Chris, thank you. The explanation below is very helpful in answering my
question regarding predicativity in logics with FO semantics and HO
syntax. (01)
I enjoyed your paper [1] and recommend it highly for anyone who wants to
understand the design goals of CL. (02)
--
Rick (03)
[1] http://cmenzel.org/Papers/Menzel-KRTheWWWAndTheEvolutionOfLogic.pdf (04)
On Fri, 2011-08-19 at 14:56 -0500, Christopher Menzel wrote: (05)
> > In CL, the statement equivalent to Russell's paradox
>
> I'd say: The sentence that *generates* Russell's paradox...
>
> > has a stable truth value: false. No paradox.
>
> Right. In more detail: the chief culprit in Russell's paradox is the
> so-called "naive" comprehension principle that, for any formula φ there
> is a set containing (or a property true of) all the things that are φ.
> In the CL dialect CLIF, this principle is expressed schematically as:
>
> NC (exists (p) (forall (q) (iff (p x) φ))).
>
> In CL, where self-predication is permitted, an instance of NC is:
>
> R (exists (p) (forall (q) (iff (p x) (not q q)))).
>
> A contradiction follows from R immediately. For let r be one of the
> things p said to exist by R:
>
> (1) (forall (q) (iff (r x) (not q q)))).
>
> But CL is type-free, so r itself is among the things being quantified
> over; so by universal instantiation:
>
> (2) (iff (r r) (not r r)),
>
> contradiction. So, as John notes, in CL, sentence R is false; more
> exactly, it is *logically* false; its negation is a theorem of CL.
>
> -chris
>
>
>
>
>
>
>
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