Re: Aristotelian syllogistic and monadic FOL

In article <1117816445.335401.20450@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Klaus" <logic@xxxxxxxxxxxx> wrote:

> Impossibility of transcribing Aristotelian logic into monadic FOL
>
> Let x, y, z, ... be the variables of a FOL L, and let S, P, M, ...
> denote the monadic predicate symbols of L.
>
> Let us define a monadic FOL - transcription of Aristotelian syllogistic
> by choosing four arbitrary closed wffs A(S,P), E(S,P), I(S,P), and
> O(S,P), all of which contain exactly two predicate symbols S and P.
>
> Theorem.
> There is NO transcription of Aristotelian syllogistic into monadic FOL,
> for which all the following wwfs Con, Sub, Obv, and IdA are theorems
> of FOL:
> Con:= (A(S,P) <-> neg O(S,P)) and (E(S,P) <-> neg I(S,P))
> Sub:= A(S,P) -> I(S,P)
> Obv:= A(S,neg P) <-> E(S,P)
> IdA:= A(S,S)
>
> I recently proved this theorem, but I am not sure whether it is really
> new. Please let me know if you have read this result before. Give me
> also a note in case you think that the theorem is trivial, or
> irrelevant, or if you have a counterexample...
>
> Klaus


Many years ago I sorted this out (inspired by G. E. Hughes & D.G.
Londey, "The elements of formal logic" pp.316-349), then found that the
same result using a very similar argument had been published by
Stanislaw Jaskowski in Polish in 1950 and later in English translation:
"On the interpretations of Aristotelian categorical propositions in the
predicate calculus," Studia Logica XXIV (1969) 161-172.

To get an interpretation which verifies all of traditional
syllogistic, it's not necessary to assume existential import for A, E, I
or O propositions individually, but only a non-empty universe of
discourse. This appears in the translation of your last item
IdA:= A(S,S) to (there exists x)(Sx or not Sx).

Here's the (almost unique - see below) translation which works.

A(S,P): ((for each x)(Sx implies Px)) and ((for some x)Px implies
(for some x)Sx) and ((for each x)Px implies (for each x)Sx).

E(S,P): (not (for some x)(Sx and Px)) and ((for some x)Sx or (for
each x)Px) and ((for each x)Sx or (for some x)Px).

Formulae for I(S,P) and O(S,P) can be obtained from those above by
negation. I'm getting tired of typing all this ascii. ;-(

That translation is not unique, as you can get another by
replacing every Sx by (not Sx) and every Px by (not Px).
However, this alternative is intuitively far more remote from the
traditionally intended meanings.

Ken Pledger.
.

Relevant Pages

  • Re: Aristotelian syllogistic and monadic FOL
    ... > Impossibility of transcribing Aristotelian logic into monadic FOL ... > O, all of which contain exactly two predicate symbols S and P. ... > There is NO transcription of Aristotelian syllogistic into monadic FOL, ...
    (sci.logic)
  • Re: Aristotelian syllogistic and monadic FOL
    ... > Impossibility of transcribing Aristotelian logic into monadic FOL ... > O, all of which contain exactly two predicate symbols S and P. ... with nonempty classes, ...
    (sci.logic)
  • Re: Aristotelian syllogistic and monadic FOL
    ... >> Impossibility of transcribing Aristotelian logic into monadic FOL ... >> O, all of which contain exactly two predicate symbols S and P. ... >> There is NO transcription of Aristotelian syllogistic into monadic FOL, ...
    (sci.logic)
  • Aristotelian syllogistic and monadic FOL
    ... Impossibility of transcribing Aristotelian logic into monadic FOL ... O, all of which contain exactly two predicate symbols S and P. ... There is NO transcription of Aristotelian syllogistic into monadic FOL, ... IdA:= A ...
    (sci.logic)
  • Re: Aristotelian syllogistic and monadic FOL
    ... >> Impossibility of transcribing Aristotelian logic into monadic FOL ... >> O, all of which contain exactly two predicate symbols S and P. ... the existence of something that is both an X and a Y. ...
    (sci.logic)
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