[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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