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I assume implication is the operator that is most of the times used in mathematics, am I right? (i.e. whenever if/then construct is used)

If yes, then I think it is known that it has some philosophical problems

Merely that implication is true if premise is false. Which doesn't make sense in real world because I could say: if pigs can fly, then we can cure any illness. But this is not true in the way we understand if/then in real life, isn't it? Because we can't cure any illness so far.

So my question is more philosophical in nature, if we agree that mathematics is using an operator implication which is problematic from philosophical point of view and doesn't very well align with the real world meaning of if/then, then doesn't this cause problems for mathematics when we try to apply some results of mathematics to real world? More particularly because the operator (implication) that was used in math may not be the one that reflects the one in real life? This means we can't use some results from math which relied on implication, in real life?

Alex Robinson
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    See similar post : Implications and Ordinary language. – Mauro ALLEGRANZA Oct 10 '19 at 15:36
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    See also Truth table of the conditional. – Mauro ALLEGRANZA Oct 10 '19 at 15:39
  • @MatthewLeingang I can't check the links right now, but please re read my question before jump to conclusion that it is duplicate, from quickly looking at those questions, I can't see where they are interested how this fact reflects on the connection of mathematics results and real life results. I don't say yet that it is not duplicate, but I can't check it right now. –  Oct 10 '19 at 15:42
  • @MauroALLEGRANZA Please pay attention to the bold parts in my question, from quickly looking at links I am not sure they answer that. But I will have to check those links more thoroughly later on –  Oct 10 '19 at 15:45
  • Most of mathematics is made of theorem deduced from axioms : we assume the axioms as TRUE and the validity of logical rules guarantee that the conlusion are TRUE also. In real life pigs do not fly; thus, the (logically valid deduced) conclusion about illness is not TRUE. – Mauro ALLEGRANZA Oct 10 '19 at 18:38
  • You are confusing the truth-value of the conditional (the "if…, then…") with the truth-value of the consequent. – Mauro ALLEGRANZA Oct 11 '19 at 06:01
  • @MauroALLEGRANZA OP here(don't have access to my original account now): Can you please elaborate on your last comment? –  Oct 11 '19 at 06:51
  • It is stunning how questions are closed as duplicated even though the bold part of the question isn't answered in the linked questions! @MauroALLEGRANZA –  Oct 17 '19 at 09:29

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Merely that implication is true if premise is false. Which doesn't make sense in real world because I could say: if pigs can fly, then we can cure any illness. But this is not true in the way we understand if/then in real life, isn't it? Because we can't cure any illness so far.

Actually, your reasoning isn't right here: just because the statement 'if pigs can fly, then we can cure any illness' would be (mathematically!) regarded as true because pigs don;t fly, does not mean that we can cure all illnesses. In fact, what this statement is doing is that because we can't cure all illnesses, it must be false that pigs fly ... which is really the point of the statement 'if pigs can fly, then we can cure any illness'

That said, however, I agree with you on the mismatch between the mathematical implication and the real-life conditional. Consider this:

'If Bob lives in Los Angeles, then Bob lives in Florida'. (and to be clear: by 'Los Angeles I mean that big city on the West coast of the U.S., in the state of California, and by 'Florida, I mean that state on the South-East of the U.S.)

Well, any normal person would consider this a false statement: If Bob lives in Los Angeles, then Bob lives in California, not Florida! However, if we treat the 'if ... then ...' mathematically, and if it turns out that Bob does not live in Los Angeles, then suddenly the statement becomes true.

It turns out that real life conditionals often do not have the truth-functional property: knowing the truth-values of its parts does not mean we automatically know the truth-value of the whole. This mismatch between the way we normally think about conditionals in real life and the way we use the mathematically defined operator of material implication sometimes leads to very counterintuitive results which collectively are called the Paradoxes of Material Implication.

And yes, what this means is that we should be very careful in our application of mathematics to the real world ... just as with any other branch of mathematics or scientific idealizations: Euclidian geometry works ... as long as we don't deal with massive objects or very high speeds. Same for Newtonian mechanics. So in the end, these are all tools and, like all tools, we should know their scope and limitations.

Bram28
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  • something is still confusing me because if in mathematics a theorem relies on such lemma say which relies say on SomeNonTrueFact->SomeOtherNonTrueFact,which according to implication is true, then if we use such theorem, in real life, it might not work, am I right? and since all math is based on implications I am confused now... –  Oct 11 '19 at 18:39
  • @dango Well ... how would you 'use' a theorem that is an implication $A \to B$? Presumably it is because you have $A$, and so then use this theorem to infer $B$. But that means that if $A$ is not true in real life, you never get to 'use' the theorem, simply because $A$ is not true. So, it is not so much that applying such a theorem 'does not work', but that it is just not applicable to real life in the first place. So, you never get to infer $B$. ... which is good, since $B$ wasn't true either. – Bram28 Oct 11 '19 at 18:59
  • also another thing I think according to this:https://math.stackexchange.com/a/48202, you (as well as me) may also be confusing the truth of the descendant with the truth of the implication itself. I think this: 'If Bob lives in Los Angeles, then Bob lives in Florida' -> says that If bob doesn't live in LA, then I haven't made any promise, so you can't falsify this implication as a whole. You should not instead conclude that if bob doesn't live in LA then he lives in florida, rather to look at it that in that case you can't reject the implication as a whole. –  Oct 11 '19 at 19:30
  • but on the other hand your example can be easily rejected as an implication as a whole because you have a counter example where you can show that p is true and q is false –  Oct 11 '19 at 19:31
  • ". But that means that if A is not true in real life, you never get to 'use' the theorem, " -> Yeah but maybe it is using some lemma in between that is relying on some false fact as p. Sorry for lengthy discussion but this topic has confused me... –  Oct 11 '19 at 19:33
  • @dango Sorry, I am not sure what you just meant ... I also question the existence of 'false facts' ... – Bram28 Oct 11 '19 at 19:37
  • Check the link in the comments I think when we say: 4>5 implies 44 is prime. This implication is true, but this doesn't mean that the result (44 is prime) is true, rather the implication as a whole is still true. Because since 4>5 is false, then this implication doesn't guarantee anything in that case -so at least we didn't prove the implication as a whole as wrong. –  Oct 11 '19 at 19:59
  • @dango Exactly correct! – Bram28 Oct 11 '19 at 20:00
  • What is still bugging me if I have implication say: if p is prime, then it is sum of two even numbers. Now, if someone inserted 44 as p, and couldn't factor it as sum of two prime numbers, then the implication as a whole would remain TRUE right? But in reality we don't know if that implication is actually true (goldbach conjecncture). So wouldn't the ___conclusion___ that that implication is TRUE be wrong? –  Oct 11 '19 at 20:09
  • @dango About the Los Angeles and Florida example. Yes, if Bob does not live in Los Angeles, then you have indeed not falsified the statement. But my point was that every normal person in real life regards 'If Bob lives in Los Angeles, then Bob lives in Florida' as False. In other words, we use and think about conditionals in real life differently from the way we use conditionals in mathematics. The link in the Comments is merely trying to make the mathematical usage a little more palatable by saying that as long as you haven;t explicitly falsified the conditional, then it is true. .cont'd – Bram28 Oct 11 '19 at 20:11
  • But, obviously, it is very easy to falsify the 'If Bob lives in Los Angeles then Bob lives in Florida': point to a Bob who lives in Los Angeles! So, just because yoyu haven;t explicitly falsified a statement does not mean that it isn't falsifiable. And here's the thing: the conditional 'If Bob lives in Los Angeles' in real life is understood as: "we don;t know whether Bob lives in Los Anegeles or not .... but if he were to live in Los Angeles, he would live in Florida" ... which we all regard as patently false. – Bram28 Oct 11 '19 at 20:14
  • Please just check my final comment above, I think that is what's left unclear to me about goldbach –  Oct 11 '19 at 20:16
  • @dango The Goldbach conjectuve claim is a universal claim (and not quite the same as what you said): "every even number greater than two is sum of two primes, ie. if a number is even and greater than $2$, then it is the sum of two primes. So, if we point to $43$ and say: well, for that number the antecedent is false, and hence the conditional is true .. well, that does not falsify the Goldbach conjecture .. but it also indeed does not prove the Goldbach, since the conditional would have to be rtue for all numbers. .. continued – Bram28 Oct 11 '19 at 20:18
  • @dango ... I think the same is true for the Bob example> When we are given the claim 'If Bob lives in Los Angeles', we are not interpreting that claim as a claim about where Bob currently lives, but as a universal claim regarding where Bob might live at any time during his life. So, him currently living in Colorado would not falsify the claim ... but that does not make the claim true ... and in fact we say it is false since it is not true that for any time $t$: if Bob lives in Los Angeles at time $t$, then Bob lives in Florida at time $t$ – Bram28 Oct 11 '19 at 20:22
  • yeah I confused goldbach - I see your point with goldbach you say it would not be regarded as TRUE unless we tried ALL numbers, because it is a general statement right? And in that process indeed every ODD number would render the implication as TRUE in general, but we would still need to test the EVEN numbers –  Oct 11 '19 at 20:26
  • @dango Exactly! – Bram28 Oct 11 '19 at 20:27
  • I think implication slowly starts to make sense, and after all, it may not be so paradoxical, isn't it? –  Oct 11 '19 at 20:28
  • @dango Well, how about this: if you do the truth-table, you'll find that $(A \land B) \to C$ is equivalent to $(A \to C) \lor (B \to C)$ ... that's certainly very counter-intuitive. The left statement seems to be saying that $C$ is true if both $A$ and $B$ are true ... but why would it follow that just one of those would be sufficient to make $C$ true? Consider: if you are both male and unmarried, you are a bachelor. True, right? But if all we know is that you are male, that's not enough to know you're a bachelor. And the same goes for being unmarried by itself. – Bram28 Oct 11 '19 at 20:32
  • I see even if A->C were true, that wouldn't mean A and B ->C is true you say. To avoid risk of further confusing myself as I am not well familiar with this field and not to consume too much of your time, maybe you can just leave some links or further pointers where I can read about such stuff where these things are clarified (if they are) - and/OR adapt your answer based on our comments. –  Oct 11 '19 at 20:39
  • i agree your last comment seems paradox (if true) - but I don't know what to make of it :) i.e. what conclusions to draw –  Oct 11 '19 at 20:41