Re: An instance of Russell's paradox?

george wrote:
> A.T. wrote:
> > Hello,
> > I have been studying Prolog for a while,
> > and stumbled across several
> > difficulties which I seem to be unable to
> > overcome on my own. Since I
> > am doing it all solely for the purpose
> > of understanding FOL
>
> Prolog is seriously not a good choice for that.
> "Negation-as-failure", "operational semantics",
> and a host of further caveats are relevant.
> On the plus side, if you actually understand
> all of these, you will understand more than
> "solely" FOL.
>
> > (and Russell's PM straight after that)
>
> Russell's PM is NOT the same thing as classical
> FOL and there IS A REASON why modern treatments
> DON'T look like PM! Seriously, unless you have
> to write a paper on the history of science or something,
> PM is simply not worth bothering with.
>
> > I have let myself choose Sci.Logic as
> > the addressee of my questions.
>
> I pity the f__....
>
> > 1. Can all algorithms be enumerated,
>
> Yes, but that depends on a definition of "algorithm"
> due to Church and Turing, NOT Prolog or PM. It also
> depends on a broad natural definition of "can" (this
> field invites a narrower technical definition).
>
> > or will that be another
> > formulation of Russell's paradox
>
> only if you can formulate the enumeration algorithmically.
> But precisely because that assumption (that there is an
> algorithm for performing the enumeration) leads to
> contradiction, you canNOT "effectively" enumerate the
> algorithms (even though it must in some sense be abstractly
> possible to enumerate them, since they are all finite and
> there are therefore only a denumerable number of them).
> In Turing-speak, they are denumerable but not "recursively"
> enumerable.
>
> > (with the list of all algorithms being
> > another way of expressing the set of all sets)?
>
> Sort of, yes. If you could list all the algorithms
> by using an algorithm, then, yes, you would have a
> Russellian paradox. But the class of all sets is
> not a set, and the list of algorithms (despite the
> fact that it exists) is not the output of one of
> those algorithms.
>
> > 2. Can all predicates be generalized as having
> > infinite arity, e.g. a
> > predicate "philosopher(socrates)." as being
> > "philosopher(socrates, ..., ...)."
>
> No. The classical paradigm admits only finite
> arities. Given that there is no finite bound
> upon these finite arities, it might
> seem that allowing denumerable arities wouldn't
> hurt anything, but it does. Even in some infinitary
> logics, arities are still restricted to being finite.

I can only excuse myself by saying that I am a proponent of Alan
Turing's views on intelligence, views that I would _necessarily neglect
unsolicited:

http://plato.stanford.edu/entries/mind-identity/

Views which if _understood, make everything else empty twaddle. And
only that.

Thank You very much indeed and I truly am very sorry.
Tom

.

Relevant Pages

  • Re: Modeling for Enumerative Coding and QI (Entropy Pump)
    ... since enumeration subtasks of EP ... part of the overall EP pattern of dividing the complete coding task by ... arbitrary algorithms, in contrast to AC+PM where 'model' is synonymous ... algorithm - 'calculating probabilities for the next symbol based on ...
    (comp.compression)
  • Re: An instance of Russells paradox?
    ... Prolog is seriously not a good choice for that. ... Can all algorithms be enumerated, ... only if you can formulate the enumeration algorithmically. ... upon these finite arities, it might ...
    (sci.logic)
  • Re: An instance of Russells paradox?
    ... Can all algorithms be enumerated, ... >Diagonalize out of your enumeration to prove that not all functions are ... list of total functions will be another total function which is ...
    (sci.logic)
  • Re: An instance of Russells paradox?
    ... Can all algorithms be enumerated, ... >>language. ... >>Diagonalize out of your enumeration to prove that not all functions are ... and there is no diagonalisation problem with that. ...
    (sci.logic)
  • Re: Matrices implementations
    ... I think you are perfectly right about the need for C/C++ algorithms. ... it is in my expectation that Prolog systems could advance ... is "extract all substrings from a string, ... extract_substring(String, SubString):- ...
    (comp.lang.prolog)