implications

The word imply comes from the Old French emplier, which itself comes from Latin implicare (“to enfold”, “to involve”, “to entangle”). It likely ultimately goes back to PIE *pleḱ- (“to plait”, “to weave”). According to Merriam-Webster, the entity that everyone agrees officially decides on the meanings of words, to imply means “to express indirectly”, “to involve or indicate by inference, association, or necessary consequence rather than by direct statement”, “to contain potentially”, or perhaps obsoletely “to enfold, entwine”. We are mostly interested in the first and second senses of this word today, although I am somewhat intrigued by the proposition of potential containment, enfolding, and entwining.

This usage of the word in this manner seems to date to circa 1400s. One of the earlier quotes I was able to turn up:

Crist hath offred Him-silf to God, Crist, etc..& te text emplieth þe same..þat it was sum tyme þe maner a-mong þe romaynes…
— Worcester Sermons

I would argue that the definitions listed by Merriam-Webster leave out one relatively common usage of the word: Wiktionary gives “to hint; to insinuate; to suggest tacitly and avoid a direct statement”.

This English word can be contrasted with an etymologically related jargon term, often specified by referring to it as logical implication or material implication. Logical implication is, of course, when

$\lnot P \lor Q$. The relationship of this to the vernacular English concept of to imply is somewhat confusing at times. Wikipedia has a whole page listing “paradoxes of material implication”. These include:

• The positive paradox: If the sky is typically blue during a sunny day with a clear sky in most parts of the world, then the Navier Stokes equations can experience finite time blowup.

• Vacuous truth: False implies false

• The principle of explosion: If $0 = 1$, then False.

I think one of the fundamental issues here is that $P \Rightarrow Q$ stubbornly insists on having a truth value even in situations where, for Gricean reasons, no one would ever particularly discuss an implication. But this often turns out to not be so bad in practice. These “perverse” examples of logical implications usually just aren’t emphasized so much in the day-to-day practice of a mathematician (though I guess principle of explosion comes up occasionally); mostly, any $P \Rightarrow Q$ statement a theorem seeks to prove is going to be one where you can at least form a solid mental model of a person who is unsure about whether $P$ is true or not. Certainly it would be very strange for someone to ever articulate a proof of a statement like $P \Rightarrow Q$ which relied mostly upon establishing $P$ to be unconditionally false or $Q$ to be unconditionally true — if you were doing that, why prove the weaker statement $P \Rightarrow Q$ in lieu of the strictly stronger statements $\lnot P$ or $Q$?

A favorite issue of mine is Raymond Smullyan’s drinker paradox. Consider a pub. The statement

There is someone in the pub such that, if they are drinking, then everyone in the pub is drinking.

is necessarily true. Either everyone is drinking, in which case $P \Rightarrow Q$ for any choice of person, or $P \Rightarrow Q$ for any choice of non-drinking person.

I would say the reason this fails is mostly because a normal person would parse it so that there is some particular fixed person, such that over time it is the case that, if they are drinking, then everyone is drinking. Or perhaps over some sort of set of counterfactuals that isn’t quite time. Counterfactuals are very confusing. But the point is that this isn’t obviously a disagreement about how $P \Rightarrow Q$ works — I think it could be a mere disagreement about what sorts of quantifiers are implied. It’s just tricky because maybe one of the quantifiers that is implied involves ranging over whatever a counterfactual is. Probably it can also be even messier and involve counterlogicals or counterpossibles or other countersomethings.

Some people on my Twitter argue

Aprii *️⃣@ApriiSR

is it, from "p implies q" and "not q", valid to conclude "not p"

yes59.5%

no34.4%

see results6.2%

2,301 votes · final results

Sep 8, 2026

that the reason people don’t consistently endorse modus tollens in polls has something to do with the mismatch between the standard English conception of implication and the logical implication. As much as I am a big fan of keeping careful track of that distinction, I’m not totally certain it applies here. Like, I suppose in the standard English usage of “implies”, $p$ could be a person rather than a proposition, and then you’d need to think carefully about when “not [person]” is a legitimate proposition. But if we are actually putting in something vaguely like a proposition, I think modus tollens basically applies to the common English understanding of the word! Sure, the premise “$p$ implies $q$” is probably somewhat weaker in that case, so it’s not really a 100% irrefutable deduction, but it’s still like, a generally valid inference¹¹ Apparently people confuse implication and inference enough that dictionaries need to warn about it!.

Overall I think when you get confused about how a phrase involving implication should be read it’s best to start thinking very carefully about the quantifiers. That usually clears it up, at least if you understand quantifiers well enough to think clearly about them.

…This is probably mostly useless, because most people who get confused about logical implication probably aren’t especially the sort of people to understand quantifiers particularly clearly. Oh well! Nevertheless.