Understanding implication

Question

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.

Answer 1

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 $\neg A \to (A \to B)$

Proof Table of $A\to B$

Another way to look at it...

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).

Answer 2

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.