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

Roger Bishop Jones rbj at rbjones.com
Mon May 25 05:59:51 EDT 2009

I am responding to Speranza on Grice and Leibniz here
and to Speranza on Grice and Carnap in my next message.

On Friday 22 May 2009 18:45:52 Jlsperanza at aol.com wrote:

>While Grice indeed seems to be treating izzing and hazzing as predicates,
>one wonders. At one point R. B. Jones mentions the 'grasp[ing] of [the
>problem]  with ' = '. At that point, I was referring to what I think rather
> loose use of '  = ' by, of all people, linguists:
>
>         'cat' = 'chat'
>
>         'Gavagai' = ????
>
>If Grice has things like
>
>a)
>
>     I(x, y) & I(y, z) ---->  x = y
>
>(I think)
>
>One may like to compare that with the usual definition of " = " as per
>Leibniz's Law of the identity of indiscernibles:

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.

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.

>b)
>
>(x)(y) Fx <----> Fy ---->  x = y
>

Putting aside my previous remarks, you should have:

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

Its 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) ((F x <=> F y) =>  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).

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

A izz B and B izz A => A = B

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 A is
an individual and B is a universal and the latter otherwise.
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).
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.

Gice's principle then becomes the familiar:

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

from set theory.

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

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

>I think there is a good hope of providing a ProofPower HOL for Aristotle's
>basics. After all, all that can be said in Aristotle's metaphysics, one
>hopes,  is retrievable (if that's the word) alla Venn -- if not
>algorithmically, at  least in some way procedurally (if that's the word).

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.

Roger Jones