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Re: [ontolog-forum] borrowing terminology

To: Christopher Menzel <cmenzel@xxxxxxxx>
Cc: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 9 Mar 2007 10:41:10 -0600
Message-id: <p06230904c2173e14b535@[]>
>On Mar 8, 2007, at 5:01 PM, Pat Hayes wrote:
>>  ...The notion of
>>  'completeness' of a logic or inference machine is
>>  well-established in logic, used in virtually all
>>  textbooks, and has a clear, simple definition. It
>>  means that if X is a logical theorem then a proof
>>  of X exists, so that a complete inference engine
>>  is one which will, in principle, find a proof of
>>  X.
>Just a quick clarification to Pat's lucid post:  Pat is using 
>somewhat nonstandard terminology here, as the term "logical theorem" 
>is usually used to refer to the provable sentences of a logical 
>system.  So understood, that would render Pat's definition trivial 
>(which is obviously not what he had in mind).    (01)

I say at least three trivial things before 
breakfast, to get my brain into gear for the day.    (02)

>  The more common term 
>in the definition of completeness is "logical truth", that is, a 
>sentence that is true simply in virtue of the meanings of its 
>component logical operators (or in logician-speak, true under all 
>interpretations of its nonlogical vocabulary).    (03)

Yes, sorry I was careless. Better still, I should 
have said, if X is entailed then...,but in this 
forum I was worried that we might get into a long 
discussion of what 'entail' means :-)    (04)

>Also, the completeness theorem is often stated in a somewhat more 
>general form whose relevance to formal ontology is a bit clearer:  A 
>logical system (or inference machine) is complete if, for any set of 
>sentences S (notably, the axioms of an ontology), if S logically 
>implies X (that is, if X is true under any interpretation that makes 
>all the members of S true), then there is a proof of X from S.    (05)

Better, yes. Thanks.    (06)

Pat    (07)

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