## A Game Theoretic Semantics for Vagueness

April 6, 2008

Ok, so there’s a risk I’m going to alienate my readers with all these wacky theories of vagueness, but here’s another one if you’re keeping track. I was thinking of trying to capture the (possibly) Fregean idea that the sense of an expression is a method for determining what the referent of the expression is. For vague expressions this method may be non-deterministic – on some ways of carrying out the method you arrive at one referent, on others, other referents. Like supervaluationism, this position views vagueness as a kind of semantic underdeterminacy, but, I shall argue, gives us a very different logic.

I’ve been considering two different ways of representing a ‘method for determining the referent’ formally: 1) to think of senses as computer programs of some sort, 2) to think of them as a game between two players. I’m going to be considering the second option here. First, let’s look at the game semantics for first order logic for those not familiar with it. Given a model, M, and an assignment of variables, v, we can define an assortment of games between two players $G_M(\ulcorner \phi \urcorner , v)$ as follows.

• $G_M(\ulcorner \phi \vee \psi \urcorner , v)$: the verifier chooses between $\phi$ and $\psi$ then the game continues with $G_M(\ulcorner \chi \urcorner , v)$ where $\chi$ is the chosen formula.
• $G_M(\ulcorner \phi \wedge \psi \urcorner , v)$: the falsifier chooses between $\phi$ and $\psi$ then the game continues with $G_M(\ulcorner \chi \urcorner , v)$ where $\chi$ is the chosen formula.
• $G_M(\ulcorner \neg \phi \urcorner , v)$: the verifier and falsifier swap roles and the game continues with $G_M(\ulcorner \phi \urcorner , v)$
• $G_M(\ulcorner \exists x \phi \urcorner , v)$: the verifier chooses an assignment $v^\prime$ that differs from v at most in its assignment to x, and the game continues with $G_M(\ulcorner \phi \urcorner , v^\prime)$.
• $G_M(\ulcorner \forall x \phi \urcorner , v)$: the falsifier chooses an assignment $v^\prime$ that differs from v at most in its assignment to x, and the game continues with $G_M(\ulcorner \phi \urcorner , v^\prime)$.
• $G_M(\ulcorner P^n_i(x_1, \ldots , x_n) \urcorner , v)$: if $\langle v(x_1), \ldots , v(x_n) \rangle \in P^M$ then the player playing the role of verifier wins. Otherwise the falsifier wins.

We can then say that a formula $\phi$ is true (in M, on v) if the player who starts off playing the verifier has a winning strategy for $G_M(\ulcorner \phi \urcorner , v)$, and say its false if the falsifier has a winning strategy for this game. To extend to a simple system with vagueness, we can say $\phi$ is supertrue (superfalse) if the verifier (falsifier) has a winning strategy for the game that starts with the falsifier (verifier) picking a precisification and continues as $G_M(\ulcorner \phi \urcorner , v)$. This is equivalent to standard supervaluationist semantics.

Note: there is a related way to do things which gives different results. If you allow games of imperfect information then not every game is determined, and you can get violations of LEM. This is relevant to vagueness. Say that a sentence $\phi$ is true (false) if the verifier (falsifier) has a winning strategy for the following game:

• A precisification is chosen at random without the verifier or falsifier knowing which. The game then continues with $G_M(\ulcorner \phi \urcorner , v)$.

In this case we get different results, for example $(p \vee \neg p)$ is neither true nor false, when p is borderline (true on some but not all precisifications.) In fact, this version will be equivalent to the strong Kleene 3-valued logic with the neutral value holding when neither the verifier or the falsifier have a winning strategy.

But the view I’m interested isn’t this one. I want to identify the sense of an atomic formula with a game $G_M(P(x_1, \ldots , x_n), v)$, in this case given by the model, M, which may or may not be determined. The idea is that vague expressions correspond to games in which neither player has a winning strategy, because the method for determining the truth value does not always land you with the same result. The method is unreliable, inaccurate or ill defined. This reflects the idea that if someone asks you to determine whether borderline balding Billy is bald there is no well defined procedure, or way to go about doing this.

Adopting the rules above except for the atomic case, which we replace with the game supplied by the model, M, we again get a non-classical logic. In this case (I think) we get the weak Kleene 3-valued logic, which is interesting, because as far as I know, no-one takes this as the logic of vagueness.