Tag: David Lewis

A Dilemma for Minimally Consistent Belief

Bas van Fraassen4 Comments

We tend to have wrong beliefs about many things.  The criteria for having a belief do not stop at introspection and so we may be wrong also about what beliefs we have.  We are not fully self-transparent, and so it may not be right to blame us for such mistakes. 

But it is still appropriate to point out debilitating forms of error, just as we would for a distracted or forgetful accountant.  After all, the success of our practical projects may depend on the beliefs we had to begin.

A Criterion for Minimal Consistency

As a most minimal norm, beyond mere logical consistency, I would propose this:

our belief content should not include any awowal of something we have definitely disavowed.  

We can avow just by asserting, but to disavow we need to use a word that signifies belief in some way. For example, to the question: “Is it raining?”,  you can just say yes.  But if you do want to demur, without giving information that you may not have, the least you must do is to say “I don’t believe that it is raining”.

Definition.  The content of someone’s beliefs is B-inconsistent if there it includes some proposition p and also the proposition that one does not believe that p.  

B-consistency  is just its opposite.  

I am modeling belief content as a set of propositions, and minimally consistent belief contents are B-consistent sets of propositions.  I will also take it that belief can be represented as a modal operator on propositions:  Bp is the proposition that encodes, for the agent, the proposition that s/he believes that p.

Normal Updating

Now the study of belief systems has often focused on problems of consistency for updating policies.  Whatever you currently believe, it may happen that you learn, or have warrant to add, or just an impulse to add, a new belief.  That would be a proposition that you have not believed theretofore.  The updating problem is to do without landing in some inconsistency.  That is not necessarily easy, since the reason that you did not believe it theretofore is because you had contrary beliefs.  So there is much thought and literature about when such a new belief can just be added, and when not, and if not, what to do.

However, responses to the updating problem generally begin by mapping out a safe ground, where the new belief can just be added.  Under what conditions is that unproblematic?  

A typical first move is just to require consistency:  that is, if a new proposition p is consistent with (consistent) belief content B then adding p to B yields  (perhaps part of) a (consistent) belief content.  I think we had better be more conservative, and so we should require that the prior beliefs include an explicit disavowal of any belief both of p and of it its contraries.

So here is a modest proposal for when a new belief can just be added without courting inconsistency of any sort:

Thesis.  If a belief system meets all required criteria of consistency, and it includes disavowal of both p and not-p, then the result of adding p while removing its disavowal, does not violate those criteria of consistency.

We might think of the Thesis as articulating the condition for a system of belief to be updatable in the normal way under the best of circumstances.

A pertinent example then goes like this:  

I have no idea whether or not it is now raining in Peking.  I do not have the belief that it is so, nor the belief that it is not so.  For all I know or believe, it is raining there, or it is not raining there, I have no idea. 

The Thesis then implies that if I were to add that it is raining in Peking to my beliefs (whether with or without warrant) the result would not be to make me inconsistent in any pertinent sense.

The Dilemma

But now we have a problem. In that example, I have expressed my belief that I do not believe that it is raining in Peking – that part is definite.  But whether it is raining in Peking, about that I have no beliefs.  Let’s let p be the proposition that it is raining in Peking.  In that case it is clear that I neither believe nor disbelieve the following conjunction:

            p & ~Bp

So according to the thesis I can add this to my beliefs, while removing its disavowal, and remain consistent.  

But it will take me only one step to see that I have landed myself in B-inconsistency. For surely I believe this conjunction only if I believe both conjuncts.  I will be avowing something that I explicitly disavow.

Dilemma:  should we accept B-consistency as a minimal consistency criterion for belief, or should we insist that a good system of beliefs must be one that is updatable in the normal way, when it includes nothing contrary, and even disavows anything contrary, to the new information to be added?

(It may not need mentioning, but this dilemma appears when we take into account instances of Moore’s Paradox.)

Parallel for Subjective Probability Conditionalization

If we represent our opinion by means of a subjective probability function, then (full) belief corresponds to probability 1.  Lack of both full belief and full disbelief corresponds to positive probability strictly between 0 and 1.

Normal updating of a prior probability function P, when new information E is obtained, consists in conditionalizing P on E.  That is to say, the posterior probability function will be

            P’:  P’(A) = P(A|E) = P(A & E)/P(E), provided P(E) > 0.

So this is always the normal update, whenever one has no full belief either way about E.

In a passage famous in certain quarters David Lewis wrote about the “class of all those probability functions that represent possible systems of beliefs” that:

This class, we may reasonably assume, is closed under conditionalizing. (1976, 302)

In previous posts I have argued that probabilistic versions of Moore’s Paradox raise the same problem for this thesis, that a class of subjective probability functions represent possible systems of belief only if it is closed under conditionalization. 

 ( “Stalnaker’s Thesis –> Moore’s Paradox” 04/20/2023; “Objective Chance –> Moore’s Paradox” 02/17/2024).

Lewis, David K. (1976) “Probabilities of Conditionals and Conditional Probabilities”.  The Philosophical Review 85 (3): 297-315.

Fitch, too, had a Paradox: (2) Actuality

Bas van Fraassen1 Comment

As it stands, Fitch’s argument is water-tight, air-tight. But as it stands, it relies on a very shallow representation of being known.

To say that something is known is to say that someone knows it.

First point: such words as “someone”, “everyone”, “anyone” are used with some range implicitly in mind. For example, if I say “Everyone is vulnerable to the Covid 19 virus”, you aren’t going to object by mentioning Napoleon and Julius Caesar. You will have understood that I meant everyone alive now. In any interpretation of a text, along the lines of our familiar ‘truth-and-reference’ semantic analysis, the quantifiers range over a specific set, the domain of discourse.

So let us look at Fitch’s argument with the relevant range made explicit. Instead of “It is known” let us write “It is known in Seattle”. Let us take it that A is true, but it is not known in Seattle. My friend David lives in Seattle, and so I can deduce at once that David does not know that A, and certainly does not know that (A and it is not known in Seattle that A).

But is it the case that David could not know that? In actual fact David lives in Seattle, but there is nothing impossible about his living in Tacoma instead. And if he were living in Tacoma, there would nothing impossible about him knowing that A is both true and unknown in Seattle.

So the equivocation that tends to give us a sense of unease with Fitch’s argument is this. It is impossible that someone knows that (A and no-one knows that A). But for each person X we can also hold, without running into contradiction It is possible for X to know that (A and no-one knows that A).

The reason is that, what ever is the range associated with the use of “someone” in this context, it is possible for X to be outside of that range.

First objection: surely there is a sense of the words “some”, “every”, in which there is nothing relevant is outside the range? For example, “everyone” could in this case be all humans.

That does not include the gods on Mount Olympus. They are not real, but they could be. If it could be the case that they know that truth (which refers to the ignorance of all humans) then that truth is knowable. There are other possibilities to consider. My friend David is human, but it would be possible for him (in purely logical possibility!) to be transformed by a witch’s spell into a frog, while retaining all his cognitive powers and resources. If he were a frog then he could know that A and that no humans know A.

Second objection: surely there is an absolute sense of the words “some”, “every”, in which there is nothing outside their range, period? Bring up any thing at all that could know something, or any thing at all for that matter, and it is covered.

That is daring. It means that in our language we can quantify over all the denizens of all possible worlds. We have to take it seriously, because David Lewis once said that this was the case for his language:

“I can. Some say they can’t. They say their understanding is limited to what can be expressed by modalities and world-restricted quantifiers. I have no help to offer these unfortunates, since it is known that the expressive power of a language that quantifies across worlds outruns that of the sort of language they understand.” (Lewis 1979, footnote on page 517) 

Second point: as Lewis himself made very clear in his article “Anselm and Actuality” (1970) we need to take seriously the logic of the word “actual” as an indexical. In unadorned simple sentences, the word “actual” is logically negligible: I actually have ten dollars if and only if I have ten dollars. But embedded in modal contexts, it takes on a dramatic role:

I wish I had more money than I actually have!

Here you cannot drop the “actually”, for it is certainly not the case that I wish for the impossible situation in which (BvF has more money than he has).

So, it is possible for my friend David to know that A is true and that actually no one knows that A. That is,

(*) there is a possible world β in which David knows that in world α it is the case that A and that no one knows that A.

That means equivalently that the following is true in our world:

(**) It is possible that David knows that Actually( it is the case that A and no one knows that A).

Very odd though! If you take possible worlds seriously, in a metaphysical sort of way, you may insist that the denizens of one world cannot know about those in another world.

(Well, maybe you will not insist; in the Appendix I’ll mention an example.)

But as a point of logic, it should count — metaphysical scruples aside. Think of this similar case (which is how Hans Kamp began): the word “now”. It may be true in 2075 that there were men on Mars years earlier, but it certainly won’t be true in 2075 that it is now the case that there were men on Mars years earlier. And there is no difficulty, of course, in someone knowing in 2075 that a certain proposition was already true in 1900 but that no one knew that even now.

So, if we had discussed this with David Lewis, what should we have said? To those of us who are a-metaphysical, the possible world machinery is just for modeling, to capture inference patterns in language. So we have no difficulties with the weirdness of (*) or (**). We can live with it.

Some say they can’t. They say their understanding is constrained by a reasonable metaphysics about the reality of other worlds. We have no help to offer these unfortunates …

So, to summarize, the impact of Fitch’s paradox is, on my diagnosis, due to an equivocation. When we say that it is possible for someone to know that something is true which no one knows, what do we mean? Is the word “actual” there implicitly or not, does the possible knowledge refer to us actual knowers, or does it refer to all possible knowers? In the first case we have a real possibility related to our world:

there are truths which are actually unknown, and they are knowable

In the second case, there are no unknown truths, but that is something that is not a fact about actual knowledge. And surely it is the actual knowledge, rather than any other sort of knowledge, that we, inhabitants of this actual world, like best.

Does this feel like a satisfying response to the Knowability Paradox?

SOURCES

David K. Lewis, “Anselm and Actuality”, Nous 4 (2) (May 1970): 175-188; “Attitudes de dicto and de se, The Philosophical Review 88 (4) (1979): 513-543.

Hans Kamp, Tense Logic and the Theory of Linear Order. Dissertation, Ucla (1968); “Formal properties of ‘now'”, Theoria 37(3) (Dec 1971): 227-273.

APPENDIX

Would someone who takes other worlds seriously, in a metaphysical way, ever allow that a denizen in one world could know things about denizens in some other world?

Yes. Quite a long time ago I came across a periodical called The Fortean Times, in which I found some disturbing news. One article broached the question: why are psychics’ predictions so often false? The answer offered was: actually their predictions are typically true! But often, what they are true about, is what is to happen in other worlds than our own.