# [hist-analytic] A posteriori knowledge of necessary truths

Baynesr at comcast.net Baynesr at comcast.net
Sat Dec 18 10:47:03 EST 2010

```Suppose p is a necessary truth. Suppose I know p is true. Do I know a necessary truth? It would, at first, appear that nothing could be more obvious. But consider the following question: In such a case, is what I know identical with what is necessary? How do we link the necessary truth with the knowing? I can think of one way: introduce a quantifier. Without the quantifier I cannot express an identity between what is necessary and what is believed. Forget, for the time being, the problem of quantifying over propositional contents. So, we have

1. (Ep)[Nec(p) and I believe p].

Notice that neither of the premises:

2. Nec(p)

3. I believe p

need be de re, neither the belief nor the necessity. But if you join these quantificationally, then you infer from these premises the de re conclusion:

4. (Ep)[Nec(p) and I believe p]

Here we are "quantifying-in," in particular into a belief context, as well as coindexing things where one occurs within an opaque domain and the other does not. My skepticism is over whether a de re statement can be inferred from such a conjunction of de dicto statements. If 'I believe p' is not de dicto, then we have an even greater problem binding the 'p' of belief and the 'p' of necessity.

If we claim that

5. I know p, aposteriori --> Nec p

Then we have it that if p is not necessary then I cannot know it a posteriori, which is palpably absurd. But (5) is exactly what is being claimed by those who argue that there is a priori knowledge of necessary truths of one particular sort: identities. Where 'p' is some proposition such as 'a=b' where 'a' and 'b' may be rigid designators, from,

6. I know p, aposteriori

I infer

7. p is true

and from (7) I infer

8. p is necessary

and from (8) I infer

9. I know, a posteriori some necessary truth, p.

But I do cannot infer (8) from (6). I must add a premise:

10. a=b --> Nec. a=b

And this is not known a posteriori. In other words I cannot infer from my a posteriori knowledge of p, alone, that p is necessary; but, on the Kantian view, I can know that a proposition is necessary if I know it a priori. Are there necessary truths such that if I know them to be true my knowledge is a posteriori; that is, known by experience alone?

We are compelled to ask whether there are necessary truths such that I know them *only if* I know them a posteriori? So, then, if my knowledge of them is not a posteriori, then they cannot be necessary. Such seems to be the case with identity statement involving proper names, for in the case of identities we have it that:

11. I know p, aposteriori --> Nec p.

But taken alone this seems palpably false, since it entails that if p is not necessary it cannot be known a posteriori. There is no corresponding problem in the a priori case: If a proposition is not necessary it cannot be known a priori. Having knowledge of necessary truth is not having knowledge of a truth that is necessary. Knowledge a posteriori will not, alone, supply knowledge of necessity, in particular knowledge that p is necessary. When Kant, and others, suggest that we have no a posteriori knowledge of necessary truths, what is meant is that there is no a posteriori knowledge of the necessity of a truth. If we say,

12.If Nec. p --> (I know p, a aposteriori --> Nec p)

We have a triviality. And this is what the advocates of a necessary a posteriori are suggesting: If 'a=b' is true it is necessary; and, since it is necessary, if I know it a posteriori, then it is necessary, giving a posteriori knowledge of a necessary truth. Otherwise, they are pretty much stuck with (11) as expressing a posteriori knowledge of necessity; but (11) is false.
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