[FOM] explicit variables
William Tait
wwtx at earthlink.net
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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