Friday, June 29, 2007

"Probabilistic Semantics for Epistemic Modals" Moritz Schulz, New College oxford

The Thesis

It is prima facie plausible that “might” and “must” express a certain kind of
epistemic modality. We shall assume that “might” expresses some kind of epistemic possibility and that “must” can be used to express a corresponding kind
of epistemic necessity. Evidence for the view that “might” and “must” provide
an interdefinable pair of modals can be gained from examples like the following:

(1) They might be away.
(2) No, they must be at home.

It seems to be intuitively plausible that (2) is the negation of (1). In what
follows, we shall thus assume that “might” and “must” are interdefinable (in
their epistemic usages). In our informal discussion, we shall be mainly concerned
with “might” because it seems to be less ambiguous than “must”. Besides, we
shall only deal with indicative or present tense usages of “might”.
A good starting point for our investigation into the semantics of “might” is
the following observation:
(Basic Observation)
We are prepared to assert “It might be that X” iff our credence in X
is positive, i.e. iff C(X) > 0.
If one is not sure that they are away, one is in a position to say “They might
be at home”. And if one is sure that they are away, one should reject that they
might be at home. Moreover, the basic observation seems to provide a good
explanation why it would be an odd thing to say
(3) They might be at home, but I am certain that they are away.
Now, assuming that the basic observation gets the assertability conditions of
“might”-statements about right, we can go on and ask what our credence in a
“might”-statement should be. Since assertability goes by high credence, it follows
from the basic observation that our credence in a “might”-statement should be
high iff our credence in the embedded statement is non-zero. Actually, I would
like to argue for a more definite thesis:
(MIGHT)
Our credence in “It might be that X” should be 1 iff our credence
in X is positive and it should be 0 iff our credence in X is 0. Thus
C(Might X) = 1 iff C(X) > 0 and C(Might X) = 0 iff C(X) = 0 for
C being any reasonable credence function.

My main reason for proposing MIGHT stems from the observation that beliefs
in “might”-statements do not come in degrees. Rather, it seems to be an all or-
nothing matter. For instance, we usually do not qualify a “might”-statement
with a phrase such as “probably” which can be used to compare the likelihood
of statements. It is rather odd to say
(?) Probably they might be at home.
Also, we do not say that one “might”-statement is more likely than another:
(?) It is more likely that they might be at home than it is that they might be
away.
All this is evidence for allowing a “might”-statement to receive only two values.
In addition, it is hard to see what kind of further evidence (over and above nonzero
credence) would be needed in order to be certain about a “might”-statement.
It seems to be enough to give the embedded statement some (subjective) chance
of being true. So, it seems that one can make a good case for MIGHT.
Given our assumption about the interdefinability of “might” and “must”, our
thesis has a natural counterpart.
(MUST)
Our credence in “It must be that X” should be 1 iff our credence
in X is 1 and it should be zero otherwise. Thus C(Must X) = 1 iff
C(X) = 1 and C (Must X) = 0 iff C(X) < 1 for C being any reasonable
credence function.

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5 Comments:

Anonymous Anonymous said...

” My main reason for proposing MIGHT stems from the observation that beliefs
in “might”-statements do not come in degrees. Rather, it seems to be an all or-
nothing matter.”


I would argue for the opposite conclusion.

“Must” expresses impossibility: it is black or white. There is no middle ground.

Something Must or Must-not be the case.

However, “Might”, according to ordinarily linguistic usage, in fact expresses possibility and therefore uncertainty.

E.g. “It might be the case that I shall be going out this evening”.

This expresses the possibility that I may go out.

But might and possibility are fairy interchangeable in ordinary linguistic usage. As such, ‘possibility’ and ‘might’ comes in varying degrees.

“I might be going out with friends this evening” expresses the bare possibility. But if you were arranging to meet someone this evening at a pub or a restaurant and you told them that you “might” be attending, I seriously doubt whether they would find this satisfactory, especially if one of you had already made a reservation at the restaurant!

Your potential dining companion would insist that you elaborate, by stating the degree of likelihood of your attendance at the said restaurant:

This would need to be modified as follows:

“It is highly likely that I might go to the restaurant”.

Or

“It is highly unlikely that I might go to the restaurant”.

The point I’m making is that “might”, on its own, is pretty redundant, both w.r.t ordinary linguistic usage as to a practicing philosopher.

Why? Because it is far too bare and ambiguous to have any practical epistemic or metaphysical import.

“Must”, on the other hand, is an absolute term, such as the term flat or black or white.

E.g. Something can be black [total absence of light], white [all the wavelengths in the colour spectrum] or flat [with zero-degree curvature]. Full stop.

“I must be going to the restaurant this evening” expresses metaphysical and epistemic certainty, as ‘must’ and ‘certainty’ go hand in hand and can be exchanged in a proposition salva veritate.

E.g. “It must be the case that I shall be at the restaurant” = “It is certain that I shall be at the restaurant”.

This is binary logic in its crudest layman form: Must = certainty = Pr(1.0).

Now here’s my real problem:

I do agree with you that “might”, on its own, without qualification is a binary term: but on its own it is an impotent word without any real use.

It merely expresses an equiprobable possibility.

E..g “It might rain” or “It might not rain” are equivalent.

So if a statement and its negation are logically equivalent, then it begs the question:

What is the point in including a term that is truth-value invariant….?

I would conclude that such a truth-value invariant term is of no use.

R.

3:49 PM  
Anonymous Anonymous said...

I might agree with R on this one, though it is probably inexpedient to use expressions of intentions as examples.
"I might go to the restuarant" is not a statement of opinion, but of intention. It is interdefinable with "I will go to the restuarant" Not "I must go to the restuarant."

However I think the general point stands. I will go further and say neither "must" nor "might" are all or nothing.
Here's why.
It seems correct in some contexts to say "They might have spontaneously combusted". Suppose it was presupposed in a discourse that "they might have spontaneously combusted"
If someone in the discourse to say
"They must be at home." It would not contradicted by
"No, they might have spontaneously combusted." Even though credence > 0 in the latter is incompatible with a credence 1 in the former.

Jonny

5:07 PM  
Anonymous Anonymous said...

To clarify that last comment, I am thinking of cases where non relevant alternatives render a must statement false according to the proposal, but in reality would fail to contradict a must statement. This would indicate a minimum credence threshold for "might ~ p" under which is does not effect the truth of "must p". The implication being that "they must be at home" is consistent with < 1 credence in "they are at home"
Jonny

2:13 PM  
Anonymous Anonymous said...

Maybe I can clarify things a little further.

Firstly, I still find it odd to say "Probably I might go to the restaurant". If people say things like this at all, I would guess they just mean "Probably I will go to the restaurant". So, I am still inclined to say that we do not qualify epistemic usages of `might' with epistemic usages of `probably' (and similar notions). This can be explained, of course, if one assumes that `might'-statements assume only two credential values. Saying `probably A' if you have credence 0 in A seems to be inadequate. And saying it if you are certain that A seems to be pragmatically misleading: you could have benn more informative.

Secondly, I don't think that `might' is redundant (and one shouldn't think this if one doesn't think that `must' is redundant, since they seem to be interdefinable in their modal usages). It is clearly a weaker statement to state `Might A' than to state `A'. But still, `Might A' is non-trivial: it is an informative statement in many situations.

Hope this helps. Best, Moritz

6:55 PM  
Anonymous Jonny said...

It's all similar to De Finetti's definition of Certain, Possible and Impossible. Since he didn't believe in objective probability, these define the range of possibility, certain = 1, impossible = 0 and possible is any other value <1 >0. "Must" can be thought of as redundant on this view if you use the assertion symbol. For example:
"He must be in the Kitchen" doesn't assert anything over an above
"He is in the kitchen."
I agree that "he probably might be in the kitchen" is odd, but the problem with analysing "Might" as straightforwardly being reducible to the speaker's credence is that when people hear new information the admit error in their previous Might statements.
eg:Alf:
"He might be in the kitchen"
Beth: "No, I just looked in there."
Alf can't then retort,
"Yes, but I was right, he might have been in the kitchen."

10:36 AM  

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