RE: SUO: RE: Re: Peirce's MS 514
The contributions of John S, Pat Hayes, Chris Menzel and
others to this thread have helped clarify a lot of
things that I thought I already understood. A short
footnote to Pat's note--
It is also amazing that there is such a large variety
of useful (not necessarily complete) inference systems,
involving various combinations of axioms,
axiom schemata, and rules of inference. If
one has only axioms, and no rules of inference, the theory
is only the axioms (using theory to mean the body of
statements that can be inferred, not the totality of
what is entailed.) In logic programming, the rules in a
program are usually regarded as axioms (insofar as we're
speaking of rules with only logical intent). In extending
prolog to a certain version of paraconsistent logic
Vladimir Lifschitz proposes instead to regard the
rules as rules of inference. This provides a completely
new perspective on the 'meaning' of the logic programs
in question.
John Velman
pat hayes <phayes@ai.uwf.edu>@ieee.org on 03/22/2001 10:43:28 AM
Please respond to pat hayes <phayes@ai.uwf.edu>
Sent by: owner-standard-upper-ontology@ieee.org
To: "West, Matthew MR SSI-GREA-UK" <Matthew.R.West@is.shell.com>
cc: standard-upper-ontology@ieee.org
Subject: RE: SUO: RE: Re: Peirce's MS 514
>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?
Incorrect. They are different notions which should not be confused
with each other. What is true of many inference systems for FOL is
that A is inferrable from B if and only if B entails A, but even that
doesn't say there is no difference. For example, the way you show
that an A is inferrable from B is by showing that a proof exists with
its premises in B; the way you show that A is entailed is by showing
that it is true in all interpretations which satisfy B.
Of all the many logical formalisms that have been devised, relatively
few have a completeness theorem proven for them, and many are known
to not be complete. Even when talking about FOL, some proposed set of
inference rules might not be complete. Having entailment and
inference line up in this precise way is a rare and delicate quality
of a logic and its inference system.
Pat Hayes
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