[hist-analytic] Discussion of Aune's ETK, Chapter Two: Modus Ponens/Tollens

steve bayne baynesrb at yahoo.com
Fri Oct 30 12:49:14 EDT 2009


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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