[hist-analytic] Aristotle's Metaphysics: The Izz and the Hazz

Jlsperanza at aol.com Jlsperanza at aol.com
Wed May 27 12:00:09 EDT 2009


R. B. Jones:

>I am responding to  Speranza on Grice and Leibniz here ...
>I'm inclined to doubt that that is  the usual definition of identity,
>though it may be the one which most  readily comes to the mind
>of philosophers.  Certainly, outside of  philosophical circles I
>don't believe it is much referred to.
>In  first order logic identity is usually taken as primitive.
>It is not  defined and does not necessarily comply with Leibniz's
>law.
>In  higher order logic identity is also likely to be primitive
>but the extra  ontological content trivialises Leibniz's law.
>This is because for every  object x there exists is a predicate
>"equal to x" which is true only of  x.

It would be good to check what the strategy was in Russell/Whitehead  (or
Whitehead/Russell, as everyone should prefer) "Principia Mathematica". I
don't know if that work has been scanned online. Should  check.

>Anyway, be that as it may, I think Grice's rule can be  reconciled
>with Leibniz's and I have attempted below an explanation  which
>may or may not be helpful!
>Before my attempted  reconciliation, some nit picking.


>>(x)(y) Fx <----> Fy  ---->  x = y
>Putting aside my previous remarks, you should  have:
>(x)(y)((F) Fx <----> Fy) ---->  x = y
>It's  hard to get out of quantifying over properties
>or predicates  here.
>If you leave the quantifier out, then, whether F is
>a  variable ranging over propositional functions
>or a syntactic variable  ranging over formulae,
>you will still need only one property shared  or
>shunned by x and y to get x=y, which is not enough.
>To spell it  out:


((F)(x)(y)((Fx <=> Fy)  => x = y)) => (x)(y) x = y

>is a theorem of higher order  logic.
>i.e. your presentation of Leibniz's principle suggests  that
>there is only one thing (of each type).

Point taken. I'm glad  it's a theorem in higher-order logic.

>The connection between  Leibniz's law and Grice's:

I(x, y) & I(y, x) ---> x =  y

[my reformulation. JLS -- :)]

>can be explained as follows.
>We  have noted before that from a modern point of view
>"izz" may be thought  to conflate

*  set membership

>and

* set  inclusion

>since it will be the former if [x] is
>an individual  and [y] is a universal and the latter otherwise.

Good. I am reluctant now  to have replaced your fine "A" and "B" by 'x' and
'y', but I kept the square  bracket to show it's my idiosyncrasy. Indeed,
from what I recall Aristotle ONLY  uses "A" and "B" in _his_ symbolisms.

>However, an alternative is to  confuse individuals
>(or identify) with their unit sets (which is what  I
>have done in my formalisation of the Grice/Code/Speranza  formulae).

Yes, this is _Very_ Good.

I think it's a _good_ Quinean  move. I recall his

"Fa"   -- Pegasus  flies
~Fa          Pegasus does not fly

But in "On What There Is" ('to be is to be the value  of a variable')

"Pegasus"  -- becomes "to  pegasise"

-- Grice discusses this in "Vacuous Names".

If we call  the horse, "Adolphus" instead

Fa

Then for Quine it becomes

Ax

(x is "Adolphus") where

x is included in the unit set {Adolphus}

I stick to "a" because  apparently logicians use "a, b, c, ..." for names
of individuals, while "F, G,  H, ..." for names for predicates. In
meta-logic, they use Greek letters (phi,  psu, I think) for _general_ predicates, but
I don't know what they use for  _general_ names. And in any case, names are
not so important (in science or  academia).

----

R. B. Jones:

>This is possible because  there are no singular universals,
>so we might as well think of an  individual as its unit set
>and treat izz uniformly as set  inclusion.

Excellent.

>Gice's principle then becomes the  familiar:


A subsetof  B & B subsetof A => A = B

>from set theory.

Very  good.

>Now set theory is a very parsimonious theory,
>it needs  only one primitive predicate (relation),
>membership (equality is  definable).

Very good. I agree that equality is best definable rather  than taken as a
primitive. But certainly it must be a problematic notion. Grice  did not
have worries in accepting his credo as being that of 'first-order  predicate
logic WITH IDENTITY" as the slogan goes. In fact, I never met anyone  whose
credo is "first-order predicate logic WITHOUT identity". I haven't read  your
post on Carnap yet, and I hope it's not _he_.

>Leibniz's  principle can then be spelled out explicitly
>without quantifying over the  predicates:

(x)(y) ((z) (x in  z <=> y in z) & (z in x <=> z in y)) => x =  y

>However, sets are extensional, so we have the following  axiom:

(x)(y) ((z)(z in x  <=> z in y)) => x = y

>Which can be rendered without the  equality as:

(x)(y) ((z)(z in x <=> z in y)) =>  ((z) (x in z <=> y in z))

>(dropping the redundant repetition of  the lhs on the right.)

Very good. I'm currently studying the use of  "Simpl." by logicians and
people in general. (e.g. "It rains and it rains" +>  I'm so tired of this
weather).

>and equality defined by Leibniz's  definition, and the result is
>that Leibniz's law agrees with  Grice's(Aristotle's?) rule for deriving
>equality from reciprocal  izzing.

Excellent.

>Aristotle is of course not dealing with a  pure set theory,

I don't know Venn's Aristotle _is_. And if Davidson can  gerrymander Hume
like that I cannot see how a distinguished Logician as John  Venn was cannot
gerrymander Aristotle. Aristotle has been often gerrymandered.  Possibly the
most idiotic gerrymandering was the object of affection of my logic  tutor:
Lukasiewicz, so we would spend precious hours doing the Polish notation  to
please the Slavs.

Another good gerrymandering of Aristotle involves  Davidson's Hume. I am
thinking (alla Strawson, "Causation and Explanation",  available online as one
chapter of his "Analysis and Metaphysics") that CAUSE  _is_ an Aristotelian
RELATION that holds between substances! (with Kant, but  against Hume and
Davidson)

Jones:

>so the rules for equality of  sets do not suffice,
>one needs also to be able to tell when the members  of
>the sets (possibly not themselves sets) are equal.

or  co-extensional. But I see what you mean.

I'll see if I can doublecheck  with Grice's 'extensionalism' in "Reply to
Richards".

In set terms, I  would think 'married bachelor' for example would be a null
set. It's the  intersection of the set "married" and "bachelor" but we know
by definition it's  empty. Oddly when J. L. Borges reviewed the Buenos
Aires edition of "Alice's  Adventures in Wonderland" he  wrote:

"An impoossible book,  almost -- as Dodgson has the intersection  of
things that weigh TWO tons and that a  child
is able to lift.   -- Symbolic Logic

-- So we shouldn't forget  that set-theory was all the rage in Oxford, and
it takes a Grice to criticise  extensionalism like that and get away with
it. Oddly, when it comes to 'cause',  Strawson, with Anscombe, think of it as
an _intensional_ relation, but I'm not  sure I understand them.

Jones:

>But nevertheless, the rule for  equality of the sets still holds,
>and is consistent with the Leibniz  law.
>The complication arising from the lack of purity
>(i.e. the  existence of non-sets) might be thought of
>as transforming the  rule:

x subsetof y  <=> (z)(z in x => z in  y)

>into:

x subsetof y <=>  (z)(z in x => (Ez') (z = z' & z' in y))

>instead of saying A is  a subset of B if every member of A
>is a member if B say that A is a  subset of B if every
>member of A is "equal to" some member of  B,
>Thus in this impure set theory, the same rule holds for  equality
>of the sets but involves an implicit appeal to a
>possibly  more complex standard of equality
>(involving more relevant predicates)  on
>the things which may be in the sets,

Good.

>I now  gather that Aristotle's theory is inconsistent, which is a bit
>of an  impediment to a proper formalisation!  This may explain  or
>contribute to the apparent incoherence of the  Grice/Code/Speranza
>formulae.
>My biggest problem however, is not  having the Code paper,
>if anyone can come up with an electronic copy I  should
>be very grateful.

Well, yes. It may be inconsistent, but I  hope Plato's is too!

Cheers,

J. L. Speranza

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