The term 'or' is a logical term, and is used to convey that the statement in which it occurs, provided it occurs as a connective between statements, is such that at least one of the two or more statements it is used to connect is true.

In formal logic, 'or' is often written as 'V' and rules that are adopted for 'V' are often these:

From (A) it follows that (AVB).
From (B) it follows that (AVB).
From (AVB) and (~A) it follows that (B).

That is: From any disjunct derived by itself one can derive a disjunction of several disjuncts involving it, and one can derive a disjunct from having already derived a disjunction and the denial of a disjunct.

In standard propositional logic, there is an equivalence for disjunction that is known since antiquity, and usually goes by the name of the English 19th Century mathematician and logician De Morgan:

(AVB) IFF ~(~A&~B)

Thus one can define 'or' in terms of 'and' and 'not'.

Some other theorems in standard propositional logic involving 'or':

T1. pVp iff p
T2. pVq iff qVp
T3. pV(qVr) iff (pVq)Vr

There are several logical problems related to 'or', that can be best discussed by reference to the above rules, and that show that 'or' is somewhat more subtle than 'and'.

First, there is the problem that on the above rules one can infer a disjunction of several statements from having just one of these, which may be somewhat counter-intuitive in case the other disjunct that is inferred in the disjunction is unknown.

Second, the rules for inferring a disjunct from a disjunction have several alternative forms, such as

From (AVB) and (~AVB) it follows that (B)
From (AVB) and (A implies C) and (B implies C) it follows that (C).

The main reasons for preferring one of these over the earlier one is that they make some proofs easier or more perspicuous, or may seem more intuitive.

Third, it is an interesting fact that it took quite a long time in logic to arrive at the so-called inclusive or written as 'V' that is defined by the following truth-table that also defines the exclusive or written as '|':

Boole set up his Mathematical Analysis of the Laws of Thought, in which first occured what is now known as Boolean Algebra with an exclusive or, taking it apparently as fully intuitive and evident that saying 'or' amounted to claiming 'either .. or .. but not both', as defined by 'p|q'.

It since has been found that an inclusive or is far more convenient, and that it is easy to render both kinds of 'or' in standard logic, but it remains an interesting fact that one of the greatest logicians intuitively thought otherwise, showing that there are subtleties involved with 'or', of which more in the following point.

Fourth, there is a fundamental problem about what it is that one believes if one believes a disjunction such as 'pVq'. By the above table, this amounts to believing either (p and q), or (p and not q), or (not p and q) and thus in any case either p or else q - but it is obvious that quite often one does not know which alternative is true, even though one firmly believes one of them is.

An example is (pV~p): Most people will agree - as a table will show - that this is true in any case, but will also insist that they often do not know which of the two disjuncts is true.

Thus formulated in terms of beliefs of a person a: It may happen that a believes that p or not-p, while disbelieving that p and disbelieving that not-p, both for the simple reason that a believes he lacks the relevant knowledge to say which obtains. It is difficult to fit this sort of rather normal intuition involving 'or' into the constraints of standard propositional logic.

Fifth, related to the previous point, there is the fact that the choices people make are perceived as choices from alternatives, so that it at least appears as if before making a choice there is such a thing as an alternative from which one may make a choice, but that only exists as alternative until the choice is made. As in the previous case, it is difficult to fit this sort of rather normal intuition involving 'or' into the constraints of standard propositional logic.