So far we have considered several internalist arguments that we need mental content, and several attempts to explain what narrow content is supposed to be; we have also considered externalist replies to these claims. This is where most of the action has centered: internalist attempts to defend narrow content, and externalist attempts to block these defenses.

But there is also at least one prominent argument for the conclusion that there is no such thing as the narrow content of a belief. This argument is to be found in ch 3 of Williamson’s Knowledge and its Limits (previously published as ‘The broadness of the mental’ in Phil. Perspectives 12 (1998)).

1 The shape of Williamson’s argument

Williamson regards the core of the internalist position as the claim that externalist mental states are really conjunctions of internalist mental states with environmental factors. His argument is for the claim that this position is false.

The argument can be seen as breaking into two parts.

1. The internalist claims that externalist mental properties are conjunctions of internalist mental properties with environmental non-mental properties. Williamson argues that we can characterize, in advance of any particular view of narrow content, what these internalist and environmental properties would have to be.

2. He argues that the externalist mental state in question can fail to obtain even when its internalist and environmental ‘halves’ obtain. What this will show is that these ‘halves’ were not really halves of the externalist property after all.

More quickly, the form of argument, applied to the case of believing p, is: if believing p had an internal and external component, they would have to be e and i; but e and i can obtain without the agent in question believing p, so believing p cannot simply consist in i and e. Hence there are no internal and external components which comprise believing p.

2 Terminology

Williamson’s argument is very clear, but some of the terminology can be confusing. Here I depart from his terminology in ways that I think may make the argument clearer to you, even though this involves making the argument a bit less concise (and being a bit sloppy in switching back and forth between talk of properties of situations and properties of agents in situations). Instead of ‘case’ I use ‘situation’, and ‘property’ instead of ‘condition.’

We can think of a situation as a possible world, with one agent specified as the agent of the situation. This does not mean that we can only consider possible worlds with one person in them; it’s just that we specify one of these people as the agent of the situation. These are are similar to what are sometimes called ‘centered worlds.’

We have already defined intrinsic duplicate.

The remaining definitions we’ll use fall into three categories:

Relations between situations

A situation is internally like another iff the agent in the first is an intrinsic duplicate of the agent in the latter.

A situation is externally like another iff the total state of the environment (external to the agent) in the first is the same as that in the latter.

Relations between properties and situations

A property is narrow iff it is shared between agents in all situations which are internally like each other.

A property is broad iff it is not narrow.

A property is environmental iff it is shared by all situations which are externally like each other.

Kinds of properties

A property is composite iff it is the conjunction of a narrow property with an environmental property.

A property is prime iff it is not composite.

In addition to these, I will use the term factor. I will say, e.g., that composite properties have two factors, a narrow factor and an environmental factor.

With these definitions in hand, we can now see how to express the claim of the internalist. The internalist is claiming that supposed externalist mental properties are composite, whereas the externalist is denying this. This is an admirably clear way to state what is going on in the debate.

We’re now in a position to examine Williamson’s argument against the internalist.

3 The argument

(The argument is of a very general form; to focus ideas I will limit it to the case of belief, with which we have been concerned.)

3.1 What the narrow and environmental factors of belief would have to be

We begin with two more definitions.

For any broad property F, we can define the property virtual-Fas follows: an agent has virtual-Fiff it is an intrinsic duplicate of an agent who has F. It should be clear that virtual-F will be a narrow property, in the above sense.

(Intutively, virtual-F will often be a disjunctive property.)

We can also define an analogous property which is environmental rather than narrow. We will say that a situation has outward-F iff it is externally like some situation in which the agent has F.

Just as virtual-F will be narrow, outward-F will be environmental.

Now let’s turn our attention to belief. Let the mental property with which we are concerned be the property of believing that tigers growl. For familiar reasons, this is plausibly an externalist property.

Using the above definitions, we can define the properties virtual-tiger-belief and outward-tiger-belief.

We will need to prove two lemmas which will be of use later in Williamson’s argument.

Argument. Let N be a narrow property entailed by tiger-belief. Since N is entailed by tiger-belief, whenever an agent has the property tiger-belief, that agent also has N. Now take any situation in which the agent has virtual-tiger-belief. We know from the definition of virtual-tiger-belief that the agent in this situation is an intrinsic duplicate of an agent who has tiger-belief. But this means that the agent is also an intrinsic duplicate of an agent who has N. But this also means that the agent has N, since N is narrow. Hence virtual-tiger-belief entails every narrow property entailed by tiger-belief.

By parallel argument, we can establish a second lemma:

The internalist thesis, recall, is that the property of believing that tigers growl is composite. This means, in our terminology, that it must have a narrow and an environmental factor.

The first main thesis that Williamson argues for is the following:

Let’s see how he argues for the truth of this conditional.

We establish (C) on the supposition that tiger-belief is a composite property; so we have also shown that if tiger-belief is composite, then (C) is true. And that is what proposition [1] above says.

(Well, it’s close enough for our purposes; we are ignoring issues about the relationship between property identity and necessary coextensiveness. These do not, as far as I can see, matter at all to Williamson’s argument.)

3.2 Virtual-belief and outward-belief without belief

To establish his conclusion, Williamson needs to establish one further claim:

If he can establish this, it will then follow from [1] and [2] that:

And this, Williamson thinks, will amount to the refutation of internalism.

Williamson’s argument for [2] is a simple one: a situation can be such that the agent of the situation is in virtual-tiger-belief and the situation is in outward-tiger-belief without the agent believing that tigers growl.

How do we do this? We come up with a situation S which is (i) internally like one situation in which the agent of the situation believes that tigers growl, and (ii) externally like another situation in which the agent of the situation believes that tigers growl, and yet (iii) such that the agent in S does not believe that tigers growl. It will follow from (i) and (ii) that S is one in which virtual-tiger-belief and outward-tiger-belief obtain; so if we can find such a situation which also satisfies (iii), Williamson will have proved his case.

See the example on pp. 71-72.

4 Free recombination

5 Internalist replies

Does Williamson assume that the internalist must hold that every agent in a given broad mental state must be in the same narrow mental state? Is this assumption legitimate?

Another way to put the same point: does Williamson’s argument cut against an internalist who holds that externalist mental properties do not decompose into narrow and environmental properties, but who holds that different instances of the externalist properties decompose differently?