Logic notation question

Question

If I have some statements $P$ and $Q$ and I have the following logic formula: "$P$ or $Q$", does that mean only one of them is true or both of them can be true?

Answer 1

Posting this here as CW in case link provided loses visibility.

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What you are getting at is the difference between the inclusive disjunction and the exclusive disjunction.

Suppose you have two arguments, $P$ and $Q$. The following is the truth table for the "inclusive or" often denote by $P\lor Q$:

$ \boxed{ \begin{array}{c|c|c} P & Q & P\lor Q \\ \hline T & T & T\\ T & F & T\\ F & T & T\\ F & F & F \end{array}} $

The following is the truth table for the "exclusive or," sometimes denoted by $P\,\dot\lor\,Q$ (other notations include $\mathsf{XOR},\oplus,\underline\vee, \dot\vee, \nleftrightarrow, \not\equiv$, etc.):

$ \boxed{ \begin{array}{c|c|c} P & Q & P\,\dot\lor\,Q \\ \hline T & T & F\\ T & F & T\\ F & T & T\\ F & F & F \end{array}} $

As you can see, $P\lor Q$ is true when either $P$ is true or $Q$ is true (or both), whereas $P\,\dot\lor\,Q$ is true when either $P$ is true or $Q$ is true (but not both).

That's really all there is to it.

Answer 2

It means that at least one of them is true (so, if both are true, then "P or Q" is also true).

Answer 3

Typically in mathematics, the standard "or" as in p or q, is inclusive. This means that it is true if and only if p and q are not both false. So it will be true if both are true. There are other "ors" that follow different definitions. For example the exclusive or requires that only one can be true.