This is another one:
A +′ p, the (non-closing) expansion of A by p is the set A∪{p}.
The symbol +’ is the least elegant symbol, where everyone else uses * for closure, + for no closure. Or is this a trick philosophers are fond of?.
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Rich Cooper
Sent: Saturday, January 03, 2015 9:02 AM
To: '[ontolog-forum] '
Cc: cg@xxxxxxxxxxxxx; 'Pavlos Peppas'
Subject: Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lindenbaum
John,
In the Plato paper on belief systems, it is written:
Abe has the basic beliefs p and q, whereas Bob has the basic beliefs p and p ↔ q. Thus, their beliefs (on the belief set level) with respect to p and q are the same.
The symbol “↔“ looks like “is equivalent to”, but it is very messily nonstandard where the usual “=” should be, or even the three hyphen equivalence symbol usually read as “defined as”. But am I missing something symbolic in that interpretation? Why would the author generate idiosyncratic symbology to explain something as normal as equivalence, if that is really what he meant by it.
But it’s a good paper anyway.
-Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Rich Cooper
Sent: Friday, January 02, 2015 10:41 AM
To: '[ontolog-forum] '
Cc: cg@xxxxxxxxxxxxx; 'Pavlos Peppas'
Subject: Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lindenbaum
Thanks John, I'm still reading the Plato paper, but when I finish it, I will read those two links as well.
But for now, you wrote:
According to Dunn's semantics for modal logic (which I recommend),
that two-way distinction can be used to define the modal operators.
Any statement implied by the laws (or by the T-Box) is defined to
be necessarily true. Any statement that is consistent with the
laws (or T-Box) is defined to be possible.
You didn’t mention “any statement that is inconsistent with the laws is …” or otherwise dispose of that remaining case. To look for inconsistencies within a system could be a useful capability. Could you complete the sentence 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: Friday, January 02, 2015 9:54 AM
To: ontolog-forum@xxxxxxxxxxxxxxxx
Cc: cg@xxxxxxxxxxxxx; Pavlos Peppas
Subject: Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lindenbaum
On 1/2/2015 11:13 AM, Rich Cooper wrote:
> That belief revision article is confusing when talking about
> “epistemic value” as a measure of “entrenchment”.
Any document that uses the word 'epistemic' is likely to be confusing.
That comment is not a slur on the people who use the word 'epistemic'.
It's just an observation that nobody has ever been able to state a
precise definition of the words 'knowledge' and 'belief' that is
consistent with the way ordinary people use those words.
However, the word 'entrenchment' as used in discussions of belief
or theory revision can be defined precisely. It just means that
you can specify a partial ordering of certain statements.
The simplest ordering assumes two kinds of statements. For example,
they may be called laws and facts. The laws are more "entrenched"
than the facts. When you're revising a theory, you would preserve
the laws and revise the facts.
Another pair of terms is T-Box (terminology) vs. A-Box (assertions).
The T-Box (AKA ontology) is more entrenched than the A-Box, and
any revisions should be made to the A-Box rather than the T-Box.
According to Dunn's semantics for modal logic (which I recommend),
that two-way distinction can be used to define the modal operators.
Any statement implied by the laws (or by the T-Box) is defined to
be necessarily true. Any statement that is consistent with the
laws (or T-Box) is defined to be possible.
In the following articles, I generalized Dunn's semantics to allow
a partial ordering of the laws by degree of entrenchment:
http://www.jfsowa.com/pubs/laws.htm
Laws, facts, and contexts
http://www.jfsowa.com/pubs/worlds.pdf
Worlds, models, and descriptions
For example, the most entrenched laws would be "logically true"
(i.e., valid or true in all models). Next would be physically
true according to known laws of physics. Next would be the laws
of various authorities, such as the Bible, the US Constitution,
your mommy, etc. -- ordered according to your personal concerns.
The meanings of the words 'necessary' or 'must' depend on which
laws are being considered in the current context.
Summary: The reasons why I prefer Dunn's semantics and Lindenbaum
lattices are (a) they're consistent with the confusing publications
philosophers produce, (b) they're compatible with the implementations
used in AI, and (c) they're much, much easier to explain to students
and ordinary human beings.
John
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J