How to understand conditional statements

Question

$p \rightarrow q$ is read as ${\rm{if}}\:p\:{\rm{then}}\:q$.
It is clear that the result of the statement when $p$ is true is $q$. So, when $p$ is true, the truth value of the statement is the same truth value of $q$.
But how should I understand the cases when $p$ is false? I don't get why the statement is true when the hypothesis is false and the conclusion is true, and when both the hypothesis and the conclusion is false.

I know that this can be made clear with some examples of a promise or a contract. But I want to understand it independently from specific examples (might not be a good idea). So how can I interpret the cases when $p$ is false only with the statement "${\rm{if}}\:p\:{\rm{then}}\:q$" without putting anything in $p$ and $q$ or using logical manipulations (converse, inverse, contrapositive)?

Answer 1

The key to understanding the truth or falsehood of an implication is to understand what it means.

The implication $p \to q$ only fails (i.e., is false) when the truth of $p$ fails to imply the truth of $q$. That's what "implication" means. So it is false precisely when $p$ is true, yet $q$ is false.

In all other cases, the implication fails to fail (i.e., is true).

Implication does not mean that the falsehood of $p$ says anything at all about the truth or falsehood of $q$, so the implication doesn't fail (i.e., is true) whenever $p$ is false (since $q$ doesn't even need to be considered in that case).

Answer 2

a binary boolean operator like $\rightarrow$ may be defined as a function $$ f: B \times B \rightarrow B $$

where $B$ is the set $\{T, F\}$ of 'truth values'. you may check that there are exactly sixteen different binary boolean operators.

it usually causes confusion if we try to understand mathematical concepts in terms of the semantics of ordinary language.

however, it is natural for us to do precisely this when attempting to understand a new term.

the genuine question here would be "why do we gloss $a \rightarrow b$ as 'if a then b'.

it may help to think in terms of computer programming. in this context the IF statement is a selector:

$$ y = 0 \\ \text{input} \quad x \\ \text{if} \quad x > 10: y = 1 $$

the third statement does not raise an error condition in the cases when x is 10 or less.