Re: Aristotelian syllogistic and monadic FOL
- From: "Stephen Harris" <stephen.p.harris@xxxxxxxxxxxxx>
- Date: Fri, 03 Jun 2005 21:58:10 GMT
"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
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