Open knowledge and changing the subject

Abstract

Knowledge is closed under (known) implication, according to standard theories. Orthodoxy can allow, though, that apparent counterexamples to closure exist, much as Kripkeans recognize the existence of illusions of possibility (IPOs) which they seek to explain away. Should not everyone, orthodox or not, want to make sense of “intimations of openness” (IONs)? This paper compares two styles of explanation: (1) evidence that boosts P’s probability need not boost that of its consequence Q; (2) evidence bearing on P’s subject matter may not bear on the subject matter of Q.

Appendix: Hempel was right

Much of what goes on in this appendix is due to Gemes (1990, 1994, 1997, 2007).

The golden age of confirmation theory begins with Hempel’s “Studies in the Logic of Confirmation.” He articulates there four possible conditions on evidential support.³³

entailment::

e confirms any h that it entails.

consistency::

If e confirms h, it does not confirm any h* contradicting h.

special consequence::

If e confirms h, it confirms what h implies.

converse consequence::

If e confirms h, it confirms that which implies h.

A fifth principle, mentioned in passing, is

converse entailment::

e confirms any h that entails e.

Hempel accepts the first three of his proposed conditions, but not the two converses. What mainly bothers him about them is that they trivialize the confirmation relation, given entailment and special consequence. For let e and f be arbitrarily chosen hypotheses.

• 1. e confirms e (entailment)

• 2. e confirms e&f ((1), converse consequence)

• 3. e confirms f ((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&f. But (2) is an instance of converse entailment, which is apt to seem obvious. If h entails e, so that $\lnot e$ precludes h, how can e not speak in favor of h, if only by removing a possible obstacle? converse entailment is backed, too, by the Bayesian analysis of confirmation: $\hbox {pr}(h | e )$ exceeds pr(h) if h entails e. special consequence, on the other hand, is from a Bayesian perspective untenable. Evidence making h likelier cannot make all its consequences likelier, and there are particular consequences whose probability is bound to go down.

How could Hempel have been so wrong? Why will he not let go of special consequence, when the problems seem obvious? Carnap suggested a diagnosis. He thought Hempel was mixing up two notions of confirmation, whose differences emerge when we consider the matter quantitatively. Let c(h, e) be the degree to which e confirms h. e confirms h incrementally if c(h,e&k) ${>}$ c(h, k), for k some body of background information. It confirms h absolutely if c(h,e&k) exceeds some chosen parameter, let’s say .98.

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 for the absolute notion, others for the incremental notion. But all of Hempel’s preferred conditions hold for the absolute notion! It is only Converse Consequence, which he rejects, that fails to hold absolutely.

The problem is that Hempel’s rhetoric and his examples, which mostly involve the confirmation of generalizations by their instances, suggest the relative notion. A black raven makes it likelier, not absolutely likely, that all ravens are black. Relative confirmation is naturally understood as positive probabilistic relevance, or probabilification. And probabilification meets neither of Hempel’s two main conditions.³⁴ Not consistency, for Rudy is a raven is positively relevant both to Rudy is a happy raven and Rudy is an unhappy raven. Not special consequence, for Rudy is a black raven relatively confirms Rudy is a black raven and all other ravens are white despite being negatively relevant to All other ravens are white.

At this point Hempel might seem refuted. His conditions hold for absolute confirmation, but that is not what he’s talking about. The standard model of relative confirmation is positive probabilistic relevance, but that interpretation does not meet his conditions. This doesn’t 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…it is up to the defender of Hempel’s instance-confirmation to produce the tertium quid (Earman 1992, p. 67).

Hempel in fact left a number of clues about this. Here he is introducing the stronger condition of which special consequence is meant to be a corollary:

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 Condition (GC): 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 1945a, 103, italics mine)

Hempel’s reasoning here is interesting. P’s consequences are confirmed by e, he says, because “any such consequence is but an assertion of all or part of the combined content of the original hypotheses.” Supposing for simplicity’s sake that $P = \{p _{1}, p _{2}\}$, Hempel thinks that any consequence of $p _{1}$&$p _{2}$ asserts part or all of the combined content of $p _{1}$ and $p _{2}$, and that this helps us to see why e’s support for $p _{1}$ and $p _{2}$ must carry through to q.

Why does Hempel insist on e confirming “every one” of the sentences in $\{p _{1}, p _{2}\}$, as opposed, say, 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 is the conjunction we focus on; f might be a free rider in e&f. The point is that general consequence would not be plausible, if e were not asked to confirm each of P’s members separately. Let’s hold onto this as it will be important later.

Given that Hempel insists in general consequence on “wholly” confirming evidence—evidence confirming both of $p _{1}, p _{2}$—why does he not also insist on wholly confirming evidence in special consequence? Any reason there might be for asking e to confirm both members of $\{p _{1}, p _{2}\}$ is surely also a reason for asking it to confirm both conjuncts of $p _{1}$&$p _{2}$! special consequence as we read it today imposes no such requirement, which to me suggests that we may be reading it incorrectly.

Again, Hempel objects to converse consequence that Rudy is black does not confirm Rudy is black & Hooke’s law holds. But, Rudy’s blackness does confirm the conjunction in the sense of probabilifying it. What it doesn’t do is confirm all of the conjunction; Hooke’s Law is not made the least bit likelier. Charity requires us to interpret him as imposing the stronger requirement: To confirm a conjunction, e must confirm both conjuncts. Bayesian confirmation lacks this property, but we can easily impose it:

(WC) e wholly or pervasively confirms h iff e probabilifies all of h’s parts.³⁵

Wholly confirming h is confirming all of it. Let’s review Hempel’s conditions with this notion of confirmation in mind.

consistency: Rudy is black and happy is incompatible with Rudy is black and unhappy, and yet both are probabilified by Rudy is black. The problem here is non-pervasiveness. To probabilify both statements and their parts, Rudy is black wouid have to make it likelier both that Rudy is happy and that Rudy is not happy.

entailment: Suppose e entails h. Then if g is part of h, e entails g by transitivity of entailment, whence pr($g | e$&k) = 1. e pervasively confirms h, then, provided only that $\hbox {pr}(g | k ) < 1$. Call a hypothesis h novel if it has no parts g such that $\hbox {pr}(g | k ) = 1$. Evidence e that entails a novel hypothesis confirms all of that hypothesis.

special consequence: If e confirms all of h, then it confirms all of h’s parts, and hence (by transitivity of part-whole) the parts of its parts. To probabilify all of the parts’ parts is the same as pervasively probabilifying each part. Let hempelized special consequence be the principle

($\hbox {SC}_{h}$):

If e pervasively confirms h, then it pervasively confirms h’s parts.

This is virtually a logical truth! Hempel’s version of special consequence has its problems, but implausibility is not one of them; triviality is more like it.

A word finally about Hempel’s positive theory. Though rejecting converse consequence, he thinks a certain kind of consequence is confirming. A generalization h is confirmed by its “development” for observed individuals, written $Dev _{I}$(h). Now, h’s development for I sounds like it should be the part of h that concerns I, that is, what h says about those particular individuals. Hempel’s positive theory would then be that a generalization is confirmed by (certain of) its parts.

If that is the intention, though, the definition doesn’t capture it. $Dev _{I}$(h) is defined by Hempel as what h says when its quantifiers are restricted to the individuals in I. This is not always even a consequence of h, let alone a part.

To see the problem, let pluralism be the theory that for all x, there exists a y that is not identical to x. Pluralism is true, I take it, hence true about every subject matter. But its development for I is false, if I contains just one individual. There is a problem in the other direction too. Let monism be the theory that for all x and y, x is identical to y. Monism is false, I take it, and there is not a lot of evidence for it.. Hempel has us gaining such evidence whenever we refuse to look at more than one individual. (There is indeed just one thing, leaving aside all the other things.)

One might think of Hempel as groping here for the notion of partial truth—truth where a certain subject matter is concerned. For h to be true about subject matter $\textsf{m}$, recall, goes with the truth outright of the part of h that concerns $\textsf{m}$. The problem we were running into above is that the part of h about $\textsf{m}$ cannot always be obtained by restricting quantifiers. This is a very small mistake on Hempel’s part! Yet it’s the only one he has made, on the present interpretation.