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Can anyone exhibit a mathematical sentence in which a conditional (not necessarily the main connective) has to be STRICTLY understood as a MATERIAL one, and would become false if the material conditional was understood as logical implication instead?

Some context:

In logic, strictly speaking, material implication (' → ') has to be carefully distinguished from logical implication (' ⇒ '). However, I have noticed that in mathematics books, the distinction is not emphasized, as if, in that field, all implications are logical implications. Is it actually the case? (Reference at Archive.org: On this distinction and on the symbols I use , Seymour Lipschutz, Schaum's Outline of Set Theory , ch. 14 " Algebra Of Propositions".)

To illustrate the difference between material and logical implication, consider the sets A={ x | x is a mathematician → x is a musician } and B={ x| x is a mathematician ⇒ x is a musician }. A is simply the set of people who (contingently) happen not to be both mathematician and non-musician, since its conditional is a material one. However, B is the set of people such that for each member, it is or would have been logically impossible for them to be mathematician without being musician; depending on one's opinion concerning the relationship between mathematics and fine arts, one will probably tend to answer either that B is either the universal set (a mathematician is necessarily a musician) or the empty set.

I think that substituting ' → ' for ' ⇒ ' cannot lead to important problems, since, if A logically implies B, then A should also materially imply B ("A ⇒ B" meaning that (A → B) is true in all possible cases, all possible "interpretations"). Here I'm asking the reverse question: is it always correct to substitue ' ⇒ ' for ' → ' in mathematics, in other words, is it correct to use always " ⇒ " in mathematics?

My question is not on symbols.

ryang
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  • Logical implication usually is material implication. – Christopher King Mar 21 '19 at 17:24
  • Here is a helpful reference on typesetting mathematical symbols. – Théophile Mar 21 '19 at 17:35
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    In mainstream mathematics, there is only one type of implication. I've never heard of the distinction you mention, but it sounds like the one type is material implication, because if it is true that Paris is the capital of France, then the implication is true. – Matt Samuel Mar 21 '19 at 17:37
  • @MattSamuel If you have never heard of it, with what authority are you saying that "there is only one type of implication"? Actually, you are wrong that the concept of material implication doesn't exist in mathematics. – user647486 Mar 21 '19 at 17:48
  • @user647486 I speak only for mainstream, nonspecialized mathematics, in which I am well versed. I obviously cannot deny the distinction exists in areas I am not familiar with, since I don't know. But the fact that I've never heard of it and I have a PhD in math suggests that at least in the areas of mathematics I've dealt with there's only one kind of implication, and this is the norm for algebra, analysis, topology, etc. I know next to nothing about mathematical logic, which is where I would expect this to come up if I had to guess. – Matt Samuel Mar 21 '19 at 18:00
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    Could shed some light on the issue https://math.stackexchange.com/questions/68932/whats-the-difference-between-material-implication-and-logical-implication – chhro Mar 21 '19 at 18:12
  • @user647486 Perhaps mainstream is the wrong term. What I mean is subjects that nearly everyone who gets a degree in math will take a course in. Nobody ever suggested a course in logic to me and it was not required. This is also true for, say, model theory and set theory. – Matt Samuel Mar 21 '19 at 18:14
  • @RayLittleRock, This one has a good example of what you're looking for. https://math.stackexchange.com/questions/1015518/in-logic-do-the-longrightarrow-and-rightarrow-signify-different-things?noredirect=1&lq=1 – chhro Mar 21 '19 at 18:22
  • Does this answer your question? What's the difference between material implication and logical implication? – user1147844 Feb 26 '23 at 00:20

3 Answers3

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The symbols you want are $\to$ (\to) for material implication and $\implies$ (\implies) for logical implication. Insofar as mainstream mathematics distinguishes them, $p\implies q$ means that $p\to q$ is (a) true in all models of a theory of interest (however, in that context we'd usually write $\models$ (\models) instead of $\implies$ to make it clear) or (b) a tautology. And in modal logic, we can rewrite $p\implies q$ as $\Box(p\to q)$ (note the use of \Box). But in practice, $\implies$ is often used in proofs to indicate an inference from what was already known.

J.G.
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  • Thanks for the clarification concerning symbols and underlying concepts. How would you symbollically translate statement such that : " if 4² is even then 4 is even ". With the (\Box) or without it? –  Mar 21 '19 at 22:26
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    @RayLittleRock Always make your statement only as strong as you intend. The statement $2|4^2\to 2|4$ will be implied whatever symbols you use, as all alternatives to $\to$ are at least as strong. If you want to make the further statement that all models yield the above, change $\to$ to $\models$; if you deem it "necessary" (whatever you take that to mean metaphysically), by all means wrap the statement in $\Box()$; if you dare claim definitions have sufficed to make it a tautology, use $\implies$. – J.G. Mar 21 '19 at 22:29
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Consider the statement

"If x is even, then x is divisible by $2$"

In 'math world' we regard this statement as true.

But this is not a logical truth. That is, logically one is allowed to interpret 'even', 'divisible by' and '$2$' in a way that would make the statement false.

So, the statement is a mathematical truth, but not a logical truth. More to the point: the 'if' part does not logically imply the 'then part. Indeed, if we were to symbolize it, we should be using the material implication, and not the logical implication.

Of course, if we are given the (normal!) definitions of 'even', 'divisible by' and '$2$', then we can logically infer the truth of the statement above as a whole. That is, the statement as a whole is logically implied by the relevant definitions.

Also, if we fill in a specific value for $x$, say $4$, then the statement becomes:

"If $4$ is even, then $4$ is divisible by $2$"

And now, given the standard definitions/axioms (let's refer to that as a set of statements $A$), we have that $A$ together with "$4$ is even" logically implies that "$4$ is divisible by $2$" ... but we still don't have that "$4$ is even" by itself logically implies that "$4$ is divisible by $2$"

Bram28
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  • Very clear and precise answer, thanks! Contrary to what I thought mathematicians hardly ever use ' ==> ' as a symbol for logical implication! –  Mar 21 '19 at 18:38
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    @RayLittleRock Right. Or to be exact: when a mathematician makes an 'if ... then .. ' claim, it is rarely a claim of logical implication. – Bram28 Mar 21 '19 at 18:44
  • This is the only answer that makes sense, even as it crucially neglects to point out that its usage of the phrase 'logical implication' does not coincide with the OP's (actually, it is unclear what exactly the OP means by 'logical implication'). $\quad$ Also, the OP's above comment is worth reiterating: in mathematics, the symbol ⟹ never means logical implication—merely implication given axioms and usually a certain context. – ryang Feb 24 '23 at 18:44
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    @ryang Yeah, well noted! The OP’s usage of logical implication in their example was rather weird, so I decided to use my own example. – Bram28 Feb 24 '23 at 22:12
  • Expanding my above comment: $\quad$ In mathematics outside of Mathematical Logic, whether → or ⇒ is their favoured symbol (I prefer the latter, as the former gets overloaded), the author generally means mathematical implication (some folks call this material implication, but it's probably more accurate to say that the material implication underlies mathematical implication), since logical implication (⊨) is not directly of interest. (By 'logical implication', we neither mean implication in a given context, nor mean just any implication that respects logical rules.) – ryang Feb 25 '23 at 05:56
  • If “If x is even, then x is divisible by 2” isn’t a logical truth, then “If P, then P” isn’t. That is, you can concoct a logic in which $P \to P$ fails, just like you can concoct definitions of ‘even’, ‘divisible’, et. al and have the implication in question fail. – PW_246 Jun 01 '23 at 19:21
  • @PW_246 True. So when we say 'logical truth' we need to be careful in spelling out what logic we are talking about. for example, $\forall x \ P(x) \lor \exists x \neg P(x)$ is a first-order logical truth, but not a truth-functional logical truth. In some logics, $P \lor \neg P$ is not a logical truth. And you're right, we can even come up with logics where $P \to P$ is not a logical truth. All of which serves to underscore the point I made that these 'truths' are all relative to whatever system you are using, and that in particular there is a difference between mathematical and logical truth. – Bram28 Jun 01 '23 at 21:04
  • @Bram28 Ok. I don’t see why “If x is even, x is divisible by 2” can’t be a logical truth in a theory that has axioms sufficient to prove it. – PW_246 Jun 01 '23 at 22:49
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    @PW_246 It would be something that I can logically derive from some set of axioms that I postulate about some domain, but that does not make it a logical truth ... it would merely be a truth-in-that-theory. A logical truth is something I can derive from no axioms at all, i.e. something that would be true-in-any-theory – Bram28 Jun 02 '23 at 14:52
  • @Bram28 so there are no logical truths? – PW_246 Jun 02 '23 at 15:04
  • @PW_246 Depends on the logical system you are using, i.e. what logical inference principles you define as valid. Every logic system I know is ok with inferring $P$ from the assumption $P$, and inferring $P \to P$ from that by discharging the assumption $P$, thus showing that $P \to P$ is a logical truth that can be derived from no assumptions at all. – Bram28 Jun 02 '23 at 15:30
  • How is that different from having a system in which, on assumption “n is even”, one can derive “n is divisible by 2”? Are you getting at that mathematical reasoning pre-supposed the Deduction Theorem? I’m not seeing the difference otherwise. Also, there are plenty of logics that validate $P \to P$, but don’t validate DT. – PW_246 Jun 02 '23 at 15:47
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    @PW_246 Logic is all about patterns. "If all gluckeroos are smibbel, then all gluckeroos are smibbel" logically makes sense. From a logic point of view, this is of the form "If P then P". On the other hand, from a purely logical point of view, "If n is even, then is divisible by 2" is of the form "If P then Q", so it is not a logical truth. You need to provide meaning and content and background assumptions and axioms (i.e. math) to see that "If n is even, then is divisible by 2" is true. – Bram28 Jun 02 '23 at 18:02
  • Isn’t “if n is even, n is divisible by 2” more like “if everyone is funny or cool, then someone is funny, or everyone is cool”? Neither of them are propositional tautologies, but given basic assumptions about their respective meanings, both of them are analytic truths. I don’t see how an analytic truth could be otherwise than a logical truth, at least in some sensible theory. – PW_246 Jun 02 '23 at 19:55
  • @PW_246 ‘If everyone is funny or cool, then someone is funny, or everyone is cool’ is indeed a logical truth… not a propositional one, but a quantificational one. But from the standpoint of pure logic, the predicate if being even is a different predicate than being divisible by 2. From a quantificational logic point of view, it would be of the form ‘ If n has property P, then n has property Q’, so still not a logical truth. I agree with you that it is an analytical truth, but meaning of predicates goes beyond pure logic. – Bram28 Jun 03 '23 at 01:20
  • Yes, but there is clearly an axiom system capable of proving the mathematical claim in question. Just as propositional logic can’t handle quantifiers, FOL can’t handle math. But, if FOL is counted as a logic in its own right, why can’t validities of at least the basic systems (like PA) be considered in the same light? What I’m saying is you don’t have to just assume “if n is even, it is divisible by 2”, but rather it can be explicitly shown. – PW_246 Jun 03 '23 at 01:23
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    @PW_246 Of course it can be shown to be true. But not by logic alone. Like you say, you need axioms that define the notions of ‘even’, ‘2’, and ‘divisible’. As soon as you do that, you go beyond pure logic. That’s what makes this a mathematical truth, not a logical truth. – Bram28 Jun 03 '23 at 02:13
  • You need axioms to define what “for all” and “there exists” mean as well. – PW_246 Jun 03 '23 at 02:31
  • Let us continue this discussion in chat. – Bram28 Jun 04 '23 at 01:43
  • @PW_246 No. You can leave that to inference rules. Indeed, logic is about inference, and if we can infer something from nothing, it is a logical truth. Another way to think about this is that logic is something that works independent of any domain. This is what makes something like $ P \to P$ a paradigmatic logical truth: it doesn’t matter what $P$ stands for, it will always be true. On yhe other hand, the statement “If n is even, then n is divisible by 2” is true in virtue of assumptions we make about a particular domain, which is mathematics. So that makes it a mathematical truth. – Bram28 Jun 04 '23 at 01:47
  • $P \to P$ is not always valid. Switch the Gödel-Tarski translation for implication from $(P \to Q)’= \Box (P’ \to Q’)$ to $(P \to Q)’= \Diamond (P’ \to Q’)$ and make the accessibility relation non-reflexive. You also can’t always infer something from nothing, e.g. In a natural deduction system that doesn’t allow one to prove the deduction theorem and which contains no axioms. – PW_246 Jun 04 '23 at 02:44
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    Well, I think we’ve come full circle. Not sure either of us will be able to convince each other. But thanks so much for pressing me on this issue… it certainly forced me to do some more thinking on this! :) – Bram28 Jun 04 '23 at 02:54
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In mathematics based on classical logic, there appears to be no difference between material and logical implication.

Consider, for example, the implication: If it is raining ($R$), then it is cloudy ($C$).

$$R \implies C$$

This implication does not mean that rain causes cloudiness, or that cloudiness causes rain. It means only that, at the moment, it is not both raining and not cloudy.

$$\neg [R \land \neg C]$$

This is often used in one form or another as The Definition of $\implies$ in introductory textbooks, but it can also be derived from other widely accepted properties of implication, conjunction and negation in classical logic:

  • Introducing and Eliminating $\land$
  • Eliminating $\neg\neg$
  • Conditional proof
  • Proof by contradiction
  • Detachment (Modus Ponens)

See my formal proof.

So, in classical logic anyway, the above "definition" would seem to apply to every implication.

Dan Christensen
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  • Neat proof. Thanks! How do you understand this equivalence of ( A-->B) and ~ (A& ~B) –  Mar 22 '19 at 17:37
  • @RayLittleRock I see it as applying in mathematics and in natural language as well, provided you are talking about a pair of unambiguous true-or-false propositions at a given moment in time. In mathematics, of course, there is no notion of the passage of time, no future or past, essentially only the present -- that which IS true. – Dan Christensen Mar 22 '19 at 17:58