The precise meaning of "or"

Nalin Pithwa

May 22

Given two sets A and B, one can form a set from them that consists of all the elements of A together with all elements of B. This set is called the union of A and B and is denoted by $A \bigcup B$ . Formally, we define

$A \bigcup B = \{ x : x \in A or x \in B\}$

But we must pause at this point and make sure exactly what we mean by the statement " $x \in A$ or $x \in B$ ."

In ordinary everyday English, the word "or" is ambiguous. Sometimes the statement "P or Q" means "P or Q or both" and sometimes it means "P or Q but not both." Usually one decides from the context which meaning is intended. For example, suppose I spoke to two students as follows:

"Miss Parijaat, every student registered for this course has taken either a course in linear algebra or a course in analysis."

"Mr. Patel, either you get a grade of at least 70 on the final exam or you will flunk this course."

In the context, Miss Parijaat knows perfectly well that I mean "everyone has had linear algebra or analysis or both.", and Mr. Patel knows I mean "either he gets at least 70 or he flunks, but not both." Indeed, Mr Patel would be exceedingly unhappy if both statements turned out to be true!

In mathematics, one cannot tolerate such ambiguity. One has to pick just one meaning and stick with it, or confusion will reign. Accordingly, mathematicians have agreed that they will use the word "or" in the first sense, so that the statement "P or Q" always means "P or Q, or both". If one means "P or Q, but not both", then one has to include the phrase "but not both" explicitly.

In the language of digital logic design, "or" means inclusive or (that is, P or Q or both) and "xor" (means P or Q but not both).

Cheers,

Nalin Pithwa

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