Always Related?

Logic Level 3

True or False?

In classical logic, for any truth values of two propositions $p$ and $q$ , at least one of the following statements is true:

$\begin{aligned} p &\implies& q \\ q &\implies& p. \end{aligned}$

False True

1 solution

Anthony Holm
Sep 8, 2016

Let us assume that p->q is false. This can only happen if p is true and q is false. Then, because q is false, q->p must be true. The reverse also applies if q->p is assumed to be false. Thus at least one of the two statements is always true.

But that's not what the question asks. It says "for any two propositions p,q". It doesn't say "For any fixed truth values of the propositions p,q".

Because the statement that is true depends on the truth values, it is not true that "one of the following is definitely true", where it is implicit that the "one statement" is fixed.

Calvin Lin Staff - 4 years, 9 months ago

I don't understand what the issue is, can you explain differently?

Anthony Holm - 4 years, 9 months ago

The previous version of the problem was

For any two propositions $p$ and $q$ , one of the following statements is definitely true:

The issues were:

The context of a Logic system that assumes the law of excluded middle (propositions are either true or false) in important, since that is what we're using in the proof
Minor issue with phrasing: It was not clear if the "one statement" is fixed. IE there is a difference between "the number of true statements in this list is (at least) one" and "
Minor issue with phrasing: It was not clear that we were allowed to have "at least one" statement. In particular, for such problems, the implicit understanding is that the count is an exact count. E.g. in problems of the form "There is one true statement in this list.", we that "one" to mean "exactly one".

Calvin Lin Staff - 4 years, 9 months ago

Always Related?

1 solution

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