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If $P \implies Q$ and $not P \implies Q$ then can we conclude that Q does not follow logically correct from P?

Someone on the MathStackExchange asked a question that 'why "Socrates is a martian and martians lives in pluto therefore 2+2=4" is logically incorrect?'

I wrote suppose the given statement is P => Q. now assume not P but since we can prove Q from the notions of basic algebra. so P=> Q and not P=> Q which means P has nothing to do in the truth value of Q... so I said Q does not follow logically from P... is that correct?

Praveen Kumaran P
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  • You can't prove Q from P, or not P, in this case. – Bertrand Wittgenstein's Ghost Jul 27 '23 at 04:16
  • This question originates from a previous question What is meant by "logically incorrect"?. – peterwhy Jul 27 '23 at 05:33
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    If "logically correct" is taken in the same sense as in your previous question, then $Q$ might well logically follow from $P$ (and maybe also from $\lnot P$). This is the basis of a proof by "distinguishing cases". Say, take $P:=x\ge 0$ and $Q:=x^2\ge 0$. If $x\ge 0$ then $x^2\ge 0$. If, however, $x<0$ then $-x>0$ so $x^2=(-x)^2>0$. Ultimately, in both cases $x^2\ge 0$. Having said that, it may not "logically follow" from either. Say both $P\implies Q$ and $\lnot P\implies Q$ are true if $P:=$"Socrates is Martian" and $Q:=2+2=4$ but $Q$ doesn't "logically follow" from either $P$ or $\lnot P$. –  Jul 27 '23 at 06:48
  • @StinkingBishop So Socrates is Martian therefore 2+2=4 does not logically follows isn't it? I said that there in the question mentioned above but my response got removed there... – Praveen Kumaran P Jul 29 '23 at 02:41

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I am editing your Question within the quotation blocks below instead of directly above, in case I've misunderstood your query.

If $P \implies Q$ and $\lnot P \implies Q$ then can we conclude that $Q$ does not follow logically from $P$?

Whether given P⟹Q or given (P⟹Q) and (¬P⟹Q), it is accurate to assert (literally) that Q follows from P.

Suppose that the given statement is $P \implies Q$. Now assume $\lnot P.$

So, $P\implies Q$ and $\lnot P\implies Q.$

But ¬P⟹Q is not a consequence of (P⟹Q) and ¬P.

This means that $P$ has nothing to do in the truth value of $Q.$

(P⟹Q) and (¬P⟹Q) means precisely that Q is a logical validity; as such, Q's truth value is not generally dependent on P's truth value. Even so, P and Q need not have “nothing to do” with each other: they can even stand for the exact same sentence, say 1=1.

In this case, is this correct: $Q$ does not follow logically from $P? $

$Q$ not logically following from $P$ means precisely that P⟹Q is not a logical validity; if P and Q are arbitrary sentences, then Q does not in general logically follow from P. On the other hand, P or Q does logically follow from P and Q.

ryang
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  • ohh ohh I got that in many cases Q may follow logically even if $P \implies Q$ and $\neg P \implies Q$,... as one gave in the example above $x>0 \implies x^2>0$... so we can't say Q may not follow logically from P... now my question is how can we abstractly say that in "Socrates is a martian and martians live in pluto therefore 2+2=4" 2+2=4 does not logically follow from "Socrates is a martian and martians live in pluto" – Praveen Kumaran P Jul 29 '23 at 02:55
  • @PraveenKumaranP 1. P⟹Q literally means Q follows from P. $\quad$ 2. Say, you assign specific meanings to P and Q such that Q follows from P; I would still refrain from saying that that Q logically follows from P, because some readers understand "logically follow" to have a stronger meaning than "follow". – ryang Jul 29 '23 at 03:09
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    @PraveenKumaranP 3. 2+2=4 indeed does not logically follow from Socrates is a martian and martians live in pluto; as explained in this answer, you can also just say that Socrates is a Martian and Martians live on Pluto, therefore 2 + 2 = 4 is an invalid argument. – ryang Jul 29 '23 at 03:09