in other words, why are statements of the form “If A then B” always true if A is false?
In: 8
This is called Material Implication and is a very weird concept, but a necessary one. I’m hoping someone else will give a better answer but I’ll give it a shot. I took a logic course last spring and we spent 4 or so of the 15 weeks on this topic – it’s very confusing until it clicks in your brain.
You’re talking about one rule, implication, which only works in one direction. If you have A, then we know that we have B (“we have” means “is true” in this situation). This DOES NOT mean that if we don’t have B, we don’t have A, or the converse that if we have A then we have B – neither of these are true.
This also has to do with the B being the necessary part, the conditional (A) is not within the statement. And implication is different than biconditional. Biconditional is closer to what most people would call equality; if A then B and if B then A.
I’ll try to give another example, the basic one we used in our class: if all of the Frenchmen in the club are wearing hats, and there is no one in the club, is the first statement false? You can’t find an example to show me of a Frenchman wearing a hat, and you can’t find me a Frenchman *not* wearing a hat. Thus we can’t say for sure that the statement is false, and if there were 1+ Frenchmen in the club (with the information we know) they would be wearing a hat. I might be wrong on this but my impression is that thus: we can’t say it is false so we’ll accept it as true. And it works out that way.
I hope this helps some, I don’t know if I’ve succeeded in any way… This is a weird concept which is very hard to understand, and it makes a lot more sense as you use it (aka not great for eli5). Sorry if this hasn’t helped.
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