Re: Aristotelian syllogistic and monadic FOL
- From: "Klaus Glashoff" <klaus@xxxxxxxxxxxx>
- Date: 14 Jun 2005 13:55:22 -0700
Stephen Harris wrote:
> "Klaus" <logic@xxxxxxxxxxxx> wrote in message
> news:1117816445.335401.20450@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> > 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
> >
>
> You were in the previous discussion which to me centered around a
> philosophical assumption of the existence of a null class which isn't
> going to be proven, it will be tautologically assumed, IMHO. Have you
> resolved what appear to be the philosphical issues mentioned below?
>
> Chris Menzel writes:
>
>
> | "What more is there to it than existential import? I'm *no*
> | kind of Aristotle scholar, but that's the usual intro text
> | characterization of the difference; Aristotle assumes that
> | all the class terms in a categorical argument denote
> | nonempty classes. Adding this assumption to the usual
> | predicate logic representations of the four categorical
> | statement forms makes the "missing" valid Aristotelian forms
> | classically valid."
>
> Keith Ramsay wrote:
> "The explanation on http://plato.stanford.edu/entries/square
> comes off sounding pretty reasonable to me. It claims the
> author knew of few pre-19th century sources that assumed
> all classes are nonempty.
>
> In the square of opposition, the AEIO forms are arranged
> like so:
>
> A---E
> |\ /|
> | . |
> |/ \|
> I---O
>
>
> in order to describe the relationships. "A" is the one
> usually expressed something like "Each X is Y".
>
>
> The way the vertical relationships are described
> ("subalternation") is what I think we would tend to call
> implications, i.e. A=>I and E=>O. The diagonal relationships
> ("contradictories") are described as holding between things
> that can't both be true and can't both be false, i.e. they
> are described as negations, E=~I and O=~A. The horizontal
> relationships, contraries and subcontraries, indicate that
> A and E can't both be true and I and O can't both be false.
> (Given the diagonals and either the horizontal or vertical
> relationships, the other follows.)
>
>
> The author plausibly argues that the "A" form, each X is Y,
> is meant in a way that assumes there are some X's, although
> see the caveat below. The "I" form means what one would
> think, the existence of something that is both an X and a Y.
> "E", no X is Y, is then no special problem either. Finally,
> though, "O" or "not every X is Y" has to be considered
> vacuously true if there are no Xs. The author thinks this
> is how it was taken, historically.
>
>
> I managed to convince myself, however, that the list of
> valid syllogisms isn't enough to distinguish between this
> and the assumption that ever class is nonempty. If there
> are no Xs, then the AEIO relationships that X holds with
> another class Y can be preserved while adding instances
> of Xs that aren't Ys (or Zs or...). So if a prospective
> syllogism is invalid, it can be shown to be using an example
> with nonempty classes, and the two interpretations coincide
> for nonempty classes.
>
>
> [...]
> |What's philosophically unsatisfying about that? Or was Aristotle's
> |account subtler somehow? I really have no idea.
>
>
> Allegedly Aristotle made a distinction between different
> senses in which "each X is Y" could be meant. In one sense,
> for example, "each unicorn is an animal" would be true, in
> spite of and not because of the fact that there are no
> actual instances of unicorns, but because being an animal
> is in a sense part of the concept of being a unicorn. By
> the same token, allegedly "some X is Y" can (depending on
> context) sometimes mean something weaker than the actual
> existence of a bona fide X that's also a Y.
>
>
> As I said in my other posting last week, it appears to me that
> this is enough like expanding the domain of quantification
> to include certain types of hypothetical instances to make
> the deductions that are valid under such an interpretation
> stay the same as the ones that are valid under the less subtle
> one, A = "(Ex) X(a) & (Ax)X(x)->Y(x)", E = "~(Ex) X(x)&Y(x)",
> I = "(Ex) X(x)&Y(x)", and O = "(Ex)X(a) -> (Ex) X(x)&~Y(x)". "
>
>
> Regards,
>
> Stephen
>
>
>
>
>
> begin 666 dot_clear.gif
> K1TE&.#EA`0`!`( ``/___P```"'Y! $`````+ `````!``$```("1 $`.P``
> `
> end
My opinion about "existential import" (from my paper "Limits of
transcribing Aristotelian logic into predicate logic", 2005):
"The problem of existential import developed along with the
development of modern symbolic logic during the nineteenth
century. The problem is peculiar to the standard predicate
calculus. There never was a real problem of existential import
within the traditional syllogistic logic - it was placed there in
retrospect by the modern logicians." (Nedzynski)
This misreading of Aristotle's logic by modern logicians from
Frege and Russell to Quine, Lemmon and all those who blindly
adopted their arguments is, from a historical standpoint,
the most fatal example of what Gyula Klima
called 'paradigm-straddling'. We agree with his following words
which serve as an appropriate conclusion of the present paper:
"When we engage a historical author by simply applying our
own modern concepts in interpreting his claims, rather than
trying to acquire his concepts, then there is always the
serious danger of misinterpreting the author, who was thinking in
a radically different conceptual framework. Indeed, this approach
becomes especially precarious when the exposition turns into
criticism."
(Comment: Never trust any logician who talks about "existential import"
- this has absolutely nothing to do with Aristotelian logic which is a
term logic!)
Klaus
.
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- Aristotelian syllogistic and monadic FOL
- From: Klaus
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- From: Stephen Harris
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