasionally counterintuitive:

• •It is commutative and associative, so the order and grouping of the parts do not matter
• •It distributes over conjunction and conjunction distributes over it, which is the basis of rearranging complex conditions
• •Negating it turns it into a conjunction of negations, which is one of the two rules named after De Morgan
• •A statement or its negation is always true, which is the law of excluded middle and is rejected by some approaches
• •Adding a further alternative to a true disjunction keeps it true, which is valid and feels wrong
• •Knowing a disjunction is true does not tell you which part is true, which is what makes reasoning from one useful and awkward

The standard method for using a disjunction is proof by cases, which assumes each alternative in turn and shows that the desired conclusion follows either way, establishing it without ever determining which alternative holds. That is a powerful technique and it appears throughout mathematics, since many arguments proceed by splitting into an exhaustive set of possibilities. The complementary move eliminates alternatives, since knowing that at least one of two things holds and that one of them does not establishes the other, which is the pattern behind a great deal of ordinary detective reasoning. Both require that the alternatives genuinely exhaust the possibilities, which is where the reasoning most often fails outside mathematics, since a false dilemma presents two options as though they were the only ones and the argument is then valid while its premise is not.

The connective has a direct physical implementation and that is where most people encounter it without noticing. A logic gate implementing it outputs a signal when either input carries one, and gates of this kind together with the other basic connectives build every arithmetic and control circuit in a processor. The exclusive version has its own gate and is the workhorse of binary addition, since adding two bits without a carry is exactly that operation. Search queries use the same logic, with terms joined by it broadening a search and terms joined by conjunction narrowing it, and understanding the difference is most of what distinguishes an effective search from an ineffective one. Database queries, spreadsheet conditions and the filters in nearly every piece of software are built from the same small set of connectives, so the rules governing how they combine are among the most practically applicable results in logic.

Certain consequences of the standard treatment sit badly with intuition and have generated genuine philosophical work. A disjunction is true whenever either part is, so asserting one when you know which part is true is technically correct and conversationally misleading, since a listener reasonably infers that you did not know. That gap between what is true and what is appropriate to say is one of the central examples in the study of how implication works in conversation, and the explanation is that speakers are expected to make the strongest claim they can support. A further oddity is that from any true statement one may validly infer its disjunction with anything at all, including nonsense, which is valid and useless. Non-classical logics that reject the law of excluded middle handle disjunction differently, requiring that asserting one involves being able to say which part holds.

The inclusive reading is true when at least one part is, including both, and the exclusive reading excludes both, with English using either without marking which. Logic standardised on the inclusive version. Proof by cases assumes each alternative in turn and establishes a conclusion without determining which holds. Asserting a disjunction you could have made stronger is true and misleading, which is a matter of conversation rather than logic.