This post discusses what Kant means by ‘synthetic a priori knowledge.’ We will first discuss knowledge, then the a priori, and finally the synthetic.
For Kant, there are two stems of knowledge, viz. sensibility and understanding (B29). Through sensibility, we are presented with ‘objects’ — this can be thought of as perceptual experience. Through understanding, we think, compare, and combine our representations of these objects, and ultimately gain ‘knowledge of objects’ (B1). Understanding, or reason, supplies the rules of thought (B25), and determines how we can relate the items we are presented with in sensibility. Kant, however, acknowledges that all knowledge begins with experience in the sense that our acquaintance with objects gets our cognitive machinery started by affecting our understanding so that we might think or know what we get through sensibility (B1). Sensibility provides you with the ‘raw material’ for knowledge; understanding provides you with the ability to manipulate the raw material. If there is a brown table before me, sensibility provides me with the brownish and tablish features in experience, but understanding allows me to think that ‘there is a brown table before me’ or imagine the brown table being red. So it looks like knowledge is a special kind of relation between one’s representations acquired through sensibility and one’s pure understanding. Understanding endorses some representation(s) as true. Not all our mental representations will be true. But we do know some of them to be true, and the fact we know means we must be able to point to some kind of justification.
There are two ways that knowledge can be justified, viz. a priori or a posteriori. A priori knowledge is ‘any knowledge that is…independent of experience’ (B2). Kant distinguishes this from empirical or a posteriori knowledge, which is dependent on experience (like knowing that most swans are white). By ‘independent of experience,’ Kant means epistemic independence. It is knowledge that never receives its justification from a particular empirical experience, or even from a generalization of particular empirical experiences. An example may help (B2). You see a person digging a big hole beneath their house. A big enough hole will collapse the house. You know before the particular experience of the house collapsing that this person will collapse their house. Your knowledge, nevertheless, is not a priori because knowing that a big hole beneath a house collapses it is knowledge that could only ever be gained through experience. You must have investigated the world before you gained the knowledge that the house would fall. Moreover, Kant must mean epistemic dependence because he recognizes that all knowledge begins with experience (B1) — so the a priori must be independent of experience in some other modality, namely epistemically, not psychologically.
Two criteria for identifying a priori knowledge are (1) that the judgment is necessary and (2) that the judgment carries strict universality (B4). This is tantamount to saying that a priori knowledge brooks no counterexample. It is not possible for a priori knowledge to have been false. Because the a priori is not empirical, a priori judgments/knowledge is generated from the pure understanding or our faculty of knowledge (B5). The knowledge that ‘all bachelors are unmarried men’ is a priori because its justification is absolutely independent of experience. You know the proposition is true in virtue of knowing the meaning of the word ‘bachelor,’ you do not need to empirically investigate the world, checking each bachelor to make sure that he is unmarried. So a priori knowledge is the endorsement of a judgement whose justification does not depend on any empirical investigation.
The analytic/synthetic distinction applies only to judgments or knowledge that admits of a subject/predicate structure, for instance ‘All A’s are B’s’ (B11). In ‘all bachelors are unmarried men,’ the predicate ‘unmarried men’ is (covertly) contained in the concept ‘bachelor,’ making this an analytic judgment. In ‘all bodies have weight,’ the predicate ‘has weight’ is not contained in the concept of ‘body,’ making this a synthetic judgment. So a judgment is analytic if the concept of the predicate is contained within the concept of the subject; if not, then the judgment is synthetic. It is not clear, however, what Kant means by ‘containment.’ He provides some clues, namely that analytic judgments are those which connect subject and predicate through the law of identity, that the rest entirely on the principle of contradiction (Pro. 17) regardless of whether their concepts are empirical, but what is the law or principle operating on?
It cannot be identity of extension. Consider two sets: (1) the set of all creatures with hearts and (2) the set of creatures with livers. These two sets are coextensive. If extensional identity was all Kant had in mind, then the judgment ‘all creatures with hearts have livers’ would be analytic. But recall that all analytic judgments are a priori. We could imagine a counterexample, namely a creature that has a heart and no liver, but then this would contradict the definition of a priori. But Kant does not admit analytic a posteriori judgments, so analytic judgments based on the law of identity are not based on identity of the extensions of the predicate and the concept.
If B is not contained in A in virtue of their extensions, then perhaps B is contained in A in virtue of their intensions. Recall that knowledge is going to consist in some relation between our representations and our understanding. We might think of the intension of ‘creature with a heart’ as something like our completed mental representation of hearted-creatures. The essential features will be the concept of ‘heart’ and concept of ‘creature’ somehow united in our understanding. So the intension of ‘creature with a liver’ will be something else. We’ll have a mental representation that unites the concepts of ‘liver’ and ‘creature’ in understanding. So a proposition like ‘all bachelors are unmarried men’ is analytic (and a priori) in the sense the mental representations of ‘bachelors’ and of ‘unmarried men’ are identical — that is to say the judgment is explicative, the predicate adds no content the cognition of the concept; they are one and the same.
Synthetic propositions are not analytic or explicative. They are ampliative in that the predicate adds content to the cognition of the concept; the predicate extends our knowledge of the concept beyond what is merely ‘thought in’ or ‘contained in’ the concept. Indeed Kant puts it, ‘we are required to add in thought a particular predicate to a given concept’ (Pro. 19). That creatures with hearts have livers extends our knowledge of creatures with hearts.
So synthetic a priori knowledge will amount to the following. It is the endorsement of the truth of a mental representation (like a judgment), where the justification of the endorsement is epistemically independent of experience, and the predicate of the judgment is not intensionally contained within the concept.