Knowledge By Necessity
We can know that a proposition is true and we can know that a proposition is necessary, but can we know that a proposition is true by knowing that it is necessary? Consider a simple tautology like “Hesperus = Hesperus”: don’t you know this is true by seeing that it is necessary? If someone asks you why you think it is true, you will answer, “It couldn’t be otherwise, so it has to be true” or words to that effect. The sentence is clearly necessary, so you can infer that it must be true. You treat the modal proposition as a premise to derive the non-modal proposition. The former proposition acts as the ground of your knowledge of the latter proposition. You can tell just from the form of the proposition that it must be true, and thus it is true. You derive an “is” from a “must”. You really can’t help seeing that the sentence expresses a necessity, given that you grasp its meaning, and truth trivially follows. We can call this “necessity-based” knowledge: knowledge that results from, or is bound up with, modal knowledge. How else could you know the proposition to be true—not by empirical observation, surely? You know it by analysis of meaning: the meaning is such as to make the sentence necessary. The sentence has to be true in all possible worlds, given its meaning, and so it is true in the actual world—truth is a consequence of necessity. It is immediately obvious to you that the sentence is necessary—and so it must also be true. If someone couldn’t see that “Hesperus = Hesperus” is necessary, you would wonder whether he had understood it right. Maybe someone could fail to see that necessity entails truth and hence not draw the inference; but how could he fail to see that “Hesperus = Hesperus” is a trivial tautology, in contrast to “Hesperus = Phosphorous”? The sentence is self-evidently a necessary truth.
It thus appears that some knowledge of truth is necessity-based: knowledge of the truth involves knowledge of the necessity, with the latter acting as a premise. Sometimes people believe things to be true because they perceive them to be necessary. You know very well that Hesperus is necessarily identical to Hesperus—how could anybody not?—and so you are entitled to believe that “Hesperus is Hesperus” is true. For analytic truth generally the same epistemic situation obtains: you can see the sentence has to be true given what it means, so it follows that it is true. Even if the move from necessity to truth is not valid in every case (e.g. ethical sentences), it is in some cases. We can thus derive non-modal knowledge from modal knowledge. But clearly not all knowledge is like this—mostly you can’t come to know truths by perceiving necessities. You can’t come to know the truth of “Hesperus = Phosphorus” that way: here you have to investigate the empirical world. The sentence is necessary, but you can’t use this necessity to decide that the sentence is true. You may know that if it is true then it is a necessary truth, but you don’t know that it is true just by understanding it, so you can’t use its necessity as a premise in arguing that it is true. You need to appeal to observation to show that the sentence is true—as you do for any other empirical proposition. Here your knowledge is observation-based not necessity-based—observable facts about planetary motions not the analysis of meaning. You won’t cite tautology as a reason for truth in this case, but you will in the other case. You won’t argue that there is no alternative to being true for “Hesperus = Phosphorus”. Clearly you can’t argue that “Hesperus = Phosphorus” follows from “Possibly Hesperus = Phosphorus”, but that is the only modal truth you have at your disposal in your current state of knowledge, unlike the case of “Hesperus is Hesperus”. So you can’t take a short cut to knowledge of truth by relying on an evident necessity—you have to resort to arduous empirical investigation. You may wish you knew that the sentence is necessary, so as to spare yourself the epistemic effort, but that is precisely the knowledge you lack in this case, since the expressed proposition in question refuses to disclose this fact. We resort to observation when our modal sense cannot detect necessity, which is most of the time. Necessity-based knowledge is quick and easy, unlike the other kind.
I have been leading up to the following thesis: a priori knowledge is knowledge by necessity while a posteriori knowledge is not knowledge by necessity. Here we define the a priori positively and the a posteriori negatively, unlike the traditional definition in terms of knowledge by experience versus knowledge not by experience. This gives us a result for which we have pined: a positive account of the nature of a priori knowledge. The two definitions map onto each other in an obvious way: knowledge by necessity is not knowledge by experience, and knowledge by experience is not knowledge by necessity. That is, we don’t come to know necessities by experiencing them, and necessities are no use to us in the acquisition of empirical knowledge. Necessity plays a role in acquiring a priori knowledge, but it plays no role in acquiring a posteriori knowledge. To have a crisp formulation, I shall say that a priori knowledge is “by necessity” and a posteriori knowledge is “by causality”—assuming a broadly causal account of perception and empirical knowledge. We can also say that a priori knowledge is knowledge grounded in our modal faculty, while a posteriori knowledge is knowledge grounded in perception and inference—thus comparing different epistemic faculties. But I think it is illuminating to keep the simpler formulation in mind, because it directs our attention to the metaphysics of the matter: modality in one case and causality in the other. The world causally impinges on us and we thereby form knowledge of it, and it also presents us with necessities that don’t act as causes—thus we obtain two very different kinds of knowledge. The mechanismis quite different in the two cases—the process, the structure.
Is the thesis true? This is a big question and I shall have to be brief and dogmatic. There are two sorts of case to consider: a priori necessities and a priori contingencies. I started with an example of a simple tautology because here the necessity is inescapable—you can’t help recognizing it. Hence knowledge of necessity is guaranteed, part of elementary linguistic understanding. But not all a priori knowledge is like that, though tautology has some claim to be a paradigm of the a priori. What about arithmetical knowledge? If it is synthetic a priori, then we can’t say that knowledge of mathematical necessity results from linguistic analysis alone. Nevertheless, it is plausible that we do appreciate that all mathematical truths are necessary; we know that this is how mathematical reality is generally. When we come to know that a mathematical proposition is true we thereby grasp its necessity: a proof demonstrates this necessity. Mathematics is arguably more about necessity than about truth: we can doubt that mathematical sentences express truths (we might be mathematical fictionalists), but we don’t doubt that mathematics cannot be otherwise—it has some sort of inevitability. We might decide that mathematical sentences have only assertion conditions, never truth conditions, but we won’t abandon the idea that some sort of necessity clings to them (though we may be deflationary about that necessity). Modal intuition suffuses our understanding of mathematics, and this can function in the production of mathematical knowledge. I see that 3 plus 5 has to be 8, so I accept that 3 plus 5 is 8. Mathematical facts are inescapable, fixed for all time, so mathematical truths are bound to be true: I appreciate the necessity, so I accept the truth. The epistemology of mathematics is essentially modal and this plays a role in the formation of mathematical beliefs: in knowing necessities we know truths—and that is the mark of the a priori. 
Much the same can be said of logic, narrowly or widely construed. You cannot fail to register the necessity of a logical law, and you believe the law because you grasp its necessity. Nothing could be both F and not-F, and so nothing is. The necessity stares you in the face, as clear as daylight, and because of this you come to know the law—the knowledge is by necessity. Accordingly, it is a priori. It isn’t that you can believe in the truth of the law and remain agnostic about its modal status (“I believe that nothing is both F and not-F, but I’ve never thought about whether this is necessary or contingent”). Your belief in the law is bound up with your belief in its necessity; thus logical knowledge is a priori according to the proposed definition. The same goes for such propositions as a priori truths about colors: “Red is closer to orange than blue”, “There is no transparent white”, etc. Here again the necessity is what stands out: we know these propositions to be true because we perceive their necessity—not because we have conducted an empirical investigation of colors. Accordingly, they are a priori. In all the cases of the a priori in which the proposition is necessary this necessity plays an epistemic role in accepting the proposition; it is not something that lies outside of the epistemic process. It is not something that is irrelevant to why we accept the proposition. We recognize the necessity and that recognition is what leads us to accept the proposition. If we accepted the proposition for other reasons (testimony, overall fit with empirical science), then our knowledge would be a posteriori; but granted that our acceptance is necessity-based the knowledge is a priori. Being known a priori is being known by necessity: the involvement of modal judgment is what defines the category.  By contrast, a posteriori knowledge does not involve modal judgment—you could achieve it and have no modal faculty at all. The basis of your knowledge is not any kind of modal insight, but observation and inference (induction, hypothesis formation, inference to the best explanation). You don’t have modal reasons for believing that the earth revolves around the sun, but you do have modal reasons for believing that red is closer to orange than blue—viz. that it couldn’t be otherwise. Since things couldn’t be otherwise, they must be as stated, and so what is stated must be true. The modal reasoning is not a mere add-on to knowledge of a priori truths but integral to it.
It may be thought that the contingent a priori will scuttle the necessity theory, since the proposition known is not even a necessary truth; but actually it is not difficult to accommodate these cases with a little ingenuity. One line we could take is just to deny the contingent a priori, and good grounds can be given for that; but it is more illuminating to see how we could extend the necessity theory to cover such cases. Three examples may be given: reference fixing, the Cogito, and certain indexical statements. What we need to know is whether there are necessities that figure as premises in these cases, even if these necessities are not identical to the conclusion drawn from them. Thus in the case of fixing the reference of a name by means of a description (e.g. the meter rod case) we can say the following: “No matter what the length of this rod may be the name ‘one meter’ will designate it”. If I fix the reference of a name “a” by “the F”, then no matter which object is denoted by that description it will be named “a”. This doesn’t imply that the object named is necessarily F; it says merely that the name I introduce is necessarily tied in its reference to the description I link it to. Because we recognize this necessity we can infer that a is the F(no matter who or what the F is). We don’t need to undertake any empirical investigation to know that a is the Fsince it follows merely from the act of linguistic stipulation—and that act embodies a necessary truth (“the person designated by ‘a’ is necessarily the person designated by ‘the F’”).
In the case of the Cogito it is true that the conclusion is not a necessary truth (since I don’t necessarily exist), but there is a necessary truth lurking behind this proposition, namely “Necessarily anyone who thinks exists”. It is a necessary truth that thinking implies existence (according to the Cogito), but it is not a necessary truth that the individual thinker exists—he might not have existed. I know that I exist because I know that I think and I know that anything that thinks necessarily exists. Thus I use a modal premise to infer a non-modal conclusion: from “Necessarily anything that thinks exists” to “I exist”. That is my ground for believing in my existence, according to the Cogito, and it is a necessary truth. Thus the knowledge derived is a priori, according to the definition. I don’t make empirical observations of myself to determine whether I exist; I rely on a necessary truth about thought and existence, namely that you can’t think without existing. I know that I exist (contingent truth) based on the premise that anything that thinks exists (necessary truth), so my knowledge essentially involves the recognition of a necessity.
Thirdly, we have “I am here now”: this expresses a contingent truth whenever uttered but is generally held to be a priori. I know a priori that I am here now, but it is contingent that I am here now. But again there is a necessary truth in the offing, namely: “Anyone who utters the words ‘I am here now’ says something true”. By knowing this necessary truth I know that I must be speaking the truth when I utter those words, but my utterance expresses a contingent truth. So I rely on a necessary truth to ground my belief in a contingent truth. Without that necessary truth I would not know what I know, i.e. that my current utterance of “I am here now” is true. Again, the case comes out as a priori according to the definition; we just have to recognize that the modal premise need not coincide with the conclusion. We can have a priori knowledge of a contingent truth by inferring it from a distinct necessary truth. So we have found no counterexamples to the thesis that all a priori knowledge is knowledge by necessity.
I have assumed so far that the type of necessity at issue is metaphysical necessity, not epistemic necessity. This is the kind of necessity we recognize when we come to know something a priori. But we could formulate the main claims of this essay using the concept of epistemic necessity. For simplicity, just think of this as certainty, construed as a normative not a psychological concept—not what people are actually certain of but what they ought to be certain of. Then we could say that when I am presented with a tautology I recognize that it is certain and infer from this that it must be true, and similarly for other cases of a priori knowledge. This approach converges with the account based on metaphysical necessity, because certainty and necessity correlate (more or less) in cases of a priori truth. But I prefer the metaphysical formulation because it connects an epistemic notion with a metaphysical notion—a priori knowledge with objective necessity. When I know something a priori I know it by recognizing the objective trait of necessity not a psychological trait of certainty (however normatively grounded). Thus the epistemological distinction has a metaphysical correlate or counterpart. To know something a priori is to know it by detecting an objective fact of necessity, though we may also be certain of what we thereby know. In contrast, to know something a posteriori is not to know it by necessity detection but by perception and inference (by causality). This is a deep and sharp distinction, and it at no point relies on a purely negative characterization of what we are trying to define. We really do know things in two radically different ways: by apprehending necessity or by registering causality.
 Perhaps part of the attraction of the view that mathematics consists of tautologies is that it comports with the idea that our knowledge of mathematics involves knowledge of necessities. The necessities occupy the epistemic foreground.
 Given this account of a priori knowledge, it is doubtful that animals have it, because they lack modal sensitivity—they don’t perceive that propositions are necessary. If you present an animal with a tautology, it will stare at you blankly. They may have innate knowledge, but they don’t have a priori knowledge. Not even the most intelligent ape has ever thought that water is necessarily H2O or that the origin of an ape is an essential property of it. Animals have no knowledge of metaphysical necessity. This explains their lack of a priori knowledge.