On 10/23/2015 5:25 PM, Thomas Johnston wrote:
> But I don't know enough about accessibility relational among S1 - S5
> possible worlds, and I would like to learn a bit more before I dismiss
> Kripke. (01)
Dunn did not dismiss Kripke. In K's semantics, a possible
world is an undefined element of a set of worlds. K made no
assumptions about the worlds or any structure in them or on them.
K's accessibility relation among worlds is the *only* source of
any information about the worlds, and it is assumed as primitive. (02)
Kripke's major contribution was to show how various constraints
on the accessibility relation are related to the S1 -- S5 axioms.
Since K did no assume any structure on the worlds, he only
developed the axioms for modal propositional logic. (03)
Hintikka had earlier proposed a version that represented each world
represented by a "model set" that consisted of all the propositions
that are true of that world. He sets of propositions could be stated
in propositional logic or in full FOL. Hintikka's name for K's
accessibility relation was "alternativity relation". (04)
But Hintikka did not show how constraints on that relation were
related to the S1 - S5 axioms. That was K's contribution. (05)
What Dunn contributed was a combination of Hintikka + Kripke
plus one important innovation: For each Kripke world or each
Hintikka model set, Dunn distinguished a privileged subset of
the propositions called *laws*. That implies, (06)
1. The accessibility (or alternativity) relation is no longer
primitive: it is derivable from the choice of laws for
each world. But the set of worlds and the accessibility
relation among them is isomorphic to Kripke's version.
Every one of K's theorems is valid in Dunn's version. (07)
2. If every world has exactly the same laws, you get S5. (08)
3. If the accessibility relation from w1 to w2 implies that
every law in w1 remains true in w2 (but might not be a law
in w2), you get S4 (which is more realistic for most
applications -- it enables you to update the laws). (09)
4. Dunn's version also allows further innovations (which I
discuss in my worlds.pdf article). For example, you can
mix multiple modalities by letting some laws be alethic,
some deontic, some epistemic, etc. (010)
5. With option #4, each subset of laws gives you a different
accessibility relation: Ralethic(w1,w2), Rdeontic(w1,w2),
Repistemic(w1,w2). If you want to obey all the constraints,
you get a combined accessibility relation R:
Ralelthic(w1,w2) & Rdeontic(w1,w2) & Repistemic(w1,w2). (011)
John (012)
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