The answer: rationalism.
First some definitions:
Empiricism is the doctrine that observation is the sole basis of any given instance of knowledge.
Rationalism is the doctrine that at least some knowledge has a strictly ratiocinative (reason-based) component.
Rationalism is correct.
No knowledge is strictly observation-based. There is always a ratiocinative component. Knowledge is analytic when there is no empirical component to it; otherwise empirical.
The fact that no knowledge is strictly observation-based is a consequence of the fact that it is propositions that are known, not objects. One observes objects. One does not observe truths. In other words, one does not observe true propositions. One observes assemblages of spatiotemporal entities in virtue of which truths hold, but one does not observe truths per se.
This is a corollary of the fact, cited earlier, that truths are sets of properties: one observes instances of properties, not properties per se. One observes particular instances of roundness, tallness, redness, etc.; one does not observe the corresponding properties.
Also, given any one empirical proposition, there is no one sort of observation that warrants acceptance of that proposition; and given any observation, or observation-class, there is no one sort of proposition whose acceptance is thereby warranted. Consider the proposition John smokes. There is no one sort of observation that warrants acceptance of that proposition. And any observation, or class of observations, that warrant acceptance of that proposition also warrant acceptance of many other propositions.
Also, one observes only what is, not what isn’t; nor a fortiori what cannot possibly be. When I look inside a room, I see what is in the room, not what isn’t. I don’t see John’s absence. I see various presences and, on that basis, deduce that John is absent. In general, negative truths are not known strictly on the basis of observation.
The same is true of disjunctive truths---truths of the form either p or q. It may be that my visual perception of the new car in John’s driveway warrants acceptance of the proposition:
Either John bought a new car or that car belongs to a houseguest.
But if so, that is because my perception directly warrants acceptance of some non-disjunctive proposition---some proposition along the lines of there is a new car in John’s driveway---which, in its turn, warrants acceptance of the aforementioned disjunction.
There cannot possibly be any strictly observational basis for acceptance of statements of the form:
If q follows from p,
p is incompatible with not q,
which in turn means:
The joint truth of not q and p is an impossibility.
Since one cannot observe what isn’t, let alone what cannot possibly be, observation cannot, at least not by itself, warrant acceptance of any statement of the form if p, then q.
Also, belief, and therefore knowledge, necessarily involve classification.
To believe that John smokes is to believe that John belongs to the class of smokers. To believe that Chester is green is to believe that Chester belongs to the class of green things.
But sense-perception cannot itself result from an act of classification, since such an act would presuppose awareness not just of the object being classified, but also of its having such properties as to warrant its being classified in a given way. Therefore, a sense-perception could result from an act of classification only if there were some other, prior state of awareness from which that perception and of whose content its own derived. But in that case, that other awareness would be the sense-perception. A sense-perception is, by definition, an awareness of the external world that is non-inferential. Classification is a form of inference, and sense-perception therefore precedes inference.
Thus, it isn’t that some truths are known strictly on the basis of observations and others are not; it is rather that no truths are known strictly on the basis of observation. No knowledge is strictly empirical. Some knowledge is partly empirical.
What we call ‘knowledge’ is conceptually articulated awareness. It is one thing for me to be aware of the uncomfortableness tightness of my shoes. It is quite another for me to know that my shoes are uncomfortably tight. I cannot know the latter without having various concepts---shoe, tight, uncomfortable, foot, etc. But I can certainly experience an uncomfortable feeling in my feet without having those concepts---and, indeed, without having any concepts. My having knowledge of that truth involves my conceptualizing my feelings of discomfort---it involves my understanding those feelings in terms of the aforementioned concepts.
Raw observation gives me the feeling, but it is up to me to use the concepts that I have to articulate that feeling into something that can be known. This is because to articulate an observation is to classify its contents, and classification observation precedes classification, as we just saw.
Therefore, observation cannot tell me how to use the concepts that I have to articulate the deliverances of observation. Observation could tell me this only if observations themselves were already conceptually articulated, which they are not, for the reason just stated.
Therefore, nothing empirical is known unless it is known how interpret observations, and observation cannot in all cases tell me how it is to be interpreted. Given on ability on my part to interpret my observations, observation can obviously improve my ability to interpret observation; but it couldn’t tell me this unless I could already interpret it. Since, consequently, the interpretive principles involved in the interpretation of observation are not always known on the basis of observation, it follows that at least some knowledge is analytic. Indeed, this follows from the very fact that we have observation-based knowledge: given that we have empirical knowledge, it follows that we have non-empirical knowledge. This follows because it is by using non-empirical knowledge to interpret sensory observations that we acquire empirical knowledge.