# [FOM] explicit variables

Thu Jun 1 11:39:09 EDT 2006

```There is this difference between \$F\$ and \$F(x)\$: In compositional
semantics, \$F\$ denotes a function from \$D\$ to \$D\$, where \$D\$ is the
domain of individuals. \$F(x)\$ denotes a function from \$(V-->D)\$ into
\$D\$, where \$V\$ is some set of variables containing \$x\$.

Of course, variables can in principle be eliminated entirely. In the
case of first-order logic, Weyl did this in *Das Kontinuum*. In the
case of abstraction terms, Schoenfinkel did it in "On the building-
blocks of mathematical logic" A general treatment, building on
Schoenfinkel's, which in particular covers predicate logic of finite
type, is in my paper "Variable-free formalization of the Curry-Howard
type theory." (On my website.)

Regards,

Bill Tait

On May 31, 2006, at 2:12 AM, Thomas Forster wrote:

>
>  On Mon, 29  May 2006, Arnon Avron wrote:
>
>>
>> Sorry for the stupidity, but what is wrong with or missing from
>> the usual Tarskian semantics for formulas with free variables?
>>
>>
>> Arnon Avron
>
>
>  Nothing at all, but that wasn't the implication of my question!
>
>  What i was wondering was: is there anywhere in the logical literature
>  any discussion of the possible significance of the difference between
>  writing
>
> 	\$F\$
>
>  and writing
>
>  	\$F(x)\$  	(to signify that `\$x\$' is free in \$F\$)
>
>  in - for example - presentations of the rule of UG (and suchlike).
>
>           tf
>
>   URL: www.dpmms.cam.ac.uk/~tf   Tel: +44-1223-337981
>   (U Cambridge); +44-7887-701-562 (mobile)
>
>
>
>
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```