Is it possible to find a counter model for epistemic closure in Nozick's system?

Question

The epistemic closure is that:

If S knows (if p then q) then (If S knows p then S knows q).

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In Nozick's Truth-Tracking Analysis

S knows p if and only if

• p is true
• S believes that p
• If p were false, S would not believe that p
• If p were true, S would believe that p

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The counter model here I mean something might includes

• W (a nonempty set of worlds),

• R (a relation over W), and

• I(S×W) (where I is the interpretation function I: S×W → {0,1}, where S is the set of sentense letters. This should tells us whether a sentense letter is true or not true in a world.)

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To show the epistemic closure is not valid under Nozick's system, we simply find a counter model with world w s.t. the negation of epistemic closure is true in this world.

I tried to do this by myself, but I didn't figure out Nozick's semantics by reading his philosophical explansion

for example

• How did Nozick define the valuation of the belief operator
• For the relation of worlds, what rules they must obey? (are they transitive, or symmetric etc.)
• What's the valuation of Nozick's counterfactual conditionals

(I know Nozick's system of counterfactual conditionals is different from Stalnaker and Lewis.)

What Nozick said on page 173 is that

This point is brought out especially clearly in recent 'possible-worlds' accounts of subjunctives: The subjunctive is true when (roughly) in all those worlds in which p holds true that are closest to the actual world, q also is true (Examine those worlds in which p holds true closest to the actual world, and see if q holds true in all these.)

To me, this really sounds like Stalnaker's account of subjunctive conditionals:

That p subjunctive implies q is true in w if and only if for all worlds x, [if p true in x and for all worlds y (if p true in y then x sees y under relation Rw)], then q true in x.

Could someone explain Nozick's semantics to me in details?

Any help or suggestion would be appreciated.

Answer 1

Nozick's account of knowledge is presented in the context of his treatment of skepticism. He proposes to defend the possibility of knowledge from various kinds of radical skepticism. In particular, he wants to overcome challenges such as the infamous 'brain-in-a-vat' (BiV) argument, which purports to show that we cannot know anything because we cannot know that we are not a BiV with all of our experiences being simulated by a supercomputer.

Nozick gives the four conditions for knowledge that you quote:

1. p is true
2. S believes that p
3. If p were false, S would not believe that p
4. If p were true, S would believe that p

Nozick does not offer any definition of belief (or truth for that matter) - he just takes it as primitive. He also takes the logic of the subjunctives as a given. While he refers to the possible world accounts of conditionals given by David Lewis and Robert Stalnaker, he is not committed to these being authorititive. In fact, in footnote 8 on page 680, he diverges from Lewis by adopting a structure under which a conditional "if A were true, B would be true" is not automatically true when A and B are both true in the actual world.

Nozick's account steers round the Gettier cases, but in order to avoid the BiV type skepticism, he has to reject the principle of closure of knowledge under known entailment. The reason this principle can fail on his account (albeit in exceptional circumstances) is that Nozick wants to say, I know I am in Boston, but I don't know that I'm not a BiV. The subjunctive conditionals correctly track the truth of the first belief but not the second. Nozick is a competent judge of whether he is in Boston or not, and if he were not in Boston he would not believe he was. But he is not a competent judge of whether he is a BiV, so if he were, he would believe he was not. But if he is a BiV then he is not in Boston (and he knows this), so we have

• Nozick knows he is in Boston; and
• Nozick knows that being in Boston entails he is not a BiV;

but not:

• Nozick knows he is not a BiV.

In possible world terms, we might explicate this as follows: If Nozick is indeed in Boston, then the closest relevant worlds in which he is not in Boston are those in which he is well aware that he is not. He is at a conference in Chicago, perhaps, or on vacation. A world in which he is a BiV is remote from the actual world and doesn't contribute to the truth of the tracking conditional. But if Nozick is in fact a BiV, then the closest relevant worlds to that possible world are ones in which he believes he is in Boston, because he is being deceived.

The issue of the accessibility relation between possible worlds is not important here: you may as well take it (as Lewis does) to be Euclidean. The critical point is that on the possible worlds account of the semantics of conditionals, the conditional is variably strict. Some possible worlds contribute to its truth, while others do not, depending on their closeness, not their accessibility.

Answer 2

Regarding your your claim "this really sounds like Stalnaker's account of subjunctive conditionals", here Nozick is just invoking the usual variable strict conditional analysis invented by David Lewis, and Stalnaker's treatment is a little different from Lewis's according to reference here

Stalnaker's account differs from Lewis's most notably in his acceptance of the limit and uniqueness assumptions. The uniqueness assumption is the thesis that, for any antecedent A, among the possible worlds where A is true, there is a single (unique) one that is closest to the actual world. The limit assumption is the thesis that, for a given antecedent A, if there is a chain of possible worlds where A is true, each closer to the actual world than its predecessor, then the chain has a limit: a possible world where A is true that is closer to the actual worlds than all worlds in the chain. (The uniqueness assumption entails the limit assumption, but the limit assumption does not entail the uniqueness assumption.) On Stalnaker's account, A > C is non-vacuously true if and only if, at the closest world where A is true, C is true. So, the above example is true just in case at the single, closest world where he ate more breakfast, he does not feel hungry at 11 am. Although it is controversial, Lewis rejected the limit assumption (and therefore the uniqueness assumption) because it rules out the possibility that there might be worlds that get closer and closer to the actual world without limit. For example, there might be an infinite series of worlds, each with a coffee cup a smaller fraction of an inch to the left of its actual position, but none of which is uniquely the closest. (See Lewis 1973: 20.)

Thus Stalnaker's proposal to evaluate the truth condition of subjunctive conditional is like: “P -> Q” is true if and only if, in the closest possible world in which P is true, Q is true. Nozick’s rejected this proposal on the ground there're some truly random events like QM so we cannot establish the subjunctive conditions at possible worlds correctly for certain. Nozick’s proposal is like “P -> Q” is true if and only if, in every closest and almost-closest possible world in which P is true, Q is true to create a stronger foundation.

Now regarding your "How did Nozick define the valuation of the belief operator", here Nozick just uses belief same way as common justified true belief which can be expressed as modal operator K in epistemic logic.

Regarding your "For the relation of worlds, what rules they must obey (are they transitive, or symmetric etc.)", only closeness relation is relevant here to track truth. We don't track truth in a remote world very different from ours by Nozick's approach.

Regarding your "What's the valuation of Nozick's counterfactual conditionals", Nozick insisted a connection between one's belief of a fact and the fact itself. For him any belief must be not only actually true but also true in slightly different circumstances satisfying his four conditions in all close PWs as p-neighborhood. Nozick conceded that we can never know whether we are really brain-in-a vat ourselves because we cannot track the truth of this proposition in any close PWs assuming the simulation technology is so great. But it by no means we cannot track the truth of other external propositions and know them so long as his two counterfactual conditions are met, but very unintuitively following this line we have to give up the epistemic closure principle as the most vulnerable link since the conclusion's truth value may not be trackable even its premises usually can be externally reliably tracked and verified. This is very similar to modern reliabilism as a type of modest foundationalism which may contain fallible beliefs due to reasons such as the rejection of epistemic closure principle here.