Why do we still use analogies like "long in the tooth" which harken back to our
horse trading days? (01)
Sent from my iPhone (02)
On Aug 19, 2011, at 6:53 PM, Rick Murphy <rick@xxxxxxxxxxxxxx> wrote: (03)
> Chris, thank you. The explanation below is very helpful in answering my
> question regarding predicativity in logics with FO semantics and HO
> syntax.
>
> I enjoyed your paper [1] and recommend it highly for anyone who wants to
> understand the design goals of CL.
>
> --
> Rick
>
> [1] http://cmenzel.org/Papers/Menzel-KRTheWWWAndTheEvolutionOfLogic.pdf
>
> On Fri, 2011-08-19 at 14:56 -0500, Christopher Menzel wrote:
>
>>> 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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