[FOM] Odd Thought About Identity

Paul Hollander paul at paulhollander.com
Thu May 21 19:33:32 EDT 2009

```    On 5/13/09 4:04 PM, rgheck wrote:
>     However, such an axiomatization is not "schematic" in what I
>     take to be the standard sense of that term, viz, the sense used by Vaught.

I assume Richard is referring to Vaught's 1967 "Axiomatizability by a
Schema," in which surprisingly the word "schematic" does not even
appear, and nowhere in that paper do I find Vaught even defining what he
means by "schema." This observation is confirmed by John Corcoran in
Corcoran, J., 2006, ‘Schemata: the Concept of Schema in the History of
Logic,’ Bulletin of Symbolic Logic, 12: 219–40.

Interestingly, John writes in that paper,

Vaught, in his inﬂuential 1967 paper [49] “Axiomatizability by a
Schema”, deﬁnes the unitary expression ‘is axiomatizable by a
schema’ without deﬁning ‘schema’ at all, even though his remarks
using the word make perfect sense read in the senses in focus in
this paper. However, none of the logicians mentioned, in fact none
that I know of, directly raise the issue of whether a schema is or
is not to be regarded simply as a string of characters. It is worth
explicitly noting that there can be no objective criticism of
identifying a schema with a schema-template — this is a matter of
terminology and nothing more. (p. 222)

I think the fact that this came up for discussion indicates that
ambiguity results from identifying a schema with a schema-template and
failing to take the side condition into account.

Those who accept Alex Blum's side condition:

>     If x=y, then any free occurences of x in a wff F may be replaced
>     by y (if y does not become bound where x was free) and any free
>     occurences of y in the formula may be replaced by x (if x does not
>     become bound where y was free).

will see

(x)(y)(x = y --> Rxy <--> Ryx)

as an instance of

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

but those who accept Richard Heck's side condition, that F(x)

>     is any formula you like, subject to the usual sorts of restrictions, viz, x is free for y.

will regard it as not being an instance of the schema.

The schema-template by itself is ambiguous.

Cheers,

-paul
```