UNIVERSITY OF PITTSBURGH
PITTSBURGH, PENNSYLVANIA

August 21,1967

Professor Ruth Barcan Marcus
Philosophy Department
University of Illinois at Chicago Circle
Chicago, Illinois 60680

Dear Ruth,

I have been reading Feys on modal logic. I am puzzled about

i(x = y) ↔ I(x,y)
We (or, at least I) want to say things like
x = y · ◊(x ≠ y)
Clearly in some state descriptions belonging to a semantical system S
a = (’x) fx
holds, whereas in others it does not, e.g. where
∼fa
holds.

I have been pondering the proof to see if I can locate the source of the above counter-intuitive result. But first some ground work. The notion of 'equivalence' is clearly connected with that of truth. With respect to a semantical system, S, we wish to articulate and relate the concepts of truth, possibility and necessity. As I see it, the concept of truth (in S) involves selecting one of the state descriptions, Di, to be that which 'holds' in the system; to be, in Carnap's sense, the RS of the system. To be true in the system S is to 'hold' in RS. Propositions which hold in other state descriptions, but not in RS, are false in the system, though, as holding in these state descriptions they can be said to be 'true (in Di)' . We thus distinguish 'true (in Di)' from 'true (in S).'

Now I submit that in its primary sense

x = y
is to be defined with respect to 'holds (in RS)'. Thus
x = y (in S)
has roughly the sense of
'(f) fx ↔ fy' holds in RS of S.
Where '↔' stands for material equivalence. If so,
x = y (in S)
pertains to the system as a whole, and hence, in a sense, 'holds for', every state description, though it does not 'hold in' them. After all the RS specifies how matters actually stand in S with the items with respect to which we are exploring alternatives. Thus if RS includes
fa
then if Di( ≠ RS) contains
∼fa
we can paraphrase the situation by saying that a, which is actually f (in S) might not have been f (in S).

Thus I suggest that we distinguish between

x = y (in S)
and
x = y (in Di) (roughly, '(f) fx ≡ fy' holds (in Di))
A special case of the latter would be
x = y (in RS)
which must be distinguished from, though it entails,
x = y (in S).
Given these definitions it follows that
x = y (in S)
is compatible with
(∃Di) x ≠ y (in Di)
Now consider the following argument:
  1. ixy → [x ∈ û(Ixu) → y ∈ û(Ixu)]
  2. ixy (hyp)
  3. x ∈ û(Ixu) ↔ y ∈ û(Ixu)
  4. x ∈ û(Ixu)
  5. y ∈ û(Ixu)
  6. ixy → Ixy
Using the concepts explicated above, the argument would begin
(1') x = y (in S) → [(Di) x = x (in Di) → (Di) x = y (in Di)]
Granted this premise, the argument goes through. But all we are entitled to, unfortunately, is
(1'') x = y (in S) → [x = x (in S) → x = y (in S)]
which is trivially true and gets us nowhere.

So much for the 'case' or 'state description' approach. Now consider the argument:

  1. x = y → (x ∈ ûN[x = u] ↔ y ∈ ûN[x = z])
  2. y = u (hyp)
  3. x ∈ ûN[x = u] ↔ y ∈ ûN[x = u]
  4. x ∈ ûN[x = u]
  5. y ∈ ûN[x = u]
  6. x =y → N[x = y]
Now, I am all in favor of quantified modal logic, but I find (6) hopelessly paradoxical. I would like to take a modified Fregean line, according to which in modal contexts we quantify over senses. But the distinctive feature of my approach (though it may be commonplace) is the idea that the so-called 'transparent' cases, where prima facie a term occurs 'non-obliquely' or in 'purely referential position,' are actually special cases of 'oblique' contexts. In this respect they are like
That Tom is tall is true
where the 'truth move' takes us from an oblique context to the direct context
Tom is tall.
I shall use 'IND', 'PROP' and 'ATT', respectively, for individual, propositional and attributive senses. I shall use brackets instead of my usual dot quotes. Thus, leaving aside the niceities,
'[a]' stands for a certain IND
'[p]' stands for a certain PROP
'[f]' stands for a certain ATT.
Now the modal statement in sensu composito
N[9 > 4]
entails
(∃IND) N[IND > 4]
also
(∃INDi) (∃INDj) N[INDi > INDj]
and
(∃PROP) N[PROP].
What of the modal statement represented by Quine (Word and Object, p. 199) as
x[x > 4] is necessary of 9
On my view it should be represented as follows:
(∃IND) [IND = 9] is true · N[IND > 4]
Notice that whereas Quine takes the '9' in his formula to have a referential use, I take it to have an oblique use in which, however, it is governed by the degenerate modality 'true.'

Now since we have

The number of planets (NP) = 9
and hence
[NP = 9] is true
we can derive
(∃IND) [IND = NP] is true · N[IND > 4]
which, in Quine's paraphrase, becomes
x [x > 4] is necessary of the number of planets
and generates puzzlement. In my 'canonical notation', however, the statement is harmlessly true.

Let me use this apparatus to comment on the classical proof of

x = y → N[x = y]
an unperspicuous formulation of the first premise might be:
  1. x = y → (x is necessarily equal to x ↔ y is necessarily equal to x)
If 'x is necessarily equal to x' is created as a case of the form 'Fx' the proof moves along. Suppose, however, we try, with Quine
  1. x = y → (z[x = z] is necessary of x ↔ z[x = z] is necessary of y.
This, also, would carry one along, if
z[x = z] is necessary of y
is taken to authorize
N[x = y]
But (as I see it) perspicuously formulated the argument becomes:
  1. x = y → ((∃IND) [IND = x] is true · N[x = IND] ↔ (∃IND) [IND = y] is true · N[x = IND])
  2. x = y (hyp)
  3. (∃IND) [IND = x] is true · N[x = IND] ↔ (∃IND) [IND = y] is true · N[x = IND]
  4. N[x =x]
  5. (∃IND) [IND = x] is true · N[x = IND]
  6. (∃IND) [IND = y] is true · N[x = IND]
  7. x = y → ((∃IND) [IND = y] is true · N[x = IND])
The argument is valid, but the conclusion., although it can be paraphrased as
The property of being necessarily equal to x belongs to y
or
The property of being equal to x is necessary of y
is unparadoxical.

One final point in another, though not unrelated, area. Quine thinks that if one quantifies over predicates, one is (purporting) to mention attributes. Thus, in effect, he paraphrases

(f) fa ↔ fb
as
For all attributes, f-ness, a has f-ness ↔ b has f-ness.
I regard this as absurd.

Thus I do not believe that the RHS of

x ∈ y (fy) ↔ (∃g) (z) fz ↔ gz · gx
"commits one to attributes" in the sense of mentioning attributes. Thus if, in a sense, it enables me to generate extensions out of intensions, it does not involve a simple hypostatization of intensions. Nor do I regard 'û(fu)' as a singular term. Quine reads it, roughly, as 'the class of objects, x, such that fx.' I prefer to read it as 'an f, and treat it as a predicate with '∈' being, in effect, the copula 'is.'

I hope all this isn't just confused.

As ever,

Wilfrid Seliars

WS/esr


Transcribed into hypertext by Andrew Chrucky, August 14, 2004.