[hist-analytic] Discussion of Aune's ETK, Chapter Two: Modus Ponens/Tollens
Danny Frederick
danny.frederick at btinternet.com
Fri Oct 30 15:44:23 EDT 2009
Hi Steve,
I denied that the example we have considered is a counter-example to modus
tollens. Bruce denies it too in his Chapter 3. But I am open to the general
possibility that modus tollens has a counterexample (or loads of them). I
think dialetheists accept counter-examples to modus tollens, since they
think some contradictions are theorems and, since these contradictions are
both true and false and implied by true axioms, they generate exceptions to
modus tollens (MT). I am not quoting here: this is just what I think I
recall from some reading I did a few months ago (Graham Priest). Similarly,
with other logical rules like modus ponens and conjunction elimination:
there are serious and competent logicians who have denied them in order to
try to solve difficulties with standard logic. What all this means is that
it is not self-evident that there are no counter-examples to standard logic:
the latter is not a priori true/valid.
What I provided, once again, is a 'proof' that the argument (the supposed
counter-example to MT) is VALID and thus not a counter-example to MT. But,
as you say, this leaves it open as to whether MT has any other
counter-examples. I did not understand your last sentence.
Incidentally, the reason I put 'proof' in quotes all the time is that we can
never be sure that a proposed proof is really a proof. We can give a 'proof'
in standard logic, say; but the 'proof' is questionable because standard
logic is questionable. The same applies to every other logical system.
Whether or not something is a proof is something we can only guess at.
Best wishes,
Danny
_____
From: hist-analytic-manager at simplelists.com
[mailto:hist-analytic-manager at simplelists.com] On Behalf Of steve bayne
Sent: 30 October 2009 16:49
To: 'hist-analytic'
Subject: RE: Discussion of Aune's ETK, Chapter Two: Modus Ponens/Tollens
Danny,
Ok, now it is clear. Yes, vaid. However, I was under the impression that you
were challenging Bruce on whether MT actually has a counter instance. You
seem to disagree with him at this point.So how does this impact the alleged
counterexample, or its possibilty.
I saw this earlier, but opted for my formulation since it leaves the
consequent of the first premise a logical truth, which makes it not just
invalid but counterintutitive in the sense that obviously (?) it is not a
logical truth!
So what you have provided is a proof that the argument is invalid, not that
MT does or does not have a counter instance. My point was that there is no
counter instance here because the first premise is not a conditional; it is
in fact something of a conjunction. That would dipsose of the counter
example, which of course our proof does not actually do.
Regardsd
STeve
--- On Fri, 10/30/09, Danny Frederick <danny.frederick at btinternet.com>
wrote:
From: Danny Frederick <danny.frederick at btinternet.com>
Subject: RE: Discussion of Aune's ETK, Chapter Two: Modus Ponens/Tollens
To: "'hist-analytic'" <hist-analytic at simplelists.co.uk>
Date: Friday, October 30, 2009, 10:51 AM
Hi Steve,
You seem to have misunderstood me. Here is the 'proof' of inconsistency
spelt out (I am a bit rusty on this formal stuff, but here goes).
[1] If (Ex)(Rx & Yx) then ~ (Ex)(Rx & Hx & Yx) [Premise]
[2] (Ex)(Rx & Hx & Yx) [Premise]
[3] Ra & Ha & Ya [from 2 by EI]
[4] Ra & Ya [from 3 by conjunction elimination]
[5] ~(Ex)(Rx & Yx) [from 1 and 2 by modus tollens]
[6] For all x, ~(Rx & Yx) [from 5 by the rule for passing 'not' through
the quantifiers]
[7] ~(Ra & Ya) [from 6 by UI]
[8] ~Ra or ~Ya [from 7 by De Morgan]
[9] if Ra then ~Ya [from 8 by the conversion rule for 'or' and
'if-then']
[10] Ra [from 4 by conjunction elimination]
[11] ~Ya [from 9 and 10 by modus ponens]
[12] Ya [from 4 by conjunction elimination]
[13] Ya & ~Ya [from 11 and 12 by conjunction introduction]
No violation of the rules for EI, which was used only once.
Cheers.
Danny
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