2006/09/01

A thought or two, to pass the time.

The first two sections of Richard Cartwright (1979) are a discussion of various formulations of Leibniz’ Law and similar indiscernibility principles. These are not to be confused with identity principles or substitutivity principles. The going theory seems to be that
(I) ∀x∀y∀z(x=y→[z a property of x → z a property of y])
is a schema, or at least a principle, of which all indiscernibility principles are instances: one could call it the indiscernibility principle. (In the article, it’s (1).) Each identity principle has an associated comprehension principle (pp294-5):
Suppose φ and φ' are expressions in which α, α' respectively appear as free variables and are identical except that each place of α in one is taken by α′ in the other. Suppose φ'' is an expression like φ except it has variable β in the positions α took in φ. Suppose γ is a bound variable ≠β in φ''. Then some universal closure of
+∃γ∀β(γ a property of β ↔ φ'') + [here, '+' replaces corner notation, or "Frege quotes"] is a comprehension principle for relevant universal closures of the schema
+∀α∀α′(α=α' → (φ→φ')+
for example, the comprehension principle of

∀ζ∀χ∀υ(χ=υ → (ζ a property of χ→ζ a property of υ))

is

∀χ∃ζ∀υ(ζ a property of υ ↔ χ=υ);

which is to say, there is some particular property which obtains in υ if and only if υ and χ are one and the same. One must specify such a property!


Such a property is peculiar to χ and to υ and to no other entities, and χ≣υ (strict equivalence between what is denoted by χ, υ).
The indiscernibility principle is supposed to be something like
“(3) (x)(y)(x=y→(Fx→Fy))” (pg. 293).

One could also write (3) not in second order logical notation as
(II) (F)(x)(y)(x=y→(Fx→Fy)),
which looks a lot like (I). [This alternative notation comes from Cartwright's footnote 3.]


But now one might pause, having a "third-man moment": can’t we also say something like
(III) (F)(x)(y)(x=y→[(P)((P(x,F)↔P(y,F))])
i.e. for every property, if two objects are identical than if one has it the other has it; or alternatively, any property x possesses is one y possesses when x=y, and neither x nor y has any other property?
Metalinguistically we could also point out that in
(IV) ∀α∀α'([α names x in Ln ∧ α names y' in Lm ∧ x=y]→∀β∀β'[(β and β' are expressions in Ln identical save for containing α in the one and α' in the other at the same places, respectively)→(β(α)≡β'(α')])
β(α) and β'(α') won’t be cognitively equivalent (≡cog) for all Ln speakers. This is failure of substitutivity.
(V) Suppose
i. δn(x*) is an expression (translation, interpretation) in the metalanguage Lm of βn(x) s.t. Tr(βn(x),δn(x)),
ii. β(x)∈〈γ1,…, γn〉= the set of all expressions containing x in Ln ordered and δ(x) a member of the companion ordering of a set of expressions {ε1,…,εm} in Lm, and
iii. Σ is an Lm speaker:
If (IV) then if Σ knows
∀β∀β'∀x∀y[Tr(β(α),δ(α*)) ∧ Tr(β'(α'),δ'(α'*)) ∧
(β(α)≡β'(α') ∨ δ(α*)≡δ'(α'*))]
then (≡cog(Σ,α,α') ∨ =cog(Σ,α*,α'*)

This implies also a mapping function from each βi to one and only one δi. Assuming nonambiguity. Call this our translation manual. …even though it is likely that (V) is false @.

Question: how to characterize “companion ordering”?
This is a serious problem in that, although we can characterize the metalinguistic cognitive equivalence principle (V) abstractly, we need a way to specify how it comes out that cardinality of Γ={γ1,…,γn} (n) and Ε={ε1,…,εm} (m) are the same (n=m), and that the relevant ordered multiples 〈γ1,…,γn〉,〈ε1,…, εm〉 come out with expressions of the same meaning (whatever the hell that is) for each pair (γk, εk) in any actual linguistic situation.

Or in other words, how do we know that Tr came out “right”?


But this is to return us to the problem of translation addressed by Quine, Davidson, et al. ad nauseam in the last century.
Suppose you did by hypothesis have an effective mapping function Tr. Then one could say Tr maps sets of utterances onto meanings. E.g. for Davidson this is truth-condition, for Quine a set of utterance occasions, for Frege it would be a Sinn. —But then suppose we give the indiscernability criterion within some language L, then we characterize Tl as a function that maps multiples of expressions within L onto "meanings." And taking it, as with φ,φ' above, that some of these expressions are otherwise identical except for terms appearing free at certain loci, one can say we use the related function Tt to map multiples of terms onto somethings (?!).

It is at this point that the discerning critic will notice how Cartwright's moves here are designed to trip this view up–at least to the extent that it needs to be modified in the face of obvious inadequacy to the task for which it is intended: to explain indiscernibility in terms of property-set coextensionality. I am not suggesting that any conclusions about haecceitism need be drawn from this sketch of a popular notion of indiscernibility.

However, with a view to a theory of identity for all and only indscernibles (that is, something is indiscernible from something if and only if they is identical), we ought to be suspicious of theores that can gnerate implications in only one direction. And I take it such an equivalence theory is useful because it subserves ordinary notions of identity without prima facie denying the possibility of numerically distinct identicals. One would do well to remember the original direction of Leibniz' formulation, and ponder what significance it has, both in his and our theories.

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