SUO: RE: Re: Peirce's MS 514

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

Dear John,

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.

Presumably there are cases where something can be proved through
inference, but is not entailed. Could you give an example of
that please?

Regards  
      Matthew
===============================================================
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> -----Original Message-----
> From: John F. Sowa [mailto:sowa@bestweb.net]
> Sent: 21 March 2001 15:13
> To: James L Piat; arisbe@stderr.org; jawbrey@oakland.edu;
> PORT-L@LISTSERV.IUPUI.EDU; arisbe@dns1.stderr.org; 
> semiocom@listbot.com;
> sowa@bestweb.net
> Cc: cg@cs.uah.edu; standard-upper-ontology@ieee.org
> Subject: SUO: Re: Peirce's MS 514
> 
> 
> 
> James Piat wrote:
> 
> >....  But what else?  My guess is that the division between
> >the two [axioms and rules of inference]
> >also reflects some fundamental unrecognized meta-axioms and or
> >meta-rules of inference upon which the proposed logic rests. 
> >Unrecognized assumptions that may come back to haunt the project.
> 
> Yes, indeed.  Logicians usually make a distinction between
> "provability" and "semantic entailment".  Suppose you have
> a set of statements A and another statement p.  Then you can
> define the two notions as follows:
> 
>  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.
> 
> This distinction is as old as Aristotle, but it was most
> clearly formulated by the medieval Scholastics.  In particular,
> William of Ockham's _Summa Totius Logicae_ is a very clear
> statement of this distinction.  Aristotle, Ockham, and everybody
> who read them, including Peirce, took semantic entailment as
> the more fundamental notion and used it to justify their
> rules of inference.
> 
> The situation was muddied at the end of the 19th century by
> two prominent logicians who didn't do their homework:  Gottlob
> Frege and Bertrand Russell.  Frege was an independent thinker,
> who happened to invent the first complete system of inference
> for first-order logic.  But he hadn't read Ockham or the other
> logicians, and his approach was highly idiosyncratic.
> 
> Instead of taking semantic entailment as the fundamental notion,
> Frege took provability as fundamental and even claimed that
> "truth" was too vague a notion to be used as a foundation
> for logic.  Bertrand Russell fell hook, line, and sinker for
> Frege's point of view, which he preached to the masses.
> 
> In 1935-36, Alfred Tarski introduced (or actually reintroduced)
> semantic entailment as the more fundamental notion.  Tarski's
> approach, which is usually called "model-theoretic semantics"
> is equivalent to what Ockham and Peirce used as the foundation
> for logic.  Tarski, in fact, did quote Aristotle as a precedent
> for his approach, because he was fighting an uphill battle
> against the Frege-Russell propaganda, which had taken firm root
> in the early 20th century.
> 
> For an indication of how hard Tarski had to argue for his
> position, see the "polemical remarks" in a paper that Tarski
> wrote in 1944 to support his position:
> 
>    http://www.bestweb.net/~sowa/misc/tarski.htm
> 
> I also summarize some of those points in my commentary:
> 
>    http://www.bestweb.net/~sowa/peirce/ms514.htm 
> 
> JP>.... I'm not so sure rules of inference (for example) are
> >operations.  Maybe they are simply definitions of what 
> constitute things
> >--the properties of things.  Something like that.  Does any 
> of this make
> >sense?  I guess I'm just questioning the notion that logic 
> is (or should
> >be) an account of how the transformation of things must 
> necessarily (in
> >all possible worlds) be.  Maybe logic should be an account 
> of what the
> >notions of "transformation" and "things" presuppose.
> 
> Aristotle, Ockham, Peirce, and Tarski would be sympathetic
> to your concerns.  They developed their systems of logic
> for the purpose of characterizing the more fundamental notion
> of semantic entailment, which is based on the way the world
> itself happens to be.
> 
> Frege and Russell, who had not read the literature, claimed
> that logic was more fundamental and that no independent notion
> of truth or semantic entailment was necessary.
> 
> Today, logicians have gone back to the Aristotelian-Scholastic
> point of view, but they usually credit it to Tarski, who waged
> the hardest battles against the mistakes of Frege and Russell.
> 
> John Sowa
>