Sunday, January 24, 2010

Inference and evidence

Just a quick post on a familiar topic. I've been working on a paper where I say that one of the problems with E=K is that, intuitively, it does not seem that inference is a process for gathering or acquiring new evidence. Inference is a way of extending our knowledge by using old evidence to settle new questions. If I know p and deduce p v q and then deduce (p v q) v r, I don't think I've just added some new evidence to my stock of evidence but I know things I mightn't have not known before. And, if I deduce ~ p --> q, that's not more evidence but it's something I know that I've not known before.

One objection to this is that I'm just working with a very narrow conception of evidence, W is working with a broader conception of evidence, and there's little here that gives us reason to revise E=K. Fair enough. The problem cannot be that there's redundant evidence on E=K, there's going to be redundant evidence on views that restrict our evidence to things we know non-inferentially because we can know the same thing non-inferentially in different ways. Moreover, there is a perfectly good sense in which things we know via competent deduction are reasons for belief. It's because I knew p v q that I could knowingly deduce (p v q) v r and deduce ~p --> q. There's no real difference between bits of evidence and reasons for belief, so there's really nothing to the objection.

Okay, but I still think we ought to distinguish evidence (or ultimate evidence) from derivative reasons for belief. Suppose I see that there's been a fox in the garden and suppose I then see that the fox in the garden ate blueberries off of the bushes. I then read that there's never been an observed case of male foxes that eat blueberries, but there's solid evidence that suggests that female foxes just love the things. So, let's say I have good inductive evidence, e, for the belief:

(1) There's been a female fox in the garden.

Now, suppose that my inductive evidence is strong enough that I can be said to know (1). Because I know that it is analytic that female foxes are vixens, I can knowingly deduce from (1):

(2) A vixen has been in the garden.

Now, I deduce that since (1) is deducible from (2), (1) is at least as well supported by what I know as (2) is.

Here's what I'd like to say. I'd like to say that the evidential probability of (2) is neither less nor greater than (1). I'd like to say that the evidential probability of (1) prior to believing (2) is less than 1. I don't think I can say these things if E=K is true. If among the propositions included in my evidence is (1), then (2) has the evidential probability of 1. So, what about (1)? If I know full well where (1) came from, the evidential probability for that should be lower than 1 and remain that way, even when I realize that (1) is a deductive consequence of (2) and so realize that (1) is deducible from (2) and so as well supported by what I know as (2) is.

One of the advantages of the view that limits evidence to what you know non-inferentially is that you can say what I'd like to say. If our evidential probabilities are determined by conditionalizing on our evidence and our bodies of evidence include everything E=K says, you get the unfortunate result that the negation of (1) is inconsistent with your evidence even if your evidence for (1) consists of a circular argument that cites (1) itself and inductive grounds that are consistent with (~1).


Christopher Cloos said...

Hi Clayton...Interesting idea. I might, however, caution against making the blanket statement that, “intuitively, it does not seem that inference is a process for gathering or acquiring new evidence.” While this might seem intuitively true in the case of bottom-up reasoning, it does not seem intuitively true in the case of top-down reasoning. When using abductive reasoning to move from true observational evidence to the best explanation of that evidence [i.e. (1) in the fox example] it seems intuitively true that no new evidence has been generated; we have simply moved from what we knew to what we didn’t know (or, what we knew has been summarized, explained, or extended in a new way). But, this isn’t the only direction inference can proceed.

It's possible to generate new (potential) evidence from a hypothesis. Imagine Fred is a field biologist surveying a plot of land to determine the frequency of female foxes in the area. A reasonable hypothesis for Fred to postulate based on prior survey data and what he knows of the area is that:

(A) There’s been a female fox in the meadow.

Fred notices a bush of wild blueberries and knows that: (i) female foxes love blueberries, and (ii) there’s never been an observed case of male foxes eating blueberries. Fred inspects the bush and observes that a large portion of the blueberries have been eaten. He reasons:

(B) A female fox has eaten blueberries off the bush.

Fred looks around the bush and identifies footprints in the dry dirt that are unequivocally fox prints. Fred reasons:

(C) There’s been a fox near the blueberry bush.

Putting (B) and (C) together Fred reasons that:

(D) There’s been a female fox eating blueberries off the bush.

Based on potential evidence (D) Fred hides at a safe distance and monitors the bush using binoculars. Sure enough, after time has gone by, Fred spots a fox eating blueberries off the bush. Fred begins taking a count of female foxes on this plot of land to predict and forecast the density of the female fox population in the area.

Intuitively, it seems (D) is new evidence. That is, using (A), new evidence (D) has been generated. It seems unintuitive to say that inference from (A) to (D) is not a way of generating new evidence, especially when (D) provides evidence that Fred uses to predict the density of the female fox population in the area. Top-down reasoning is not simply telling Fred something he didn’t know before (though it’s doing that as well), it’s also providing Fred with new evidence that increases the likelihood of his hypothesis (A) and provides Fred with a tool for making forecasts and predictions.

Shalla said...

It seems to me if Fred were a biologist, Fred would not reason (B). He would reason, “It is probable that (B),” and hypothesize (B). Fred would reason (C), and putting "it is probable that (B)" and (C) together, Fred would hypothesize (D).

If Fred spots a fox eating blueberries off the bush, Fred still cannot reason that the fox eating blueberries of the bush is a female. That is, it would be a rookie mistake to assume that, because a male fox has never been seen eating blueberries, male foxes do not and will not eat blueberries off the bush. If Fred's count of female foxes (based on determination of gender by some criterion other than whether or not they are seen eating blueberries) on this plot is markedly high compared to male foxes on this plot of land such that the number of male foxes on this land is statistically negligible, Fred can then promote (D) from a hypothesis to a theory. He still, however, cannot accept hypothesis (D) because he cannot effectively reject the null hypothesis.

If Fred were to use (D) as evidence to "predict the density of the female fox population in the area," i.e., assume that every fox eating blueberries off the bush is female solely based on the fact that they are eating blueberries off the bush and male foxes have not been seen doing so, then Fred is a poor biologist. Fred has, at best, an extrapolation of data based on an assumption, i.e., a shaky guess.

Anonymous said...

A vixen has definitely been in the garden.

Christopher Cloos said...

The general point you raise is a version of the problem of induction: there’s no guarantee that the future will reflect the past. I don’t think the problem of induction needs to be solved before Fred can make reasonable inferences. It is reasonable for Fred to infer that a fox eating blueberries is a female even though it is possible that Fred is the first person to be witnessing a male fox eating blueberries. It is also reasonable for Fred to infer that missing blueberries (coupled with other evidence/background knowledge) indicates the prior presence of female foxes. Even if male foxes only eat blueberries off bushes when no one is watching Fred’s estimation is made in light of his background knowledge. It would be odd (irrational?) to require Fred to adjust his credence in a proposition according to an event that has never been witnessed nor could ever be observed.

Your more specific point is that (D) is not really evidence: it’s just another hypothesis, hunch, or extrapolation of data. I’m OK with saying the real evidence is the empirical observation and (D) is a hypothesis supported by the
evidence. One way of understanding evidence is in terms of what increases the probability of a hypothesis. Rejecting the revision made above would require saying: Fred perceiving a fox eating blueberries off the bush is not evidence (i.e. does not increase the probability) that there’s been a female fox eating blueberries off the bush. Even if the increase in probability is slight it seems reasonable to say that Fred’s observation is evidence, which returns me to my general point: that new evidence can be discovered using top-down reasoning--moving from hypotheses to the discovery of new data. Then, after data (evidence) is discovered, the bottom-up reasoning can begin anew.