RE: SUO: RE: Re: Peirce's MS 514

["West, Matthew MR SSI-GREA-UK" <Matthew.R.West@is.shell.com>]

Dear John,

Thank you for this.

So the reason I have not been able to distinguish inference and entailment
is that in FOL there is none. Correct?

Regards  
      Matthew
===============================================================
Matthew West                    http://www.matthew-west.org.uk/

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> -----Original Message-----
> From: John F. Sowa [mailto:sowa@bestweb.net]
> Sent: 22 March 2001 01:40
> To: West, Matthew MR SSI-GREA-UK; John F. Sowa; James L Piat;
> arisbe@stderr.org; jawbrey@oakland.edu; PORT-L@LISTSERV.IUPUI.EDU;
> arisbe@dns1.stderr.org; semiocom@listbot.com; cg@cs.uah.edu;
> standard-upper-ontology@ieee.org
> Subject: Re: SUO: RE: Re: Peirce's MS 514
> 
> 
> Matthew West wrote:
> 
> >I'm glad you posted this, because I have been puzzled by the 
> >difference between inference (provability) and entailment 
> >for some time.
> >
> >Below you give the difference by:
> >
> >>  1. Provability:  p is said to be _provable_ from A in some
> >>     logical system L iff there are rules of inference that
> >>     allow p to be derived by performing some operations on
> >>     the statements of A to generate the statement p.
> >> 
> >>  2. Entailment:  p is said to be _semantically entailed_ by
> >>     A iff in any state of affairs (or universe of discourse,
> >>     or possible world) for which all the statements of A are
> >>     true, the statement p is also true.
> >
> >Well I can see some difference, and that entailment should be
> >stronger, but I don't really understand the difference.
> 
> The basic point is that semantic entailment depends on the
> subject matter, but provability depends on your choice of
> notation for logic and the rules of inference associated with
> it.  Provability can be defined in terms of the syntax of the
> notation, but entailment depends on the semantics of whatever
> subject is under consideration.
> 
> >Presumably there are cases where something can be proved through
> >inference, but is not entailed. Could you give an example of
> >that please?
> 
> There are two terms that are used to characterize inference
> methods:
> 
>   1. Soundness:  A proof procedure is said to be _sound_
>      iff everything that is provable from some starting
>      set of statements A is also semantically entailed by A.
>      This point is often summarized by saying that a rule of
>      inference is sound, if it preserves truth:  if you start
>      from true statements all the deductions will also be true.
> 
>   2. Completeness:  A proof procedure is said to be _complete_
>      iff every statement that is semantically entailed by a
>      set of statements A is provable from A by that procedure.
>      This point may be summarized by saying that everything
>      that is true by semantic entailment is also provable.
> 
> First-order logic has the very nice property of being both
> sound and complete:  everything that is provable is also true
> by semantic entailment, and everything that is semantically
> entailed is also provable.
> 
> With his famous incompleteness theorem, Kurt Goedel proved that
> higher-order logic is not complete:  there are true statements
> of arithmetic that are not provable.  He gave some examples
> of such statements, including an encoding of a statement of
> the following form into an arithmetical proposition:
> 
>    "This statement is not provable."
> 
> It turns out that statements of this form (in arithmetic)
> can be true, but there is no way to prove them.
> 
> For most purposes, soundness is extremely desirable, since
> one would not like to have a logic whose conclusions are
> unreliable.  However, there are some systems of plausible
> reasoning (called "nonmonotonic logics") for which inference
> rules are allowed that are usually, but not always correct.
> Such logics are "unsound", and their conclusions must be
> used with caution.  Sometimes, it is better to have plausible
> information that is more likely right than wrong; however,
> it is wise to be prepared for possible surprises.
> 
> John
>