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Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lind

To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
Cc: cg@xxxxxxxxxxxxx, 'Pavlos Peppas' <pavlos@xxxxxxxxxx>
From: "Rich Cooper" <rich@xxxxxxxxxxxxxxxxxxxxxx>
Date: Fri, 2 Jan 2015 08:13:10 -0800
Message-id: <!&!AAAAAAAAAAAYAAAAAAAAAAb3x6NyrzVKo6ReWvn+7BjCgAAAEAAAAGwffExMV6RNpd0AaAXuVL8BAAAAAA==@xxxxxxxxxxxxxxxxxxxxxx>

Dear John,

 

That belief revision article is confusing when talking about “epistemic value” as a measure of “entrenchment”. 

 

Quoting that section from that Plato URL:

http://plato.stanford.edu/entries/logic-belief-revision/#His

 

“This was the basic idea behind Peter Gärdenfors's proposal that contraction of beliefs should be ruled by a binary relation, epistemic entrenchment. (Gärdenfors 1988, Gärdenfors and Makinson 1988) To say of two elements p and q of the belief set that “q is more entrenched than p” means that q is more useful in inquiry or deliberation, or has more “epistemic value” than p. In belief contraction, the beliefs with the lowest entrenchment should be the ones that are most readily given up.

 

The following symbols will be used for epistemic entrenchment:

 

p≤q : p is at most as entrenched as q.

 

p<q : p is less entrenched than q ((p≤q)&¬(q≤p))).

 

p≡q : p and q are equally entrenched ((p≤q)&(q≤p)).

 

Gärdenfors has proposed the following five postulates for epistemic entrenchment, that are often referred to as the standard postulates for entrenchment:

 

 

Transitivity: If p≤q and q≤r, then p≤r.

 

Dominance: If p q, then pq.

 

Conjunctiveness: Either p≤(p&q) or q≤(p&q).

 

Minimality: If the belief set K is consistent, then p K if and only if p≤q for all q.

 

Maximality: If q≤p for all q, then p Cn().

 

It follows from the first three of these postulates that an entrenchment relation satisfies connectivity, i.e. it holds for all p and q that either p≤q or q≤p.

 

An entrenchment relation ≤ gives rise to an operator ÷ of entrenchment-based contraction according to the following definition:

 

q K÷p if and only if q K and either p < (p q) or p Cn().

 

Entrenchment-based contraction has been shown to coincide exactly with transitively relational partial meet contraction.”

 

There is no discussion in this article about what “entrenchment” means in any intuitive sense.  I can read the definition, but how is “value” established to determine what is more or less entrenched than which. 

 

Can you enlighten me a bit about what “entrenchment” does, and how the value is established and calculated for entrenchment please?

 

-Rich

 

Sincerely,

Rich Cooper

EnglishLogicKernel.com

Rich AT EnglishLogicKernel DOT com

9 4 9 \ 5 2 5 - 5 7 1 2

 

-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
Sent: Thursday, January 01, 2015 7:58 PM
To: ontolog-forum@xxxxxxxxxxxxxxxx
Cc: cg@xxxxxxxxxxxxx; Pavlos Peppas
Subject: Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lindenbaum

 

On 1/1/2015 2:07 PM, David Whitten wrote:

> what are the AGM operations?  Do I know them by another name?

 

They're named after the three authors of the classic paper:

 

Alchourrón, C.E., P. Gärdenfors, and D. Makinson, 1985, “On the Logic

of Theory Change: Partial Meet Contraction and Revision Functions”,

Journal of Symbolic Logic, 50: 510–530.

 

The two basic operators determine walks through the lattice of theories.

 

  - Contraction:  Removing an axiom (or belief) from a theory.  This

    moves up the lattice to a more general theory.  This is always safe

    because it can never create an inconsistency.

 

  - Expansion:  Adding an axiom (or belief).  This moves down the

    lattice.  If the new axiom is inconsistent with the others, it

    causes a drop to the inconsistent theory (AKA the absurd theory)

    at the bottom of the lattice.

 

I prefer to add a third operator:

 

  - Relabeling:  Systematically renaming one or more names (of functions,

    relations, or constants) to names that are otherwise unused.  This

    jumps to a theory in another branch of the lattice that is isomorphic

    to the original.  Logicians usually ignore it because it doesn't add

    anything new, but it is very useful for metaphors and analogies.  It

    is also guaranteed to preserve consistency.

 

Other operations are defined in terms of these three.

 

  - Revision:  Replacing one or more axioms.  This is a sideways move,

    equivalent to contraction followed by expansion.

 

  - Consolidation.  Restoring consistency to a theory that has become

    inconsistent because of some unsafe expansion.  This process usually

    involves some contraction and/or relabeling and possibly expansion.

 

  - Merging.  Expanding one theory by adding all the axioms of another

    theory.  This theory is a common specialization of both theories

    that were merged.  If the starting theories were inconsistent,

    the merger drops to the absurd theory at the bottom.  In that case,

    consolidation is required.

 

For an intro to the axioms of belief (or theory) revision and related

issues, see http://plato.stanford.edu/entries/logic-belief-revision/

 

Following is a 42-page survey of the field by Pavlos Peppas.  He gives

a thorough survey with applications and 111 references:

http://pavlos.bma.upatras.gr/paperFs/8.pdf

 

See his web site for more recent papers about belief revision and

applications:  http://pavlos.bma.upatras.gr/

 

John

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