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

Danny Frederick danny.frederick at btinternet.com
Fri Oct 30 10:51:51 EDT 2009


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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