Confirmation and Relative Frequency

In “Two Concepts of Probability,” Carnap writes: “It is clear that from a probability1 statement [i.e. a statement about degree of confirmation] a statement on frequency can never be inferred, because the former is purely logical while the latter is factual.” (526) For instance, consider the probability1 statement that the hypothesis that any given throw of a die will be a six has a 1/6 probability on the evidence that the die is symmetrical, or P1(h|e) = 1/6. Carnap says that, from this probability1 statement, we cannot infer that the limit of the relative frequency of the six-event is 1/6, or P2(h) = 1/6. My intuition is that this is, strictly speaking, true, but that given both e and P1(h|e) in an adequate inductive logic, the proposition that P2(h) ≈ 1/6 is thereby confirmed to an acceptable degree. If we take an “inference” in inductive logic to be the premises’ conferral of an acceptable degree of confirmation on the conclusion, then, given e and P1(h|e) in an adequate inductive logic, we can infer P2(h) ≈ 1/6. This is not a deductive inference of P2(h) from P1(h|e), but it is enough to suggest that Carnap might have been a little bit confused about the relation between confirmation and relative frequency.

Nevermind, for now, how to extend the basic moral here to cases in which the “≈” won’t work. Am I missing something?

"Grue" and "Third Child" Are Sometimes Projectible

The observation that some batch of emeralds is green is evidence for the hypothesis (K) that all emeralds are green, and is not evidence for the hypothesis (H) that all emeralds are grue. But this need not be the case. Suppose we lived in a world in which the spectral reflectance curve of everything that comes into existence, after a certain length of time n, shifts some specified amount. In particular, the shift is such that, after a duration of n, everything that comes into existence as a green object becomes blue. Suppose, further, that no emeralds naturally form in this world. Then, at time t – n, Jack synthesizes the first batch of emeralds ever to come into being in this world. Jack observes that the emeralds are green.

Isn’t it the case, here, that Jack’s observation is evidence for H, but not for K? Isn’t it the case, here, that “grue” is projectible over objects created at t – n?

A similar and perhaps slightly less ridiculous imaginary case would be a nearby possible world in which birth order strongly impacted the way we form friendships, so that people with the same number of older siblings were much more likely to be friends. Then if friends are more likely to be amongst one another than with others, then the observation that five people in a room of thirty are third children is evidence for the hypothesis that (at least most of) the other twenty five are also third children.

What are the consequences of this? Well, if “grue” and “third child” are projectible when we have certain background information or given certain background conditions* and unprojectible when our information or conditions are otherwise, then we sometimes need certain information about our own information or the condition of the world in order to determine whether a hypothesis or predicate is projectible. If we want to maintain that claims about the projectibility of a hypothesis or predicate are a priori, then we will have to relativize “projectible” to a specification of the condition of the world or our own information (and even that might not work). Otherwise, we will have to say that claims about the projectibility of a hypothesis or predicate are a posteriori, and that the information about our own background information or about the condition of the world is somehow evidence for the projectibility claim. Either way, the most important consequence is that the best theories of projectibility cannot distinguish the projectible from the unprojectible solely on the basis of necessary or irreducibly monadic features of given hypotheses or predicates.

Note also that, in Jack’s world, in populations of objects existing for a duration of n or longer, then we can only project normal color predicates (e.g. “green”/“blue” – not “grue”/“bleen”) from samples. So, in this case, even within a given world, sometimes “green” will be projectible and sometimes “grue” will be projectible. This seems to have nothing to do with how entrenched “green” and “grue” are in the world.
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* - I say “background information or given certain background conditions” because it is not clear what is making “grue” and “third child” projectible in these scenarios – that they have certain weird features or that we know they have the weird features.

An Unnecessary Criterion of Adequacy for a Definition of "Confirmation?"

One of Carl Hempel's criteria of adequacy for a general definition of "confirmation" in his "Studies in the Logic of Confirmation" (1945) is that it must apply to sentences of any logical form. According to Hempel, a general definition of "confirmation" cannot state only what sorts of sentences confirm, say, universally quantified sentences with a single variable.

Maybe I'm missing something, but I think this is not a good criterion. As far as I can tell, as long as we define “confirmation” for, say, universally quantified sentences of single variable (with negation), then, if we accept Hempel’s “equivalence condition”*, we have a general definition of “confirmation.”

Can’t we eliminate any existential quantifier in terms of a universal quantifier and negation by means of the logical equivalence (Ex)(Fx) <=> ~(Ax)~(Fx)?

Isn’t it true that, for any n-adic predicate R, for any place pi in the predicate, for any ordered set of objects {x1, x2, …, xi-1, xi+1, …, xn} occupying the other places in the predicate, we can define a monadic predicate R`, such that R`xi <=> Rx1,x2,…xi-1,xi,xi+1,…xn?

And, at least in languages without empty names, isn’t it true that Fa <=> (Ex)(x = a & Fa)?

So, if we allow ourselves to define somewhat contrived predicates and help ourselves to languages non-free logics, isn’t every sentence logically equivalent to some sentence in universal form with one variable? This is not, of course, to say that we could actually get by in science or daily life without sentences of more complex logical forms; this is a claim only about what a definition of “confirmation” needs to be truly general. As usual, I reserve the possibility that I’m totally mistaken about the logic here.
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* - The equivalence condition states that "If an observation report confirms a hypothesis H, then it also confirms every hypothesis which is logically equivalent with H."

Does the Possibility of Discovering Novel Experiences Have Serious Meta-Ontological Consequences?

It sure seems to me as if we can discover new sui generis properties, and that this has some important consequences for the approach to ontology, which I am very sympathetic to, outlined in Carnap’s “Empiricism, Semantics, and Ontology.” Some examples of discoveries of new sui generis properties: that feeling I had when I was writing the verses of Sophie thinking about Alyssa; Dennett's example (from “Quining Qualia”) of discovering that the sound of the open E on a guitar is, in part, composed of the sound of the E harmonic at the 12th fret might also work (especially if "sounds like open E" was a sui generis property before, and so the structure of my linguistic framework is more severely changed around by the introduction of "sounds like the E harmonic at the 12th fret"); also, the first time one tries any really distinctive kind of food, esp. a raw ingredient. Really, any time we use demonstratives like “this experience” because other words just wouldn’t do the trick. (One thing: how do I know the properties are sui generis right off the bat? Well, I don't, especially since I take calling them "sui generis" to amount to a claim about the place of the predicate for the property in the L-structure [i.e., undefined and probably also undefinable]. But I discover them and add them at least to the model at once [even if I only pick out the properties demonstratively], and I can try to reassure myself that they are sui generis later.)

What exactly do I mean by "discover"? I discover a property when I acquire a concept of or else actually refer to a property that was not previously in my ontology, in the model of the language I was using. This property is introduced all at once, not after subtle reflection and meta-reflection on previous experience and theory. It is precisely because it is all at once that I do not know that the property is sui generis, though I think I have my suspicions from the get-go.

What exactly are the consequences? Well, that the changes to an L-structure that we can effect are sometimes completely determined by what seem to be universally adopted methodological norms. It seems like, just in the face of these novel experiences, I have to add the new property to the model of my linguistic framework. There is no room for conceptual engineering at this point (though that can come later - perhaps "sounds like E-12th fret harmonic" later becomes a defined term). Also, a sort of realism about properties - not that we need to quantify over predicate symbols, but that (at least some sui generis) properties are in some sense "out there". They are not necessarily "out there" in that our norms should be such that the property should have been in the model all along; this is not a claim about the inadequacy of our norms, but about the sensitivity of linguistic change to extra-linguistic forces.

At least that’s what I want to say the consequences are, but maybe I’m wrong. In ESO, Carnap says only that in a “linguistic framework” the use of existential quantifiers must be rule-governed. But my discovery of these new properties of my experience (if it was as I describe it) was quite rule-governed, in the sense that I did not violate or change any rules of English semantics in pronouncing it. Perhaps adding a property to a model does not change the rules of a language. It seems to depend on what counts as a “linguistic framework.” If numerically identical frameworks can have different universes of discourse (perhaps relative to a single context) or words (say, predicates coined ad hoc to pick out newly discovered property), then I do not necessarily adopt a new linguistic framework whenever I denote new sui generis properties. I have been thinking of linguistic frameworks as interpreted L-structures, but perhaps this is too rigid. Perhaps we might conceive of linguistic frameworks in such a way that they might be constituted, in whole or in part, by rules for introducing new terms, or even new “kinds” of terms.

I guess I still don’t really have an answer to the question, at first a bit inchoate, that motivated this post: Does the apparent fact that, in our linguistic framework – scientific English? –, we can denote (what we sometimes correctly take to be) entirely new sui generis properties somehow entail that ontology is not guided by the convenient choice of a linguistic framework?

In other words: sometimes we introduce altogether new (sometimes sui generis) properties into our ontology by a process that apparently does not necessitate commitment to any sort of heavy background theory. The introduction, and the need for the introduction, presses on us with a feeling of utter immediacy. Clearly, we are following some norm here – both in feeling the need for the introduction, and in actually executing it (an act perhaps beyond our control). This norm, whatever it is, is meta-ontological. It dictates how and when to introduce new kinds of terms and objects. The question is: can this norm be stated within or somehow compassed by the meta-ontology of ESO, say as a peculiar fact about the rules that constitute the linguistic framework of present-day scientific English? It seems hard to square the utter non-verbalness of the need to follow the norm and ESO’s (perhaps deceptive) appearance of making all ontological norms essentially semantical rules, and meta-ontological norms reasons for changing semantical rules. It all seems to depend on the nature of linguistic rules and linguistic frameworks.

A Little Cicero, and Why Even Bad Friendships Might Be a Little Good

But I must at the very beginning lay down this principle - friendship can only exist between good men. I do not, however, press this too closely, like the philosophers who push their definitions to a superfluous accuracy. They have truth on their side, perhaps, but it is of no practical advantage. Those, I mean, who say that no one but the "wise" is "good." Granted, by all means. But the "wisdom" they mean is one to which no mortal ever yet attained. We must concern ourselves with the facts of everyday life as we find it - not imaginary and ideal perfections. Even Gaius Fannius, Manius Curius, and Tiberius Coruncanius, whom our ancestors decided to be "wise," I could never declare to be so according to their standard. Let them, then, keep this word "wisdom" to themselves. Everybody is irritated by it; no one understands what it means. Let them but grant that the men I mentioned were "good." No, they won't do that either. No one but the "wise" can be allowed that title, say they. Well, then, let us dismiss them and manage as best we may with our own poor mother wit, as the phrase is.
- Cicero, De Amicitiae, I.5

The principle here is that only good people can be friends. Strictly speaking, this must be false. If Hitler spends a lot of time together with Joe, if they enjoy one other’s company, if each readily sees the good qualities of the other, if they are prepared to defend one another at some risk to their own personal comfort or well-being, then Hitler and Joe must be friends, no matter how terrible a person Hitler is in other respects. Bad people can be friends.

Maybe Cicero’s (really, the speaker Laelius’) statement of his principle is an imprecise formulation of the view that friendship necessarily consists, at least in part, in treating one’s friends as good people treat one another. Even if it is bad of Joe to treat Hitler as a friend, Joe does treat Hitler, in many respects, as a good person treats another good person. In many instances, it is good, if difficult, to defend somebody else at risk to oneself. In many instances, it is good, if difficult, to be ready to see the good qualities in another person. In many instances, it is good, if difficult, to be happy to be with other people. Even if one picks one’s friends poorly, friendship would seem to be a sort of training ground for treating or dealing with non-friends in these difficult but good ways. This is an empirical proposition – I guess I should look through the social psych literature to see whether it’s true. But if it is, it’s one reason to think that friendship of any sort is, in at least one respect, instrumentally good.

Goodman on Relevant Conditions

Counterfactual conditionals aren’t truth-functions of their antecedents and consequents. If the consequent counterfactually follows from the antecedent, in general, it would not be because there is an entailment from A to C, but because there is a sort of entailment from A conjoined with other propositions that are, in some sense, implicitly assumed along with A. In “The Problem of Counterfactual Conditionals” (in Fact, Fiction, and Forecast, 4th ed.) Goodman calls these other propositions the “relevant conditions” of the counterfactual, and says that one of the main philosophical problems with counterfactuals is the explicit specification of relevant conditions. I think I agree thus far.

Goodman also says, however, that the sense in which the relevant conditions are implicitly assumed is that they are taken to be true in the actual world. In Goodman’s words: “in asserting the counterfactual, we commit ourselves to the actual truth of the statements describing the requisite relevant conditions” (8). This part I’m not sure about. My worry is that this stand on speakers’ attitudes towards relevant conditions cannot account for certain counterfactuals related to utterly non-actual states of affairs. Take Irish Jim, who occupies a world in which the Irish can fly. In this world, the Irish are unlike other human beings in this respect, and they can fly strictly in virtue of their Irishness. Now consider:

(1) If Jim weren’t Irish, he couldn’t fly.

This counterfactual seems as true to me as any other counterfactual, and it sure looks to me like one of its relevant conditions is that:

(2) All Irish people can fly.

But, of course, I don’t believe (2). Since it seems that my beliefs about (1) and the powers of the Irish don’t seem to be inconsistent, it seems that the sense in which (1) implicitly assumes (2) is not that it assumes that (2) is actually true.

But perhaps (2) is not a relevant condition of (1), and (1) doesn’t implicitly assume (2) in any sense at all. Perhaps the relevant condition that I wanted to capture by (2) is something more like:

(3) In Jim’s world, all Irish people can fly.

Now, (3) I actually do believe (to the extent that I believe in facts about non-actual worlds), and (3) seems to do all the work that (2) can do in making (1) come out true. And, strictly speaking, I guess that (3) is true just in case (3) is true in the actual world – i.e., just in case, in the actual world, it is true that, in Jim’s world, all Irish people can fly. Furthermore, in general, I suppose, one believes that (3) is true just in case one believes that (3) is true in the actual world. So, if (3) is a relevant condition of (1), it looks like a speaker of (1) assumes (3) to be actually true. So, strictly speaking, maybe Goodman is right about speakers’ attitudes towards relevant conditions – they assume that relevant conditions are actually true. But if, as Goodman suggests at a certain point (6f), we want the relevant conditions to be non-modal, then (3) won’t do. In that case, I am convinced that we would do better to say that (2) is a relevant condition of (1), and rethink the nature of speakers’ attitudes towards the relevant conditions of counterfactuals that they utter.

I want to say that the attitude of the speaker towards the relevant conditions is one of something like provisional acceptance, or acceptance for the sake of argument. I can’t think of a good reason not to endorse this view, and I think it clearly has important consequences (different from those of Goodman’s view) for how we should approach both the problem of specifying relevant conditions and the problem of understanding the illocutionary force of utterances of counterfactual conditionals.