The question that has bothered me for a while has been answered and closed here (Implication in mathematics - How can A imply B when A is False?) and probably many other posts. Although all the answers are accurate and correct in their own way of explaining (or went above my head), those still didn't "click" for me so I kept trying to find a more specific example. Here's what I came up with and would like folks to review and comment.
First the question:
In classical logic, why is (p->q i.e. p implies q) True if both p and q are False? I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go. In the truth table, where A->B we obtain this result:
A | B |A->B ------------ T | T | T T | F | F F | T | T F | F | TNow, my confusion here is regarding why A->B given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.
My conclusion and probably the mistake I've been making to understand here is that this truth table is not about when the result will be true if you use the equivalent (~A V B) logic. What this truth table represents is the fact that if you have a data set (or situations) that results in a false value of (~A V B) then your assumption that A implies B is violated (or is not correct). In simpler words, the true values in the truth table are for the statement A implies B. Conversely, if the result is false that means that the statement A implies B is also false. And now as I read it, I guess, I'm stating the obvious but let's try it with an example.
Let's say we have an assumption that the State of California is the only State with a city named Los Angeles. So we setup two tests; 'A' (City=Los Angeles) and 'B' (State=CA). Now, based on our assumption, whenever we find an address with city of Los Angeles, we can infer that the State must be California (if A is true it will imply that B is also true). However, if we were not correct in making that earlier assumption, and there is another State which has a city of Los Angeles, in that case the test (~A V B) will result in false thus proving that the assumption "City=Los Angeles implies State=CA" is wrong.
$\endgroup$ 53 Answers
$\begingroup$My conclusion and probably the mistake I've been making to understand here is that this truth table is not about when the result will be true if you use the equivalent $(\lnot A \lor B)$ logic. What this truth table represents is the fact that if you have a data set (or situations) that results in a false value of $(\lnot A \lor B)$ then your assumption that $A$ implies $B$ is violated (or is not correct). In simpler words, the true values in the truth table are for the statement “$A$ implies $B$”. Conversely, if the result is false that means that the statement “$A$ implies $B$” is also false.
Bear in mind that
$(A\to B)$ is just a truth function whose lookup table is defined as $(\lnot A \lor B)$'s truth table.
“$A$ implies $B$” means that $(A\to B)$'s truth table's second row has been eliminated.
$A$ implying $B$ (e.g., <I won the game> implies that <I scored higher than Jane>) doesn't mean that $A$ causes $B.$
The $(F,F)$ and $(F,T)$ cases are perhaps unintuitive to grasp, but the alternative would make even less sense: if we let $F\to T\equiv F,$ then $$\forall n\in\mathbb Z \,\big(n \text{ is a multiple of }4\, \to \,n \text{ is even}\big)$$ would be a false statement (try $n=6$).
Let's say we have an assumption that the State of California is the only State with a city named Los Angeles. So we setup two tests; '$A$' (City=Los Angeles) and '$B$' (State=CA). Now, based on our assumption, whenever we find an address with city of Los Angeles, we can infer that the State must be California (if $A$ is true it willimply that $B$ is also true).
However, if we were not correct in making that earlier assumption, and there is another State which has a city of Los Angeles, in that case the test $(\lnot A \lor B)$ will result in false thus proving that the assumption "City=Los Angeles implies State=CA" is wrong.
✔ ‘Every LA is in CA’ $\implies$ $(A\implies B)$
✔ ‘Not every LA is in CA’ $\implies$ $(A\kern.6em\not\kern -.6em\implies B)$
$\endgroup$ $\begingroup$It seems to me you are confusing a material implication, that is just a sentence, and a reasoning based on a material implication + some extra information as premisse. "A materially implies B" does not mean " A therefore B".
In a propositional logic formula, names ( city names, state names) cannot be substituted for letters (A, B, C, etc). Letters represent propositions ( declarative sentences).
I think there are two ways to grasp logical implication :
(1) consider all possible functions from the set TT, TF, FT, FF ( a 4 elements set) to the set T, F ( a 2 elements set); one of these functions will map TT to T, TF to F, FT to F and FF to F ; call this truth function " material implication" ( if you like) and you are done ; here the question as to " why do we get T in the FT and the FF case? " is simply : " because this particular truth function exists and that's the way it is".
(2) a more intuitive road is to define material implication as an abbreviation for " its not the case that A is true and B is false" ; suppose A is false ; in that case you do not have both A true and B false, since you already do not have A true ( in the same way, if you've never been to Japan, a fortiori you've never been both to Japan and to France) ; with this definition at hand, you can reconstruct the truth table.
Note : one could also use , as a defnition : $(A\rightarrow B )\equiv ( \neg A\lor B)$; this definition would result in the same truth table.
- A third thing is not to confuse material implication and logical implication : when you say that "A materially implies B" you simply assert that you are, factually, on line (1), or (3) or (4) of the truth table , that is, a situation in which you are not on line (2); this is why ( given the factual truth values in the actual world of the propositions involved ), " Biden is President in 2021" materially implies " Paris is the capital of France" ; but logical implication only holds when not only we do not have $(A\&\neg B)$ , but when it is not logically possible to have A true and B false. So to say, logical implication is a claim that involves all possible worlds, not only the actual world.
To truly understand why, in classical logic, an implication with a false antecedent must always be true, you have to understand conditional proofs and proofs by contradiction. Until then, you pretty much have to accept the usual truth table as The Definition of Implies. If or when you understand these basic methods of proof, have a look at my blog posting on this topic at There, among things, I effectively derive each line the truth table in question using a form of natural deduction.
$\endgroup$
