RE: SUO: RE: Re: Peirce's MS 514
When thinking about notions of inference and entailment (and when reading
various authors), it's also important to keep in mind considerations about:
1) Inference (or entailment) from infinite sets of assertions vs. that from
finite sets.
2) Inference (or entailment) from sets of assertions which have free
variables vs. that from sets of assertions without free variables.
Logical systems differ in these regards. Results about entailment and
provability will vary with them, in general.
Jay
-----Original Message-----
From: pat hayes [mailto:phayes@ai.uwf.edu]
Sent: Thursday, March 22, 2001 10:43 AM
To: West, Matthew MR SSI-GREA-UK
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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