In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?

Provided we have this truth table where “$p\implies q$” means “if $p$ then $q$”:

p&q&p\implies q\\ \hline

My understanding is that “$p\implies q$” means “when there is $p$, there is q”. The second row in the truth table where $p$ is true and $q$ is false would then contradict “$p\implies q$” because there is no $q$ when $p$ is present.

Why then, does the third row of the truth table not contradict “$p\implies q$”? If $q$ is true when $p$ is false, then $p$ is not a condition of $q$.

I have not taken any logic class so please explain it in layman’s terms.

Administrative note. You may experience being directed here even though your question was actually about line 4 of the truth table instead. In that case, see the companion question In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False? And even if your original worry was about line 3, it might be useful to skim the other question anyway; many of the answers to either question attempt to explain both lines.


If you don’t put any money into the soda-pop machine, and it gives you a bottle of soda anyway, do you have grounds for complaint? Has it violated the principle, “if you put money in, then a soda comes out”? I wouldn’t think you have grounds for complaint. If the machine gives a soda to every passerby, then it is still obeying the principle that if one puts money in, one gets a soda out.

Similarly, the only grounds for complaint against $p\to q$ is the situation where $p$ is true, but $q$ is false. This is why the only F entry in the truth table occurs in this row.

If you imagine putting an F on the row to which you refer, the truth table becomes the same as what you would expect for $p\iff q$, but we don’t expect that “if p, then q” has the same meaning as “p if and only if q”.

Source : Link , Question Author : user701510 , Answer Author : JDH

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