# At what point in the history of mathematics, and why, did mathematicians come to say "A implies B" to mean "not A or B"?

Here is what one respondent to my previous question says:

A big part of the problem here lies with interpreting the word ‘implies’, which is ambiguous in English. Unfortunately, mathematicians get very sloppy with this word. They are usually taught to call the material conditional material implication and to read P → Q as “P implies Q”. This is really rather misleading. It blurs the important distinction between a conditional in the object language and a consequence relation in the metalanguage. It is important not to fall into this trap. If you’d like, ask a separate question on this topic and I’ll explain it in more detail. - Bumble

Perhaps a few comments on what Bumble says here.

The notion of implication emerged in natural language well before mathematicians got interested in it in the 19th century, and I don't see any objective reason to say that the word "imply" is at all ambiguous in English:

"A implies B" just means that if A is true, then B is true.

And that is totally unambiguous and all we need to know.

Still, as is clear from previous answers to my several questions concerning logic, mathematicians have come to say "A implies B" to mean, broadly, "not A or B", which is also called variously a "material implication", a "material conditional", or even a "horseshoe". So, if there is an ambiguity, it is entirely the responsibility of mathematicians.

I also don't believe that there are different species of conditionals. All conditional expressions can we reworded using the IF-THEN structure.

The expression "material conditional" is just a faux nez for the horseshoe, which on the face of it is not a conditional.

I also don't believe that there is any logical distinction to be made between language and metalanguage.

Given these remarks, and to follow Bumble's suggestion, my question is as follows:

At what point in the history of mathematics, and why, did mathematicians come to say "A implies B" to mean "not A or B"?

Thank you for any scholarly reference.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Apr 15 at 15:41
• Boy, it would be great if people could just use the plain meaning of everyday words! Why must they redefine things to an unrecognizable form? Make up a new word if you want a new meaning, leave the existing words alone! Commented Apr 19 at 23:10
• Do you have any examples of mathematicians talking at the meta level (as your titular question) with “(not A) or B”? Commented Apr 20 at 7:06
• @JKusin "any examples of mathematicians talking at the meta level" Sorry, I don't do meta. You could give me an example first, but I would probably argue that there is in your example no logical difference to be made between meta and non-meta. - 2. Still, all mathematicians would say something like "If A, then B", or "A implies B", when they have to explain a mathematical proof in natural language. Yet, don't they mean "not A or B" in any such case? Commented Apr 20 at 10:42
• @Speakpigeon a problem with that line of thinking is we could then just make horrendously long symbolic “equivalents” to A implies B and say mathematicians mean them. No one for that matters appends 500 pages worth of tautologies to their real life/meta level statements. Commented Apr 20 at 15:53

The word 'implies' is definitely ambiguous. Saying that "it just means if A is true, then B is true" does not explain anything. John Corcoran wrote a paper ("The Meanings of Implication" Dialogos 25, pp 59-76, 1973) describing 13 different senses of implication without claiming that this is an exhaustive list.

Usually in ordinary language when we say A implies B we are expressing the thought that B follows from A in some way, Implication between A and B expresses some kind of relation and there are many different relations that might qualify. For example, if A is true and B is true, does it follow that "if A then B" is true? It depends. What kind of relation are we trying to express? If it is just what logicians call material, then "if A then B" is true whenever A and B are both true. If we mean that A entails B then not necessarily: we would need some kind of connection between A and B that acts as a guarantee that B always holds whenever A holds. That guarantee might be a syntactic derivability relation, or a semantic or conceptual relation. Or we might have some modal connection in mind. There might be an unvoiced, "necessarily, if A then B".

These differences were known to the stoic philosophers. Philo described a conditional that is the same as what we now call the material conditional: it is not the case that A is true and B is false. Chrysippus described something more like what we now call a strict conditional: it is impossible for A to be true and B false. Some medieval logicians used the material conditional without explicitly describing it. Frege used it in his logic because it solved a problem in expressing universally quantified expressions. It allowed "All S is P" to be represented as "for any x, Sx → Px" where → is the material conditional. Frege did not claim that all conditionals are material, only that it is a connective that is useful for certain purposes.

Russell gave this connective the name material implication, which is a pity, because calling it implication is potentially misleading. It is simply a conditional within the object language. Many people have pointed out that it is better to call it the material conditional. Quine even called it a use/mention error to describe the material conditional as an implication.

The material conditional serves as a conditional under the assumption that "if A then B" is a bivalent, dyadic, truth function. To say it is bivalent means that "if A then B" has a truth value that is always either true or false. To say it is dyadic means that its truth depends only on A and B, not on any third parameter, e.g. some background or context. To say it is a truth function means that its truth value depends only on the truth values of A and B and not on any other properties of A and B. Under these assumptions it is easy to prove that the material conditional is a conditonal. I did so in my answer to this question.

So the material conditional is a conditional. It is a very crude approximation to the meaning of 'if' in ordinary language. but it is extremely useful in formal logic. Logicians wouldn't use it if it weren't. The material conditional plays an essential role within classical logic. However, there are many ways in which conditionals do not behave like the material conditional. I listed some of them in my answer to this question.

Because of the complexity of understanding conditionals, there has been a great deal of study of their logic. It is a complex subject with an enormous literature. The curious thing is that mathematicians have made little contribution to our knowledge of conditionals in the last 50 years. I would say that the serious study of conditionals began roughly in the late 1960s. Since then there have been thousands of published papers and scores of books on conditionals. But this literature has come from (1) logicians within the philosophy community, (2) linguists, (3) cognitive psychologists, and (4) AI researchers working on knowledge representation. Hardly any of it has been contributed by mathematicians. For whatever reason, mathematicians are content to use the material conditional and are usually not even aware of how huge a subject the logic of conditionals is.

To amplify the paragraph that you quoted from another answer of mine...

You seem to be falling into a common mistake of confusing a conditional within the object language with the logical consequence relation. The material conditional is a connective within the object language. It is a truth function, like conjunction and disjunction. It is often represented by any of the symbols ⊃ → ⇒. In the same way that A ∧ B is true under some interpretations and false under others, A → B is true under some interpretations and false under others. You plug in the truth values for A, B and this determines the truth value for A → B.

Logical consequence, also called entailment, is a metalevel relation between a set of sentences Γ and a sentence A. It is often written as Γ ⊨ A to mean that Γ has the semantic consequence A, and Γ ⊢ A to mean that A is formally deducible from Γ. If Γ ⊨ A holds then under any interpretation in which the sentences within Γ are all true, A is also true under that interpretation. Sometimes it is also handy to have a metametalevel consequence relation.

Unfortunately, the word ‘implies’ is ambiguous between these meanings. In particular, mathematicians are often taught to call the material conditional material implication and to read A → B as "A implies B". This is potentially misleading, because it leads to confusing the connective with the entailment relation. What makes the confusion even worse is that many mathematicians use the symbol ⇒ without being clear whether it is supposed to be the material conditional or entailment or some kind of metalevel consequence relation. It doesn’t help that different textbooks use ⇒ differently. For example, Mendelson uses ⇒ for the material conditional, but Enderton uses → for the material conditional and ⇒ for a metalevel consequence relation. Some texts use ⇒ to indicate a sequent. It's quite a mess.

In case you think I am the only one complaining about this, let me quote you a passage from Peter Smith’s book, Introduction to Formal Logic.

There is an unfortunate practice that - as we said - goes back to Russell of talking, not of the 'material conditional', but of 'material implication', and reading something of the form (α → γ) as α implies γ. If we also read α ⊨ γ as α (tauto)logically implies γ, this makes it sound as if an instance of (α → γ) is the same sort of claim as an instance of α ⊨ γ, only the first is a weaker version of the second. But, as we have just emphasized, these are in fact claims of a quite different status, one in the object language, one in the metalanguage. Talk of 'implication' can blur the very important distinction. (Even worse, you will often find the symbol ⇒ being used in informal discussions so that α ⇒ γ means either α → γ or α ⊨ γ, and you have to guess from context which is intended. Never follow this bad practice!)

To learn about conditionals, a good start is to read some of the articles in the Stanford encyclopedia:

https://plato.stanford.edu/entries/conditionals/

https://plato.stanford.edu/entries/counterfactuals/

https://plato.stanford.edu/entries/logic-conditionals/

Some other references are:

• Jonathan Bennett, Conditionals: A Philosophical Guide (2003).
• David Sanford, If P then Q (2003).
• Ernest Adams, The Logic of Conditionals (1975).
• David Lewis, Counterfactuals (1973).
• Robert Stalnaker, several papers collected in Knowledge and Conditionals (2019).
• Dorothy Edgington, “On Conditionals”, Mind, Vol. 104, pp. 235–329, (1995).
• Angelika Kratzer, Modals and Conditionals (2012).
• Nicholas Rescher, Conditionals (2007).
• Michael Woods, Conditionals (1997).
• Igor Douven, The Epistemology of Indicative Conditionals (2015).
• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Apr 13 at 10:28
• Couldn't we have a less "crude approximation to the meaning of 'if' in ordinary language"? How about a really precise approximation, and call it implication? Commented Apr 19 at 23:18
• And what would a precise approximation be? Relevant implication? Strict implication? The Stalnaker conditional? Adams' probabilistic conditional? David Lewis' counterfactual? A connexive conditional? A nonmonotonic conditional? Commented Apr 20 at 9:52
• Precise would be: if A and B are true, then the If is true. If A is false, then you stop considering the If. That's how computers work. If that didn't work, then computers wouldn't work and we wouldn't be having this conversation. (see what I did there?) Commented Apr 20 at 12:09
• Well, for one thing, what you are describing is not the material conditional, since that is true when A is false. A gappy conditional that has no value when its antecedent is false would be unable to justify inferences such as modus tollens. Also, that conditional does not express any kind of connection between its antecedent and consequent, so it is not typical of how implicational relations work. Commented Apr 20 at 18:34

The core of the matter can be contemplated by understanding the difference between the tautology

1. (P → Q) ∨ (Q → P)

and the absurdity

1. (P implies Q) or (Q implies P).

The first is true for any P and Q that have a truth value. The second need not be true if P and Q are chosen to be independent axioms.

It's a hint at the difference between language and meta language.

• Nice! I actually don’t see why the first is apparent a tautology though. Would you care to add a deduction of this theorem? Commented yesterday
• Actually, I do. “(not P or Q) or (not Q or P)” should be equivalent to “P or not P or Q or not Q”, I think, by associativity and commutativity. Commented yesterday
• I find there is a lot more to be thought about here though. Commented yesterday

Among the moderns, Bocheński mentions Frege first. History of Formal Logic p. 311:

Frege [41.12: Begriffsschrift 5 f.] then introduces the Philonian concept of implication, though, unlike Peirce (41.14 [CP 3.280]) he knows nothing in this connection of Philo or the Scholastics. It is remarkable that he proceeds almost exactly like Philo.

20.07 [Sextus Empiricus, Adversus Mathematicos, VIII 113 f.]
Philo said that the connected (proposition) is true when it is not the case that it begins with the true and ends with the false.

This is exactly what we call material implication today:

20.071

Antecedent Consequent Connected proposition
true true true
false false true
false true true
true false false
• "Frege introduces the Philonian concept of implication" Alright, but did he actually call it an implication? Commented Apr 20 at 10:25
• @Speakpigeon The Greeks called it a "connected (proposition)" (not "condition", as that concept came later). Frege discusses it under the heading "Conditionality". The OED gives B. Russell in American Journal of Mathematics vol. 28 202 (1906) as the earliest usage of "implication" in the logic sense: "The subject which comes next in logical order is the theory of formal implication." Commented Apr 20 at 17:38
• "The OED gives B. Russell in American Journal of Mathematics vol. 28 202 (1906) as the earliest usage of "implication" in the logic sense" Well, that is clearly not true. You are clearly over-interpreting the OED. - 2. ""The subject which comes next in logical order is the theory of formal implication."" Yes, but the use of the word implication in the logical sense is already well-established when Russell makes his first move. Commented Apr 21 at 9:13

I just found this on Wikipedia which I think relates to a lot of your questions:

C. I. Lewis took issue with Russell’s idea of material implication, and sought to amend it with the use of modal logic:

Lewis studied logic under his eventual Ph.D. thesis supervisor, Josiah Royce, and is a principal architect of modern philosophical logic. In 1912, two years after the publication of the first volume of Principia Mathematica, Lewis began publishing articles[citation needed] taking exception to Principia' s pervasive use of material implication, more specifically, to Bertrand Russell's reading of a→b as "a implies b." Lewis restated this criticism in his reviews[citation needed] of both editions of Principia Mathematica. Lewis's reputation as a promising young logician was soon assured. Material implication (the rule of inference which claims that stating "P implies Q" is equivalent to stating "Q OR not P") allows a true consequent to follow from a false antecedent (so if P is not true still Q may be true since you only stated what a true P implies, but did not state what is implied if P is untrue). Lewis proposed to replace the usage of material implication during discussions involving logic with the term strict implication, by which a (contingently) false antecedent, which is false but could have been true, does not always strictly imply a (contingently) true consequent, which is true but could have been false. The same logical result is implied, but in a clearer and more explicit way. Stating strictly that P implies Q is explicitly not stating what the untrue P implies. And therefore if P is not true, Q may be true, but may be false as well.[14] As opposed to material implication, in strict implication the statement is not primitive - it is not defined in positive terms, but rather in the combined terms of negation, conjunction, and a prefixed unary intensional modal operator, ◊ {\displaystyle \Diamond }. The following is its formal definition: If X is a formula with a classical bivalent truth value (which must be either true or false), then ◊ {\displaystyle \Diamond }X can be read as "X is possibly true".[14] Lewis then defined "A strictly implies B" as " ¬ ◊ {\displaystyle \neg \Diamond }(A ∧ ¬ {\displaystyle \land \neg }B)".[15] Lewis's strict implication is now a historical curiosity, but the formal modal logic in which he grounded that notion is the ancestor of all modern work on the subject. Lewis' ◊ {\displaystyle \Diamond } notation is still standard, but current practice usually takes its dual, the square notation ◻ {\displaystyle \square }, meaning "necessity", which is stating a primitive notion, while the diamond notation, ◊ {\displaystyle \Diamond }, is left as a defined (derived) meaning. With square notation "A strictly implies B" is simply written as ◻ {\displaystyle \square }(A→B), which states explicitly that we are only implying the truth of B when A is true, and we are not implying anything about when B can be false, nor what A implies if it is false, in which case B can be false or B can just as well be true.[14]

https://en.m.wikipedia.org/wiki/C._I._Lewis#Philosophical_work

• Tnx. Valuable . Commented 23 hours ago

The first one to make a similar assumption is Parmenidis (600bc), who started with the principle of identity ( A is by necessity A, it cannot be A and B at the same time), which was later "renamed" as the principle of non-contradiction. He was the first to conceive this thought and considered it as the basis for logic thinking. From then on, logic and mathematics started to develop. It's not exactly what you are asking, but I hope this helps.

• You're getting downvoted because your answer has nothing to do with the question
– TKoL
Commented Apr 20 at 8:14