in other words, why are statements of the form “If A then B” always true if A is false?
In: 8
Because phrases like that are silent on not(A). Effectively, If not(A) then <anything can happen>. There is no conflict in a system which has one “If A” statement and another “If not(A)” statement, in fact this is the sort of construction you’d expect.If you leave off the “If not(A)”, then the default is <anything can happen> which is always true.
What you know is that when A is true, B is true. That is *all* you know from that statement. If A is false then nothing is said about B, it might be true or false.
So if A is false then any state of B would conform to the logical statement and so can always be true.
The rule doesn’t apply if Bob doesn’t drink. If he doesn’t drink then everyone else could still drink or they could not drink.
Because a logical statement in the form of “If A then B” does not provide a way for the statement to be disproven if A is false. Let’s look at your example. There are 4 possible scenarios here:
1. Bob drinks and everyone else drinks.
2. Bob drinks and everyone else doesn’t drink.
3. Bob doesn’t drink and everyone else drinks.
4. Bob doesn’t drink and everyone else doesn’t drink.
Of those 4 scenarios, only #2 breaks the original statement, making it false. If Bob doesn’t drink, then no matter what everyone else does, it doesn’t disprove the original statement. Since the statement isn’t proven false, according to the rules of logic, it must be true.
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.
This sounds like the drinker’s paradox.
This statement remains true when you look at the two possibilities.
1) Bob is drinking:
As that statement says, if Bob drinks, everyone drinks. If A, then B. Bob is a part of everyone, so if Bob drinks, it is not contradictory that everyone might drink. So, that statement could be true; it is not necessarily wrong.
2) Bob is not drinking:
In logic, there is something called the contrapositive. It is basically the logical reverse of a statement. You find the contrapositive by reversing the order of the argument and reversing the positive or negative quality of the parts of the argument. So, the original state is “if A, then B”. The contrapositive is therefore “if not B, then not A.” Both these statements are logically identical.
The contrapositive of the original statement about Bob is “if not everyone is drinking, Bob isn’t drinking.” Again, this statement could be true. Since Bob is a part of everyone, Bob’s absence of drinking is a sign that not everyone is drinking.
So, we say that “If Bob drinks, then everyone drinks” is true if Bob doesn’t drink” because when you look at both sides of the coin, the logic does not contradict itself. It’s true in the sense that it could be true. This kind of truth is called vacuous truth. It’s “true” because it isn’t untrue.
You used the math flair, so I will assume you understand basic logic. A conditional of the form “P → Q” is equivalent to “¬P V Q”, where ¬ is the negation operator and V is a stand-in for the proper OR operator (I’m on my phone right now).
If we then take your example:
P is “Bob drinks”,
¬P is “Bob doesn’t drink”,
and Q is “everyone drinks”.
Thus, “If Bob doesn’t drink, then everyone drinks” can be symbolically represented as “¬P → Q”, which becomes “¬(¬P) V Q”. ¬¬P = P, so we have P V Q, which is True V True
Just adding on previous answers, a equivalente way to write “if A then B” (A -> B) using just the basic operators (not, or, and) is “not A or B” (~A v B). Since ~A will be true in case A is false, the expression will always be true.
“If Bob drinks, then everyone drinks.”
In formal logic, this can be alternatively worded as:
“In the event that Bob drinks, then I know for certain that everyone must also drink.”
So now you suppose, “So what happens in the event that Bob *does not* drink?” The answer is: we do not have enough info to know. Nothing about the initial phrase told us about what happens in that scenario.
It’s tempting to *infer* that the opposite of the phrase (in strict logic terms, the *inverse*) must also be true, i.e. “If Bob does not drink, not everyone drinks.” We do that a lot in common speech. Like, if I told you, “If it is Tuesday, I will be gone”, you may reasonably gather from context the inverse, “If it is not Tuesday, I will be here”. That works in day-to-day speech. But in formal logic, that doesn’t fly. I did not explicitly tell you what happens when it is not Tuesday. Maybe I will be here. Maybe I won’t. For all you know, my presence and the day of the week are not related at all on all days that are not Tuesday.
If you do want to explicitly say the inverse is also true without adding another statement, you can say “if *and only if*” instead of just “if”, e.g. “If *and only if* it is Tuesday, I will be gone.”
EDIT: Reread the question.
In general, “if A is true, then B is true” can still be a true statement even when A is not true. You never *said* “A is true”, you said, “*if* A is true”.
If I said, “If you turn right at the intersection, then you will go to the grocery store”, but you turn right instead, what I said doesn’t stop being true. *If you did* turn left, you *would have* gotten to the grocery store. But you didn’t that time.
Shit like this is why I struggled to get a D in community college logic. I’ve read the top explanations like three times and they still don’t make sense to me.
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