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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
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T | T | T 
T | F | F 
F | T | T 
F | F | T

Now, 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.

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Sep 19, 2021 at 2:00
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    $\begingroup$ I think you answered your own question. $\endgroup$ Sep 19, 2021 at 2:00
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    $\begingroup$ @Community : "As it's currently written, it's hard to tell exactly what you're asking" : no it isn't. $\endgroup$ Sep 19, 2021 at 2:06
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    $\begingroup$ It seems to me that you are trying to blur the distinction between truth table evaluation and real world (non mathematically oriented) causation. Consider the statement : "If it doesn't rain today, then I will go to the store today". To a non-mathematician, this makes sense. The good weather is causing you to go to the store. Now consider the statement "If I don't go to the store today, then it will rain today. To a mathematician, this is equivalent to the 1st statement. To a non-mathematician, the statement is nonsense - how can staying indoors cause rain to start? $\endgroup$ Sep 19, 2021 at 2:10

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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.

  1. $(A\to B)$ is just a truth function whose lookup table is defined as $(\lnot A \lor B)$'s truth table.

  2. $A$ implies $B$means that $(A\to B)$'s truth table's second row has been eliminated.

  3. $A$ implying $B$ (e.g., <I won the game> implies that <I scored higher than Jane>) doesn't mean that $A$ causes $B.$

  4. 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 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 $(\lnot A \lor B)$ will result in false thus proving that the assumption "City=Los Angeles implies State=CA" is wrong.

$Ax$:  City $x$ is Los Angeles
$Bx$:  City $x$ is in California

Every LA is in CA. $\qquad Ax\implies Bx\qquad\forall x\;(Ax\implies Bx)$
Not every LA is in CA. $\qquad Ax\kern.6em\not\kern -.6em\implies Bx\qquad\exists x\;(Ax\land\lnot Bx)$

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    $\begingroup$ I have seen a lot of explanations. But (P: n is a multiple of 4 -> Q: n is even) and using P: n = 6, to have that implication mean it's false, is precisely why (F,T) and (F,F) should be defined to have P => Q mean true! Thank you. $\endgroup$
    – nz_
    Aug 24, 2023 at 0:06
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  • 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.
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Maybe one simple way is to consider as follows:

  • A: x>8
  • B: x>4

Then we will have three cases as follows:

  • Case1 or x>8: if statement A is true, then statement B cannot be false or statement B must be true (first line of the truth table)

  • Case2 or 4<=x<=8: if statement A is false, then statement B can be either true or false (third line of the truth table). This case implies that the value of x may be selected in a way (x=7) that the statement A becomes false, but statement B becomes true.

  • Case3 or x<4: if statement B is false, then statement A will be false too or statement A must be false (last line of the truth table)

The second row of the truth table implies that it is impossible we find a value which meets statement A, but simultaneously does not meet statement B.

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A MORE INTUITIVE EXPLAINATION:

The conditional statement $P \to Q$ does not make any assertions about the actual truth values of $P$ and $Q$. Instead, it merely claims that if $P$ is true, then $Q$ must also be true. In other words, it makes an assumption about the truth value of the antecedant and then, as a result, concludes something about the truth value of the consequent.

So, $P$ and $Q$ can both be false and the conditional can still be true. In other words, $P$ and $Q$ can be false and I can still say something like "well, if $P$ was true (even though it is not), then $Q$ would also be true (even though it is not)." Again, this is because the conditional statement asserts that one statement follows from the assumption that another statement is true. This reasoning holds for every configuration of truth values except the case where $P$ is true and $Q$ is false. The conditional statement simply cannot hold if your assumption $P$ is correct (i.e. true) but the conclusion you draw from your assumption $Q$ is wrong (i.e. false).

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EXAMPLE

Consider the implication, it is raining ($R$) implies it is cloudy ($C$)

$$R\to C$$

This does not mean that rain causes cloudiness. It also does not mean that it is always cloudy when it is raining (e.g. so-called sunshowers). It simply rules out the possibility that it is currently both raining AND not cloudy (present tense). This is entirely consistent with the usual truth table for implication in classical logic **:

enter image description here

Note the following:

  • When $R$ is true and $R\to C$ is true (line 1), then $C$ is true. (The Rule of Detachment)
  • When $R$ is false (lines 3-4), then $R\to C$ is true regardless of the truth value of $C$. (The Principle of Vacuous Truth.) This form of argument is rarely if ever used in daily discourse since we rarely consider that the implications of a proposition known to be false. It is, however, routinely used in very technical arguments, e.g. in mathematical proofs.

** Text version of truth table:

R C R=>C

T T T

T F F

F T T

F F T

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