Which one is true?

Logic Level 4

Which of the following statements are true?

If $n$ is a prime number and a square number, then $n$ is a triangular number.
If $n$ is a prime number and a square number, then $n$ is not a triangular number.

Assumption:

Assume classical logic, in particular the law of excluded middle: $p \vee \neg p$ is a tautology.

Notes:

A prime number is a positive integer greater than $1$ that is only divisible by $1$ and itself.
A square number is an integer that can be expressed in the form $n^2$ for some integer $n$ .
A triangular number is an integer that can be expressed in the form $\frac{n(n+1)}{2}$ for some integer $n$ .

Both 1 and 2 Neither 1 nor 2 Only 1 Only 2 Not enough information

1 solution

Ivan Koswara
Aug 20, 2015

Both statements are true, despite the appearance that they contradict each other!

First, observe that there is no square number that is also a prime. $0$ and $1$ aren't prime numbers by definition. Consider a square number $n^2 > 1$ . Then $|n| \ge 2$ , thus $n^2$ is divisible by $|n|$ . But this means it is not prime either.

This means the antecedent (the "if" part) is always false. In classical logic, conditional implication in the form "if A then B" can only be false if A is true and B is false. In this case, A is always false, so "if A then B" is always true. Thus both statements are true! This also means both statements are meaningless (except that they state A is false).

A statement in the form "if A then B" where A is always false is called a vacuous truth . Another form would be "every A is B", where A doesn't exist (for example, "every prime, square number is triangular"). A statement in the form "if A then B" where B is always true is called a trivial truth .

Moderator note:

Great! Statements about the empty set are always true, regardless of what the conclusion is. This is why if you believe a lie, you will then believe anything.

My whole problem was that from the start I knew that a square number cannot be a prime number so I just said neither.

Austin Griner - 5 years, 9 months ago

A prime number indeed cannot be a square number. This means the "if" part will never be satisfied. The fault is assuming that since the if part is not satisfied, then the whole statement is false; this is incorrect (it would make "if A then B" equivalent to "A and B"). The opposite is true; since the if part is never satisfied, the whole statement is true since you can show no counterexample exists.

Ivan Koswara - 5 years, 9 months ago

Wow......a really good one....especially the part where you both proved them to be true as well as meaningless

Arnav Das - 5 years, 9 months ago

Great! Statements about the empty set are always true, regardless of what the conclusion is. This is why if you believe a lie, you will then believe anything.

Calvin Lin Staff - 5 years, 9 months ago

can u tell this to my teacher? i lost marks for a problem similar to this

Parker Prathyoom - 5 years, 7 months ago

what a tautology!

Akash singh - 5 years, 9 months ago

Which one is true?

1 solution

Moderator note:

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