Second-order 'indescirnibles' and "Pegasus = Pegasus" ('Vacuous Names', p. 133)

By JLS for the GC

This may also relate to Jones's point in 'Identity and predication'. Grice is interested in formula where 'a' (or 'a' and 'b' -- e.g. Bellerophon and Pegasus) are NOT dominant. In the object-language, this is transparent enough. On p. 133 he cares to provide

a second-order

formula which will

generate the non-dominant formula of the object-language.

Specifically:




(F3)(F1a3 )(2 F1B4).

------

I.e. for all property F(3), a has property F if and only if a has property F.

As a matter of fact, Grice has 'b' above, because he is wanting to examine the consequences of his system for things like

---- "Should Bellephoron be identical with Pegasus?" (Both names are vacuous).

But if we replace, and we have, in the apodosis of the 'iff' formula above, not 'b', as Grice has it, but 'a' repeated, we have a case that would relate.

It would be the formula, in the second order, that would generate:

Pegasus = Pegasus

in the first-order (rather than object-language. It is unnecessary to think of an order as a level of a language at this point, and indeed confusing and possibly wrong to do so).

So, we want to say:

Pegasus = Pegasus.

And this we look in a sot of 'equivalent' formula in the second-order.

For any property (or feature, F) that a (Pegasus) has, Pegasus has it iff Pegasus has it.

----


Grice notes the distinction between the horseshoe and the 'iff' at this point:

If = is defined, in the second-order, in terms of the horseshoe:

"any affirmative identity-formula involving

a vacuous individual constant will be false"


He does not like that.

So he is suggesting that SOME formulae

involving a vacuous individual constant, such as

'Pegasus' will be true. E.g. Pegasus is identical with the horse imagined by the Greek to be winged and being ridden by Bellerophon.

----

Grice is suggesting that since Pegasus does not exist,


Pegasus = Pegasus

is true:

"the sentence 'Pegasus is identical

with Pegasus' is TRUE because

Pegasus does NOT exist" (p. 133).



--- This is Kripke with a vengeance, as when he says, "The Queen of England is bald" holds for counteparts where 'the Queen of England' refers. Or something.


---

Grice is wanting to call

Pegasus = Pegasus

a 'weak' identity formula. Where 'weak' is a rather weak sense. Not to be confused with 'weak' without 'existential commitment', I would think. It's rather 'weak' BEYOND existential commitment?

---

Grice writes of the first extension Q' of Q (he'll later have a second extension Q") to

'contain such formula as [a2=3[a2]] ('weak' identity formulae)."

----


Grice points to the ATOMIC character of

a = a

Pegasus = Pegasus.

It contains no 'device' of a truth-functional kind. He seems to regard ~p as a molecular formula.

Thus he writes:

"To allow individual constants to
be NON-dominant in a formula
which is NOT molecular will be a ...
departure from the practice ... adopted
in Q".

He adds:

"In view of the possibility of

defining '=' in a SECOND-ORDER

calculus ... one may perhaps

regard this departure as justified."



----




So, I think the kernel is in the INTRODUCTION rule for '=' as defined on p. 134. I will use Grice's precise subscript formulation here. He is using omega as a variable for a variable.

He is having Lambda as anything, I think

He has

Lambda /- (i.e. entails)

(Aomega-n+m wn-l =n wn-l

what he calls "a weak identity law".

In better symbols:


Λ ⊢ ∀ω_n+mω_n-l=_nω_n-l

Reading:

Note that n+m is the highest subscript, which attaches to the universal quantifier which is the quantifier symbol, the inverted A, followed by the variable, omega.

Qua typical intruduction rule, the main symbol, =, is given a given subscript, n.

Given that n gets n, Grice needs to note that the subscript for the variable for which '=' holds has a lower subscript, which he formulates as 'n - l', where 'l' is the letter, not the number. In this way, it is a 'weak' identity law, since the variable is by definition NOT made dominant with respect to the identity-sign.

Or something.


Incidentally, the ELIMINATION of the = sign, as it stands, is sill open to J's type of counterexample, where Watson smokes a pipe (since Watson is identical with Sherlock Holmes).

For the elimination of the = sign, Grice holds, holds btween

phi entailing another formula, phi', where 'b' replaces 'a', such that 'a' and 'b' are said to be held in the weak notion of 'identity'.

So, if Sherlock Homes = Watson, and Sherlock Holmes smoke a pipe, Watson smoked a pipe.

But Grice is aware of that, when he adds that we still have not introduced 'it is believed that...':

"To develop a representation of an

INTERESTING weak notion of

identity, one such that

Pegasus will be identical with

Pegasus but NOT with Bellerophon,

... one would need a system

within which such psychological

notions as 'it is believed that'

are represented", which is what

his final section is meant to provide for.

Or something.