[FOM] Odd Thought About Identity

[Paul Hollander]

    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