7/17/2018

'Extra-Weak' Underdetermination

I’ll start briefly with a few reminders and fix a little terminology. I then introduce a new(?) sub-species of underdetermination argument, whose premises are logically weaker than existing underdetermination arguments, but still(?) deliver the anti-realist’s conclusion.

Underdetermination arguments in philosophy of science aim to show that an epistemically rational person (= someone who weighs their available evidence correctly) should suspend belief in the (approximate) truth of current scientific theories, even if such theories make very accurate predictions.

A scientific theory T is strongly underdetermined = T has a genuine competitor theory T*, and T and T* make all the same observable predictions. (So the two theories’ disagreement must concern unobservable stuff only.)

A scientific theory T is weakly underdetermined = T has a genuine competitor theory T*, and all the observable data/ evidence gathered thus far is predicted equally well by both T and T*.(†) (So collecting new data could end the a weak underdetermination situation.)

Anti-realists then argue from the purported fact that (all/most) of our current scientific theories are undetermined, to the conclusion that an epistemically rational person should suspend belief in (all/most) of our current scientific theories.

Realists can reasonably respond by arguing that even weak underdetermination, in the above sense, is not common: even if one grants that there is an alternative theory T* that is consistent with the data gathered so far, that certainly does not entail that T and T* are perfectly equally supported by the available data is unlikely. There is no reason to expect T and T* would be a perfect ‘tie’ for every other theoretical virtue besides consistency with the data. (Theoretical virtues here include e.g. simplicity, scope, relation to other theories, etc.) The evidential support for a hypothesis is not merely a matter of the consistency of that hypothesis with available data.

At this point, anti-realists could dig in their heels and simply deny the immediately preceding sentence. (The other theoretical virtues are ‘merely pragmatic,’ i.e. not evidential.) But that generates a standoff/stalemate, and furthermore I find that response unsatisfying, since an anti-realist who really believes that should probably be a radical Cartesian skeptic (yet scientific anti-realism was supposed to be peculiar to science).

So here’s another reply the anti-realist could make: grant the realist’s claims that even weak determination is not all that common in the population of current scientific theories, and furthermore that typically our current theory T is in fact better supported than any of the competitors T1, T2, ... that are also consistent with the data collected so far. The anti-realist can grant these points, and still reach the standard underdetermination-argument conclusion that we should suspend belief in the truth of T, IF the sum of the credences one should assign to T1, T2, ... is at least 0.5.

For example: suppose there are exactly 3 hypotheses consistent with all the data collected thus far, and further suppose
 Pr(T) = 0.4,
 Pr(T1) = 0.35, and
 Pr(T2) = 0.25.
In this scenario, T is better supported by the evidence than T1 is or T2 is, so T is not weakly underdetermined. However, assuming that one should not believe p is true unless Pr(p)>0.5, one should still not believe T in the above example.

I call such a T extra-weakly underdetermined: the sum of rational degrees of belief one should have in T’s competitors is greater than or equal to the rational degree of belief one should have in T.

We can think about this using the typical toy example used to introduce the idea of underdetermination in our classes, where we draw multiple curves through a finite set of data points:
We can simultaneously maintain that the straight-line hypothesis (Theory A) is more probable than the others, but nonetheless deny that we should believe it, as long as the other hypotheses’ rational credence levels sum to 0.5 or higher. And there are of course infinitely many competitors to Theory A, so it is an infinite sum. The realist, in response to this argument, will thus have to say that that infinite sum will converge to less than 0.5.

The above argument from extra-weak underdetermination is clearly related to the ‘catch-all hypothesis’ (in the terminology above, ~T) point that has been discussed elsewhere in the literature on realism, especially in connection with Bayesian approaches (see here and the references therein). But I think there is something novel about the extra-weak underdetermination argument: as we add new competitor theories to the pool (T3, T4… in the example above), the rational credence level we assign to each hypothesis will presumably go down. (I include ‘presumably,’ because it is certainly mathematically possible for the new hypothesis to only bring down the rational credence level for some but not all of the old hypotheses.) So the point here is not just that there is some catch-all hypothesis, which it is difficult(?) to assign a degree of rational belief to (that's the old news), but also that we increase the probability of something like the 'catch-all' hypothesis by adding new hypotheses to it. (I have to say 'something like' it, because T1, T2... are specific theories, not just the negation of T.)

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(†): Note that, unlike some presentations of underdetermination, I do not require that T and T* both predict ALL the available data. I take "Every theory is born refuted" seriously. And I actually think this makes underdetermination scenarios more likely, since a competitor theory need not be perfect -- and the imperfections/ shortcomings of T could be different from those of T* (e.g. T might be more complicated, while T* has a narrower range of predictions).

3/13/2018

cognitive impenetrability of some aesthetic perception

For me, one of the interesting experiences of getting older is seeing, from the internal first-person perspective, many of the generalizations one hears about 'getting older' come true in my own life. One of the most obvious/ salient ones for me is about musical tastes. I love a lot of hip hop from the early-to-mid 90's. (This is probably still my favorite hip hop album of all time.) I do also like some of the stuff that is coming out now, but on average, the beats in particular just sound bad to me. In particular, the typical snare sound -- I can't get over how terrible and thin it sounds.

But on the other hand, I know full well that, as people get older, they start thinking 'Young people's music today is so much worse than when I was a kid!' And that I heavily discounted old people's views about new music when I was in school.

Yet this makes absolutely no difference to my perceiving the typical trap snare sound today as really insubstantial and weak -- just ugly. The theoretical knowledge makes zero difference to my experience.

This reminded me of Fodor's famous argument from the Müller-Lyer illusion* for the cognitive impenetrability of perception. No matter how many times I am told that the two horizontal lines are the same length, no matter how many times I lay a ruler next to each line in succession and measure them to be the same length, I still perceive one line as shorter than the other. My theoretical knowledge just can't affect my perception. In a bit of jargon, the illusion is mandatory.

My experience of the typical hip hop snare sound today is similarly mandatory for me, despite the fact that I know (theoretically/ cognitively) that, as an old person, I should discount my aesthetic impressions of music coming out today.

This seems like it could make trouble for a Fodorian who wants to use the mandatoriness of illusions as an argument that perception is unbiased/ theory-neutral -- in a conversation about the best hip hop albums of all time, my aesthetic data would extremely biased towards stuff that came out between 1989-1995.

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*(Have you seen the dynamic Müller-Lyer illusions? Go here and scroll down to see a few variants.)

12/01/2017

Morals and Mood (Situationism vs virtue ethics, once more)


Given how much has been written in the last couple of decades about the situationist challenge to virtue ethics, I'm guessing someone has probably already said (something like) this before. But I haven't seen it, and I'm teaching both the Nicomachean Ethics and the section in Appiah's Experiments in Ethics on situationism vs. virtue ethics now, so the material is bouncing around in my head.

First, a little background. (If you want more detail, there are a couple nice summaries of the debate on the Stanford Encyclopedia of Philosophy here (by Alfano) and here (by Doris and Stich).) The basic idea behind the situationist challenge to virtue ethics is the following: there are no virtues (of the sort the virtue ethicist posits), because human behavior is extremely sensitive to apparently minor -- and definitely morally irrelevant -- changes in the environment. For example, studies show that someone is MUCH more likely to help a stranger with a small task (e.g. picking up dropped papers, or making change for a dollar) outside a nice-smelling bakery than a Staples office supply store, or after finding a dime in a pay phone's coin return, or in a relatively quiet place than a relatively loud one. The Situationist charges that if the personality trait of generosity really did exist, and was an important driver of people's behavior, then whether people perform generous actions or not would NOT depend on tiny, morally irrelevant factors like smelling cinnamon rolls, finding a dime, or loud noises. A virtue has to be stable and global; that behavior can change so much in response to apparently very minor environmental changes suggests that there is no such stable, global psychological thing contributing significantly to our behavior. That's the basic Situationist challenge.

Defenders of Virtue Ethics have offered a number of responses to this situationist challenge (the SEP articles linked in the previous paragraph describe a few). Here is a response that I have not personally seen yet: a person's mood is a significant mediator between the minor situational differences and the tendency to help a stranger. When we describe the experimental results as a correlation between a tiny, apparently unimportant environmental change and a massive change in helping behavior, then the experimental results look very surprising -- perhaps even shocking. But it would be less surprising or shocking, if instead of thinking of what's going on in these experiments as "The likelihood of helping others is extremely sensitively attuned to apparently trivial aspects of our environment," we rather think of what's happening as "All these minor environmental changes have a fairly sizeable effect on our mood." For to say that someone in a particularly good mood is much more likely to help a stranger is MUCH less surprising than "We help people outside the donut shop, but not outside the Staples." In other words: if mood is a major mediator for helping behaviors, then we don't have to think of our behaviors as tossed about by morally irrelevant aspects of our environments. That said, we would have to think of our behavior as shaped heavily by our moods -- but I'm guessing most people would probably agree with that, even if they've never taken a philosophy or psychology class.

Now, you might think this is simply a case of a distinction without a difference, or "Out of the frying pan, into the fire": swapping smells for moods makes no real difference of any importance to morality. I want to resist this, for two reasons; one more theoretical/ philosophical, and the other more practical.

First, the theoretical reason recognizing that mood is a mediator in these experiments matters: I don't think a virtue ethicist would have to be committed to the claim that mood does not have an effect on helping behaviors. Virtue ethicists can agree that generosity should not be sensitive to smells and dimes per se. However, the fact that someone who is in a better mood is (all else equal) more likely to help strangers than someone in a worse mood is probably not devastating evidence against the virtue ethicist's thesis that personality traits (like generosity) exist and play an important role in producing behavior.

Second, more practically: one concern (e.g. mentioned by Hagop Sarkissian here) about the Situationist experimental data in general is that often times we are not consciously aware of the thing in our situation/environment that is driving the change in our behavior (I may not have noticed the nice smells, or the absence of loud noises). But mood is different: I have both better access to my mood, and a baseline/ immediate knowledge that my mood often affects my behavior. Whereas given the situationist's characterization of the data, I often don't know which variables in my environment are causing me to help the stranger or not. So if I am in a foul mood, and realize I am in a foul mood, I could potentially consciously 'correct' my automatic, 'system-1' level of willingness to help others.

Of course, on this way of thinking about it, i.e. mood as mediating, I often won't know what is CAUSING my good mood. But that's OK, because I will still be able to detect my mood (usually -- of course, sometimes we are sad, or angry, or whatever, without really noticing it. But my point is just that we are better detectors of our current mood than we are of the various elements of our environment that could potentially be influencing our mood positively or negatively).

So in short: I think the situationist's challenge to virtue ethics is blunted somewhat if we think of mood as a mediator between apparently trivial situational variables and helping behaviors.

6/22/2017

Tarski, Carnap, and semantics

Two things:

1. Synthese recently published Pierre Wagner's article Carnapian and Tarskian Semantics, which outlines some important differences between semantics as Tarski conceived it (at least in the 1930s-40s) and as Carnap conceived it. This is important for anyone who cares about the development of semantics in logic; I'd been hoping someone would write this paper, because (a) I thought it should be written, but (b) I didn't really want to do it myself. Wagner's piece is really valuable, in my opinion. And not merely for antiquarian reasons: many today have the feeling that semantics in logic (roughly: model theory) is the natural/ inevitable way to come at semantics in logic. But how exactly to pursue semantics was actually very much up for debate and in flux for about 20 years after Tarski's 1933 "On the Concept of Truth in Formalized Languages." And the semantics in that monograph is NOT what you would find in a logic textbook today.

2. I am currently putting the finishing touches on volume 7 of the Collected Works of Rudolf Carnap, which is composed of Carnap's three books on semantics (Introduction to Semantics, Formalization of Logic, and Meaning and Necessity). There is a remark in Intro to Semantics that is relevant to Wagner's topic, which Wagner cited (p.104), but I think might be worth trying to investigate in more detail. Carnap writes:

our [= Tarski's and my] conceptions of semantics seem to diverge at certain points. First ... I emphasize the distinction between semantics and syntax, i.e. between semantical systems as interpreted language systems and purely formal, uninterpreted calculi, while for Tarski there seems to be no sharp demarcation. (1942, pp. vi-vii)

I have two thoughts about this quotation:
(i) Is Carnap right? Or did he misunderstand Tarski? (Carnap had had LOTS of private conversations with Tarski by this point, so the prior probability I assign to me understanding Tarski better than Carnap does is pretty low.)
(ii) If Carnap is right about Tarski on this point, then (in my opinion) we today should give much more credit to Carnap for our current way of doing semantics in logic than most folks currently do. We often talk about 'Tarskian semantics' today as a short-hand label for what we are doing, but if there were 'no sharp demarcation' between model theory and proof theory (i.e. between semantics and syntax), then the discipline of logic would look very different today.

3/30/2017

Against Selective Realism (given methodological naturalism)

The most popular versions of realism in the scientific realism debates today are species of selective realism. A selective realist does not hold that mature, widely accepted scientific theories are, taken as wholes, approximately true---rather, she holds that (at least for some theories) only certain parts are approximately true, but others parts are not, and thus do not merit rational belief. The key question selective realists have grappled with over the last few decades is: which are the 'good' parts (the "working posits," in Kitcher's widely used terminology) and which are the 'bad' parts (the "idle wheels") of a theory?

An argument against any sort of philosophical selective realism just occurred to me, and I wanted to try to spell it out here. Suppose (as the selective realist must) there is some scientific theory that scientists believe/ accept, and which according to the selective realist makes at least one claim (call it p) that is an idle wheel, and thus should not be rationally accepted.

It seems to me that in such a situation, the selective realist has abandoned (Quinean) methodological naturalism in philosophy, which many philosophers---and many philosophers of science, in particular---take as a basic guideline for inquiry. Methodological naturalism (as I'm thinking of it here) is the view that philosophy does not have any special, supra-scientific evidential standards; the standards philosophers use to evaluate claims should not be any more stringent or rigorous than standards scientists themselves use. And in our imagined case, the scientists think there is sufficient evidence for p, whereas the selective realist does not.

To spell out more fully the inconsistency of selective realism and methodological naturalism in philosophy, consider the following dilemma:
By scientific standards, one either should or should not accept p.

If, by scientific standards, one should not accept p, then presumably the scientific community already does not accept it (unless the community members have made a mistake, and are not living up to their own evidential standards). The community could have re-written the original pre-theory accordingly to eliminate the idle wheel, or they could have explicitly flagged the supposed idle wheel as a false idealization, e.g. letting population size go to infinity. But however the community does it, selective realism would not recommend anything different from what the scientific community itself says, so selective realism becomes otiose ... i.e., an idle wheel. (Sorry, I couldn't help myself.)

On the other hand, if, by scientific standards, one should accept p, then the selective realist can't be a methodological naturalist: the selective realist has to tell the scientific community that they are wrong to accept p.
I can imagine at least one possible line of reply for the selective realist: embrace the parenthetical remark in the first horn of the dilemma above, namely, scientists are making a mistake by their own lights in believing p. Then the selective realist would need to show that there is a standard operative in the scientific community that the scientists who accept p don't realize should apply in the particular case of p. But that may prove difficult to show at this level of abstraction.

2/11/2017

Confirmation holism and justification of individual claims

How do epistemological(/confirmation) holists think about the justification of the individual claims that compose the relevant ‘whole’?

Epistemological holism or confirmation holism, I take it, holds that sentences cannot be justified or disconfirmed in isolation. In other words, we can only fundamentally justify or disconfirm sufficiently large conjunctions of individual claims. What counts as ‘sufficient’ depends on how big a chunk of theory you think is needed to be justifiable: some holists allow big sets of beliefs to be confirmed/justified even if they are proper subsets of your total belief-set. (I will use 'individual claim' to mean a claim that is 'too small' i.e. 'too isolated' to admit of confirmation, according to the holist.)

I’m guessing confirmation holists also think that the individual claims that make up a justified whole are themselves justified. (If holists didn’t think that, then it seems like any individual claim a holist made would be unjustified, by the holist’s own lights, unless they uttered it in conjunction with a sufficiently large set of other utterances.) The individual claims are justified, presumably, by being part of a sufficiently large conjunction of claims that are fundamentally/ basically justified. Individual claims, if justified, can only be derivatively justified.

Presumably, if one believes that ‘A1 & A2 & … & An’ (call this sentence AA) is justified, then that person has (or: thinks they should have) a rational degree of belief in AA over 0.5.

But now I have questions:

(1) How does a holist get from the degree of belief she has in AA, to the degree of belief she has in a particular conjunct? There are many, many ways consistent with the probability calculus to assign probabilities to each of the Ai’s to get any particular rational degree of belief (except 1).

(2) We might try to solve that ‘underdetermination’ problem in (1) by specifying that every conjunct is assigned the same degree of belief. This seems prima facie odd to me, since presumably some conjuncts are more plausible than others, but I don’t see how the holist could justify having different levels of rational belief in each conjunct, since each conjunct gets its justification only through the whole. (Perhaps the partial holist can tell a story to be told about claims participating in multiple sufficiently large conjunctions that are each justified?)

(3) Various ways of intuitively assigning degrees of belief to the individual conjuncts seem to run into problems:

(i) The holist might say: if I have degree of belief k in AA, then I will have degree of belief k in each conjunct. Problem: that violates the axioms of the probability calculus (unless k=1).


(ii) Alternatively, if the holist wants to obey the axioms of the probability calculus, then the rational degree of belief she will need to have in each conjunct must be VERY high. For example, if the degree of belief in AA is over 0.5, and each conjunct is assigned the same value (per (2)), and there are 100 individual conjuncts, then one’s degree of belief in each conjunct must be over 0.993. And that seems really high to me.


(iii) One alternative to that would be to say that each conjunct of a large conjunction has to be over 0.5. But then you would have to say that the big 100-conjunct conjunction is justified when your rational degree of belief in it is anything above 7.9x10-31. And that doesn’t sound like a justified sentence.


Two final remarks: First, it seems like someone must have thought of this before, at least in rough outline. But my 10 minutes of googling didn’t turn up anything. So if you have pointers to the literature, please send them along. Second, for what it's worth, this occurred to me while thinking about the preface paradox: if you think justification only fundamentally accrues to large conjunctions and not individual conjuncts, then it seems like you couldn’t run (something like) the preface paradox, since you couldn't have a high rational degree of belief in (an analogue of) the claim ‘At least one of the sentences in this book is wrong.’

9/08/2016

Defining 'inductive argument' should not be this difficult

Carnap, give me strength. I cannot define ‘inductive argument’ and ‘fallacious argument’ in a way that correctly captures the intuitive boundary between inductive and fallacious arguments.

Like (almost) everyone else, I define ‘deductively correct,' i.e. 'valid' as follows:
An argument A is deductively correct (valid) =
If all of A’s premises are true, then A’s conclusion must be true =
If all of A’s premises are true, then it is impossible for A’s conclusion to be untrue.

Now, there are two (or maybe three) definitions of ‘inductive argument’ that follow this pattern of definition.

(Definition 1: probable simpliciter)
An argument B is inductively correct =
If all of B’s premises are true, then B’s conclusion is probably true =
If all of B’s premises are true, it is unlikely that B’s conclusion is untrue

(Definition 2: more probable)
An argument C is inductively correct =
If all of C’s premises are true, then C’s conclusion is more likely to be true =
If all of C’s premises are true, the probability of the conclusion’s untruth decreases

In other words:
Definition 1: Pr(Conclusion | Premises) > 0.5
Definition 2: Pr(Conclusion | Premises) > Pr(Conclusion)

(If you think >0.5 is too low, you can pick whatever higher cutoff you like. My current problem is different from picking where to set that threshold number.)

Now I can state my problem: it looks like neither definition makes the correct classifications for some paradigm examples of fallacies.

On Definition 1, any argument whose conclusion highly probable regardless/independent of the truth or falsity of the premises, will count as inductively correct. That is, any non sequitur whose conclusion is probably true will count as inductively correct. (This is the inductive analog of the fact that a logical truth is a deductive consequence of any set of premises. But it just feels much more wrong in the inductive case, for some reason; maybe just because I've been exposed to this claim about deductive inference for so long that it has lost its un-intuitiveness?)

On Definition 2, hasty generalization (i.e. sample size too small) will count as inductively correct: suppose I talk to 3 likely US Presidential voters and all 3 say they are voting for Clinton. It is intuitively fallacious to conclude that Clinton will win the Presidency, but surely those 3 responses give some (small, tiny) boost to the hypothesis that she will win the Presidency.

But non sequiturs and hasty generalizations are both paradigm examples of fallacies, so neither Definition 1 nor Definition 2 will work.

I said above that there might be a third definition. This would simply be the conjunction of Definitions 1 and 2: If the premises are true, then the conclusion must be BOTH probable simpliciter (Def. 1) AND more probable (Def. 2). It seems like this would rule out both non sequiturs (because the truth of the premises increases the probability of the conclusion) and hasty generalizations (because the conclusion wouldn’t be probable simpliciter).

Problem solved? I don’t think it is, because there could be a hasty generalization for an argument whose conclusion is probable even if the premises are all false. Given our current background information (as of Sept. 6 2016) about the US Presidential race, the above example probably fits this description: ‘Clinton will win’ is more likely to be true than not, and the sample of three voters would boost a rational agent’s confidence in that claim (by a miniscule amount). That said, I will grant that a reasonable person might think this example is NOT a fallacy, but rather just an inductively correct argument that is so weak it is ALMOST a fallacy.

Before signing off, I will float a fourth candidate definition:

Definition 4: Pr(Conclusion | Premises) >> Pr(Conclusion)
put otherwise:
Definition 4: Pr(Conclusion | Premises) > Pr(Conclusion)+n, for some non-tiny n>0.

You could also conjoin this with Definition 1 if you wanted. This would take care of hasty generalizations. But does it create other problems? (You might object “That n will be arbitrary!” My initial reaction is that setting the line between inductive and fallacious at >0.5 [or wherever you chose to set it] is probably arbitrary in a similar way.)

Defining 'inductive argument' should not be this difficult

Carnap, give me strength. I cannot define ‘inductive argument’ and ‘fallacious argument’ in a way that correctly captures the intuitive boundary between inductive and fallacious arguments.

Like (almost) everyone else, I define ‘deductively correct,' i.e. 'valid' as follows:
An argument A is deductively correct (valid) =
If all of A’s premises are true, then A’s conclusion must be true =
If all of A’s premises are true, then it is impossible for A’s conclusion to be untrue.

Now, there are two (or maybe three) definitions of ‘inductive argument’ that follow this pattern of definition.

(Definition 1: probable simpliciter)
An argument B is inductively correct =
If all of B’s premises are true, then B’s conclusion is probably true =
If all of B’s premises are true, it is unlikely that B’s conclusion is untrue

(Definition 2: more probable)
An argument C is inductively correct =
If all of C’s premises are true, then C’s conclusion is more likely to be true =
If all of C’s premises are true, the probability of the conclusion’s untruth decreases

In other words:
Definition 1: Pr(Conclusion | Premises) > 0.5
Definition 2: Pr(Conclusion | Premises) > Pr(Conclusion)

(If you think >0.5 is too low, you can pick whatever higher cutoff you like. My current problem is different from picking where to set that threshold number.)

Now I can state my problem: it looks like neither definition will make the right classifications for some paradigm examples of fallacies.

On Definition 1, any argument whose conclusion highly probable regardless/independent of the truth or falsity of the premises, will count as inductively correct. That is, any non sequitur whose conclusion is probably true will count as inductively correct. (This is the inductive analog of the fact that a logical truth is a deductive consequence of any set of premises. But it just feels much more wrong in the inductive case, for some reason; maybe just because I've been exposed to this claim about deductive inference for so long that it has lost its un-intuitiveness?)

On Definition 2, hasty generalization (i.e. sample size too small) will count as inductively correct: suppose I talk to 3 likely US Presidential voters and all 3 say they are voting for Clinton. It is intuitively fallacious to conclude that Clinton will win the Presidency, but surely those 3 responses give some (small, tiny) boost to the hypothesis that she will win the Presidency.

But non sequiturs and hasty generalizations are both paradigm examples of fallacies, so neither Definition 1 nor Definition 2 will work.

I said above that there might be a third definition. This would simply be the conjunction of Definitions 1 and 2: If the premises are true, then the conclusion must be BOTH probable simpliciter (Def. 1) AND more probable (Def. 2). It seems like this would rule out both non sequiturs (because the truth of the premises increases the probability of the conclusion) and hasty generalizations (because the conclusion wouldn’t be probable simpliciter).

Problem solved? I don’t think it is, because there could be a hasty generalization for an argument whose conclusion is probable even if the premises are all false. Given our current background information (as of Sept. 6 2016) about the US Presidential race, the above example probably fits this description: ‘Clinton will win’ is more likely to be true than not, and the sample of three voters would boost a rational agent’s confidence in that claim (by a miniscule amount). That said, I will grant that a reasonable person might think this example is NOT a fallacy, but rather just an inductively correct argument that is so weak it is ALMOST a fallacy.

Before signing off, I will float a fourth candidate definition:

Definition 4: Pr(Conclusion | Premises) >> Pr(Conclusion)
put otherwise:
Definition 4: Pr(Conclusion | Premises) > Pr(Conclusion)+n, for some non-tiny n>0.

You could also conjoin this with Definition 1 if you wanted. This would take care of hasty generalizations. But does it create other problems? (You might object “That n will be arbitrary!” My initial reaction is that setting the line between inductive and fallacious at >0.5 [or wherever you chose to set it] is probably arbitrary in a similar way.)

3/12/2016

the no-miracles argument may not commit the base-rate fallacy

Certain philosophers argue that the No-Miracles Argument for realism (Colin Howson, Peter Lipton), the Pessimistic Induction against realism (Peter Lewis), or both arguments (P.D. Magnus and Craig Callender) commit the base-rate fallacy. I am not sure these objections are correct, and will try to articulate the reason for my doubt here.

I need to give some set-up; many readers will be familiar with some or all of this. So you can skip the next few paragraphs if you already know about the base-rate objection to the No-Miracles Argument and the Pessimistic Induction.

I suspect many readers are familiar with the base-rate fallacy; there are plenty of explanations of it around the internet. But just to have a concrete example, let’s consider a classic case of base-rate neglect. We are given information like the following, about a disease and a diagnostic test for this disease:

(1) There is a disease D that, at any given time, 1 in every 1000 members of the population has: Pr(D)=.001.

(2) If someone actually has disease D, then the test always comes back positive: Pr(+|D)=1.

(3) But the test has a false positive rate of 5%. That is, if someone does NOT have D, there is a 5% chance the test still comes back positive: Pr(+|~D)=.05.

Now suppose a patient tests positive. What is the probability that this patient actually has disease D?
Someone commits the base-rate fallacy if they say the probability is fairly high, because they discount or ignore the information about the ‘base rate’ of the disease in the population. Only 1 in 1000 people have the disease. But for every 1000 people who don’t have it, 50 people will test positive. You have to use Bayes’ Theorem to get the exact probability that someone who tests positive has the disease; the probability turns out to be slightly under 2%.

In the context of the No-Miracles and Pessimistic Induction arguments, the objection is that both arguments ignore a relevant base rate. For example, the No-Miracles argument says:

(A) Pr (T is empirically successful | T is approximately true) = 1

(B) Pr (T is empirically successful | ~ (T is approximately true)) <<1. Inequality (B) is supposed to capture the ‘no-miracles intuition’: the probability that a false theory would be empirically successful is so low that it would be a MIRACLE if that theory were empirically successful. Hopefully you can see that (A) corresponds to (2) in the original, medical base-rate fallacy example, and (B) corresponds to (3). Empirical success is analogous to a positive test for the truth of a theory, and the no-miracles intuition is that the false-positive rate is very low (so low that a false positive would be a miracle). The base-rate objection to the No-Miracles argument is just that the No-Miracles argument ignores the base rate of true theories in the population of theories. In other words, in the NMA, there is no analogue of (1) in the original example. Without that information, even a very low false-positive rate cannot license the conclusion that an arbitrary empirically successful theory is probably true. (And furthermore, that base rate is somewhere between extremely difficult and impossible to obtain: what exactly is the probability that an arbitrary theory in the space of all possible theories is approximately true?)

OK, that concludes the set-up. Now I can state my concern: I am not sure the objectors’ demand for the base rate of approximately true theories in the space of all possible theories is legitimate. Why? Think about the original medical example again. There, we are simply GIVEN the base rate, namely (1). But how would one acquire that sort of information, if one did not already have it? Well, you would have to run tests on large numbers of people in the population at large, to determine whether or not they had disease D. These tests need not be fast-diagnosing blood or swab tests; they might involve looking for symptoms more ‘directly,’ but they will still be tests. And this test, which we are using to establish the base rate of D in the population, will still presumably have SOME false positives. (I’m guessing that most diagnostic tests are not perfect.) But if there are some false positives, and we don’t yet know the base rate of the disease in the population, then—if we follow the reasoning of the base-rate objectors to the NMA and the PI—any conclusion we draw about the proportion of the population that has the disease is fallacious, for we have neglected the base rate. But on that reasoning, we can never determine the base rate of a disease (unless we have an absolutely perfect diagnostic test), because of an infinite regress.

In short: if the NMA commits the base-rate fallacy, then any attempt to discover a base rate (when detection tools have false positives) also commits the base-rate fallacy. But presumably, we do sometimes discover base rates (at least approximately) without committing the base-rate fallacy, so by modus tollens, the NMA does not commit the base-rate fallacy.

NMA does not commit the base rate fallacy, because it does not ignore AVAILABLE evidence about the base rate of true theories in the population of theories. In the medical example above, the base rate (1) is available information; under-weighing generates the fallacy. In the scientific realism case, however, the base rate is not available. If we did somehow have the base rate of approximately true theories in the population of all theories (the gods of science revealed it to us, say), then yes, it would be fallacious to ignore or discount that information when drawing conclusions about the approximate truth of a theory from its empirical success, i.e. the NMA would be committing the base-rate fallacy. But unfortunately the gods of science have not revealed that information to us. Not taking into account unavailable information is not a fallacy; in other words, the base-rate fallacy only occurs when one fails to take into account available information.

I am not certain about the above. I definitely want to talk to some more statistically savvy people about this. Any thoughts?


2/15/2016

huh

wait, what?

I'm not sure I can articulate why, but this makes me want to stop blogging...