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While it's kind of counter-intuitive, we know that the statement "If A, then B" is always correct as long as A is false. For example, both

  • If $1 = 2$, then $e + \pi$ is irrational
  • If $1 = 2$, then $e + \pi$ is rational

are true, because $1 \neq 2$.

In real life, many statements are based on a false hypothesis. For example,

World War I will never break out without the assassination of Archduke Franz Ferdinand.

My history teacher tells me that this is incorrect, because the assassination is a prelude, rather than the cause. However, from a mathematical perspective, this statement is true, because the assassination did happen, right?

This is weird. Could anyone explain it to me?

Trying to justify this post:

I think this question essentially focuses on the comparison between classic logic and the logic in natural languages, which can help people understanding the former. As I know, many people find it hard to intuitively understand why false implies everything in classic logic, presumably because they are thinking in a natural language, and reading answers under this question would benefit them. According to the help center, questions about "Understanding mathematical concepts and theorems" are welcome on this site.

I admit that it's not entirely about math, so I've tagged it with soft-question. However, I would argue that it's not particularly related to a specific area, say, history. Here WWI is merely used as an example, and it can be anything else, like rational choice theory in economics and virtually every theory with an assumption.

Plus, based on the positive response from the community (votes and comments), I believe this question would receive some interesting answers if it's re-opened.

Mauro ALLEGRANZA
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nalzok
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    This might be better for a philosophy site, since it's not exactly mathematical logic. – Nate Eldredge Feb 10 '18 at 16:32
  • Probably better for a philosophy or history forum. – mathreadler Feb 10 '18 at 16:36
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    @mathreadler I don’t really care what the cause of WWI is; this statement is just used as an example to show what I mean... – nalzok Feb 10 '18 at 16:38
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    If you define if A then B as $A \implies B$ the way mathematicians use it, then you are correct. You should ask your teacher how (s)he defines "if A then B". – user388557 Feb 10 '18 at 16:39
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    This question isn’t really math, but it might be an infinite generation of reputation machine if it doesn’t get closed. Congratulations! – Stephen Feb 10 '18 at 16:40
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    https://en.wikipedia.org/wiki/Counterfactual_conditional#Philosophical_treatments – Micah Feb 10 '18 at 16:41
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    @SunQingyao - See also here, here and here for previous discussions related to your question. – Taroccoesbrocco Feb 10 '18 at 16:46
  • I think the right framework to study problems of this kind is not classical logic (where, indeed, false premises imply everything) but rather a temporal logic, where causality can be explored in a semantics based on possible worlds. – Andrés E. Caicedo Feb 10 '18 at 16:51
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    "If Franz Ferdinand was not assassinated then World War 1 would not have broken out" is not straightforward to express mathematically because we do not have a logical connective for the counterfactual "would not have". However, "If Franz Ferdinand was not assassinated then World War 1 did not break out" is mathematically true, and has the same logical structure as "If you're an Olympic-level athlete, then I'm the King of Siam". –  Feb 10 '18 at 17:11
  • Classical logic works perfectly for logical propositions that are unambiguously either true or false in the moment, something that may not be the case for propositions in much of casual discourse (natural language). See my answer at https://math.stackexchange.com/questions/2596766/arguments-pro-material-implication/2598406#2598406 – Dan Christensen Feb 11 '18 at 05:47
  • "This is weird. Could anyone explain it to me?" Yes. Your history teacher is a historian and not a mathematician. Your math teacher is a mathematician and not a historian. That's all the explanation you need. "I think this question essentially focuses on the comparison between classic logic and the logic in natural languages" Mostly. But I think there's also a difference between causality and conditionalism (unfortunately math has the same vocabulary for both). If x then y, can be interpretted as x will cause/be the reason for y. Or it can mean "it is not the case that y and not x". – fleablood Feb 11 '18 at 05:59
  • @fleablood: Well, you hope his teachers are trained in their professions. – Asaf Karagila Feb 12 '18 at 08:03

3 Answers3

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The "logic" of the connective $\to$ used in formal logic and mathematics to formalize "if..., then..." is not causal.

Thus, when we assert "if $1=2$, then $1$ is odd" we do not presuppose that there is a causal link between the antecedent and the consequent.

The assertion of your teacher about:

"World War I will never break out without the assassination of Archduke Franz Ferdinand"

is that, historically, the assassination did not cause the war (assuming that the causes of war are deeper one: economical, social, etc.)

Thus, we may rephrase his assertion as follows:

"The assassination of Archduke Franz Ferdinand did not cause World War I."

Rephrased this way, it is an assertion about an historical fact, that is perfectly intelligible without any use of the conditional.

See Causal Processes as well as Counterfactual Theories of Causation for useful details.

See also Scientific Explanation as well as Methodological Holism in the Social Sciences

The debate between methodological holists and methodological individualists concerns the proper focus of explanations in the social sciences: to what extent should social scientific explanations revolve around social phenomena and individuals respectively?

Mauro ALLEGRANZA
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The confounding factor here is that the statement is meant counterfactually — the propositions aren't meant to be interpreted as referring to actual world history, but instead to a hypothetical, fictional world history drawn from a class of those closely resembling our own.

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The conditional $\rightarrow$ in mathematical logic serves certain needs, and the standard truth-table provides exactly what is needed. The most important need is making Modus Ponens work: from $P\rightarrow Q$ and $P$, infer $Q$. You can see how the standard truth-table does this. If $Q$ is true, then inferring it is fine. If $P$ is false, then we don't get to apply Modus Ponens, regardless of the truth-value of $Q$, so again everything is A-OK.

But the notion of causal or necessary implication seems natural, and so logicians have tried to formalize using so-call modal logic. The idea is to add an operator to the symbolism, $\Box$, where $\Box P$ reads, "$P$ is necessarily true". So you would represent necessary implication by $\Box(P\rightarrow Q)$. So in modal logic, we could say that your example counterfactual implications are "true in our world", but they are not necessarily true; $P\rightarrow Q$ holds, but $\Box(P\rightarrow Q)$ does not.

Actually many modal logics have been studied. The one I just mentioned is called alethic logic (alethic=necessity). Even for alethic logic, different inequivalent axiomatizations have been proposed. You can find a nice introduction in Modal Logic at the Stanford Encyclopedia of Philosophy.

Some other responses have mentioned "possible worlds". The most famous semantics for alethic logic is so-called Possible Worlds semantics, introduced by Saul Kripke. The Modal Logic entry has more about this, with further links.

Michael Weiss
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