Re: Aristotelian syllogistic and monadic FOL

In article <1118785122.676016.172290@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Klaus Glashoff" <klaus@xxxxxxxxxxxx> wrote:

> .... IF, for showing the "Aristotelicity" of your
> transcription, you have to assume "(there exists x)(Sx or not Sx)",
> then you have to assume this for ANY predicate S, and this is exactly
> what we call "existential import"....


No. To require non-empty models I simply wrote "(there exists x)"
followed by a tautology, which doesn't assume existential import for any
term/predicate. You could instead specify a non-empty universe of
discourse in any other way you preferred.


In article <1118938723.188118.311680@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Klaus Glashoff" <klaus@xxxxxxxxxxxx> wrote:

> Ken, I veryfied that your transcription is identical to Jaskowski's.
> As I said before, this transcription does NOT render Sub:= A(S,P) ->
> I(S,P) a theorem of predicate logic and is therefore NOT a
> counterexample to my theorem. This is astonishing in view of J.'s
> Theorem 4.19, (a1) and (a2) ( |- PaP and |-CSaPNSaP' in the kryptic
> Lukasiewicz notation).


I think there's a mistake somewhere. In the notation of your
paper (and thank you for sending it :-), I'll write

p for (for some x)(Sx and Px),

q for (for some x)(Sx and (not Px)),

r for (for some x)((not Sx) and Px),

s for (for some x)((not Sx) and (not Px)),

and abbreviate basic formulae, so for example

p and (not q) and (not r) and s

will be written pq'r's. (Actually I used Lewis Carroll's biliteral
diagrams for these, and had a lot of insight from geometrical symmetries
of those diagrams. But never mind that now.)

Jaskowski's (and my) best interpretation, written in disjunctive
normal form, represents A(S,P) by

pq'r's or pq'r's' or p'q'r's or pq'rs

and represents I(S,P) by

pq'r's or pq'r's' or p'q'r's or pq'rs or pqrs or pqr's' or p'q'rs or
pq'rs' or p'qr's or pqr's or pqrs'

where you can see the that first four disjuncts in the second formula
are just the first formula. So there's no problem with subalternation.

Ken Pledger.
.

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