[FOM] explicit variables

[William Tait]

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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