I’ve written a paper proposing the following constraint on knowability a priori and arguing that it has some wide-ranging implications for the study of the a priori. You can see the paper for more on that. In this post, I’ll be concerned to motivate the constraint and to defend it against some looming objections. Here is the constraint:

(AK) For anyp,pis knowable a priori only if, for any presuppositionqofp,qis knowable a priori.

Here are several arguments for (AK), or something close enough.^{[1]}

**1. Presuppositions and entailments **

The following principle relating presuppositions and entailments falls out of a fairly standard understanding of the notion of presupposition.

(PE) For any proposition

pand presuppositionqofp: necessarily, ifpthenq(i.e.,pentailsq).

The “fairly standard” understanding of presupposition is this: a sentence *S* presupposing *q* expresses a proposition *p* that is a *partial function* from worlds to truth-values. *p* is partial in virtue of being *defined* *only* for *q*-worlds; in every world in which it is not the case that *q*, *p* is undefined. In every world over which *p* is defined, *q* is true. So in every world which *p* takes to true, *q*. Therefore: necessarily, if *p*, then *q.*

(AK) would follow directly from (PE) and the following plausible epistemic principle:

(AN) For any

p,pis knowable a priori only if, for any entailmentqofp,qis knowable a priori.

(AN) and (PE) entail that for any *S* and any presupposition *q *of *S*, the proposition that *S* is true (i.e., *p*) is knowable a priori only if *q* is knowable a priori. This is (AK).

Unfortunately **(AN) is false**, so long as we define entailment as strict implication. For any *p* and any necessary truth *N*, *p* strictly implies (entails)* N*. But it is not the case that every necessary truth is knowable a priori. So if (AN) is right, then nothing is knowable a priori. Obviously (AN) needs to go.

Modifying it is a cinch, however:

(AN*) For any

p,pis knowable a priori only if, for any contingent entailmentqofp,qis knowable a priori.

This doesn’t get us all the way to (AK), but it gets us close enough:

(AK*) For any

p,pis knowable a priori only if, for any contingent presuppositionqofp,qis knowable a priori.

(AK*) is close enough because it still rules out Kripke’s meter stick example as a case of a priori knowledge (see the paper for more on this argument). Still, we can do better. Everyone, so far as I know, accepts the following constraint on knowability a priori (compare (AN)):

(AE) For any

pand entailmentqofp, if it is knowable a priori thatpentailsq, thenpis knowable a priori only ifqis knowable a priori.

(AE) obviously gets around the problem I raised for (AN). Combine (AE) with (PK) and (AK) follows straightaway.

(PK) For any

pand presuppositionqofp, it is knowable a priori thatpentailsq

(PK) strikes me as extremely plausible, for any number of reasons, the most obvious of which is that it seems possible to establish (PE) by a priori argument. Let me qualify this somewhat by noting that if Eric Swanson is right about the existence of non-entailing factives, (PE) — and (PK) by extension — are obviously problematic. (I’m very skeptical that he is right, but that’s an issue for another post.)

**2. The concept of a priori knowledge **

As my old professor Al Casullo taught me, the concept of a priori knowledge is merely the concept of nonempirical knowledge — i.e., knowledge justified entirely through nonexperiential means. For this reason, the expressions ‘knows through experience alone’ and ‘knows a priori’ are directly analogous. That is to say, ‘knows a priori’ means just knows through non-experiential means alone. Now clearly one doesn’t know *p* through experience alone if even one of one’s warrants for believing *p* is non-experiential in character. But if *p* presupposes *q*, then one’s warrant for believing *p* will include a warrant for believing *q*. So if one’s warrant for believing *q* is non-experiential in character, then one cannot know* p* through experience alone. It easy to construct a perfectly analogous argument for (AK).

Somewhat more formally, (AK) is entailed by (α) and (β), each of which strike me as quite plausible:

(α) For any

Kandp, it is knowable a priori thatpforKonly if there is a set of possible epistemic warrantsWs.t.: (1)Kcould haveWforp; (2) ifKwere to believep,K’s havingWwould be sufficient forKto know thatp; (3) no warrant inWis a posteriori (empirical, experiential) in character.(β) For any

K,p, presuppositionqofp, and set of warrantsW, ifK’s havingW(conjoined with the truth ofp) would be sufficient forKto know thatp, there is a warrant or warrantsw∈Ws.t.: (1)Khaswforqand (2) ifKwere to believeq,K’s havingwwould be sufficient forKto know thatq.

(β) is bound to be the controversial one of these, but if knowledge is closed under (known) entailment, than why wouldn’t it be closed under (known) presupposition?

Let me bookend this argument with a note about the kind of modality relevant to (α) and (β) (and, by extension, to (AK)). *K* could, in my sense, have *W* for *p* iff the kind of cognitive capacities required to have *W* are of the sort that *K *— qua human cognizer — could have. I.e.,* K* could have *W* for *p* iff having the kind of cognitive capacities required to have *W* would not make *K* a radically different kind of cognitive being than *K* actually is. I use this restricted notion of modality because I’m interested only in what cognitive creatures *reasonably like us* could know a priori.

**3. An objection and response**

In personal correspondence, my friend Anders has objected to (AK) by arguing that it rules out the a priori knowability of (6)-(11).

(6) Cicero is Cicero.

(7) Unicorns are unicorns.

(8) Tigers are tigers.

(9) Bachelors are unmarried.

(10) Stallions are horses.

(11) The shortest spy is a spy.

(6) is indeed not knowable a priori, according to (AK). This is because proper names carry existential presuppositions; a use of the name ‘Cicero’ presupposes at least that there is someone in the contextually restricted domain so-named. Since it is not knowable a priori that this is the case, (AK) rules out the a priori knowability of (6).

I do think (AK) gives the right result for (6). Here’s an intuitive argument. The property of being self-identical is something that only *actual objects* have. To know that Cicero is self-identical, you need to know that Cicero exists. So you can’t know (6) a priori.

(7)-(10) are trickier. Is the right reading of (7) as follows?

(7*) Every unicorn is a unicorn.

If so, then (7) is indeed knowable a priori, since (7*) doesn’t presuppose that there are unicorns. In fact, (7*) is vacuously true: it’s true iff {*x*: unicorn(*x*)} is a subset of itself, which it of course is, even if the set is empty. Likewise for (8)-(10).

But maybe the right reading of (7) isn’t (7*); maybe the right reading of ‘unicorns’ is just as a generic. I don’t know much about generics, but because they seem to have some kind of universal force, my intuition is that they don’t carry existential presuppositions. But if they do, then (AK) says that you can’t know (7)-(10) a priori. But you could still know (7*)-(10*) and (7’)-(10’) a priori.

(7*) Every unicorn is a unicorn.

(8*) Every tiger is a tiger.

(9*) Every bachelor is unmarried.

(10*) Every stallion is a horse.(7’) ∀

x[unicorn(x) → unicorn(x)]

(8’) ∀x[tiger(x) → tiger(x)]

(9’) ∀x[bachelor(x) → unmarried(x)]

(10’) ∀x[stallion(x) → horse(x)]

And I suspect that’s all that’s wanted anyway.

I think I need to bite the bullet on (11). It is not knowable a priori, since it presupposes something that is not knowable a priori (namely, that there is some spy, and that she is shorter than all the other spies).

(11) The shortest spy is a spy.

Following Kaplan (“Quantifying In”, fn.14), what **is** knowable a priori is (11*), since, in (11*), all the empirical presuppositions of the consequent have been locally accommodated.

(11*) ∃

x[x= ιy[shortest spy(y)]] → the shortest spy is a spy

This is extremely intuitive for me, but I may be in the minority. Anyway, here’s an argument borrowed (roughly) from Kaplan for why (11) is not knowable a priori. According to Kaplan in “Quantifying In”, (11) **entails** (11’). Since Kaplan’s argument is an a priori argument, it seems that it is knowable a priori that (11) entails (11′).

(11’) ∃α∃

x[Δ(α,x) & *α is a spy* is true].

(The asterisks stand in for Quinean quasi-quotes.) Here’s a rough paraphrase of (11’): there is an expression α and object *x* such that α denotes (refers to)* x*, and the result of writing down α followed by ‘is a spy’ is true. Since (11’) is not knowable a priori, and since it is knowable a priori that (11) entails (11’), by (AE) (11) is not knowable a priori.

(AE) For any

pand entailmentqofp, if it is knowable a priori thatpentailsq, thenpis knowable a priori only ifqis knowable a priori.

The nice thing about this argument is that it appears to sidestep all of the worries Anders raised for the argument that relied on (PK).

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[1] Much of this material is reproduced from a series of four posts on my own blog, *de crapulas edormiendo*. Since according to my stat counter, almost no one besides me reads my blog any more, I thought it might be useful to do some proselytizing for (AK) here.

I think that (AN*) is false. Let r be some a posteriori necessity, and p be some a priori contingency, and let q be the conjunction of p and r. This p and q provide a counterexample to your:

(AN*) For any p, p is knowable a priori only if, for any contingent entailment q of p, q is knowable a priori.

q is a contingent entailment of q; it is contingent because one of its conjuncts is contingent. But it is a posteriori because of of its conjuncts is a posteriori.

Hi Jonathan. Right you are. I suppose it might be possible to further gerrymander (AN*). Maybe (AN**):

(AN**) For any p, p is knowable a priori only if, for any contingent entailment q of p s.t. that q does not entail p, q is knowable a priori.

I’d just as soon toss it overboard, though.

I think you should abandon (AN*) and (AN**) both.

There’s a much more elegant way of representing it:

For any p, p is knowable a priori only if, for any entailment q of p, both q and the fact that p entails q are knowable a priori.

I think this is really what you were trying to say with “contingent,” but in a form sufficient to remove these problems.

No, that’s too strong, isn’t it? Now I can’t say that anything is knowable a priori.

I think there’s a problem with the first step: there’s no reason to reduce presuppositions to mere entailments. We may not *want* to say that every entailment (contingent entailment, etc.) of an a priori proposition is also a priori, but it may still be possible to motivate the claim that every presupposition is.

Patrick: it seems like you’re trying to articulate the (AE) principle.

It’s true that presuppositions aren’t

merelyentailments (if that were the case then every entailment would be a presupposition), but they are entailments, on a standard understanding of the notion.Of course they are entailments; what I mean is that it is probably significant that they aren’t *merely* entailments—i.e. that part of motivating AK will involve differentiating presuppositions from non-presuppositional entailments.

Patrick: didn’t Nate already account for presuppositions being more than merely entailments by specifying that they assert partial functions? What more do you have in mind?