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Stephen Yablo

Print publication date: 2014

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Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691144955.001.0001

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Confirmation and Verisimilitude

Confirmation and Verisimilitude

Chapter:
(p.95) 6 Confirmation and Verisimilitude
Source:
Aboutness
Author(s):

Stephen Yablo

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691144955.003.0007

Abstract and Keywords

This chapter argues that logical subtraction has a role to play in confirmation theory via the notion of surplus content. Subject matter does, too, via the notion of content-part. Content-part lets us define a new type of evidential relation; E pervasively probabilifies H if it probabilifies “all of it,” meaning H and its parts. This helps with the tacking and raven paradoxes. Equivalent generalizations can be about different things, which affect their evidential relations. Inductive skeptics do not care about confirmation, but they derive some benefit too, for they care about verisimilitude—one theory having more truth in it than another—and the truth in a theory is made up of its wholly true parts.

Keywords:   truth, versimilitude, confirmation theory, surplus content, subject matter, content-part, aboutness, logical subtraction, verisimilitude

Inquiry aims at the truth. What is it for one belief state to be closer to the truth than another? There are two dimensions to this. One relates to the kind of attitude we adopt. If A is true, our attitude toward it should be as close as possible to full belief. The other is to do with the attitude’s content. If the content of our belief is A, then A should be as truthlike or verisimilar as possible. Confirmation theory is directed at the first goal. The theory of verisimilitude is directed at the other.

6.1 Surplus Content

Imagine that we are investigating a hypothesis H, when we learn that a certain consequence E of H is true. E rules out certain ways H might be false: the ones that require E too to be false. Eliminating opportunities for falsity is confirmation of a sort. Suppose we are flipping a fair coin; E is The coin just came up heads and H is It always comes up heads. E confirms—makes it likelier that—The coin always comes up heads, by eliminating a possible counterexample.1 What it does not provide is evidence for the rest of H—for, let us say, H–E. Positive instances make a generalization likelier even if they are irrelevant to, even in fact if they count against, the rest of the generalization. To come at it from the other direction: no matter how much E counts against X, E counts in favor of a hypothesis that entails X, namely X&E.

Call that basic or simple confirmation. A second and more demanding notion is obtained by asking E to bear favorably also on the rest of H—its surplus content relative to E. The coin came up heads does not in the more demanding sense confirm It always comes up heads, for it says nothing about other tosses. Still less does No one has ever run a three-minute mile confirm Today will be the first three-minute mile ever, though it makes that hypothesis overall likelier.

(p.96) The distinction between “mere content cutting” (Gemes (1994)) and, let’s call it, inductive confirmation goes back at least to Goodman, who used it to characterize lawlike, as opposed to accidental, generalizations. All Fs are Gs is lawlike, Goodman suggested, if it is inductively confirmed by its instances. All ravens are black is lawlike to the extent that a raven observed to be black counts in favor of other ravens’ being black; the tested part of the hypothesis reflects favorably on the untested. Nothing like that occurs with Everyone was born on a Thursday; the generalization is thus accidental. But, where Goodman focuses on generalizations, our notion of inductive confirmation is meant to be completely general: to inductively confirm H, E should bear favorably on H’s surplus content relative to E, whatever form that surplus content might take. So, for instance, that the planets have roughly elliptical orbits should count in favor of each of Newton’s three laws of motion and the law of gravitation along with the auxiliary hypotheses needed to derive it from those laws.

Inductive confirmation is tied up with surplus content; so views about what H’s surplus content with respect to E is will guide one’s thinking about when inductive confirmation occurs. Popper and Miller claim, in a 1983 letter to Nature, that it never occurs. H is likelier given E than without it, we assume; E basically confirms H. To test for inductive confirmation, we need to isolate H’s surplus content. When this is done, they say, we find that E lowers its probability:

  1. 1. H is logically equivalent to (HE)&(H∨¬E)

  2. 2. The first conjunct HE simplifies to E, since E is entailed by H

  3. 3. The surplus content over E of E&(H∨¬E) is H∨¬E = EH

  4. 4. E makes E→H less likely: p(H∨¬E|E) < p(H∨¬E)2

  5. 5. E does not inductively confirm H3

Line 3 says that the surplus content over E of E&(H∨¬E) is H∨¬E. What is the argument for this? Popper and Miller seem to be conceiving logical remainders on the model of numerical remainders. To find m − n, (p.97) one looks for a y such that m = n +y; m −n is that y. To find H–E, we are to look, apparently, for a Y such that H is equivalent to E&Y.

But the cases are not really analogous. The equation m =n + y determines a unique y for each m and n, while the “equation” HE&Y does not. It’s wet ought surely to be a candidate for what It’s cold and wet adds to It’s cold. Popper and Miller don’t allow this, however. They think that C&W adds CC&W to C, and more generally that HE = EH.4 Line 3 assumes that the surplus content is one, relatively complex, thing when it could just as easily be another, much simpler thing. The analogue of line 4 for the simpler thing is quite likely false, and in cases of interest, the opposite of the truth. E will indeed make HE likelier if H adds a conjunct to E, and E is positively relevant to that conjunct.

Granted that it is not the only solution to HE&Y, could EH be the best solution? Hempel seems to be backing it when he says that E→H “has no content in common with E since its disjunction with E is a logical truth” (Hempel 1960, 465). He relies here on an idea already considered (section 1.3), namely that A and B both say inter alia that AB, giving them nontrivial common content unless A and B subcontraries. We’ll be reviewing this at length in chapter 8, but two points can be made now. The first is that Snow is white does not in any sense whatsoever share content with Charlemagne was Holy Roman Emperor. Second, the idea that H–E is EH overreacts to the (correct) point that H–E should not be false just because E is false, by making it true when E is false. E should really be as far as possible independent of H–E.5

6.2 Conditions on Confirmation

The golden age of confirmation theory began with Hempel’s enunciation (in Hempel 1943, 1945a, 1945b) of four conditions onevidential (p.98) support: ENTAILMENT, CONSISTENCY, SPECIAL CONSEQUENCE, and CONVERSE CONSEQUENCE.

  • EN E confirms any H that it entails.

  • CO If E confirms H, E does not confirm anything contradicting H.

  • SC If E confirms H, it confirms whatever H entails.

  • CC If E confirms H, it confirms whatever entails H.

A fifth principle, mentioned in passing, is

  • CE E confirms any H that entails it.

He accepts the first three conditions, but not the two CONVERSES. His objection to CONVERSE CONSEQUENCE is that it trivializes the confirmation relation, given ENTAILMENT and SPECIAL CONSEQUENCE. To see why, let E and H be any statements you like.

  1. 1. E confirms E (ENTAILMENT)

  2. 2. E confirms E&H ((1), CONVERSE CONSEQUENCE)

  3. 3. E confirms H ((2), SPECIAL CONSEQUENCE)

This objection has been found puzzling. Why put the blame on CONVERSE CONSEQUENCE? Its contribution is only to get us to (2): E confirms E&H. But (2) may seem plausible in its own right. Also (2) follows directly from CONVERSE ENTAILMENT, which seems on solider ground than CONVERSE CONSEQUENCE. (If H entails E, then ¬E precludes H. E removes the threat that ¬E poses to the truth of H.) CONVERSE ENTAILMENT is backed, too, by the Bayesian analysis of confirmation: pr(H|E) almost always exceeds pr(H) if H entails E. SPECIAL CONSEQUENCE, on the other hand, is from a Bayesian perspective completely untenable. Evidence making H likelier cannot make all its consequences likelier; there are consequences like E→H whose probability is bound to go down.

Why is Hempel so attached to SPECIAL CONSEQUENCE, when the problems nowadays seem obvious? Carnap thought that Hempel might have been mixing up two notions of confirmation. Let c(H, E&K), a real number between 0 and 1, be H’s likelihood given E, relative to some body K of background information. E confirms H incrementally, relative to K, (p.99) if c(H,E&K) > c(H,K). It confirms H absolutely if c(H,E&K) exceeds 1- for some suitable .

Absolute and incremental confirmation should definitely not be confused, but is Hempel confusing them? One would expect Carnap to argue that some of Hempel’s conditions hold only for the absolute notion, others only for the incremental notion. But all of Hempel’s conditions hold for the absolute notion! It is CONVERSE CONSEQUENCE and CONVERSE CONSEQUENCE, which he rejects, that fail to hold absolutely.

The problem is that his rhetoric and his examples, which tend to involve generalizations and their instances, suggest the incremental notion. A black raven makes it likelier, not absolutely likely, that all ravens are black. The incremental notion is naturally understood as positive probabilistic relevance, or probabilification. And probabilification satisfies neither of Hempel’s two principal conditions. Contra CONSISTENCY, Rudy is a raven is positively relevant both to Rudy is a happy raven and Rudy is an unhappy raven. Contra SPECIAL CONSEQUENCE, Rudy is a black raven incrementally confirms Rudy is a black raven and all other ravens are white despite being negatively relevant to All other ravens are white.

6.3 A Third Way

Hempel doesn’t have a leg to stand on, it seems. His conditions hold for absolute confirmation, but that is not what he means to be talking about. Incremental confirmation, which is something like probabilification, does not meet his conditions. This does not entirely settle the matter, however, for a reason noted by Earman:

There may be some third probabilistic [notion of] confirmation that allows Hempel to pass between the horns of this dilemma. But it is up to the defender of Hempel’s instance-confirmation to produce the tertium quid.

(Earman 1992, 67)

Hempel left, in fact, a number of clues suggesting what the third probabilistic notion might be. Here he is introducing the stronger condition of (p.100) which SPECIAL CONSEQUENCE is meant to be a corollary Hempel (1945b, 103 emphasis added):

an observation report which confirms certain hypotheses would invariably be qualified as confirming any consequence of those hypotheses. Indeed: any such consequence is but an assertion of all or part of the combined content of the original hypotheses and has therefore to be regarded as confirmed by any evidence which confirms the original hypotheses. This suggests the following condition of adequacy:

GENERAL CONSEQUENCE: If an observation report E confirms every one of a class P of sentences, then it also confirms any sentence [Q] which is a logical consequence of P.

Hempel’s reasoning here is interesting. Any consequence Q of P—let P be, for simplicity, {P1, P2}—“is but an assertion of all or part of the combined content” of P1 and P2 (103). Q “has therefore to be regarded as confirmed by any evidence which confirms” P1 and P2. The implicit assumption is that E’s support for P must be regarded as carrying through to its parts.

The support carries through, if E confirms “every one” of the sentences in P. Why does Hempel want E to confirm both of P1, P2, as opposed to either of them, or their conjunction? If one says either, then E confirms any F that you like, by virtue of confirming a member (the first) of {E, F}. Similar difficulties arise if it’s the conjunction we focus on; E might confirm E&F entirely by way of its first conjunct. Hempel asks E to confirm each of P’s members separately because otherwise GENERAL CONSEQUENCE would not be plausible.

But now, having insisted in GENERAL CONSEQUENCE on “wholly” confirming evidence—evidence confirming both of P1, P2—should he not also require wholly confirming evidence in SPECIAL CONSEQUENCE? Any reason there might be for wanting E to confirm both members of {P1, P2} is a reason for wanting it to confirm both conjuncts of P1&P2. SPECIAL CONSEQUENCE as we read it today imposes no such requirement. This could be an oversight on Hempel’s part. Or, it could be that the requirement is imposed by SPECIAL CONSEQUENCE as he understands it.

Consider another objection he makes to CONVERSE CONSEQUENCE; it has Rudy is black confirming All ravens are black & force = mass × acceleration. This is puzzling on the standard interpretation, since Rudy’s blackness (p.101) does “basically” confirm the conjunction; it does make it likelier. The objection has got to be that Rudy doesn’t confirm all of the conjunction, because it’s irrelevant to whether F = ma. To fully confirm a conjunction, Hempel is thinking, E must confirm both conjuncts. In Bayesian terms, E must probabilify the conjuncts—or, to avoid syntacticizing the notion, the parts—together and separately.

  • (FC) E fully confirms H iff E probabilifies H and its parts.6

E must pervasively probabilify H, the thought is, to fully confirm it.

Now we return to a problem noted in the last section. Hempel has three conditions on confirmation: ENTAILMENT, CONSISTENCY, and SPECIAL CONSEQUENCE. Two of the three fail for the kind of confirmation that he is supposedly talking about. How could he have missed this? Let me list the conditions again, first as traditionally understood, and then in modified form, with full confirmation (confirmationF) put in for basic confirmation.

Entailment

  1. (B) If E entails H, then E basically confirms H.

  2. (F) If E entails H, then E fully confirms H.

ENTAILMENT holds, we know, in its basic form, but full entailment is stronger. To see that it too is correct, suppose that E entails H, and let X be part of H. E entails X by transitivity of entailment, so pr(X|E) = 1. But then pr(X|E) > pr(X) unless pr(X) = 1. This is what it means for E to fully confirm H.

Consistency

  1. (B) If H contradicts Y, then E does not basically confirm both.

  2. (F) If H contradicts Y, then E does not fully confirm both.

Here is a typical counterexample to CONSISTENCY in its original version. E basically confirms H = E&F and Y = EF, since both entail E and a statement’s consequences make it more probable. Are both fully confirmed

(p.102) by E, that is, does E enhance the likelihood of H and Y and their parts? Certainly not, for it would then have to probabilify both F and ¬F.7

Special Consequence

  1. (B) If E basically confirms H, it basically confirms H’s consequences.

  2. (F) If E fully confirms H, then it fully confirms H’s parts.

Suppose that E fully confirms, or pervasively probabilifies, H. Then it probabilifies H’s parts, and hence (by transitivity of inclusion) the parts of its parts—which is the same as pervasively probabilifying the parts. If this is how Hempel understands SPECIAL CONSEQUENCE, then one sees why he finds it obvious. That an E confirming H and its parts confirms, too, their parts, is virtually a logical truth.

A word finally about Hempel’s positive theory of evidential support, which is related to the hypothetico-deductive model of confirmation, and also to the CONVERSE ENTAILMENT condition, which he rejects. A hypothesis is not always confirmed by its entailments, Hempel thinks, but a certain kind of hypothesis—a generalization—is, it seems, confirmed by a certain kind of entailment, what he calls its “development” for a particular class of individuals.

Now, G’s development for I sounds like it should be the part of G that concerns the relevant individuals. Hempel’s positive theory would then be that a generalization is confirmed by certain of its parts. But the definition he gives is this:

G’s development for I

= G with its quantifiers restricted to the individuals in I.

This way of doing it sometimes delivers a part. All ravens are black, with its quantifiers restricted to birds in the backyard, is All ravens in the yard are black. But not always. G’s development for I is not always even a consequence of G, much less included in it.

Let pluralism be the theory that for all x, there exists a y that is not identical to x. Pluralism is true, let’s suppose. But its development for one-element domains is false; there is indeed just one thing, leaving aside all (p.103) the other things. Pluralism’s development for one-element domains is not even a consequence, then, of the theory itself. G’s development for I may not support G even if G entails it. Let monism be the denial of pluralism: every x is identical to every y. Monism’s development for {Chicago} says that everything identical to Chicago is identical to everything else with that property. This is a truth entailed by monism. But it hardly sounds like a point in favor of monism that Chicago is only one thing.8

6.4 Bayes and Hypothetico-Deductivism

Hempel thought that qualitative confirmation theory should be developed first, followed by comparative confirmation—E favors H over H′—and then quantitative. That is certainly not the view today; quantitative confirmation has stolen the spotlight. Bayesians are sometimes willing to share the spotlight with Hempel and company, if only for motivational purposes. A typical textbook begins by isolating the grain of Bayesian truth in, say, the hypothetico-deductive model of confirmation, or inference to the best explanation. Not everything can be saved, but that itself is instructive. The feeling seems to be that what was right in the qualitative tradition is explained by Bayes, and what was wrong is refuted by Bayes.

One tests a hypothesis, according to the hypothetic-deductive model, by seeing whether its consequences check out. False consequences count definitively against H; true consequences confirm it.

(HD) E confirms H if H entails E; it confirms H relative to background information K if H&K entails E, and K alone does not.9

Bayesianism seems to vindicate the HD model, since if H entails E, then p(H|E) ≥ p(H). This may not be entirely to its credit, however, for hypothetico-deductivism has some surprising implications.10 Our focus will be on the so-called ‘tacking paradoxes.’

(p.104) TACKING BY DISJUNCTION If H entails E, then it entails EE as well. The class of confirmers of a given hypothesis is thus closed under the operation of tacking on a random disjunct. All emeralds are green, if confirmed by This emerald is green, is confirmed also by Either this emerald is green, or no emeralds are green.11

TACKING BY CONJUNCTION If H entails E, then E is entailed also by H&H′. The hypotheses confirmed by a given piece of evidence are closed, then, under the operation of tacking on random conjuncts. A green emerald comes out confirming All emeralds are red, apart from this green one.12

Surely there is something right, though, about the idea that a theory is to be judged by its consequences. Not all its consequences, perhaps, given the tacking by disjunction problem; not all theories with the given consequences, given the tacking by conjunction problem. But if E is the right kind of consequence, and H the right kind of implier, then, it seems, the relation should hold (Gemes 1998).

Fine, but what is the right kind of consequence? That H with a random disjunct tacked on is the wrong kind suggests that, of its consequences, H is better, or more reliably, confirmed by those that are parts.13 And what is the right kind of E-implier? That H with a random conjunct tacked on is the wrong kind suggests that, of its parts, H is better confirmed by those that probabilify its other parts.

6.5 Bayes and Instance Confirmation

Hempel’s paradox of the ravens has four elements:14 three plausible-looking premises and a nutty-looking apparent consequence of those premises.

  • (p.105) Nicod’s Criterion: All Fs are Gs is confirmed by its instances.

  • Equivalence Condition: Logical equivalents are confirmationally alike.

  • Equivalence Fact: All Fs are Gs is equivalent to All non-Gs are non-Fs.

  • Paradoxical Result: Ravens are black is confirmed by nonblack nonravens.

If there is a standard response to this, it’s to embrace the paradoxical result. A nonblack nonraven does confirm that all ravens are black. But, it confirms it just the teeniest little bit—not as much as a black raven does. The idea was apparently first suggested by the Polish logician Janina Hosiasson-Lindenbaum (1940). A randomly chosen item is likelier nonblack than a raven, hence we sample a larger portion of the space of possible counterexamples by looking at ravens. Hempel in his response to Hosiasson-Lindenbaum asked an obvious question, which has never been addressed to my knowledge: “[I]s this last numerical assumption [that non-black things greatly outnumber ravens] warranted in the present case and analogously in all other ‘paradoxical’ cases?” He is worried that the paradox could still arise if a randomly chosen item were just as likely a raven as nonblack.

This is hard to imagine, so consider a different, silly, made-up example. H says that Charged particles lack spin—they are, to have a positive term for spinlessness, “poised.”15 The numerical assumption is, a randomly chosen particle is likelier neutral than poised.

Well, but it’s our example; we can stipulate that the assumption is false, that there are exactly as many charged particles as spinny ones. Doesn’t it still seem that a charged, poised particle confirms Charged particles are poised more, or better, than a spinny neutral one? One can even imagine that Charged particles are poised and Spinny particles are neutral are distinct laws, each with its own physical underpinnings. The first is like Cheaters never prosper, on the theory that there is something about cheaters that prevents prosperity—they are found out and ostracized, say. The second is like The prosperous never cheat, on the theory that there is something (p.106) (else) about prosperers that keeps them honest—they have no motive for cheating, maybe, on account of their prosperity.

If the paradox still arises when a generalization’s contrapositive is statistically indiscernible from it, and just as lawlike, then we need an approach that does not require us to pick winners. Hempel mentions one briefly:

Perhaps the impression of the paradoxical character of [these cases] may be said to grow out of the feeling that the hypothesis that All ravens are black is about ravens, and not about non-black things, nor about all things.

(Hempel 1945a, 17).

One generalization is about charged particles. How is an uncharged particle supposed to tell us about them? The most it can accomplish is to take itself out of the running for the role of counterinstance; counterinstances have to be charged. The other is about spinny particles. A poised particle can serve, again, only as a thwarted potential counterexample: it at least, does not witness the possibility of a spinny particle that is charged.

How these subtleties could matter to confirmation is not obvious. Confirmation is to do with probability, and statements true in the same circumstances are equiprobable. The answer is that the statements’ parts may not be true in the same circumstances, and inductive evidence has got to probabilify parts. No Fs are Gs and No Gs are Fs differ inductively, by differing mereologically; they differ mereologically, by differing in what they’re about.

6.6 Parts and Instances

Rudy supposedly confers likelihood on the parts of Ravens are black, but not the parts of Nonblack things are not ravens. If the parts are distinct, and have their likelihood controlled by different factors, it is hard to see how the wholes can remain equiprobable—as they must given their logical equivalence. This puts pressure on Ravens are black to share its parts with its contrapositive—which destroys the proposed explanation of confirmational differences in terms of mereological differences. I reply that the two generalizations have matched parts, agreeing in probability but not inductive significance16.

(p.107) What is said by All ravens are black? One could treat it as the first-order generalization ∀x (RxBx), equivalently, ¬∃x (RxBx). But that confuses the role played by something’s non-raven-hood in the truth of All ravens are black with that played by Rudy’s blackness. Nonravens help to determine what it takes for all the world’s ravens to be black. They bear on the content of the demands made by a sentence that attributes blackness only to ravens. Black ones are relevant not to the content of the demands but how far the world goes toward meeting them.

A semantic analogue of Belnap’s conditional assertion operator was sketched in section 4.5. Bx supposing that Rx, written RxBx, is true in the same worlds as RxBx. But the reasons differ. One is true either because x is not a raven or because x is black. The other is true, should x be a raven, because x is black. Otherwise it is vacuously true—true not because its demands are met, but because it doesn’t make any demands. Explicitly, RxBx is

{ true for the reason(s) Bx is true, should Rx and Bx be true false for the reason(s) Bx is false, should Rx be true and Bx false true for no reason at all, should Rx be false

Corresponding to the two ways for RxBx to be true, there are two ways it might gain in probability. One kind of evidence lowers pr(Rx), thus boosting the chances of Rx↗Bx being vacuously true. Another kind leaves pr(Rx) unchanged while lowering that of pr(RxBx), thus boosting the chances of RxBx being substantively true. There is a corresponding distinction at the level of generalizations. White socks increase pr(∀x (RxBx)) by making more of its content trivial—by cutting into what it (nontrivially) says. A black raven does it by making the substantive part(s) more probable.

The problem was this: Ravens are black needs, on the one hand, to have different parts than Nonblack things aren’t ravens. Otherwise we can’t explain their inductive differences in the way proposed. They should on the other hand have the same parts, lest their probabilities be driven apart by evidence bearing on the parts only of, say, Ravens are black.

With virtual parts distinguished from real ones, we can understand the situation as follows. RxBx (for a given x) has a counterpart ¬Bx¬Rx that (i) agrees with it in probability, but (ii) with substantive and trivial likelihood interchanged. As the chances rise of Rudy being (substantively) (p.108) black if a raven, they rise as well of his being (trivially) a nonraven if nonblack. As the chances rise of Betty being (substantively) not a raven if not black, they rise of her being (trivially) black if a raven. Rudy’s effects on the probabilities of ∀x (RxBx) and its contrapositive are the same. Its the mechanism that is different. Rudy confirms what ∀x (RxBx) says, while (in a manner of speaking) changing what ∀xBx↗¬Rx) says.17

6.7 Verisimilitude

Confirmation is tied up with the aims of science; we want our beliefs to be as close as possible to the truth, and believing confirmed hypotheses is supposed to help us achieve this. But proximity to the truth has another aspect that confirmation theory is blind to. We want to maximize the amount of truth we believe and minimize the amount of falsehood.

Popper was famously pessimistic about the first aim; he emphasized the second.18 Science progresses, not when our theories are better confirmed, but when they achieve greater verisimilitude. His initial definition, with X and Y ranging over theories, and X’s truth-content XT (false-content XF) defined as the set of its true (false) consequences, is this.

X is at least as truthlike as Y iff

YTXT : any truth implied by Y is implied also by X, and

XFYF : any falsehood implied by X is implied also by Y

X is more truthlike than Y if in addition Y is not as truthlike as X, that is, one of the above-mentioned inclusions is strict.

(p.109) Popper’s definition has some desirable features. Among true theories, verisimilitude goes with logical strength. A false theory can never be as close to the truth as its truth-content.19 But it has as well some truly horrible features.

Suppose that X and Y are false and that neither implies the other. Then each has truth-content the other lacks; X alone implies YTXT, and Y alone implies XTYT (Tich`y 1974, Miller 1974). False theories are thus left completely unranked by Popper’s proposal. They are not in most cases even ranked lower than their negations, which are true. (Proof: Let N be the negation of X. Suppose that N is true, and let Z be a truth that it does not imply. N does not imply NZ either, but N’s negation does imply it, by virtue of contradicting the antecedent. NZ is thus a truth implied by the falsehood ¬X but not the truth X.20)

Attention has turned in recent years from content-oriented theories, like Popper’s, to the likeness approach: X has greater verisimilitude than Y to the extent that it holds in worlds closer to actuality. But now, what does it mean for the X-worlds to be closer to actuality than the Y-worlds? There are any number of ways to combine the distances of individual worlds from ours into a measure of how far the set of them is from our world (Niiniluoto 1987). If one thinks of the worlds as each casting a vote, it becomes an aggregation of judgment problem and limitative theorems from voting theory suggest there may be no fully satisfactory way of doing it (Zwart and Franssen 2007).

A second look at the content approach seems in order. Popper went wrong, in identifying X’s truth-content with its true consequences (Gemes 2007a). Suppose we define it rather as the sum of X’s true parts. His definition then becomes

X is at least as truthlike as Y iff

Y’s true parts are all implied by true parts of X, and

X’s false parts are all implied by false parts of Y

(p.110) This doesn’t quite work, however. Suppose X = P&Q and Y = Q, where P is true and Q is false. X should come out ahead since it adds a true conjunct. But it has a false part not implied by Y, Gemes observes, namely itself. This seems a merely technical problem. What’s false about P&Q is Q, and Q is part of Y. A theory’s false-content should be seen as made up of its wholly false parts—the ones with no nontrivial true parts buried within. (True parts are wholly true automatically.) The proper definition is

X is at least as truthlike as Y iff

Y’s wholly true parts are implied by wholly true parts of X, and

X’s wholly false parts are implied by wholly false parts of Y

A kind of verisimilitude that this perhaps misses involves differences in accuracy. Light travels at a hundred miles per hour is further from the truth than Light travels at a million miles per hour. Does the second underestimate of light’s speed have a true part not implied by the first underestimate? Light travels at no less than a million miles per hour has the right sort of flavor; but it may be doubted whether traveling at least n miles per hour is included in traveling exactly n miles per hour. (This is in fact the tip of a scary iceberg that I would rather avoid just now.)

Another interesting feature of our account is that logically equivalent hypotheses can be at different distances from the truth. All men are mortal has plenty of truth in it. It contains, for instance, the truth that Socrates is mortal, supposing him to be a man. Immortals are never men has very little truth in it. Certainly it does not contain anything to imply the aforementioned truth about Socrates. All men are mortal is thus apparently more truthlike than Immortals are never men, though the two hold in the same worlds. I am not sure if this is the right result.

6.8 Summing Up

Logical subtraction has a role to play in confirmation theory, via the notion of surplus content. Subject matter does, too, via the notion of content-part. Content-part lets us define a new type of evidential relation; E pervasively probabilifies H if it probabilifies “all of it,” meaning H and its parts. This helps with the tacking and raven paradoxes. Equivalent generalizations can (p.111) be about different things, which affects their evidential relations. Inductive skeptics don’t care about confirmation, but they derive some benefit too, for they care about verisimilitude—one theory having more truth in it than another—and the truth in a theory is made up of its wholly true parts.

Notes:

(1) If H entails E, 0 < pr(E) and pr(H) < 1, then pr(H|E) exceeds pr(H).

(2) Provided p(H|E) ≠ 1 ≠ p(E).

(3) Popper and Miller could have made an even stronger objection to inductivism. H entails E, so H∨¬E is equivalent to ¬E. E not only fails to confirm the surplus content H∨¬E, it positively contradicts it! Something is wrong with your theory if what H adds to E is coming out to be ¬E.

(4) This was a not uncommon view at the time. “Th[e] ‘new’ information contained in H is expressed by the sentence H∨¬E. (For H is equivalent to (HE)&(H∨¬E))” (Hempel 1960, 465). “The purely scientific utility of adding H to E is m(H∨¬E)/m(E)” (Bar-Hillel and Carnap 1953, 150).

(5) See Gemes (1994, 1997). A good many of the ideas in this chapter are due to Gemes.

(6) Strictly, those of its parts that are not already certain. pr(X|E) > pr(X) for each XH such that pr(X) = 1.

(7) I assume that F and its negation are parts (not only conjuncts) of H and Y.

(8) Insofar as developments were intended by Hempel to be parts, our objection is only this: the part of G that concerns a population-based subject matter cannot always be obtained by restricting G’s quantifiers to the relevant population.

(9) Christensen (1997) goes into some of the complications.

(10) There is a huge literature on these paradoxes: I’m drawing particularly on Gemes (1990), Gemes (2005), Grimes (1990), Schurz (1994), Moretti (2006), and Sprenger (2011).

(11) Bayesianism backs the HD model up on this; it too has This emerald is green or no emeralds are green confirming All emeralds are green.

(12) Bayesianism backs the HD model up on this; it too has This emerald is green confirming All emeralds are red apart from this green one.

(13) Alternatively, by those of H&K’s parts that are not included in K alone.

(15) “[The] claim that ‘All nonblack things are nonravens’ is not projectible needs a closer look. Even granting that the predicates here are ill entrenched, this seems to illustrate no general principle. Surely ‘nonmetallic,’ ‘noncombustible,’ ‘invisible,’ ‘colorless,’ and many other privative predicates are well entrenched. Furthermore, it should be noted that a privative predicate will be as entrenched as any of its coextensive predicates” (Scheffler and Goodman 1972, 83).

(16) I rely here on the treatment of quantifiers in section 4.4

(17) Zsa Zsa Gabor is supposed to have found a way to keep “her husband” young and healthy: remarrying every few years (Hare 2007). No individual is made younger by this process, rather “her husband” acquires younger referents. Betty’s way of making “the hypothesis that ravens are black” likelier is similar: “the hypothesis” becomes likelier by acquiring a likelier referent.

(18) “I intend to show that while we cannot ever have sufficiently good arguments in the empirical sciences for claiming that we have actually reached the truth, we can have strong and reasonably good arguments for claiming that we may have made progress toward the truth” (Popper 1972, 58).

(19) If Y is false, it is further from the truth than YT. Proof: Y’s truth-content is included in that of YT, because it is YT. Y’s false-content strictly includes that of YT, for YT’s false-content is empty; truths don’t imply falsehoods.