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I have read so many posts about the implication, but I am still confused.

The statement "if A, then B" is always true when A is false. But I think that when A is false, nothing can be concluded.

Moreover, I do not understand the example, "If today is Saturday, then tomorrow is Monday." As today is Wednesday, both sentences are false so the implication should be true, but it seems in daily life that this implication is false.

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    $\begingroup$ Can you say more about why all the existing discussions of this topic are not helping you? $\endgroup$
    – Karl
    Sep 20 at 20:41
  • $\begingroup$ I do not why they said that if the premise is false, any conclusion can be true. $\endgroup$
    – mnmn1993
    Sep 20 at 20:51
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    $\begingroup$ I like examples like "for all integers $n$, it is the case that if $n$ is a multiple of $4$, then $n$ is a multiple of $2$". At least, it motivates why we define it like this - I hope you agree that what I wrote is true. What I wrote says "for all $n$, the statement $P(n)$ is true", where $P(n)$ is the implication "if $n$ is a multiple of $4$, then $n$ is a multiple of $2$". If you disagree that for instance $P(3)$ is true, you must disagree with what I wrote! I think a point of confusion is often when people say $A$ implies $B$ in real life, they secretly mean "for all $x$, $A$ implies $B$". $\endgroup$
    – Izaak van Dongen
    Sep 20 at 20:53
  • $\begingroup$ Take a look here: en.m.wikipedia.org/wiki/Paradoxes_of_material_implication $\endgroup$
    – Bram28
    Sep 21 at 2:26
  • $\begingroup$ $A\to B$ means that $B$ is true whenever $A$ is. Another way to say that is either $A$ is false or $B$ is true. This is because $A$ is either true or false and whenever $A$ is true, so is $B$. Hence, $A\to B$ is the same as $\lnot A\lor B$. That's why the statement is true when $A$ is false. Propositions are either true or false. We can't have an indeterminate evaluation of a logical proposition. $\endgroup$
    – John Douma
    Sep 21 at 3:52

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It seems you want to prove that $\neg A \to (A \to B)$. In words: If proposition $A$ is false, then it must be true that $A\to B$ for any logical proposition $B$, be it true or false.

It can be proven with a truth table or with a formal proof.

Truth table for $\neg A \to (A \to B)$

enter image description here

Proof Table of $A\to B$

Another way to look at it...

enter image description here

Note that when the antecedent $A$ is false (lines 3-4), the implication $A\to B$ is true regardless of the truth value of the consequent $B$.

Formal Proof of $\neg A \to (A \to B)$

(1) $\neg A~~~$ (Assume)

(2) $A~~~$ (Assume)

(3) $\neg B~~~$ (Assume)

(4) $\neg A \land A~~~$ (Join 1, 3)

(5) $\neg \neg B~~~$ (Discharge 3, 4)

(6) $B~~~$ (Eliminate '$\neg \neg$' 5)

(7) $A \to B~~~$ (Discharge 2, 6)

(8) $\neg A \to (A \to B)~~~$ (Discharge 1, 7)

This form of argument is rarely if ever used in daily life since we seldom give much consideration to the implications of a proposition that is known to be false. It is, however, often used in very technical arguments, e.g mathematical proofs (the so-called principle of vacuous truth).

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The statement if A, then B is always true when A is false.

Suppose that A is false. Then as you have pointed out if A, then B is indeed true regardless of B's truth value; putting it another way, this conditional being true does not allow us to say anything about B's truth.

Summing up:

  • when A is false, we can conclude nothing about B.

And this certainly remains a fact:

  • when A is false, we can conclude that if A, then B is true.

These two bullets are consistent with each other.

But I think that when A is false, nothing can be concluded.

In your mind, what exactly is the conclusion that you refer to? Don't mix up concluding about B's truth and concluding about if A, then B's truth! It is the same when proving the theorem X⟹Y: we can consider either the thesis statement X⟹Y itself as the conclusion, or X⟹Y as an argument form with premise X and conclusion Y.

Similarly, don't be conflating the truth of the implication X⟹Y (the entire conditional) with the truth of its consequent Y (which, unhelpfully, is informally sometimes called "the implication of X").

In short, I feel that the confusion that you are raising stems largely from the differing ways we frame conditionals/implications/arguments in natural language; here and here are more examples.

"If today is Saturday, then tomorrow is Monday." As today is Wednesday, both sentences are false so the implication should be true, but it seems in daily life that this implication is false.

  1. Hmm, could you elaborate on how/why this implication feels false in everyday life? Could it be due to the listener, while processing this implication, not firmly bearing in mind that today is actually not Saturday?

    How about the implication “If pigs fly, then tomorrow is Monday”? Noting that today is Wednesday, in everyday life, does this implication feel true or feel false to you? Like the previous implication, this is a false→false statement; however, here the antecedent/premise is more blatantly false, so hopefully this implication feels more obviously true, even in everyday English?

  2. Or, are you perhaps misreading

    • if today is Saturday, then tomorrow is Monday

      (this statement is true on a Wednesday)

    as

    • if some day is Saturday, then the following day is Monday

      (this statement is false every day)

    instead? The latter, called a universal implication, can be rewritten as

    • on each day, if that day is Saturday, then the following day is Monday

      $\forall x\;\Big(\text{Sat}(x)\to \text{Mon}(x)\Big),$

    and is the most common type of implication in mathematics and natural language (implicit universal implications are commonplace). On the other hand, the former feels lacking of “logical force” as it is merely a synthetic statement instead of a statement analysing hypotheticals.

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I think the easiest way to intuitively appreciate the definition of implication is to not think of statements that are true and false, but of events that "happen" and "don't happen"

So if we think of $A$ and $B$ as events, then we would think of the statement

$A\implies B$

As saying "Every time $A$ happens, $B$ also happens"

Then false statements correspond to events that never happen, and then it becomes clear why a "false" statement implies anything, for example the truth of the implication

"The moon turns green" $\implies$ "Cats fall from the sky"

Is now much more apparent, since we are saying that every time the moon turns blue, cats start falling from the sky, and this is true in the most basic sense of the word, relating to empirical evidence, because in all our documentation of existence, every time that the moon turned green (which is $0$ times) cats fell from the sky, and we can safely say that every time in the future, when the moon turns green, cats will fall from the sky, and no one will ever be able to prove us wrong, because the moon will never turn green.

This analogy has some drawbacks, mainly that it doesn't really allow you to combine different implications using implication itself, but it gives a feeling for why we chose this definition.

You are also correct that if an implication is true, but the premise is false, then we can't conclude anything from the implication, it is only when you combine a true implication with a true premise that you can conclude the conclusion

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  • $\begingroup$ Thank you so much for the answer. But I think that "no one will ever be able to prove us wrong, because the moon will never turn green" seems like no one can prove this is true as well, so I think we do not know the implication is true or not. Anyway, it seems that "if A then B" is true when A is false almost comes from the definition of implication, is it correct? $\endgroup$
    – mnmn1993
    Sep 20 at 22:17
  • $\begingroup$ It depends on the framework that you're working in, but in many, yes it comes straight from the definition...as for whether or not we know if its true or not, perhaps you can rephrase the analogy to say "every time $A$ happened, $B$ also happened". Then you don't have to worry about what it means for us to never be able to verify something, you're just verifying statements about the past $\endgroup$
    – Carlyle
    Sep 21 at 6:34
  • $\begingroup$ But Carlyle, your "every time A happens, B also happens" characterisation of A⟹B in fact incorrectly explains the OP's given true statement If today is Saturday, then tomorrow is Monday (noting that today is Wednesday) as false! $\quad$ @mnmn1993 $\endgroup$
    – ryang
    Sep 21 at 11:33
  • $\begingroup$ @ryang, I don't quite agree...every time it happens on a Wednesday that it is a Monday, it does also happen that tomorrow is Saturday. The normal definition of implication also fails us if we don't know what day it currently is, so it is entirely reasonable to include that as "part of the event" when we evaluate it using this characterisation. But regardless, it isn't meant to be a characterisation, but rather a motivation for why we defined implication the way we did $\endgroup$
    – Carlyle
    Sep 21 at 13:19
  • $\begingroup$ @Carlyle The normal definition of implication also fails us if we don't know what day it currently is, so it is entirely reasonable to include that as part of the event $\;$ I don't really know how you mean by "fails us", but no the context/interepretation (which truth is always relative to) is always tacit in the background and never part of the proposition itself. $\quad$ regardless, it isn't meant to be a characterisation, but rather a motivation $\;$ Ok: "simplification" instead of "characterisation", then. $\endgroup$
    – ryang
    Sep 21 at 13:41

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