Why is the "implication" operation in mathematical logic called "implication"?

Question

I can perfectly understand its definition in terms of other operations and its truth table. Still, I'm completely lost on why "implication" was chosen as the way to refer to it, when it is so disconnected from how we use the words "implication" and "imply" on a logical, day-to-day basis - in the same way we understand "sufficiency" to mean.

Yes, nothing was ever said about $A$ being false having anything to do with $B$ being either true or false when we say "A implies B" in a casual conversation, but the way $A\implies B$ is defined to be strictly true whenever $A$ is false creates a gap between how this word is used in mathematics and in verbal communication in a very striking way. Does this word encapsulate something I'm not seeing? Or is it just an unfortunate historical development, and I should just abstract away that this specific word was used?

Answer 1

There actually is colloquial language corresponding to "false implies anything," sort of, e.g. sayings like "that'll happen when pigs fly" or "when Hell freezes over."

Most of the time we don't bother stating "false implies anything" implications in colloquial language because, by assumption, their antecedents are impossible, so most of the time these statements aren't very useful. However, in mathematics we do things like proof by contradiction where we assume a statement precisely to show that it is false by deriving other obviously false conclusions from it. So in mathematics we actually do need to consider the effects of assuming things that are impossible.

Anyway, as far as I'm concerned the primary justification for talking about material implication is modus ponens: $p \to q$ is the statement such that, given $p$ and $p \to q$, we can deduce $q$. And the material implication has this property. And this is what we need to write proofs, including proofs by contradiction. So we use that.

There's a different and more psychological or perhaps philosophical question of how to really capture the nuances of how implication gets used in colloquial language. One might argue that in colloquial language "$P$ implies $Q$" has something to do with considering possible counterfactual worlds in which $P$ is true, and examining whether $Q$ is true in those worlds; thinking about this more leads to modal logic, strict implication, and Kripke semantics. I don't know that these formalisms really say anything about the nuances of what humans would consider "plausible" vs. "implausible" counterfactual worlds, though.

The SEP article on logical consequence might also be a relevant read.

Answer 2

[T]he way $A\implies B$ is defined to be strictly true whenever $A$ is false creates a gap between how this word is used in mathematics and in verbal communication in a very striking way.

If proposition $A$ is false, then the implication $A \implies B$ is true regardless of the truth value of $B$. This principle of vacuous truth is rarely if ever used in our daily discourse because we are rarely concerned about the implications of propositions that are known to be false, but it is a legitimate method of proof in mathematics.

We can verify that $\neg A \implies (A \implies B)$ in a truth table:

Source: https://www.erpelstolz.at/gateway/TruthTable.html

We can also formally prove it from "first principles" using a form of natural deduction (screenshot from a freeware proof checker):

Answer 3

I wrote an answer to a very similiar question written just a few hours before yours. It should be able to answer your question as well, just keep in mind that in my answer I was refering to "implication" as "conditional", because that is how the question was phrased.

Here is the link: https://math.stackexchange.com/a/4534223/858891

Feel free to ask follow-up questions in the comments!

Answer 4

I agree that implication is hard to make sense of in a propositional context; the way we talk about implication seems to be talking about variable truth values rather than fixed ones. Really, I think that when we talk about implication, there’s implicitly a universal quantifier. I.e. most implications we talk about are of the form $\forall x \;P(x)\to Q(x)$. Now, to check the truth value of a universal quantifier, we check that the inner proposition is true for each $x$. In this framing, I think the truth table for implication makes sense: for each instance we’re checking, we don’t care if $P(x)$ is false, so in this case we should set the truth value to true, because that will have no bearing on the truth if the universal quantifier. The only thing that can make the whole thing false is if there is an $x$ so that $P(x)$ is true and $Q(x)$ is false, so that is the only case in which we need to define implication to be false.

Answer 5

As is has developed, formal mathematical logic is based on an abstract simplification of our natural process of reasoning. It classifies all statements as either true or false. This procrustean approach works well in mathematics, but it doesn't work so well in the natural world. There are instances where we may agree that something must be either true or false, but cannot tell which is the case, and there are others where distinctions are matters of degree. Furthermore, on a daily basis we deal with doubtful conditionals such as "If I do x, then y will happen",and our methods for dealing with them are informal and intuitive. We have not yet developed satisfactory formal methods for dealing with such cases, although outside mathematics we deal with the uncertain and unknown all the time. Even our formal methods of dealing with modal logic, the logic of the necessary, possible, and contingent, which is as old as Aristotle, are not entirely satisfactory.

For instance, in mathematics, we are accustomed to dealing with well-defined sets, in which we can tell whether a given object is or is not an element of a set, and how many there are, or at least whether there is a countable or an uncountable infinity of them. The sets of every day life are often fuzzier. We may not know how many objects are in a set, or whether a given object is or is not an element of a set.

The material conditional in formal logic does correspond to the notion of implication as used in natural language, although the limitations of classical logic mean that the correspondence is imperfect. In colloquial discourse we are normally not interested in technically valid but unsound arguments from false premises, although formal logic must account for these as well as the more usual techniques of sound deduction from true premises and valid inference.

Answer 6

The best way to understand the material conditional P->Q is just to say “if you assume P, then Q follows.” This might be true because other assumptions are involved. The way we usually think of implication is more akin to strict implication, which says “if you assume P, then from no other assumptions but the axioms and rules of your proof system, Q follows.” This version of implication has used the fishhook symbol P⥽Q, but it’s more common to see the modal logic version □(P->Q). The advantage to the modal version is that it’s defined in terms of the material conditional, but has modal machinery in order to preserve the difference between the material conditional and strict implication.