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Re: [ontolog-forum] web-syllogism-and-worldview

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Fri, 17 Apr 2009 14:36:13 -0500
Message-id: <55298E25-6AF6-4DD3-984B-DCE45D1D595F@xxxxxxxx>
> CM > I am not understanding some of your terminology here.  As  
> standardly
>> (and pretty much universally) defined in logic, syllogisms are
>> arguments with two premises and a conclusion satisfying a certain
>> general form in which the notion of recursion plays no role whatever
> Sorry Chris, I should have referenced the following threads better:
> AW >> The problem with the "syllogisms will never work" argument
>>> is that, if you allow them to be recursive, they have Turing
>>> machine power.  That means that they can compute anything that
>>> can be computed.
> JFS > Every major programming language (e.g., FORTRAN, COBOL, LISP,
>> ALGOL, PL/I, C, C++, C#, Ada, Java, Python, Ruby, PHP, etc.)
>> has the following two properties:
>> 1. They have the power of a Turing machine.
>> 2. It's undecidable whether an arbitrary program written in
>>   any of those languages will terminate.
> My response was to the argument that if we allow syllogisms to be
> recursive, they will become useless.  JFS's point, I believe, was that
> this since programming language constructs are based on turing
> machines, which are recursive, there's no issue here.    (01)

If I can restate: Your claim was that if, in a logical framework  
capable of expressing syllogisms, syllogisms are allowed to be  
recursive, then the resulting framework will be useless because it  
will have the expressive power of a (universal) Turing machine.   
John's response is that all programming languages have that power and  
are, obviously, useful.  John's response therefore seems to be a  
counterexample to your claim.    (02)

> My point was, apologies for not stating it clearly, this would rid  
> of completeness, for example in FOL.    (03)

I'm afraid I still do not understand.  It is just a fact that FOL is  
complete.  There's nothing you can do such that it would no longer be  
a fact.    (04)

> CM> Second, what do you mean by a "proper recursive function"?  A
>> recursive function is simply a mathematical function from (n-tuples
>> of) natural numbers to natural numbers..... A "stop condition" (if
>> I'm understanding you) is a programming construct that has nothing to
>> do with functions per se.
> I meant it as a purely programming construct. Some of the original
> arguments about syllogisms, I felt, were not always applicable to
> implementation. I mentioned that ontologies are practical things for
> us to use. The same goes for programming languages.  So if we can get
> around the unsatifiability of turing machines in programming
> languages, why not ontologies.    (05)

Again a terminological issue.  What do you mean by "the  
unsatisfiability of Turing machines in programming languages"?  Every  
(standard) programming language is capable of expressing exactly the  
functions calculated by a Turing machine.    (06)

> CM>For this appears to mean that,
>> for any recursive function, a program that calculates that function
>> has to terminate when executed.  And that, of course, is false.
> You're correct.  I wasn't referring to the a program that calculates
> any recursive function, satisfiable or not.  Here, a "proper recursive
> function" is one that does terminates, and the calling program will
> terminate, or not terminate for an "improper function" accordingly.    (07)

Well, I think your terminology here will lead you to be  
misunderstood.  The idea of a *function* terminating or not is a  
category mistake; *programs* terminate (on this or that input).  If  
program P expresses a function F, then P terminates on input  
representing the number n if and only if F is *defined* on n; that is,  
if and only if F(n) has a value.  A recursive function that is defined  
on every natural number is said to be *total*; one that is defined  
only on a subset of the natural numbers is said to be *partial*.  I  
think your "proper" and "improper" terminology is problematic.    (08)

-chris    (09)

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