Could someone here who is good at logic explain to me why the conditional (->) is true in the case where both the antecedent and the consequent are false? It makes no intuitive sense to me, and it seems to even translate pretty badly to English. Consider the following proposition:
[math]P \to (P \wedge \neg P))[/math]
This is false when P is true, and true when P is false. Let's use say "I am alive", as P. So let's say P is true. Then:
"It is the case that if I'm alive then I am both alive and not alive."; obviously false.
But let's see the same case where P is false, e.g. "It is the case that if I'm not alive then I am not alive and I'm alive."; somehow true.
Anyone have any sort of explanation?
>>7992820
if
2<1
-> 4<2
axiom wrong but logic correct
>>7992820
One interpretation is that it is not the case you're not alive, so it doesn't matter what the consequent is; the statement is still valid.
However, it's best to think of -> as a formal symbol that doesn't correspond exactly to our every day english usage of "if x then y" or "implies". It's just a mathematical symbol that is defined a certain way. The confusion comes from logic classes in philosophy departments that try to link it to natural language interpretations.
You can read more about this here:
https://en.wikipedia.org/wiki/Material_conditional#Philosophical_problems_with_material_conditional
>>7992820
Well, what you're doing is treating the truth value of the term in the parentheses as stemming from the semantical interpretation of its constituent letters. this is your problem.
What you have to do is consider the rules for conditionals,
T->T = T
T->F = F
F->F = T
F->T = T
For conjunctions
F^T = F
F^F= F
T^F = F
T^T = F
When considering P->(P ^ –P), treat the term in the parentheses as one letter, for example q. Treat P as P. This yields
P->Q
Then you just evaluate it as if this is your sentence. The reason your sentence is true when P is false is because a conditinal that is false->false is true, because it is always true to say "If P is false, then P is false".
does this explanation make sense?
>>7992914
>what you're doing is treating the truth value of the term in the parentheses as stemming from the semantical interpretation of its constituent letters.
No I'm not. I understand that we define False -> False as true, I'm asking why. The "semantics" or examples I gave just illustrated my point in that it's a bit absurd. I don't have a problem with say, vacuous truth, or that P->Q will always be true when Q is true, regardless of P, as most "problems" with the conditional seem to be in this article:
>>7992884
>https://en.wikipedia.org/wiki/Material_conditional#Philosophical_problems_with_material_conditional
It's simply the last rule of the conditional, that False -> False is true, that confuses me.
>>7992820
>why the conditional (->) is true in the case where both the antecedent and the consequent are false?
When we were introduced to propositional logic my professor simply told us that the idea was that you could expect anything from a false statement, so any implication where the thesis is false will be true.
From that I visualize it as a characteristic of trivial truths. Take your example.
>"It is the case that if I'm not alive then I am not alive and I'm alive."
This is trivially true, you can't be in both states at the same time.
>>7992932
>This is trivially true, you can't be in both states at the same time.
You do realize it says that it is TRUE that I can be both alive and not alive, if I'm not alive. That's why I find it strange.
>>7992940
Don't you realize what you just said is tautological? In order for "I'm dead" to be false you must be alive. So stating the false antecedent "I am dead" implies you are both dead and alive.
>>7992932
>>This is trivially true, you can't be in both states at the same time.
there is nothing trivial in the principle of explosion.
in fact, it can be weakened and you can still do math .
>>7993000
[math]P \to (P \wedge \neg P))[/math]
is not tautological. It is true in the case where P is false, or as an example, if I'm dead then I can be both dead and alive.
It does make intuitive sense, it just feels kind of useless.
Suppose I promise you "if it's raining on Monday, I'll have my umbrella". If it isn't raining, by default I haven't broken my promise, because the entry condition wasn't fulfilled.
If that entire expression took the value False, it would mean that I broke my promise, which I clearly didn't.
Therefore,
https://en.wikipedia.org/wiki/Relevance_logic
