Category Archives: Philosophy of Science
Evidence of Evidence is Evidence, or is it?
Branden Fitelson (forthcoming) provides counterexamples to Richard Feldman’s principle that Evidence of Evidence is Evidence (EEE). Here’s the principle in its initial (naïve) form:
(EEE1) If E (non-conclusively) supports the claim that (some subject) S possesses evidence which supports p, then E supports p. (Fitelson forthcoming: 1).
Fitelson’s counterexamples to (EEE) work by presupposing the “positive relevance” (i.e., increase-in-probability) notion of evidential support. In footnote 6 he indicates a more substantive principle of evidential support might be wielded in defending (EEE). In this post I want to explore this possibility, specifically in relation to the notion of propositional justification. Consider the following principle of propositional justification:
S is justified in believing that p iff S’s total evidence sufficiently supports p (Neta 2007: 197).
Though there are many issues that could be raised with this formulation of propositional justification, let’s see if a less demanding iteration of the principle could be used to resist Fitelson’s counterexamples to (EEE). Neta’s principle suggests the following notion of evidential support:
(1) E (evidentially) supports p iff S’s total evidence includes E and S’s total evidence (necessarily) supports p.
The counterexample to (EEE1) involves drawing a card c at random from a deck. All the evidence we are given regarding c is as follows:
(E1) c is a black card.
(E2) c is the ace of spades.
(p) c is an ace.
Imagine a guy named John knows what card c is, and the evidence above constitutes all the facts about the case. This means the following is the case:
(2) E1 supports the claim that John possesses evidence (E2) which supports p.
Positive relevance creates a problem for (EEE1) because (E1) doesn’t raise the probability of (p). (E1) alone is probabilistically irrelevant to (p); so, even though (E1) supports (E2), the second conjunct in (EEE1) is false (i.e., E1 doesn’t support p).
How does the counterexample fare under principle (1) instead of positive relevance? John’s total evidence includes (E1), and John’s total evidence (E1 and E2) necessarily supports (p). (E1) alone doesn’t necessarily support (p), but it also doesn’t support (not-p), and when coupled with (E2) it does necessarily support (p). In fact, (E2) entails (p). John’s total evidence might not sufficiently support (p), but his total evidence does necessarily do so. The next iteration of (EEE) runs as follows:
(EEE2) If E1 supports the claim that S possesses evidence E2 which supports p, then the conjunction of E1 and E2 supports p (Fitelson forthcoming: 2).
This seems like the defense I just gave for (EEE1), assuming (1). Didn’t I just claim the conjunction of (E1) and (E2) supports (p)? If so, then, assuming evidential support principle (1), it looks like the next counterexample will sink (EEE2). However, I think (EEE2) and (1) escape unscathed. Fitelson’s counterexample to (EEE2) is about a guy named Joe:
(E1) Joe has a full head of white hair.
(E2) Joe is over 35 years of age.
(p) Joe is bald.
The example works given the positive relevance principle because the conjunction of (E1 and E2) fails to raise the probability of (p). Being over 35 years of age might raise the probability that one is bald, but having a full head of white hair doesn’t raise the probability one is bald. (E1) supports (not-p), so the conjunction of (E1) and (E2) refutes (p). What about in relation to principle (1) instead of positive relevance?
(E1) is part of the total evidence. But, does the total evidence support (p)? More specifically, is the notion of total evidence equivalent to the conjunction of all the (relevant) evidence? I would argue it is possible to equate the total evidence with the conjunction of all relevant evidence, as I did in defending (EEE1) against a counterexample, but it is not necessarily the case that the total evidence must be regarded as all of the evidence conjoined. There is a probabilistic consideration in favor of this point.
The probabilistic consideration is that the total evidential support for (p) is not determined by simply conjoining the individual probabilities that (E1) and (E2) afford (p). Returning to the card counterexample, the probability that the card is a black card (E1) supports the claim the card is an ace (p) is the probability that: if the card is black, then the card is an ace. Only the ace of clubs and the ace of spades satisfy this condition, so the probability is 2/52 (approx. 4%).
The probability (E2: the card is the ace of spades) supports the claim the card is an ace (p) is 1, as the entailment takes on the maximum value. The fact that (E2) entails (p) means Pr(p|E2) = 1. The amount that the total evidence supports (p) is not simply the product of the probabilities of (p) given (E2) and (p) given (E1). This would yield a probability of Pr(p|2/52 x 52/52) = 104/2704 or about 4%.
A better estimate is attained through subtraction of the two probabilities. This measures the degree to which one bit of evidence lessens the impact of another bit of evidence on the target proposition. This better approximates the total impact of (E1 and E2) on (p). This makes it the case that Pr(p|52/52 – 2/52) = 50/52 or 96%. The actual probability, though, would be 1 or 100% because the probability the card is black (E1) is swamped by the probability the card is the ace of spades (E2) in relation to the probability the card is an ace (p). As such, (E1) can be disregarded. Again, the correct probabilistic impact of the total evidence on the target proposition is not determined by a conjunction of all the evidence. Let’s apply this back to the second counterexample about Joe’s hair or lack-thereof:
Pr(p|E1) = 0
Pr(p|E2) = .20 (estimate of men over 35 who are bald)
Conjoining the total evidence, again, doesn’t refute the target proposition. Suppose John knows the age of Joe and (2) is true. That is, (E1) supports the claim that John possesses evidence (E2) that supports (p). Does the conjunction of (E1 and E2) refute (p), as Fitelson urges? Multiplying the probabilities, as given above, would yield a probability of 0. It would indeed refute (p). However, this case is dissimilar from the card drawing case because it uses vague terms. Being “bald” is not defined as having “no hair”. Most men who are “bald” still have some hair on their head. Being bald is defined in relation to “male pattern baldness.” This is a progressive condition and, much like the term “a heap”, is wrought with vagueness. The fact that Joe has a full head of white hair (E1) and Joe is over 35 years of age (p) doesn’t make it the case that there is zero probability that Joe is bald. Joe may have hair loss as a result of male pattern baldness, yet to a casual observer (like John) he may appear to have a full head of hair. This is especially the case for someone who has all white hair because the threshold for counting as having a full head of hair is plausibly lower when all of one’s hair is white. Due to vagueness in the terms in (E1) and (p), the fact that Joe is over 35 years of age (E2) is not swamped by (E1).
A better probability estimate is attained by (i) allowing Pr(p|E1) = .05 as a correction on boundary vagueness in the terms, then (ii) subtracting the probabilities to yield the impact of the total evidence on the target proposition: Pr(p|.20 – .05) = .15 or 15%. The total evidence still supports (i.e., does not outright refute) (p) even though the total evidence makes it more likely that (not-p) than (p) is the case.
References
Fitelson, Branden (forthcoming). “Evidence of Evidence is not (Necessarily) Evidence.” Analysis.
Neta, Ram (2007). “Propositional Justification, Evidence, and the Cost of Error.” Philosophical Issues.
Workshop: Logic and Methodology
There’s a great workshop at Stanford this weekend on logic, epistemology and science. This workshop features tutorials and talks at the intersection of those domains. In addition, the tutorials and talks draw on a number of formal methods (e.g., dynamic epistemic logic, learning theory and probability). Click here for more details on the workshop.
Contrastive Bayesianism
One way to argue against Elliot Sober’s Contrastive Empiricism is to claim that the approach violates an important principle of confirmation theory. Branden Fitelson (2010) has done this in a recent paper by providing a counterexample to Sober’s Law of Likelihood (LL) for ‘favoring’ relations. According to (LL) evidence E favors hypothesis H1 over hypothesis H2 if and only if H1 confers greater probability on E than H2 does. You compare the likelihoods of alternative hypotheses. Fitelson argues against (LL) by proposing a counterexample. The counterexample includes the following facts:
- E: The card is a spade.
- H1: The card is the ace of spades.
- H2: The card is black.
This makes the probabilities run as follows: Pr(E|H1) = 1 and Pr(E|H2) = 1/2. So, on (LL), E favors H1 over H2. However, this seems wrong. The truth of E guarantees (or necessitates) the truth of H2, but the truth of E doesn’t guarantee the truth of H1. This makes E conclusive evidence for H2 but not H1. So, it seems E favors H2 over H1 despite the likelihoods pointing the other direction. For Fitelson this suggests the following confirmation principle, called Conclusive Evidence (CE), a principle which (LL) violates:
- (CE) If E constitutes conclusive evidence for H2, but E constitutes less than conclusive evidence for H1, then E favors H2 over H1.
In email correspondence with Fitelson I argued against (CE) by claiming that it causes probability and entailment to come apart. In this post I want to propose a different idea.
Carnap (1962) proposes two independent confirmation measures. The first is confirmation as firmness. Confirmation(f) tracks both truth and entailment. The second is confirmation as increase in firmness. Confirmation(i) tracks entailment in addition to an intuitive relevance. I think this suggests a Relevance (R) principle:
- (R) If E constitutes conclusive evidence for H2, but E constitutes less than conclusive evidence for H1, then E is more relevant to H2 than H1.
If Fitelson’s use of ‘conclusive evidence’ is to work with regard to confirms(i) it should work with regard to relevance. In the card-drawing counterexample is the fact that the card is a spade (E) more relevant to the fact that the card is black (H2), or is (E) more relevant to the fact that the card is the ace of spades (H1)?
For one thing, relevance cannot be assimilated into entailment. This is because confirms(i) tracks entailment AS WELL AS relevance, according to Fitelson. If E is conclusive evidence for H, then Pr(H|E) = 1. In the counterexample, E is conclusive evidence for H2, as E establishes H2 with certainty, yet Pr(E|H2) = 1/2 and Pr(E|H1) = 1. This means H1 entails E, but H2 doesn’t. However, H2 seems more relevant to E than H1 even though H1 entails E. That the card is black is more relevant to the claim that the card is a spade than that the card is the ace of spades is relevant to the claim that the card is a spade. The latter claim is redundant whereas the former claim delimits, and adds to, the semantic content on the table. “The card is black, and by the way, the card is a spade not a club.” On this account, relevance provides non-redundant semantic information in helping the inquirer delimit the space of logical possibilities.
By contrast, it is true that if the card is a spade, then the card is black. But this is running things in the opposite direction from what I stated above. This makes it the case that the card is a spade (E) is more relevant to the claim that the card is the ace of spades (H1) than (E) is relevant to the claim that the card is black (H2). In this direction the color of the card in terms of meaning is already bound-up in the card being a spade. A spade in a standard deck is by definition a black card. My basic idea is that (intuitive) relevance is a measure of the degree to which something doesn’t participate in semantic redundancy.
In conclusion, I claimed (R) is false. Intuitive relevance, when it comes to conclusive evidence, works in the opposite direction of probabilistic strength and entailment. This means (CE), which implies (R) in order to qualify under a confirmation(i) measure, is also false. If confirmation(i) only tracks entailments, and relevance is dropped from the measure, then confirmation(i) can be assimilated into confirmation(f). This is because confirmation(i) involves necessary preservation of truth. If confirmation(i) isn’t different than confirmation(f) by virtue of tracking a type of relevance, then confirmation(i) is merely tracking truth-preservation. That is, confirmation(i) is merely tracking truth and, as such, is not independent from confirmation(f).
Updated Paper: Against the Total Evidence Requirement
I just updated my Papers page with a revised version of a paper arguing Against the Total Evidence Requirement. Here’s the abstract.
ABSTRACT. The Requirement of Total Evidence (RTE) asks an agent to make her confidence in a belief proportional to the support it receives from her total evidence. This paper examines (RTE) as a norm of epistemic rationality and argues that it is problematic. Looking at the work of Peter Achinstein (2001) on the notion of evidence it becomes clear that (RTE) endorses a view of the constitution of evidence that is neither necessary nor sufficient for something to count as evidence. To overcome this and other deficiencies associated with (RTE) a move is made to an objective view of evidence. This move aligns epistemic rationality with scientific rationality in seeking to capture veridical evidence. It also leads to a new norm of epistemic rationality—the Proper Subset Evidence Requirement (PSER).
Click on the following link to access a presentation on the paper.
What Ardi Reveals About the Syntax of Scientific Findings
Big in the news right now is Ardi — the oldest known hominid skeleton (see the news here). This finding is thought to cast new light on early ancestors to humans and the upright origins of humankind. Without getting into a discussion on evolution I would like to use Ardi as a case study in scientific syntax. My wife is a chemist. She has often said that reporting research involves a great deal of massaging the syntax. How things are worded is important in reporting scientific findings. If things are not worded correctly findings can be overstated or understated. For example, if evidence e shows hypothesis h is probable one would not want to say that the evidence is conclusive in support of the hypothesis (unless the probability surpasses some threshold of conclusivness pre-established or generally understood by that scientific community). Scientific syntax needs to be properly hedged — words need to be properly chosen and arranged — to communicate semantics that are true to the findings. Syntax can even, dare I say, be used to get the findings to say things the evidence does not support.
Below are some quotes from the scientific findings as reported by the scientists in the magazine Science (2 October 2009 Vol. 326). I will place quotation marks around syntax of interest and briefly comment on the quote.
Despite its small cranial capacity, there is “tantalizing evidence” for advanced cranial based flexion in Ar. ramidus. (68e6)
It is interesting that an emotive word like “tantalizing” was used. Here is another quote that utilized a similar emotive word (“anxiously”).
More fossils “will” further advance our understanding of the CLCA, and we “anxiously await” their discovery. (74e7)
The emotive word choice makes the authors seem like they are excited about receiving more fossils, which “will” advance their understanding. There is a presumption in favor of evidence fitting theory and that what is found “will” bolster understanding. A critic might wonder whether there is some “making evidence fit theory” going on.
Now I will highlight the use of hedging in reporting scientific findings.
One of the instructive aspects of adaptive suites is the demonstration of what “must almost always” be a complex network of character interactions, even in reptiles and amphibians. “More often than not“, such interconnectivity is “likely to far exceed” relatively simplistic arguments such as somatic budgeting. (74e7)
What does “must almost always” mean? Is this like Brian Fantana in the movie Anchorman remarking about the effectiveness of Sex Panther cologne: “60% of the time, it works every time”? Also, what does “more often than not” interconnectivity is “likely to far exceed” mean? Does this mean greater than 50% of the time interconnectivity is “probably” going to outperform somatic budgeting. It is difficult to see what this hedging amounts to. Other classic hedging syntax includes: the records “suggest” X, it is now “equally clear” that Y, our comparative analyses of P “suggests” that this “probably” reflects Z. It is hard to track double-qualifications of likelihood and once identified it makes me wonder how much of the syntax is smoke-and-mirror methodology (i.e., purposely not showing one’s full cards). Another possibility is the double-hedging indicates lack of certainty on behalf of the scientists. This is more often than not probably what is going on (lol). Interpreting these findings over a period of years will determine what the findings really mean. It is ultimately the consensus of the scientific community that will settle the meaning of the evidence for various hypotheses about evolution.
Another thing that is clear from looking at the syntax of the Ardi findings is that a great deal of inferences occur in unearthing the fossils, putting together the skeleton, revising the skeleton until the scientists are happy with the reconstruction, building digital reconstructions of the entire skull, pelvis, and limbs to fill in the gaps and generate a fleshed-out virtual model and then from this virtual model drawing inferences about what hypotheses the evidence supports. For example:
The “digitally reconstructed” [Ardi] skull further allows “a variety of inferences” about African ape and hominid evolution. Cranial capacity…was “probably slightly smaller” than….The [Ardi] skull lacked the masticatory specializations of later Australopithecus, consistent with the dental evidence for an omnivore/frugivore niche lacking emphasis on hard and/or abrasive diets. Finally, comparisons of [Ardi] and extant African apes “suggest” that each is unique in aspects of its cranial anatomy. (68e6)
It would be interesting to calculate the probability of the inferences reported in the research. From the evidence the scientists infer that certain conclusions are correct or that certain hypotheses are confirmed. Many of the inferences are based on digital reconstructions. How accurate are the digital models? How likely are the models to reflect the actual creature? Is Ardi representative of that genus of animals? The sample size is so small that it makes me wonder if it is possible to infer from Ardi a conclusion about skull size in relation to Lucy (Australopithecus), especially because the sample size is so small in both cases. In my estimation the probability of these inferences from the evidence is mitigated by so many factors that the probability must be “reported” as small if the findings are to match the probability of the hypotheses conditional on the evidence. However, maybe I’m just playing with the syntax.
New Paper – Williamson on Evidence Neutrality
I just posted a new paper on my papers page. It is a critique of Williamson’s notion of Evidence Neutrality. Here’s the abstract:
- This paper looks at Timothy Williamson’s formulation of the thesis of Evidence Neutrality (EN). I motivate and argue for an upgraded version of EN by showing that changing one’s assumption about the nature of evidence (i.e. fallibility vs. factivity) generates a different verdict on EN. Then, I show how Williamson’s interpretation of EN is incomplete in light of a principle that guides his complete understanding of the nature of evidence. I reformulate EN to overcome deficiencies in Williamson’s interpretation of EN, and, lastly, I use cases from philosophy and science to show that reformulated-EN promotes better practices in both domains while, at the same time, it avoids psychologizing evidence.
