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The truth table for a logical disjunction shows that there is only one situation where the result can be false, being when both statements are false. As long as one statement is true, the result is also true.

This makes perfect sense to me, but I am wondering why this is the case. Could it not have been equally correct that the false would override the true. A true statement seems to have some governing power for no particular reason.

For example, given the propositions:

p.I play for a baseball team (Assigned to False)

q.I play for a soccer team (Assigned to True)

$$ p \lor q$$

would not be seen as a true or false compound statement in English, so what is the reasoning for this precedence in propositional logic?

Any help is greatly appreciated.

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  • $\begingroup$ In logic, or is the inclusive or, meaning p or q or both. If one wants the exclusive or that's a different symbol, sometimes a plus sign circled. $\endgroup$ – coffeemath Jan 14 '15 at 19:39
  • $\begingroup$ I think one thing to also consider is that "or" sentences can be taken one of two ways in language. Like the sentence, "Did you drink tea or water?" could be answered, "yes" or "tea". $\endgroup$ – turkeyhundt Jan 14 '15 at 19:42
  • $\begingroup$ English is not logic. The precision required for proofs means that the meaning of words used in logic must be specified more closely. $\endgroup$ – Joffan Jan 14 '15 at 19:42
  • $\begingroup$ @turkeyhundt Your explanation gave me a clear understanding of the situation. Thank you to you all. $\endgroup$ – mrybak83 Jan 14 '15 at 19:45
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You are right that there is no reason to have this precedence in propositional logic.

Therefore, we have both the definitions of conjunction and disjunction:

$$p \lor q \leadsto \text{At least one of the following is }\color{red}{\text{true}}: p, q.\\ p \land q \leadsto \text{At least one of the following is }\color{red}{\text{false}}: p, q.$$

Note that the false overrides the true in the conjunction. As you can see, the pair disjunction/conjunction together does not display any preference for the true or the false.

Pedantic linguistic constructions like the above can seem superfluous, but they help to be precise in what a symbolic expression means, especially in potentially confusing cases like disjunction.

Disjunction is a bit special because of the so-called exclusive or construct:

$$p \oplus q \leadsto \text{Either $p$ or $q$ is true.}$$

which expressly excludes the possibility that $p$ and $q$ are both true. The inclusive nature of $\lor$ takes some getting used to; however, after a while, you may find yourself answering "Did you drink tea or water?" with "Yes", as indicated by coffeemath in the comments.

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True OR False is True in every language. Simply because every statements are TRUE OR False; at least in the two valued logic.

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