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In mathematical logic, implication has the following truth table

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where $\phi$ and $ \psi$ are statements, and $1$ represents true and $0$ false.

From this table, an implication such as

"$1>2 \ $ implies $ $ vegetables are usually healthy"

is true. (Unless, I'm mistaken.)

But surely, this implication is nonsense, so how can we even begin to talk about its veracity? Doesn't mathematical logic bother with the "meaning" behind an implication? What is the rationale behind the above truth table?

Thanks

Stephen
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  • You might be interested in non-classical logic. – Shaun Aug 10 '18 at 22:32
  • @Shaun I'll check it out - thanks. – Stephen Aug 10 '18 at 22:39
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    The result that a false premise implies anything is sometimes referred to as the Principle of explosion. – JMoravitz Aug 10 '18 at 22:39
  • @JMoravitz I see, but what bothers me is the fact that a false statement can imply something. Is this just a convention? Shouldn't false statements imply nothing at all? as opposed to anything we want? – Stephen Aug 10 '18 at 22:48
  • Implication in the mathematical sense is not quite the "if-then" of the English language. As a general rule, they coincide fairly well whenever the truth value of the premise is in some way variable or uncertain, and very poorly when it is not. – Micah Aug 10 '18 at 22:51
  • One additional link... your complaint seems to rely on specifically how such phrases are used in common English. There is a common usage of the phrase monkey's uncle as in "If such and such is true then I'm a monkey's uncle" which in use is meant to mean that we the speaker do not believe the premise to be true (since we obviously know that we are not a monkey's uncle). – JMoravitz Aug 10 '18 at 22:56
  • @Micah I see, so I should be interpreting the mathematical "implication" as the definition resulting from the above truth table without trying to link it to the everyday meaning of the word implication? – Stephen Aug 10 '18 at 22:57
  • Pretty much. More precisely, I think it's a useful exercise to think about when they do and do not coincide. But when they don't coincide, it's better to think of that as indicating a difference between the two concepts than as a defect in the mathematical definition. – Micah Aug 10 '18 at 23:09
  • Thank you all. You've been very helpful (and patient). – Stephen Aug 10 '18 at 23:12
  • @Micah Using what I call elementary, common-sense properties of everyday logic and implication, it is possible to derive each entry in the truth table for implication. See my blog posting at http://www.dcproof.com/IfPigsCanFly.html – Dan Christensen Aug 11 '18 at 14:45

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You write (in response to a comment):

so I should be interpreting the mathematical "implication" as the definition resulting from the above truth table without trying to link it to the everyday meaning of the word implication?

I wouldn't quite go that far, but I would say that you shouldn't try too hard to make it match the natural-language behavior of implication. Rather, you should ultimately convince yourself that the truth table above is the best way to approximate natural-language implication as it is used in mathematics with a classical (= true or false) truth functional (that is, the truth value of an implication should depend only on the truth values of the hypothesis and conclusion).

To help motivate my claim above, let me give a concrete example of a situation where we definitely want a false statement to imply something:

If $104723$ divides $918273643635\times 85943204812$, then either $104723$ divides $918273643635$ or $104723$ divides $85943204812$ (or both).

The crucial fact here is that $104723$ is prime, and this means that for any $a, b$, if $104723$ divides $a\cdot b$ then it divides either $a$ or $b$ (or both). So we can say confidently that the statement above is true, without digging into those bigger numbers' details.

But in fact we can check - with work (or a computer) - that $104723$ does not divide $918273643635\cdot 85943204812$. So the hypothesis of the statement is false. At this point we're forced to conclude one of two things:

  • We were wrong above when we claimed that this implication held; and in particular, we can't use theorems like "If a prime divides the product of two numbers, it divides one of the numbers (or both)" in the natural way.

  • Or, having a false hypothesis doesn't make an implication false.

The latter is adopted for classical logic, simply because it better matches actual mathematical reasoning. You can argue that the former is better suited to natural language, and we have various nonclassical logics which try to address that, but it's important to recognize the value of being able to mathematically prove that an implication is true without showing that the hypothesis is true.

EDIT: Let me end by saying a couple words about a few other ways to approach implication.

  • First off, we could cook up a logic where $p\implies q$ is false whenever $p$ is false. As far as I know, this isn't done anywhere, but there's no real barrier - I just don't know that anyone's found it interesting enough to pursue (I think because "$p\implies p$" is something we pretty much all agree should be true, regardless of $p$).

  • However, we could try to draw a distinction between certain "vacuous" (= false-hypothesis) implications: some (like my example above, or "$1<2\implies 3<4$") have a kind of logic to them, while others (like that of your title) are just ... "silly." We could demand that a vacuous implication be true only if the hypothesis is somehow "relevant" to the conclusion. In fact, once we go down this road there's no need to pay attention only to vacuous implications: something like "If $2+2=4$, then Fermat's last theorem is true" is also fishy! This leads to relevance logic (and related concepts).

  • Another tack we could take is to think about contingency. For example, consider the sentence "If it were raining right now, the ground outside would be wet." This is something we probably agree is true, but - at least where I am, currently - the hypothesis is false. We can resolve the situation while remaining skeptical of vacuous implications by noticing that "it is raining right now" is false but not absurd. One way of expressing this is to say that there is a possible world where it is true; there are other approaches, but I like this one at least at first. Considerations like these will ultimately lead to modal logic.

And there are others, but hopefully the above helps at first. Incidentally, if you want to dive straight into the deep end you might check out the magnifigargantuamazing book The connectives. It's $1512$ pages long - that's not a typo - and hardback, so probably not a good idea to read without close-toed shoes (you may find this review entertaining).

Noah Schweber
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  • Another example would be equivalence of two false statements, e.g. $1<2 \iff 2<3$. We can get each statement by either adding or subtracting $1$ from the other statement on both sides, which should be an operation that doesn't change the validity. – Sellerie Aug 10 '18 at 23:03
  • @Sellerie True, but I'm not sure that would be as convincing to the OP; I wanted to pick an implication that is obviously true and apparently meaningful. But you're quite right. – Noah Schweber Aug 10 '18 at 23:03
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    Very helpful, thanks. The examples you gave make total sense, and indeed illustrates quite well why we have such a definition, and also how to think about them. – Stephen Aug 10 '18 at 23:22
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You can think of it more like "whenever $\phi$ is true, $\psi$ must also be true". Then it becomes clear that any false statement implies every other statement, no matter if true or false, since the "whenever" part never actually applies.

Sellerie
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That is because any false assertion implies whatever you want, since you cannot find a counter-example, i.e. an example with the assertion true and the conclusion false.

Bernard
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  • If $\phi$ is false, then indeed we can't find a case in which it's veracity implies a false statement, agreed. But, this is not satisfying. Instead of saying a false statement implies anything, isn't it more factual to say that a false statement does not imply anything at all? – Stephen Aug 10 '18 at 22:38
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    @Stephen No, it isn't. Classical logic actually models rather decently standard mathematical reasoning. You want it to model natural language, and find it wanting. In a sense, that is not entirely surprising, since really this was not the goal. Luckily, we now have a large array of nonclassical logics that serve as alternatives. – Andrés E. Caicedo Aug 10 '18 at 22:49
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    @Stephen I think it's natural at the very least to assert that a false statement implies itself ... – Noah Schweber Aug 10 '18 at 23:05