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I read almost all posts on material implication and vacuous truths on the site.
I understood that it was introduced for mathematical convienience use and it does not have to perfectly align with our natural language intuition of the expression. I understood all the "breaking promise" analogy and the "subset" analogy.
What I don't understand is the reason why we choose the "if p, then q" construct insead of sticking to the "Not p or q"?
To me it seems that the second one is far more intuitive and resolves pretty much all the paradoxes that material implication arise.
Let's examine an example:

If pigs can't fly, then I can't walk on water

What would you answer if someone was to ask you, "is the above statement true or false"?
I honestly would answer, "I don't know, it really doesn't seem true"
But what about...

Either pigs can fly or I can't walk on water

This seems right, seems true to me in an intuitive way.

Another example can be:

If 2+2=5, then 2+2=6

False?True? What would you choose? Doesn't seem easy! Look it this way...

It's not true that 2+2=5 or it's true that 2+2=6

Seems intuitive, clear.

The only reason I can think about is that with the "If p, then q" construct you underline a causal relationship between the antecedent and the consequent, which altough is useful in mathematical's contexts ( It's more immediate), it gives rise to vacuous truths and some paradoxes where there is no causal relationship. I will soon provide other examples for what I mean, if it's not clear!

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    $\begingroup$ To me, "If $p$, then $q$" is far more intuitive then $p$ or not $q$. $\endgroup$ – 5xum Nov 2 '17 at 12:40
  • $\begingroup$ To me, if I know $p$ and $q$, the sentence $p \rightarrow q$ seems a lot more intuitive and easier to understand than $\neg p \vee q$, even though they are equivalent. $\endgroup$ – Anguepa Nov 2 '17 at 12:43
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    $\begingroup$ It doesn't appear intuitive to me to infer $\Gamma\vdash \neg p\lor q$ (instead of $\Gamma\vdash p\to q$) from $\Gamma,p\vdash q$. $\endgroup$ – Hagen von Eitzen Nov 2 '17 at 12:44
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    $\begingroup$ We're not always doing classical propositional logic. Most commonly we're actually doing classical predicate logic, and dealing with things like $(\forall x) \: P(x) \Rightarrow Q(x)$. It is cumbersome to think about situations like this in terms of the definition of material implication. Moreover it is considerably less common to encounter vacuous truths when you slap quantifiers onto implications. There are also more restrictive logics whose notion of implication requires more of them than material implication. $\endgroup$ – Ian Nov 2 '17 at 12:57
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    $\begingroup$ "What would you answer if someone was to ask you, "is the above statement true or false"? I honestly would answer, "I don't know, it really doesn't seem true"" My usual remedy to this is to rewrite to the contrapositive. To me that's a more intuitive translation, and if the original sentence is difficult to unravel, then the contrapositive usually gives a sentence that is more easily decided. In this case "If I can walk on water, then pigs can fly", which is actually not far from many common ways of stating "I can't walk on water" in everyday speech. $\endgroup$ – Arthur Nov 2 '17 at 13:10
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I think it's because the formulation $\phi\rightarrow\psi$ is more natural when it comes to conditional proofs and modus ponens. In a conditional proof you assume $\phi$ and from there prove $\psi$ then when discharging the assumption you conclude $\phi\rightarrow\psi$ - after all you have actually proven that $\psi$ follows from $\phi$.

In the case of modus ponens you have that $\phi$ and $\phi\rightarrow\psi$ and conclude $\psi$. Here in effect you're using that $\phi$ implies $\psi$. However some persons would be fine with modus ponens in the alternate form, that is we know that $\phi$ and $\neg\phi\lor\psi$ and from that conclude $\psi$.

Of course there are merits in the other formulation too. For example it might be easier to manipulate logical formulae if you only have two or three operators (that is $\neg$, $\land$ and $\lor$) to bother about. You need it to achieve disjunctive and conjuctive normal forms.

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Psychologically, it's just easier to think about $A\implies B$ than it is to think about not $\neg A \lor B$.

As for any apparent paradoxes when we have both $A\implies B$ and $\neg A$, I just reason that we cannot really infer anything about $B$ from these two statements alone. Without any possibility of inconsistency, B could be true or $B$ could be false, as in the truth table for $A \implies B$.

Example: We know that if it is rainy, then it must be cloudy. Based on this knowledge alone, if it is not raining, it may or may not be cloudy.

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  • $\begingroup$ That's right but is it only that?...I mean why do we decide a definition that appeal less to our intuition (give rise to paradoxes) if we have a valid alternative? (stackexchange is full of posts regarding question about implication....this is evident). Moreover, we could define "if p, then q" as a subset of the "not p or q" statements when there is a casual relationship between the statements. $\endgroup$ – Gabriele Scarlatti Nov 2 '17 at 15:23
  • $\begingroup$ Which part is counter-intuitive? What "paradoxes?" There may be other notions of implication, but IMHO they are quite useless in applications, e.g. when writing mathematical proofs. In mathematics, there is no notion of causality. That is in the realm of science. $\endgroup$ – Dan Christensen Nov 3 '17 at 1:22
  • $\begingroup$ @DanChristensen I find it counterintuitive to say that any given theorem A implies any other given theorem B, but you can't start with any given theorem A as an axiom only and deduce any other given theorem B, under say detachment and substitution. Then again, to assert (A$\rightarrow$B) we really need a context, and in the same context A $\vdash$ B will usually hold also. $\endgroup$ – Doug Spoonwood Nov 5 '17 at 0:54
  • $\begingroup$ @DougSpoonwood Sorry, this makes no sense to me $\endgroup$ – Dan Christensen Nov 5 '17 at 13:44
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"What I don't understand is the reason why we choose the "if p, then q" construct insead of sticking to the "Not p or q"?"

Well doing such would cause (another) controversy among logicians. I mean, it should get noted that their exist logical systems, such as Lukasiewicz 3-valued logic, where "if p, then q" does NOT mean the same thing as "not p or q". Also, detachment with $\rightarrow$ makes for the least controversial of logical rules. Furthermore, symbol usage ends up minimized with $\rightarrow$ instead of $\lnot$ and $\lor$.

Additionally, I will note that all sound and complete systems strong enough to have a propositional calculus have vacuous truths.

That all said, there exist plenty of systems where $\lnot$ and $\lor$ get taken as primitive instead of $\rightarrow$. So, it isn't like you have to use $\rightarrow$, so if you don't want to use $\rightarrow$, then don't do so.

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  • $\begingroup$ That's a good answer because it gives multiple reasons...Anyway, I want to point out that from my little understanding vacuous truths arise just because we chose to make the indiciative conditional (the "if, then" connective) truth-functional. In fact if we chose to just use "Not p or q", then there would be no vacuous truths. The only problem I see with "Not p or q" is that is less clear in the case that the antecedent is true. In a way that reflects less the implicative relation that there is between the antecedent and the consequent (when the first is true) $\endgroup$ – Gabriele Scarlatti Nov 3 '17 at 16:25
  • $\begingroup$ "In fact if we chose to just use "Not p or q", then there would be no vacuous truths." A vacuous truth is one where the antecedent is false. So, we still have vacuous truths using "not p or q", since the antecedent "not p" can be false, while q can hold true, and thus "not p or q" can be a vacuous truth. So, no, I don't see how using $\lnot$ and $\lor$ instead of $\rightarrow$ eliminates vacuous truths. $\endgroup$ – Doug Spoonwood Nov 5 '17 at 0:50
  • $\begingroup$ But thank you for saying I had a good answer @GabrieleScarlatti $\endgroup$ – Doug Spoonwood Nov 5 '17 at 0:50

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