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The statement $(p \wedge q ) \Rightarrow (p \Rightarrow q)$ is a tautology, i.e., always true.

Now, imagine I am writing a proof and I want to show that $p \Rightarrow q$. I show that statement $p$ is always true, then I show that statement $q$ is always true, then I can end my proof and claim $p \Rightarrow q.$

This doesn't make sense to me. If I prove that $\sqrt2$ is irrational and then I prove that $7$ is odd, I don't understand how "$\sqrt2$ is irrational" $\Rightarrow$ "$7$ is odd".

Can someone help this disconnect I am having? Perhaps, you can provide information about what these logical statements actually mean and how they are used.

Rodrigo de Azevedo
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JMF9
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    The title of your question does not reflect the content of your question well, so it seems better to change the title. – Hanul Jeon Mar 19 '21 at 21:14
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    following the comment of @HanulJeon, I have edited your question slightly, hopefully that's alright :) – Atticus Stonestrom Mar 19 '21 at 21:17
  • Sometimes statements that are entirely logically correct somehow don't "sound right." In everyday discourse, implication often indicates causation, i.e. $q$ is true because $p$ is true (for example, in "if it rains, I'll take an umbrella," the idea is that I take umbrella because it's raining). And implication often has that characteristic in mathematical proofs. But it doesn't have to. The assertion "$\sqrt2$ is irrational" $\implies$ "$7$ is odd" is true simply by virtue of the fact that "$7$ is odd" is a true assertion. – Barry Cipra Mar 19 '21 at 21:23
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    $\sqrt 2$ is irrational $\implies 7$ is odd. Because $7$ IS odd. Blue mice eating green cheese implies $7$ is odd, because $7$ is odd. Every statement, whether true or false will imply a true statement. My aunt Sally love Kevin Costner movies implies the U.S. landed on the moon in 1969. – fleablood Mar 19 '21 at 21:27
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    @JMF9 One of the features of Stackexchange is that people can edit your question to improve clarity, formatting, grammar, etc. When people spend the time to make these improvements, please do not roll them back. Your original title was not informative (see the guidelines here). The title Atticus Stonestrom suggested was much better - but I think the current title is a fine compromise. – Alex Kruckman Mar 20 '21 at 04:08
  • Tarski on material implication – Rodrigo de Azevedo Mar 20 '21 at 06:26
  • Related to this, read my answer explaining vacuous truth and the meaning of first-order statements via game semantics. – user21820 Oct 01 '21 at 13:07

5 Answers5

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Note that logical implication (aka ‘material conditional’) does not translate directly into our intuitive understanding of implication. We often think of implication as synonymous with causality, whereas the material conditional exists irrespective of causality. Mathematics isn’t a science, and its truths are almost always absolute (relative to the chosen axioms), which makes mathematical truth rather than causation its object of study. You can try and shift your thinking in this direction. Try and think of statements like $a\implies b$ as “if $a$ is true, then $b$ also happens to be true” rather than “$a$ causes $b$”.

Note that the main instrument of mathematics is proof, and as used in proofs, the material implication’s purpose is not to formalise some causal connection between mathematical facts, but rather to verify that one of the statements is indeed a fact. Implications of the sort you mention, where the consequent is tautologically true, are of little interest to mathematicians, even though they are correct owing to the truth-functional definition of the material conditional.

To help you come to terms with this definition, remember that the only situation where $a\implies b$ fails to be true is when $a$ is true and $b$ is false. When doing proofs and saying that $b$ follows from $a$, what we really mean is that whenever $a$ is true, $b$ cannot possibly be false (hence is also true). The material conditional guarantees that. And it is that guarantee that is its sole purpose, and what makes making logical deductions in proofs at all possible.

Chubby Chef
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    I like this answer! Very detailed, ${+1}$. The only part I would personally disagree with is saying that Mathematical truths are absolute - I would say it still depends on the axioms you are using, right? Like - I could create two different sets of axioms that don't agree with one another, but both are valid Mathematical systems. So in a way, I'd say the truth in Mathematics is sort of halfway between relative and absolute. But this is being really picky, and it depends how you define absolute truth in the first place – Riemann'sPointyNose Mar 19 '21 at 21:47
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    Would a mathmetician ever actually use ¬p∨q in a proof to show that p→q? I get that it makes sense in propositonal logic to use this equivalency, but I mean more lke "out in the real world of math", in the context of like a proof based linear algebra or number theory course. Like does it ever actually come up where one wants to prove some implication theorem p imlies q, and one does it by showing not p holds or q holds. @Chubby Chef – JMF9 Mar 19 '21 at 23:12
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    @Riemann'sPointyNose You are 100% correct. I had opted to use a vague 'almost always' which I realise (in hindsight) is inadequate. In my defense, though, most people seem to perceive the 'absolute' truths of mathematics rather uncritically, certainly without thinking back to axioms, and aren't disadvantaged by it in any practical sense. But I will correct my post accordingly. – Chubby Chef Mar 19 '21 at 23:41
  • @JMF9 I think you are giving $\lnot p \lor q$ disproportionate attention. This is an equivalent formulation in first-order logic that may-at best-interest logicians, but as such does not hold much practical significance to mathematicians in general. How would you go about using it to prove something anyway? With implication you only need to show that whenever $p$ is true, $q$ is also true, and the rest you just don't care. With the other sentence you'd need to prove that $\lnot p$ and $q$ are never false at the same time - this just isn't practical (or even feasible) most of the time. – Chubby Chef Mar 19 '21 at 23:47
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    Speak for yourself! My understanding of implication and inference is the material reading. I never think of implication as synonymous with causality. I don't believe the main "instrument|" of mathematics is proof (mathematics is dependent on a whole host of individual and social processes other than proof). – Rob Arthan Mar 20 '21 at 00:13
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    @RobArthan No need to be emotional. I am not imposing any dogma on anybody, and the answer is mine alone, so I am speaking for myself. Statistically, however, I suspect you will find that a very significant proportion of people do associate implication with some degree of causality. As for proof being the main instrument of mathematics, this perhaps shouldn't be taken quite as literally. I didn't enclose "instrument" in inverted commas as I had thought it was clear enough from the context. – Chubby Chef Mar 20 '21 at 00:23
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    If you talk about "our intuitive understanding", you are not just speaking for yourself. If you have any substantive statistical evidence for your claims, then please provide it. – Rob Arthan Mar 20 '21 at 00:33
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    I am speaking for myself and others, whose intuitive understanding of implication is associated with causality :) As for statistical evidence (which for an issue like this is almost guaranteed to be nonexistent), I want to point out that, as far as this discussion goes, you, a researcher in mathematics and logic, would be the outlier. The vast majority of people have no idea what 'material conditional' even means (do I need to back that up with poll data too?). So if you want to play the evidence game, feel free to disprove my (necessarily anecdotal) claim - I frankly have no stakes in this. – Chubby Chef Mar 20 '21 at 00:44
  • Please see my answer below. When we are confronted with a statement $P \implies Q$ where $Q$ is absurd, then we retreat to the non-clausal reading of the implication. – Rob Arthan Mar 20 '21 at 01:04
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The meaning of the conditional $\implies$ has a different meaning than in ordinary language.

$\sqrt{2}$ is irrational $\implies$ 7 is odd

Is a true statement precisely for the reasons you claimed. In ordinary language we use "if-then" or $P \implies Q$ statements almost exclusively in situations where there is some set of "distribution" over $P$; that is, $P$ can be true or false, and then we ask whether or not in every situation where $P$ is true does it also follow that $Q$ is true. There is also often a notion of "causality."

Again; crucially, in normal language when we say $P \implies Q$ there must be at least two possible scenarios: one in which $P$ could be true and one in which $P$ could be false. Furthermore, things get a little murkier here, but $P$ being true must be the "reason" that $Q$ is true.

The fact that this is the way the statement gets used in ordinary language is why the mathematical statement feels odd. Modal logic can more closely mirror the "natural language" usage. https://en.wikipedia.org/wiki/Modal_logic

Jbag1212
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Here is an example, outside of mathematics, that I think shows how material implication is accepted and understood in ordinary human discourse. There was a court case, in which the wife of a serial killer contested claims, made in the satirical magazine Private Eye, that she had tried to profit on her husband's crimes. When, she won the case, Ian Hislop, the editor of the the magazine stated that "if that's justice, I am a banana". He was quite right and she ended up the poorer for it.

Rob Arthan
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    You are absolutely correct in this context, and I upvoted your answer for shedding light on an interesting aspect of the issue. I did however use the quantifier 'often' to refer to association with causality, so I do not consider my point disproved. If anything, I think the lack of causality is what adds to the absurd quality of the statement. Surely you will agree that a sentence like if it rains tomorrow, then I have two arms will sound bizarre, if not outright nonsensical to most people, despite being a valid implication? – Chubby Chef Mar 20 '21 at 01:18
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    I agree that examples like that sound bizarre. In mathematics, the material implication is one of those things that you just have to learn to live with (if you don't like it) and (if you are lucky) will eventually learn to love. – Rob Arthan Mar 20 '21 at 13:12
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This is one of the biggest issues with the conditional connective in propositional logic. This problem arises from the fact that $p\rightarrow q\equiv \neg p\vee q$. All of the logical connectives in the propositional calculus produce a sentence whose truth value is only determined by the truth values of the propositions it connects. In general when we think or say $p$ implies $q$ we are also implies some form of causation which is not captured by the logical connective $\rightarrow$. This means that $\sqrt(2)\notin \mathbb{Q}\rightarrow 7\text{ is odd}$. is a true sentence even though there seems to be no real connection between the two things being true.

Giorgio Genovesi
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  • Would a mathmetician ever actually use $¬p∨q$ in a proof to show that $ p→q$? – JMF9 Mar 19 '21 at 21:30
  • @JMF9 Do you mean whether someone would show $\neg p\vee q$ rather than $p\rightarrow q$? The two are just different forms of the same thing. Sometimes writing it out in one form is better than the other but it's not to important. The form $\neg p\vee q$ I think grasps the meaning of the expression better but in general conditionals better to work with – Giorgio Genovesi Mar 19 '21 at 21:43
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    Yes, all the time. In fact that's how I would have proved $(p\land q)\implies (p\implies q)$: Assume $p\land q$, therefore $q$, therefore $\lnot p\lor q$, therefore $p\implies q$. – MJD Mar 19 '21 at 21:44
  • Why not "... therefore $q$, therefore $p \implies q$"? Your digression via $\lnot p \lor q$ is unnecessary and masks the point that $(p \land q) \implies (p \implies q)$ holds in intuitionistic logic as well as classical logic. – Rob Arthan Mar 19 '21 at 23:53
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I show that statement $p$ is always true, then I show that statement $q$ is always true, then I can end my proof and claim $p \Rightarrow q.$ This doesn't make sense to me.

Using a simple form of natural deduction, we have:

(Screen shot from my proof checker)

enter image description here

Dan Christensen
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