What "THEN" means in Logical implication and Materal Implication?

Question

Up on searching on google. We can find three distinct meanings of "THEN":

1. at that time; at the time in question.

2. after that; next; afterwards; (also) in addition. and

3. In that case; therefore.

Which meaning of THEN goes with logical implication and with material implication?

Answer 1

"In that case" is the nearest.

But normally we take "if A, then C" to be equivalent to "if A, C", or "C, if A".

So the "then" is little more than helpful punctuation, indicating clearly where the consequent of the conditional starts.

(OK, English usage can be messy, and there might be cases where the "then" does a little more work, e.g. indicating temporal order, but those cases are the exception.)

Answer 2

The essence of neither material nor logical implication involves time: then 'then' part could be at the same time as the 'if' part, but it could also be later, and it could also be earlier.

For example, suppose I say: You cannot take Calculus II unless you have taken Calculus I. This means that taking Calculus I is a necessary condition for taking Calculus II: if you did not take Calculus I, then you cannot take Calculus II either. In logic: $\neg CI \rightarrow \neg CII$ .. which we know is equivalent to $CII \rightarrow CI$

Now, note that this $CII \rightarrow CI$ makes no sense if we take the $\rightarrow$ to be a temporal relationship, because in actuality we of course first take Calculus I and then take Calculus II. What we can say, however, is that if we ever meet someone who has taken Calculus II, then we can conclude that this person has taken Calculus I. In other words, the material implication is purely a truth-functional relation: 'if' [this] is true, 'then' [that] is true, is merely saying that the truth of the [this] forces the truth of the [that] ... but it does not at all say that the truth of the [that] is caused by the truth of the [this], let alone that the truth of the [that] temporally comes after the truth of the [this].

The same is true for logical implication: that one also is purely about how the truths of statements relate to each other. The difference with the material implication being that for logical implication this truth-functional relationship holds in all possible worlds, whereas the material implication expresses it for some specific world or context. For example, that you have taken Calculus I is not at all logically implied by having taken Calculus II: this may be something that is true for my school, for example, but maybe at some other place they allow students to take Calculus II without having taken Calculus I.

So, in both cases, the 3rd of your listed meanings is the closest; both meaning 1 and meaning 2 involve time, which logic abstracts away from.