Epistemic/Doxastic Possibility

Recent forays into the metaphysics and epistemology of modality require us to be quite clear about the senses  of 'possible,'  necessary,'  and the other modal words in play.  In the contexts mentioned, these words are not being used epistemically or doxastically. 

Is Joan in her office? If I ask you, you might reply, "It's possible." Or if I ask you, "Is Zorn's Lemma logically independent of the Axiom of Choice" you might reply, "It's possible." We need to clarify these uses of 'possible.'

The second example illustrates how a state of affairs can be epistemically possible but not really possible. If subject S has a limited knowledge of set theory, then it will be possible 'for all S knows' that the lemma and the axiom be independent. But as a matter of fact, they are provably equivalent. So in this case what is epistemically possible is really impossible.

 

Examples like this can be generated ad libitum.  Can a set T have a proper subset that is the same size (same cardinality) as T?  Many will say, "It's impossible."  But that response reflects their ignorance of Cantorian set theory.  Accordingly, their use of 'impossible' is epistemic.  It is impossible for all they know, but possible nonetheless.  It's possible because it's actual.

In the case of Joan, it may be possible 'for all you know' that she is in her office. But it will be really impossible if she died on her way to work. But suppose she didn't die, and suppose she is on campus near her office but the only thing preventing her from being in her office is her aversion to the place. Then, it will be epistemically possible for you that Joan be in her office, but also really possible that she be there.

So what do we mean by epistemic possibility?   The examples suggest that epistemic possibilities are possibilities parasitic upon ignorance.  I propose the following definition for discussion and possible refutation:

State of affairs X is epistemically/doxastically possible for subject S =df X is logically consistent with what S knows/believes.

My earlier talk about possibilities is to be understood as talk about real, not epistemic/doxastic, possibilities.  I don't deny that there are outstanding puzzles about real (i.e., mind-independent)possibilities, what they are and how they are known.  But the negative point that they are not to be confused with epistemic or doxastic possibilities is a step in the right direction.

Comments

[Andrew]

You define epistemic/doxastic possibility in terms of logical consistency. If logical consistency implies metaphysical compossibility, this definition not do.

For necessary falsehoods aren't metaphysically compossible with anything. And some necessary falsehoods are epistemically possible.

Example. Until I took logic, it was epistemically possible for me that no set has a proper subset of the same cardinality (call this p). But p is metaphysically impossible. And for any q, q&p is metaphysically impossible. Thus, some p is epistemically possible but not metaphysically compossible with anything (much less anything I know or believe).

[Bill Vallicella]

Andrew,

Now that's a good objection! I concede that you have refuted my definition above.

The rough idea is that what is epistemically possible for a person is that which is not ruled out by what the person knows. What if I say:

p is epistemically possible for S =df no proposition S knows entails ~p

Would this work?

[Andrew]

I don't see that this is any better.

Suppose as before that it's epistemically possible for me that no set has a proper subset of the same cardinality (call this p). Everything I know entails ~p, since p is necessarily false (I'm assuming that q entails r iff it's not possible that q be true and r false).

[Bill Vallicella]

Your point is that p is necessarily true, and so is entailed by every proposition including every proposition that you know. On your definition of 'entails' -- which is standard -- every necessary truth is entailed by every proposition, which smacks of paradox.

What if I strengthen 'entails' in my last definition to 'relevantly entails'?

[Andrew]

'Relevant entailment' is an expression that darkens counsel. But if I understand it, I think I can also see counterexamples like unto the ones given above.

Amendment: p is epistemically possible for S =df no proposition S knows relevantly entails ~p

Counterexample: The conjunction of the definitions of cardinality and a few other items relevantly entail ~p. I know these definitions, but I'm too daft to see that they entail ~p (or I just haven't had time to think the matter through); p is still a live (epistemic) possibility for me.

Amendment: p is epistemically possible for S =df no there's no proposition q such that S knows q, q relevantly entails ~p, and S knows that q relevantly entails ~p.

Counterexample: I know the definitions of cardinality and the like, and know that these definitions entail ~p. And yet, I lack the higher order belief that I know these definitions. Since it's not the case that I take myself to know these definitions (I simply haven't considered their epistemic status), p is still a live epistemic possibility for me.

[Andrew]

7 definitions of epistemic possibility are considered and rejected, incidentally, in Michael Huemer's "Epistemic Possibility" (Synthese 156 (2007): 119-42). Huemer's favored account says:

p is epistemically possibly for S iff it is not the case that p is epistemically impossible for S.

And p is epistemically impossible for S iff: a) p is false, b) S has justification for ~p adequate for dismissing p, and c) S's justification for ~p is Gettier-proof.

[Bill Vallicella]

Thanks for referring me to Huemer's article. His "Why I am Not an Objectivist" is also of interest given recent concerns hereabouts.
http://home.sprynet.com/~owl1/rand.htm

What would you say to this:

For any contingent proposition p, p is epistemically possible for subject S =df there is no proposition q such that S knows q and q entails ~p.

This formulation blocks your counterexample but at the price of a restriction to contingent propositions.

[Andrew]

An early reading of Huemer on Rand (I believe it was one of his usenet postings on the topic) saved me from becoming too enamored with Objectivism in high school.

Restricting a notion of epistemic possibility to contingent propositions will make that notion considerably less useful than it is often taken to be. Example: Kripke taught us that water must be H20. Our lingering sense that water didn't have to be H20 is often explained away as a merely epistemic possibility. If the notion doesn't extend to non-contingent propositions (propositions that are necessarily true if true or necessarily false if false), we cannot help ourselves to this explanation.

Amendment: For any contingent proposition p, p is epistemically possible for subject S =df there is no proposition q such that S knows q and q entails ~p.

Counterexample. I firmly but mistakenly (but on good grounds) believe that the man before me is Bill. It's not epistemically possible for me that Bill is 300 miles away (it seems as though I can see the man before me, after all!). But since Bill is in fact 300 miles away, nothing I know entails that Bill is not 300 miles away. Here, it's not epistemically possible for me that p (= Bill is 300 miles away), but it's not the case that I know some q that entails ~p. The relevant q will either be unknown because false (e.g., that Bill is before me) or will be known fail to entail ~p (e.g., that I am appeared to Bill-ly).

Amendment: For any contingent proposition p, p is epistemically possible for subject S =df there is no proposition q such that S justifiably believes q and q entails ~p.

Counterexample: [forthcoming]

[Bill Vallicella]

Andrew,

I'm very busy now, but I hope to respond soon. A very useful discussion. Thanks.

[Account Deleted]

Bill, Andrew,

(First Shot at a proposal for epistemic possibility.)

1) What do we mean when we say “For all I know P”? I think there are at least two thoughts wrapped up in this proposition:

(i) I do not believe P.

Why? For if I did believe P and I was aware that I believed P when I said “For all I know P”, then I would have said something much stronger than what I have actually said. I would have said instead that I believe/know that P. Also

(ii) Nothing I believe *rules out* P.

But, now, suppose that I do in fact believe Q and Q logically rules out P because Q logically contradicts P, yet for one reason or another I was unaware that I believe Q or if I was aware that I believed Q I was not aware that Q contradicts P at the time I said “For all I know P”. Therefore:

(iii) Peter believes Q.

(iv) Q (is or) entails ~P.

2) It is important to recognize that the conjunction of the propositions (i)-(iv) is not a contradiction. So I think we must admit that (i)-(iv) describes a scenario that is certainly epistemically possible about me and that it is possible in the very cases when we say things such as “For all I know ___“: namely, it is a scenario that our notion of “epistemic possibility” must somehow cover. So I submit that an adequate account of “epistemic possibility” must allow for times when they are all true for someone.

3) Now, Bill proposed a variety of definitions of the phrase “p is epistemically possible for S” (where ‘p’ is a proposition and S a person) and Andrew found cogent counterexamples to such proposals. There are several problems lurking here. First, Bill’s proposals employ knowledge rather than belief in the definians. That would not do because knowledge implies truth (by the standard definition) whereas epistemic possibility, I think, presupposes a corpus of beliefs none of which are required to be true. Second, Bill’s proposals focus exclusively upon the first-order component analogous to the traditional definition of ‘to know’, whereas the notion of “epistemic possibility” essentially includes a reflective element: namely, it includes a subject's own reflection upon the corpus of his own beliefs and it includes determining something about a relationship between this corpus and a certain proposition. I think Andrew briefly mentions this second-order element in one of his remarks. Third, Bill’s proposals do not allow such a reflection to be mistaken, whereas as we have noted above any definition of “epistemic possibility” must allow for certain sorts of mistakes at the second-level. Andrew’s counterexamples systematically exploit all of these features of Bill’s proposals.

4) Let X designate the corpus of S’s beliefs and let P be a proposition not included in X. X certainly exists. Let REF(X) be a set of statements that express S’s second-order beliefs about X (his first-order beliefs). The set REF(X) must exist for many reasons, but it certainly exists if S asserts or thinks something along the lines of "For all I know P". Some of the propositions included in REF(X) state certain logical relationship S believes hold between members of X (=his first-order beliefs) and propositions not included in X: e.g., P. Some of the members of REF(X) are false, just like some of the members of X are false. And one of the members of REF(X) might be that none of S’s beliefs entails ~P. This is something S might hold when reflecting on X because S might not be a very good logician or he might be inattentive when forming these beliefs etc. None of us are perfect logicians and none of us are perfect logicians all the time. The notion of “epistemic possibility” must allow for these facts.

5) P is epistemically possible for S =df.
(a) X is S’s corpus of first-order beliefs and REF(X) is S’s second-order beliefs;
(b) P is not in X;
(c) If R logically entails P, then REF(X) does not include “R logically entails P”;
(d) If Q is in X & Q = ~P (or logically entails the same), then REF(X) does not include “Q =~P” (or that it logically entails the same);

6) The definition in (5) (I think) accommodates normal cases as well as cases where P is logically inconsistent with something that S believes. For, unlike logical possibility which cannot allow for inconsistencies, epistemic possibility must allow for it. I am sure that (5) contains deficiencies and hopefully Bill, Andrew or others will point them out. It is a first shot at a proposal and for all I know it is totally inadequate. But I think it is worth exploring. Let’s see where it leads.

peter

[Bill Vallicella]

Peter,

Let P = No set has a proper subset of the same cardinality. P is necessarily false. P is epistemically possible for Tom, who knows little set theory. Roughly: P is possible for all Tom knows, or P is not ruled out by what Tom knows or believes. Your crucial observation is that P is not in Tom's corpus of beliefs, i.e., Tom does not believe P. That seems right. Tom neither knows that P nor believes that P.

But I am afraid I don't understand your definition. It seems entirely negative. What is the positive content of your definition of epistemic possibility?

What's wrong with this:

p is epistemically possible for S =df (i) S entertains p; (ii) for every q that S either believes or knows, p seems consistent with q.

This defn is immune to Andrew's counterexample, but it might have other problems.

[Account Deleted]

Bill,

As I was responding to your recent comment I realized a potential blunder in my clause (c) of the definition. Clause (c) is equivalent to

(c*) If REF(X) does include “R logically entails P”, then it is not the case that R logically entails P; (contrapositive)

(c*) cannot be true, because whether S's reflective corpus (REF(X)) includes that R entails P cannot be a sufficient condition for the absence of such an entailment. S may be wrong about whether such an entailment holds or does not hold. Since clause (c) is crucial, I must solve this technical problem before I even address your objections and comments.

peter

[Bradley]

Bill's proposal: p is epistemically possible for S =df (i) S entertains p; (ii) for every q that S either believes or knows, p seems consistent with q.

I think this is going to require a definition of the locution "seems consistent." It could be cashed out as "S believes that p is consistent with q," or "S doesn't know/believe that p entails ~q," or a number of other definitions that use well-defined notions. A small worry is that the temptation will be to cash out "p seems consistent with q" as "given q, it is epistemically possible that p."

Here's a two-fold worry: either (i) or (ii) has to go. (i) seems unnecessary. Surely we'd want to say that it's epistemically possible for me that I'm at my desk, but I hadn't thought about it until t, and most would want to say it's epistemically possible for me before t. It's epistemically possible for S that Joan's at her desk prior to S's entertaining it. I would think there are a host of propositions that are epistemically possible for me that I haven't entertained. If epistemic possibility is "parasitic on ignorance," isn't not considering a proposition the easiest way of ignoring it? Thus, (i) seems to rule out too much; what's the motivation for it?

But if (i) is gotten rid of, then there's this problem. Consider the conjunction of every mathematical truth about addition. Everything that I believe or know seems consistent with the conjunction, since it's necessarily true and I'm pretty good at addition. But there's no way it's epistemically possible for me, since I could never entertain it.

Maybe (i) could be modified: (i*) were S to entertain p and ~p, S would not believe that ~p. Or perhaps simply (i**) S could entertain p.

[Bill Vallicella]

Bradley,

Thanks for the useful comments. Clause (i) above now seems mistaken. A proposition can be epistemically possible for S whether or not S knows it, believes it (whether occurrently or dispositionally), or merely entertains it.

The rough idea is that a proposition p is epistemically possible for a subject S iff p is not ruled out by any q that S knows or believes. But what exactly is this relation of not being ruled out? It cannot be explained in terms of consistency or entailment. And it seems it must be some sort of objective relation.