Friday, December 9, 2011

Estimates, assertions and vagueness

I ask you to give me an estimate of how long a table is. You say "950 mm".

What did you do? You didn't assert that the table was 950 mm. Did you assert that you estimated the table at 950 mm? Maybe, but I think that's not quite right. After all, you might not have yourself estimated the table at 950 mm—you might have gone from your memory of what someone else said about it. So are you asserting that someone has estimated the table at 950 mm? No. For if someone had estimated the table at 700 mm and you could see that it wasn't (relevantly) near that, it wouldn't be very good for you to answer "700 mm", though it would be true that someone has estimated the table at 700 mm. Maybe you 're saying that the best estimate you know of is 950 mm. But that's not right either, because the best estimate you know of might be 950.1 mm.

Here is a suggestion. Giving an estimate is a speech act not reducible to the assertion of a proposition. It has its own norms, set by the context. The norm of assertion is truth (dogmatic claim): it is binary. But the norm of an estimate is not a binary yes/no norm as for assertion, but it can often be thought of as a continuous quality function. The quality function is defined by what it is that we are estimating and the context of estimation (purposes, etc.) Typically, the quality function is a Gaussian centered on the true value, with the Gaussian being wider when less precision is required. But it's not always a Gaussian. There are times when one has a lot more tolerance on one side of the value to be estimated—where it is important not to underestimate (say, the strain under which a bolt will be) but little harm in overestimating by a bit. In such cases, we will have an asymmetrical quality function. (This is also important for answering the puzzles here.) So in giving an estimate one engages in an act governed by a norm to give a higher quality result—but with a defeasibility: brevity can count against quality (so, you can say "950 mm" even if "950.1 mm" has slightly higher quality).

Moreover, what exactly is the quality function may depend on all sorts of features other than the exact value of the quantity being estimated. Thus, if you hand me a box with one cylindrical object in it and you ask me to give a good estimate of its diameter, how much precision is called for—i.e., how wide the Gaussian is—will depend on what the object is. If it is a gun cartridge, the Gaussian's width will be proportional to the tolerances on the relevant kind of gun; if it is an irregular hardwood dowel, the Gaussian's width will be significantly wider. So, in general, the quality function for an estimate that some quantity is x depends on:

  • what quantity x is being given
  • what the correct value is
  • other properties of what is being estimated
  • the linguistic context.
The second and third items can be subsumed as "the relevant bits of the extra-linguistic world".

So, here's a very abstract theory of estimates. Estimating is a language game one plays where the quality function keeps score. When one is asked for an estimate (or offers it of one's own accord), the context c sets up a function qc from pairs <x,w> to values, and one's score in the game is qc(x,@) where x is the value one gives and @ is the actual world.

Notice that this is general enough to encompass all sorts of other language games. For instance, the quantities need not be numbers. They might be propositions, names, etc. Assertion is a special case where the quantities are propositions, and qc(x,w) is "acceptable" when x is true at w and is "unacceptable" otherwise. Or consider the game initiated by this request: "Give me any approximate upper bound for the number of people coming to the wedding." The quality function qc(x,w) is non-decreasing in x: Because of the "any", saying "a googol" is just as good as saying "101", as long as both are actually upper bounds. Thus qc(x,w) is "perfect" in any world w where no more than x people come to the wedding. In worlds w where more than x people come to the wedding, qc(x,w) quickly drops off as x goes below the actual number of people coming to the wedding.

"Quantities" can be anything. They might be abstracta or they might be linguistic tokens. It doesn't matter for my purposes. Likewise, the values given out by the quality function could be numbers, utilities, or just labels like "perfect", "acceptable" and "unacceptable".

Conjecture: Assertoric use of sentences with vague predicates is not the assertion of a proposition but it is the offering of an estimate.

For instance, take as your quantities "yes" and "no", and suppose the context is where we're asked if Fred is bald. Then the quality function will be something like this: qc("yes",w) is less in those worlds w where Fred has more hair, and qc("no",w) is more in those worlds where Fred has less hair. Moreover, qc("yes",w) is "perfect" in worlds where Fred has no hair.

What if I am not asked a question, but I just say "Fred is bald"? The same applies. My saying is not an assertion. It is, rather, the offering of an estimate. We can take the quantities to be binary—say, "Fred is bald" and "Fred is non-bald"—but the quality function is non-binary.

What about more logically complex things, like "If Fred is blond, he is bald"? Well, formally treat qualities as truth values in a multivalent logic, but in the semantics, don't think that they are in fact truth values. They are quality values. So, assign qualities to sentences (keeping a context fixed), using some natural rules like:

  • qc("a or b",w) = max(qc("a",w),qc("b",w))
  • qc("a and b",w) = min(qc("a",w),qc("b",w))
  • qc("~a",w) = "perfect" − qc("a",w)
The rules may actually differ from context to context. That's fine, because this is not logic per se: this is the evaluation of quality (and that's how this approach differs from non-classical logic approaches to vagueness—maybe not formally, but in interpretation). Moreover, there may in some contexts be no assigned quality value to a particular sentence. Again, that's fine: there can be games with underdetermined rules.

In a nutshell: A vague sentence is an estimate of how the world is. Such sentences are not to be scored on their truth or falsity, but on the quality of the estimate.


Heath White said...

I like the idea of estimating as a speech act. I don't like the idea of sharply distinguishing it from assertion, because I suspect at least 80% of what we say is vague, so we'd be estimating all the time.

Alexander R Pruss said...

Assertion proper ends up being a limiting case.

Jonathan Livengood said...

Why can't we say that estimations are assertions and then say that your quality functions are the correct way to measure or evaluate how good an assertion is?

I guess I'm confused about what one's choice of measure for evaluating the quality of a speech act has to do with the nature of the speech act itself. Let me illustrate. Suppose I have two scales that measure in grams. One of them is sensitive to 0.1g. The other is sensitive to 0.001g. My friend hands me a brick. Surely the nature of the brick does not depend on which scale I use to measure its mass ... or does it?

(Incidentally, what's the difference between a quality function and a loss function?)

Heath White said...

One way to read your post is as saying that we can put the vagueness into the speech act rather than the content of an assertion. That is, rather than trying to figure out what we’re asserting when we assert “The table is about 950mm long” (small table!) we just say we are not *asserting* anything; we are *estimating* that the table is 950mm long. And then we say that assertion is estimation with zero tolerance for error.

It seems to me that this won’t work because there are any number of other propositional attitudes toward vague contents. We can believe the table is about 950mm long, wonder whether it is about 950mm long, intend the table to be about 950mm long (when cutting the parts), etc.

If there is some general way to treat propositional attitudes as having some flexibility in them (continuous quality functions, or whatever) then we should just apply that to assertion too. If there is not some general method then we have made assertion a rather odd duck.

(Analytic philosophy got its start when logicians started rethinking “the general form of the proposition,” i.e. when they stopped thinking all propositions were of the subject-predicate form and recognized relations, quantification, etc. I have a dream that someday logicians will *start* from the idea that our thought and talk is vague, not excluded-middle, and re-think logic on that basis. Who knows what might happen.)

Alexander R Pruss said...

Good point: so this suggests that one may need to generalize all sorts of other propositional attitudes. And that seems prima facie plausible to me.

But on reflection, maybe we don't need to in the case of all the other attitudes, because maybe it's already been done. For instance, take desire. If I desire to be bald, I have a utility function rather resembling the quality function of an assertion that I am bald. And utility functions are old hat, and maybe even can be defined in terms of propositions.