On Sep 12, 2008, at 1:36 PM, John F. Sowa wrote:

...PH> Indeed. And of course, change occurs in the real world and

is described by both 3-d and 4-d ontologies, but in different

ways. This is so obvious that nobody felt any need to say it.

But I certainly agree that nobody in AI mentions Dunn's work, primarily because they never heard of it. I happened to read his paper in a collection of articles in the IBM library, but Dunn himself didn't publish anything further about it or promote it.

PH> It is based on - it actually uses - inferences made in

a theory which refers directly to the worlds.

More precisely, it refers to elements of a set of undefined entities, which Kripke called 'possible worlds'. Nothing in Kripke semantics would change in the slightest if you replaced the set of worlds with a set of integers (or real numbers, if you want an uncountable set).

Well, indeed, because a structure satisfying Kripke's definition whose "worlds" are integers *is* a Kripke model. It is a bit misleading to say that the elements of the set K (as Kripke named it) of "possible worlds" in a given model are "undefined". A better term would be "arbitrary", that is, there are simply no constraints (beyond non-emptiness) on the set K. Kripke is in fact quite clear that he is only speaking *intuitively* when he calls K the set of all "possible worlds". In a Kripke model, any nonempty set will do.

Instead of using the term 'possible world', Hintikka used the term 'model set', which is a set of propositions that describe a world, but he never said much about the nature of those propositions.

That is not true, he was in fact *completely clear* about their nature. They are *formulas* in a given propositional modal language. Model sets are simply maximal, complete sets of sentences.

The important contribution that Dunn made was to distinguish a subset of those propositions called 'laws' and to derive Kripke's accessibility relation from the way the laws vary from world to world.

So Kripke's semantics is a *proper subset* of Dunn's semantics.

This is a misleading sense of "subset". Kripke models are not Dunn models. I take it what you are claiming is that you can *convert* any Kripke model (for a given language L) into a corresponding Dunn model of L in a way that preserves the logical properties of all the sentences of L. In fact, I am not certain that is true. What *is* the case is that every Dunn model can be converted (rather directly) into a Kripke model in a truth/validity/entailment-preserving way. It seems that one should be able to convert any Kripke model into a Dunn model but, off the top of my head, it's not clear to me that one can do so *uniquely*.

There is *nothing* you can do with Kripke semantics that you cannot do in exactly the same way with Dunn's semantics. It is false that Kripke's semantics is more usable than Dunn's.

Nor is Dunn's more usable than Kripke's. The two approaches are equivalent with respect at least to the standard "normal" modal systems T, S4, and S5 in the sense that a sentence is valid under Dunn's semantics if and only if it is valid under Kripke's.

PH> The laws and facts are stated explicitly as sentences in theories using Kripke-style semantics.

False. Kripke only developed his semantics for propositional modal logic.

Exsqueeze me?! I direct you to:

Kripke, S., "Semantical Considerations on Modal Logic," *Acta Philosophical Fennica* **16**, 83-94.

Moreover, in my recollection, just the opposite is the case -- Dunn's original paper was a semantics for propositional modal logic. I've never seen an extension of it to first-order modal logic.

None of the axioms, theorems, and proofs stated with Kripke's version ever used a single predicate that said anything about anything in those worlds.

Well, Dunn doesn't either. In fact, his "worlds" are just ordered pairs of the form <F,L> where F is a mapping of sentences into {0,1} and L is a submapping of F.

Those worlds were undefined points, and the only information about them was the accessibility relation that linked those points in a graph.

You don't get much more from Dunn except a very rarefied notion of "law".

PH> Check out any of the hundreds of papers on planning using a situation-calculus style of representation.

Any paper that states plans that mention anything about the things and events in any of those worlds leaves Kripke far behind. Kripke's semantics cannot handle quantified modal logic.

Oh my goodness, John. Kripke's semantics is the basis for all modern quantified modal logic. Have a look at Section 3 of my article on

Actualism in the

*Stanford Encyclopedia of Philosophy* (soon to be replaced by an extensively revised version).

To say anything about anything in a world, you need to relate facts and laws (i.e., ontologies) about that world. And that is where you need Dunn's semantics.

I just don't see it. First, I'm not sure what it means to "relate facts and laws". In Dunn's semantics, the only relation I see is the subset relation -- every law is a fact. I don't see how writing axioms doesn't capture this just as clearly, if not moreso. As Pat suggests, you can just write axioms of the form "Necessarily A" to express the laws of a world, e.g., "Necessarily, every human being is mortal" or "Necessarily, f = ma"; mere facts can be expressed by nonmodal axioms, e.g., "Socrates is a philosopher".

In fact, that is why so many of them use the axioms for S5, which Lewis and almost everybody else who studied the issues admitted is unrealistic.

In fact, S5 is still the preferred propositional system for modal logic by the vast majority of philosophers of logic/language. And S5 is valid in Lewis's own system, since there is no accessibility relation on worlds (which is equivalent to making the accessibility relation reflexive, transitive, and symmetric).

But for people who never heard of Dunn's semantics, S5 is the only one they can handle because it makes the assumption that all the worlds have exactly the same laws (or ontology).

But this is at best a sociological accident. There is nothing about Kripke's semantics that prevents one from using more modal logics that do not make that assumption.

Like most all of the stuff on your web site, these documents are quite useful and informative -- fortunately, unlike above, you don't seem to level any detailed criticisms at Kripke semantics.

You have been arguing earnestly and, occasionally, stridently for the superiority of Dunn's semantics for as long as I've known you, John, and you should in my view dial it back a bit. The fact/laws distinction is a nice little tweak of Kripke semantics, but that's all that it is, and it can be captured in languages that use Kripke semantics proper just as effectively by means of axioms. Indeed, won't you still need such axioms even if you use Dunn's semantics to make the laws explicit?