Confusion regarding implications

Question

Out of interest, I wanted to develop some basic understanding of formal logic, but I am having trouble understanding the truth table of implications.

Especially examples such as this:

A: I want a pizza

B: I go shopping

A --> B: If I want a pizza, then I will go shopping

Why is it, for example, that assuming A is false and B is false that A-->B is true? It isn't obvious to me at all that just because you don't want a pizza and you don't go shopping automatically means that if you want a pizza then you will go shopping. I mean you don't know if you will actually go shopping once you know that you want pizza just based on A and B being false, right? An intuitive explanation of this concept would be greatly appreciated

Duplicate question https://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr — nickalh, Mar 07 '24 at 17:44
Does this answer your question? In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True? — nickalh, Mar 07 '24 at 17:47

score 0 · Answer 1 · answered Mar 07 '24 at 17:42

First of all, if the presupposition is true, the conclusion must be true. This should be obvious.

If the presupposition is false, the implication will always be true. The reason for this can be explained by the principle of explosion. If you assume a false thing, you will be able to prove anything.

Consider your presupposition being $1=2$. Well, then you can subtract $1$ on both sides, so $1=2 \Rightarrow 0=1 $. This is an example of a falsehood implying falsehood. But notice that this means that subtracting $0$ from $2$ is the same as subtracting $1$ from $2$, so $1=2 \Rightarrow 1=1$. This is an example of a falsehood impliying a truth.

Being able to prove anything from a falsehood holds throughout mathematics.

Confusion regarding implications

1 Answers1