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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics
Subject: Re: Simply defining =?utf-8?Q?G=C3=B6del?= Incompleteness and Tarski Undefinability away V34 (Logical implication ERROR)
Date: Fri, 24 Jul 2020 15:47:12 -0700
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olcott <NoOne@NoWhere.com> writes:
> Logical implication
>     p  q  p ⇒ q
> (a) T  T    T
> (b) T  F    F
> (c) F  T    T
> (d) F  F    T
>
> p = "I will go to the store"
> q = "I will buy eggs at the store"
>
> (a) I will go to the store and buy eggs while I am there is true
> (b) I will go to the store and not buy eggs while I am there is false
> (c) I will not go to the store and buy eggs while I am there is true
> (d) I will not go to the store and not buy eggs while I am there is true
>
> (c) I will not go to the store and buy eggs while I am there is true
> Proves that Logical implication derives incorrect consequences

No, it doesn't.  (c) just means if it were the case that you don't go to
the the store *and* you buy eggs at the store, p ⇒ q is still true.  The
fact that that's not possible (unless you buy eggs remotely? or maybe
you were born at the store?) doesn't make the conclusion invalid,
because that's what "p ⇒ q" *means*.

The impossibility of buying eggs without going to the store is outside
the scope of the original statement, and "p ⇒ q" doesn't say either than
you can do that or that you can't.  Choosing an example that has
implications beyond the meaning of "p ⇒ q" doesn't make "p ⇒ q" invalid.

Let p be "it rains" and q be "I will carry an umbrella", so p ⇒ q is "If
it rains, then I will carry an umbrella".  (Note that "⇒" doesn't carry
the same implication of causation that the English if/then construct
sometimes does.)  If it *doesn't* rain tomorrow but I carry an umbrella
anyway (which is of course quite possible), that doesn't falsify the
statement that "If it rains, then I will carry an umbrella".

In other words, the statement
    If it rains, then I will carry an umbrella
is consistent with:
    It rains and I carry an umbrella.
    It doesn't rain and I carry an umbrella.
    It doesn't rain and I don't carry an umbrella.
but inconsistent with
    It rains and I don't carry an umbrella.
Only that last set of circumstances falsifies "p ⇒ q".

Yet again, you try to refute a statement in mathematical logic because
you dislike the way it's expressed in informal English words.

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
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