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Re: [ontolog-forum] Is modal logic first-order?

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 23 Feb 2007 02:43:01 -0500
Message-id: <45DE9B05.4000305@xxxxxxxxxxx>
Till and Chris,    (01)

By using Dunn's semantics, modal logic can be formalized in
a first-order metalanguage about a first-order object language.
After you do that, you can then collapse the two levels and
define a single-level Tarski-style model for the combined
system.  That collapsed form is less readable than the original,
but it demonstrates that you can map the modalities into a
first-order framework.  See    (02)

    Laws, Facts, and Contexts    (03)

With Dunn's semantics instead of Kripke's semantics, each
world w is replaced by a pair (M,L), where M is the set of
propositions true in w (called facts) and L is the subset
of facts (called laws) that are necessary in w.    (04)

Both versions are formally equivalent, but Dunn's approach
supports very useful generalizations that are difficult or
unwieldy with Kripke semantics:    (05)

  1. The object language (metalevel 0) is purely first-order
     and a theorem prover can ignore the distinction between
     laws and facts at the object level.    (06)

  2. Each modal operator is treated as a metalevel comment
     about provability from the laws (necessity) or consistency
     with the laws (possibility).    (07)

  3. Instead of assuming the accessibility relation as an
     undefined and undefinable primitive, Dunn's approach
     makes accessibility a derived relation from the choice
     of laws.  That can usually be done for principled reasons
     instead of plucking a mysterious relation out of thin air.    (08)

  4. The greatest advantage of Dunn's semantics comes with
     multiple modalities.  Instead of just a two-way partitioning
     of laws and facts, it is possible to have a partial ordering
     of subsets of propositions according to levels of entrenchment.    (09)

  5. For example, logical necessity would be the most entrenched.
     Next would be physical necessity according to well-established
     laws.  Then necessity according to laws that are less certain,
     economic necessity (i.e., building a colony on Mars is
     physically possible, but economically prohibitive), down to
     many levels of deontic modalities according to the laws of
     the US, Maryland, Baltimore, or the rules and regulations
     of various organizations, including the laws of your mommy.    (010)

  6. When you have epistemic logics of knowledge & belief with just
     one agent, Kripke's semantics is workable, but as soon as you
     have multiple distributed agents with different K & B operators
     for each agent, all hell breaks loose.  But with Dunn's semantics
     and the methods of belief revision, each agent can reason with
     its own partial ordering by entrenchment.  Typically, each agent
     would assume greater entrenchment for its own beliefs than for
     the beliefs it believes the other agents to have -- try doing
     that with Kripke semantics.    (011)

And by the way, I'd like to make a comment about "truly first-order 
modal logic like S5".  The fact that there are more papers published
about S5 than any other axiomatization is a very strong argument
*against* Kripke semantics.  Even C. I. Lewis said that S5 was too
strong to be realistic, and in terms of Dunn's semantics, S5 implies
that all possible worlds have exactly the same laws.  There are
many papers that apply S5 to database systems, where the laws are
the DB constraints.  But in such systems, it would be impossible
to update the constraints -- the most unrealistic assumption
imaginable.  People who publish such stuff give aid and comfort
to those who think logic is useless.    (012)

Following is another paper I wrote on this topic:    (013)

    Worlds, Models, and Descriptions    (014)

John    (015)

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