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September 04, 2010

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Mitchell Porter

Tentatively, I interpret this as being about the difference between knowing a schema and knowing an instance of a schema. For example, a+b=b+a is a schema, 5271009+196883=196883+5271009 is an instance of the schema. There are at least two reasons why you might know a schema to be true but not know that an instance of the schema is true. First, you may not have verified that the instance does conform to the schema. It takes a little checking to confirm that those numbers match up. Second, you may never contemplate a particular instance, and may even be unable to do so. If a and b are specific numbers too large for the human mind to represent, then we just can't think that instance. Your argument here turns on a different obstruction, namely, thinking about things that no-one ever thinks of. Between the limits of cognitive bandwidth and the logical impossibility of knowing something that no-one knows, it does indeed seem that the underlying phenomenon should be widespread.

Jeremy Goodman

Very cool! I'm skeptical about (5), though. If we take r to be a Russellian proposition, I think someone could entertain r without entertaining q. Suppose I introduce the name "Prop" by rigidifying the description "the negation of the first proposition entertained by Steve on Sunday." It seems I can then entertain the proposition that nobody entertains Prop iff actually nobody entertains Prop, without entertaining Prop myself and without anyone else entertaining Prop. This looks like a counterexample to (5).

Maybe Fregeans can get around this worry by stipulating more structure for r; i.e., by stipulating that entertaining r involves thinking about q as such. Very crudely, they might define r as the proposition that nobody entertains q, where the sense of the concept of q that partially constitutes r is given by the description "the proposition that is true iff q." Or something like that?

djc

Mitchell: Yes, it's plausibly a case of knowing a schema without knowing an instance. I think it differs from the cases you mention, though, in that these are all cases in which although the instance is not known, it is knowable (by a being without our resource limitations, say). Whereas in the current case, the instance is not even knowable.

Jeremy: Nice point! It's a delicate question whether you really count as entertaining ~Ep in the case you describe: that depends in part on whether or not expressions such as 'p' are understood as singular terms in these contexts. In any case I'd like to think that the interpretation of E I offer around the top of p. 2 will get around this worry.

Ralph Wedgwood

Dave – Your counterexample to the claim every instance of ‘p iff Actually-p’ is knowable a priori is very elegant and ingenious. But in a way, we shouldn’t be so surprised by it.

The deep issue behind your counterexample has to do with the connection between (a) broadly logical notions (like the notion of a provable truth, or a proposition that is true-in-all-models, or the like) and (b) broadly epistemic notions (like the notion of a priori knowledge or knowability or the like).

As Gilbert Harman has been reminding us for more than 30 years, practically every attempt to formulate a tidy connection between logical concepts and epistemic concepts runs into counterexamples. This seems to be because the properties or relations picked out by epistemic concepts (like knowledge) are subject to various constraints, like the constraint that every proposition that is known is entertained, while these constraints do not apply to logical properties and relations (like validity and logical consequence, etc.).

Still, I think that it is possible to formulate an illuminating connection here. Consider the rules of inference that license the inference from the supposition of p to the conclusion ‘Actually-p’, and from the supposition of ‘Actually-p’ to the conclusion p.

I suggest that the epistemic import of these rules of inference is simple: at least so long as you possess the concept ‘Actually’, it is always rational for you to respond to the state of considering an inference that exemplifies one of these rules by accepting that inference. (By “accepting” this inference, I mean conditionally accepting the conclusion of the inference – conditionally on the assumption of the inference’s premise.)

Given certain additional assumptions (e.g., that there are certain other rules of inference, such as conditional proof or the like, which also have the same sort of rational epistemic import), we may conclude that (at least roughly) for every proposition p that you actually entertain, it is rational for you to accept ‘p iff Actually-p’.

So, if you actually entertain the proposition ‘No one entertains q’, it is rational for you to accept the biconditional ‘No one entertains q iff Actually (no one entertains q)’. Moreover, if you entertain this proposition, then the biconditional is true – since in that case both sides of the biconditional are false. Plausibly, then, for every proposition p that you entertain,‘p iff Actually-p’ is knowable by you.

Is it “a priori”? Well, in my view, it is not fundamentally propositions or pieces of knowledge that are “a priori”. More fundamentally, it is things like rules or patterns of inference that count as “a priori”. A rule of inference is a priori if and only if the rule is “built in” to one's basic cognitive capacities (such as one's possession of some concept or one's capacity for some type of attitude) – in the sense that possessing these cognitive capacities necessarily puts one in a position to follow this rule in a rational way.

The a priori knowable propositions are precisely those that can be rationally accepted as a result of being inferred from the empty set of assumptions by means of a priori rules of inference. So it certainly looks as if it is true, for every proposition p that you entertain, that you can know ‘p iff Actually-pa priori.

djc

Ralph: Thanks for this. Your suggestion (that if p is entertained, the proposition p-iff-actually-p is knowable a priori) is an interesting one, but I don't think it can be the correct moral to draw.

To see this, note that if one holds a view on which the sentence 'p iff actually p' is semantically fragile, expressing a different proposition in each world, your suggested claim will be false. Take the standard Russellian view on which the sentence expresses p-iff-p(@) in the actual world @ [and p-iff-p(w) in other worlds w] as an illustration. Then one simply needs p such that p is actually entertained but in which p-iff-p(@) is actually not known a priori. In these cases, if one were to follow your inferences rules to come to knowledgeably accept 'p iff actually p', one will come to know not p-iff-p(@) but a different proposition p-iff-p(w), where w is the world in which one has the relevant knowledge. I think that the correct moral on the semantic fragility view is that (when p is not knowable a priori) propositions of this sort are knowable a priori iff they are known a priori (a much stronger restriction).

If on the other hand one rejects the semantic fragility view, presumably holding that 'p iff actually p' expresses the same (non-world-involving) proposition in all worlds, then one will very likely reject (2) or (3) of the original argument, and one won't need your restriction to entertained propositions.

Also, for what it's worth, my own view is that the deepest issue here is not the familiar distinction between logical and epistemic notions, but rather some somewhat different distinctions among epistemic notions: in particular, the distinction between sentential and propositional apriority, and the distinction between being knowable a priori and having an a priori justification.

[N.B. It's probably best to avoid using quotational locutions such as 'p iff actually p' for propositions here, as the distinction between sentences and propositions is crucial. Of course it remains plausible that the sentence 'p iff actually p' is always knowable a priori, by following the relevant inference rules, but it's propositions that are at issue.]

Ralph Wedgwood

Dave -- You're right. (I had a dim sense that I wasn't quite hitting the nail on the head, which is why I added the qualification 'at least roughly....'.) Thanks for pointing out my slip!

Still, the inference rules that I described sound fine to me. Whenever you follow them, you are thinking rationally. If by following them you form a belief in [p iff Actually-p], then you thereby acquire a piece of a priori knowledge.

(To comply with your recommendations, I'm follow George Bealer by using square brackets to talk about propositions. So if 'p' is a schema taking the place of a sentence, '[p]' is a schema taking the place of a term referring to the proposition expressed by the sentence.)

In general, the difficulties seem to arise when we try to formulate a claim with a modal element, involving concepts like know-able.

In other words, it is easier to formulate a claim concerning the retrospective (or ex post) epistemic notions (like "doxastic justification" or "knowledge"), which refer to an actual epistemic achievement, than with the prospective (or ex ante) epistemic notions (like "propositional justification" or "knowability"), which refer to situations in which such epistemic achievements are available.

By the way, I'd prefer not to talk about "sentential a priority", which sounds much too linguistic to me. I'd prefer to talk about "conceptual truths" (which may not be knowable at all, whether a priori or otherwise). I agree that every proposition of the form [p iff Actually-p] is a conceptual truth, but only some of them are knowable a priori.

Jeremy Goodman

Thanks Dave! I thought my case would be a counterexample to (5) on your proposed interpretation of E, but maybe I'm not understanding the interpretation. Why doesn't my thinking that nobody entertains Prop count as entertaining "a proposition of which [Prop] is a proper or improper constituent”?

Also, is there an alternative to thinking of "q" in "Eq" as a singular term? Consider the sentence (SNOW) "John entertains the proposition that snow is white." I would have thought that there were pretty compelling linguistic reasons for thinking that the occurrence of "the proposition that snow is white" in (SNOW) is a singular term. To mention two: It supports anaphora (we can say "Bill entertained it too"), and its being a singular term would explain why it's truth preserving to substitute for its occurrence in (SNOW) a name of the proposition that snow is white.

Ryan Perkins

I am inclined to accept your (very cool) proof that some instances of the schema "p iff @p" are unknowable. In fact, I think that something further is true: I think that we can argue, with relatively modest premises, that many instances of the schema in question are not only unknowable, but unthinkable - i.e. it is metaphysically impossible to even entertain them. If you're interested, I've laid out the argument in brief below.

The argument purports to show that if p is a proposition which is not in fact entertained, then it is impossible to entertain the proposition that @p. Obviously this entails the weaker claim that many instances of "p iff @p" are unthinkable (simply replace "p" with any proposition that is not in fact entertained), and hence that many such instances are unknowable (since knowledge must be entertained).

We proceed as follows. Suppose that p is proposition which no one ever actually entertains. Can someone in a non-actual possible world W entertain the proposition that @p? Evidently not. For this would in effect require the inhabitants of W to indexically refer, in the right way, to the *actual world*. And they cannot do this. Of course, the inhabitants of W could make indexical reference to their own world, i.e. W. They might use sentences like "@p" to express a proposition which holds at any world iff p holds at W. But they could not entertain the proposition which WE think when WE use the words "@p", since that would require them to indexically refer, in the right way, to our world, a world they do not inhabit, which is impossible. Hence there is no possible world where the proposition that @p is entertained. Thus, where Mp means that it is possible that p, we have:
~Ep --> ~M(E(@p)). That generates a hell of a lot of unthinkable propositions.

One could reply to the foregoing line of reasoning by holding that the proposition @p is identical to an infinitary descriptive proposition, which inhabitants of W could think (if they were smart enough): e.g., "the proposition that p is true in the world in which ... [here we have a complete description of the actual world]." But indexical propositions are certainly not identical to infinitary descriptive ones. Alternatively one could hold that the inhabitants of W could introduce a singular term, t, referring to what is in fact the actual world. Perhaps they could fix its reference by using the infinitary description described above. Then the claim would have to be that @p expresses the same proposition as "p holds in t". The problem with this is that it's not at all clear that propositions expressed by means of indexicals are the same as propositions expressed by means of coreferential singular terms, for Frege-Perry style reasons.

djc

Ralph: I agree that the retrospective notions are safe from these worries, which apply only to the prospective notions.

Jeremy: If Prop is a constituent of [no-one entertains Prop], then if you entertain the latter you'll stand in E to the former, so this won't be a counterexample to 5 (I think counterexamples are more or less ruled out on logical grounds, if we use the reformulated reading of E). Re attitude ascriptions, the dominant view analyzes them via relations to singular terms denoting propositions, but there are also views on which they involve sentential operators, multiple relations to constituents of the propositions, and various other alternatives. Also relevant here are views on which satisfying the satisfying the ascription requires being related to the proposition under an appropriate mode of presentation.

Ryan: Yes, there are at least some hard questions about how someone non-actual could entertain propositions about the actual world. Tim Williamson raises some of those questions in an old article, and Scott Soames tries to address them by a version of the descriptive-proposition move. If one thinks that Ap is a singular proposition about the actual world (as Soames does), this will require that one can introduce singular terms in this descriptive way. Of course there are issues about whether the same proposition can be presented indexically and descriptively, but these aren't obviously worse with 'A' than they are with 'I' (where some people allow that someone could know that I am a philosopher when they only pick me out using a description like 'The Australian guy who has such-and-such blog'). In any case, I think that this issue about entertainability is distinct from the issue about knowability -- the argument concerning knowability goes through whatever one thinks about entertainability.

Jeremy Goodman

Oh, so we agree that if we define "Ep" to mean “Someone entertains a proposition of which p is a proper or improper constituent," then if I entertain [no-one entertains Prop], then I stand in E to Prop. In that case, I haven't given a counterexample to (5). But I do think I've given a counterexample to the adequacy of "E"-so-defined as an analysis of entertaining, since when I entertain [no-one entertains Prop] I don't thereby entertain Prop.

Jeremy Goodman

In other words, I think I've given a case where Ep and ~ENTp, where:

Ep: Someone entertains a proposition of which p is a proper or improper constituent.
ENTp: Someone entertains p.

This suggests the following modification of your argument:

r = ~Eq

1. Ar
2. Ar → 􏱈Ar
3. 􏱈(K(r ↔ Ar) → (r ↔ Ar))
4. 􏱈(K(r ↔ Ar) → ENT(r ↔ Ar))
5. 􏱈(ENT(r ↔ Ar) → ¬r)
—————————————–
6. ¬􏱉K(r ↔ Ar)

This is stronger than your original argument, since it doesn't appeal to the (I think dubious) claim that "entertaining a proposition requires entertaining its constituents." Instead, rather than being a substantive claim, 5 comes out true by definition.

djc

Jeremy: Yes, this is the point of my remark in the article. E doesn't need to be an analysis of entertaining for the reinterpreted argument to work. The only difference between my reinterpreted version of the argument and yours is that I use the reinterpreted E in premises 4 and 5 instead of ENT. The argument works just as well either way: regarding premise 5, it's clear that if one entertains a proposition of which (r<->Ar) is a constituent, one entertains a proposition of which r is a constituent. I like my version a little better, though, in that it needs only one E symbol!

Jeremy Goodman

Thanks Dave. Yup, your version's better.

Biep

Building r requires taking a specific q, which seems to milk the axiom of choice to its utmost - unless we go for intensions, such as Jeremy's Prop.

So this exercise may end up shedding light on the extent of the axiom of choice, and the plausibility of different versions thereof.

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