Posts Tagged ‘Quantified modal logic’


Rigid Designation

October 23, 2009

Imagine the following set up. There are two tribes, A and B, who up until now have never met. It turns out that tribe A speaks English as we speak it now. However, tribe B speaks English* – a language much like English except it doesn’t contain the names “Aristotle” or “Plato”, and contains two new names, “Fred” and “Ned”.

Suppose now that these two tribes eventually meet and learn each others language. In particular tribe A and B come to agree that the following holds in the new expanded language: (1) necessarily, if Socrates was a philosopher, Fred was Aristotle and Ned was Plato, and (2) necessarily, if Socrates was never a philosopher, Fred was Plato and Ned was Aristotle.

Now we introduce to both tribes some philosophical vocabulary: we tell them what a possible world is, what it means for a name to designate something at a possible world. Both tribes think they understand the new vocabulary. We tell them a rigid designator is a term that designates the some object at every possible world.

Before meeting tribe B, tribe A will presumably agree with Kripke in saying that “Aristotle” and “Plato” are rigid designators, and after learning tribe B’s language will say that “Fred” and “Ned” are non-rigid (accidental) designators.

However tribe B will, presumably, say exactly the opposite. They’ll say that “Aristotle” is a weird and gruesome name that designates Fred in some worlds and Ned in others. Indeed whether “Aristotle” denotes Fred or Ned depends on whether Socrates is a philosopher or not, and, hence, tribe A are speaking a strange and unnatural language.

Who is speaking the most natural language is not the important question. My question is rather, how do we make sense of the notion of ‘rigid designation’ without having to assume English is privileged in some way over English*. And I’m beginning to think we can’t.

The reason, I think, is that the notion of rigid designation (and, incidentally, lots of other things philosophers of modality talk about) cannot be made sense of in the simple modal language of necessity and possibility – the language we start off with before we introduce possible worlds talk. However the answer to whether or not a name is a rigid designator makes no difference to our original language. For any set of true sentences in the simple modal language involving the name “Aristotle” I can produce you two possible worlds models that makes those sentences true: one that makes “Aristotle” denote the same individual in every world and the other which doesn’t.* If this is the case, how is the question of whether a name is a rigid designator ever substantive? Why do we need this distinction? (Note: Kripke’s arguments against descriptivism do not require the distinction. They can be formulated in pure necessity possibility talk.)

To put it another way, by extending our language to possible world/Kripke model talk we are able to postulate nonsense questions: Questions that didn’t exist in our original language but do in the extended language with the new technical vocabulary. An extreme example of such a question: is the denotation function a set of Kuratowski or Hausdorff ordered pairs? These are two different, but formally irrelevant, ways of constructing functions from sets. The question has a definite answer, depending on how we construct the model, but it is clearly an artifact of our model and corresponds to nothing in reality.

Another question which is well formed and has a definite answer in Kripke model talk: does the name ‘a’ denote the same object in w as in w’. There seems to be no way to ask this question in the original modal language. We can talk about ‘Fred’ necessarily denoting Fred, but we can’t make the interworld identity comparison. And as we’ve seen, it doesn’t make any difference to the basic modal language how we answer this question in the extended language.

[* These models will interpret names from a set of functions, S, from worlds to individuals at that world and quantification will also be cashed out in terms of the members of S. We may place the following constraint on S to get something equivalent to a Kripke model: for f, g \in S, if f(w) = g(w) for some w then f=g.

One might want to remove this constraint to model the language A and B speak once they’ve learned each others language. They will say things like: Fred is Aristotle, but they might have been different. (And if they accept existential generalization they’ll also say there are things which are identical but might not have been!)]


Counterparts and Actuality

March 4, 2008

I’ve been reading this paper by Delia Graff Fara for one of the discussion groups I’ve been going to. It’s basically a follow up to the Williamson and (Michael) Fara paper from a couple of years back, highlighting some problems counterpart theory would face if augmented by an actuality operator. I had some general methodological problems with these papers (for example, they would argue that CPT could not provide faithful interpretations of QML formulae – when CPT’s aims are to provide interpretations of English, and further, they claim, to do so more faithfully than QML). But that aside, there was one obvious response Lewis could make which neither paper seemed to consider. I mentioned it in the discussion group, but didn’t get a chance to think it through properly, so I though I might take this opportunity to expand it some more (so apologies in advance for any obvious errors!)

Consider a world of eternally recurring, qualitatively identical epochs (call it w_e.) Now Lewis wants to reconcile two things. He wants to deny a version of haecceitism that states that there can be qualitatively identical possible worlds which differ with respect to what de re possibilities they represent for some individual; while making sense of the intuitive claim that Bob, who, lets say, lives in the 17th epoch, might have lived in the 18th epoch (i.e. where is qualitatively identical twin, boB, lives.) To do this he allows the counterpart relation to hold between individuals that live in the same world. This amounts, as Graff Fara notes, to individuating possibilities more finely than possible worlds. For example, there is one possibility in which Bob lives in the 17th epoch, and one in which he lives in the 18th, yet there is only one possible world involved. In Lewis’s own words: “Possibilities are not always possible worlds. There are possible worlds, sure enough, and there are possibilities, and possible worlds are some of the possibilities.” (PoW, p230)

So why don’t we just interpret the actuality operator as being true in the actual possibility, rather than being true in the actual world? To fix ideas, let us think of a possibility as an ordered pair of a world, w, and a function from individuals in w to individuals in w. For example the first possibility we considered was w_e, with the identity mapping, taking Bob to himself, but when we considered the possibility that Bob might have lived in the 18th epoch we were considering the pair w_e and the mapping that takes Bob to boB, Bob’s 18th epoch twin, (and which is the identity elsewhere.)

Does interpreting the actuality operator like this help? For example, do we get all the inferences we usually get from it? We can show that we do by simply interpreting QML+@ (quantified modal logic augmented with an actuality operator) in terms of possibilities and it should then be clear that it will validate exactly the same inferences as classical @ would.

We let the set of states be the set of possibilities, i.e. S := \{ \langle w, \sigma \rangle \mid w \in W \wedge \sigma : Ind \rightarrow Ind(w), \sigma \subseteq C \}. Let Ind be the set of individuals from any world, Ind(w) the individuals from w, and C the counterpart relation (I’ve relaxed the constraint that the function must go from and to individuals in the world.) I’ve idealised and assumed that \sigma is a total function. We set one particular pair s^* := \langle w^*, \sigma^* \rangle to be the actual possibility. The crucial truth clauses are as follows

\langle w, \sigma, v \rangle \models Px_1, \ldots, x_n \Leftrightarrow \langle \sigma(v(x_1)), \ldots, \sigma(v(x_1)) \rangle \in [[P]]
\langle w, \sigma, v \rangle \models @\phi \Leftrightarrow \langle w^*, \sigma^*, v \rangle \models \phi
\langle w, \sigma, v \rangle \models \Diamond\phi \Leftrightarrow \langle w^\prime, \sigma^\prime, v \rangle \models \phi \mbox{ for some } \langle w^\prime, \sigma^\prime \rangle \in S
\langle w, \sigma, v \rangle \models \exists x\phi \Leftrightarrow \langle w, \sigma, \sigma \circ v^\prime \rangle \models \phi
\mbox{ for some } v^\prime \mbox{ which differs only from v in its assignment to x.}

Since the clause for @, is exactly the same as in the standard semantics where we intepret S as the set of possible worlds, and the other truth clauses are sufficiently similar – we should get exactly the same inferences for @ as in the ordinary case.

Of course, counterpart theorists don’t like to use a language with primitive modal operators like QML+@, and will, if they can, phrase it all in first order logic. Standardly counterpart theorists will need the two primitive symbols: Iwx and Cxy. I is the relation of being a part of a world, C is the counterpart relation. We shall use one primitive, Rsxy, interpreted as x = \sigma(y) where s = \langle w, \sigma \rangle. We can give a translation schema of for QML+@ as follows:

(Px_1,\ldots,x_n)^s \mapsto
\exists y_1,\ldots,y_n(Rsy_1x_1 \wedge \ldots \wedge Rsy_nx_n \wedge Py_1, \ldots, y_n)
(\neg \phi)^s \mapsto \neg(\phi^s)
(\phi \wedge \psi)^s \mapsto (\phi^s \wedge \psi^s)
(\exists x \phi)^s \mapsto \exists x \exists y(Rsxy \wedge \phi^s)
(\Diamond \phi)^s \mapsto \exists s^\prime \exists y_1\ldots y_n \exists z_1\ldots z_n(Rsz_1x_1 \wedge \ldots \wedge Rsz_nx_n
\wedge Rs^\prime y_1z_1 \wedge \ldots \wedge Rs^\prime y_nz_n \wedge \phi^{s^\prime})
(@\phi)^s \mapsto \phi^{s^*}

(Note in the \Diamond clause, x_1, \ldots, x_n are the free terms in \phi.)