Epistemic counterparts 4: Truth-conditions (first stab)

In this post, I'm going to present a first stab of a formal semantics for de re belief reports.

As I explained in the last post, I'm going to assume that for every epistemic subject at every time there is a set of doxastically accessible worlds, representing how the subject takes the world to be. I will sometimes refer to these worlds as the subject's 'belief worlds'.

On that background, we can make the guiding idea behind the Quine-Kaplan model more precise: 'S believes that x is F' is true iff there is a suitable role R such that (1) in all worlds doxastically accessible for S, whatever plays R is F, and (2) in the actual world, x plays R.

We can also clarify the notion of a role. Let's distinguish definite and indefinite roles. A definite role is occupied by at most one individual at each possible world. So it can be represented as a (partial) function from worlds to individuals – a.k.a. an "individual concept". An indefinite role can be occupied by more than one individual at a world. We'll have to allow for indefinite roles to deal with some tricky cases, but let's focus on easy cases for now.

Intuitively, a role is a way of tracking an individual at one world across other possible worlds.

There are countless ways a given individual at our world can be tracked across other worlds. Almost all of them are irrelevant to ordinary belief reports. As I explained in part 2 of this series, a "suitable" role must be simple and strong, it should make sense of the attitude report, and it should coincide with contextually salient roles for the relevant individual, if there are any.

Let's pretend for the moment that these suitability conditions ensure that there is a unique role for every individual under discussion. This is rarely true, but it simplifies the semantics.

We can now define a counterpart relation, relative to any epistemic subject S and conversational context c, in the obvious manner:

An individual i' at a world w' is a counterpart of i at w iff R(w) = i and R(w') = i' for some R that is suitable in c.

The belief operator functions as a quantifier over belief worlds. When we evaluate a sentence at a belief world, we can interpret singular terms as picking out counterparts of their actual referent. This makes the interpretation de re.

One can arguably force a de re reading by syntactically moving a singular term outside the scope of the belief operator, leaving behind a bound variable, as in 'Mary's husband is an object such that she thought that it was a bear'. Let's pretend that this is what de re reports look like in Logical Form, so that the singular terms whose reference gets shifted are always variables. (I'll explain in a minute how to do without that assumption.)

When we interpret 'it is a bear' at one of Mary's belief worlds, we want 'it' to pick out whatever plays the relevant role at that world.

Let's say that a (partial) assignment function g' is a w'-image of an assignment function g at a world w iff, for all variables x, g'(x) at w' is a counterpart of g(x) at w, or undefined if there is no such counterpart.

Since the counterpart relation is relative to a subject and a context, so are the image relations.

Now we can specify truth-conditions for a language with a belief operator. As usual in intensional semantics, truth is relative to a world w and an assignment g. The clauses for simple predications, Boolean operators, and etc. are all standard (on a formal level). The clause for belief looks as follows:

\[ [S \;\text{believes}\; \phi]^{w,g} = 1 \text{ iff } [\phi]^{w',g'}=1 \text{ for all $w' \in B_S^w$ and $w'$-images$_S$ $g'$ of $g$ at $w$.} \]

Here, \(B^{w}_{S}\) is the set of S's belief worlds at w. The subscript 'S' of 'images' indicates that the relevant image relations are those for S (in the utterance context, which I haven't specified as an extra parameter).

(If we want to drop the assumption that de re readings always involve quantification into the belief operator, we can define an extended assignment function so that for any singular term t and world w, gw(t) is the (ordinary) referent of t at w. The image relations are defined much like before: gv' is a w'-image of gv at w iff, for all terms t, gv'(t) at w' is a counterpart of gv(x) at w, or undefined if there is no such counterpart. In the semantics for belief, we replace 'g' by either 'gw' or 'g@', where @ is the "actual world" in the (pointed) model. Neither choice will be adequate for all cases, I think.)

It should be clear how this semantics fits the Quine-Kaplan model: 'S believes that x is F' is true iff there is a suitable role R such that (1) in all of S's belief worlds, whatever plays R is F, and (2) in the actual world, x plays R. Condition (1) is equivalent to saying that x's counterparts at all belief worlds are F; condition (2) is a consequence of how the counterpart relation is defined: an individual y at a world w can only be a counterpart of x here at @ if x plays the relevant role at @.

We can also answer the challenge from Hawthorne and Manley that I quoted in the first post. The challenge was to explain how the truth-conditions of 'S believes that x is F' could be compositionally derived without assuming that the sentence expresses that the subject stands in the belief relation to a singular proposition. No such assumptions figures in the above semantics.

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