# [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
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

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://rbjones.com/pipermail/hist-analytic_rbjones.com/attachments/20091030/2a6f8f44/attachment-0002.html>
```