There is a linguistic theory to the a priori which is based off the definitions of the words used. On this view, analytic truths may be better described as definitional.
Virtue of definition: If you take “All vixens are female”, one takes the definition of vixen as ‘female fox’, so by grasping the definition of the word and it being female, one sees the proposition as true. Basically instead of viewing vixen as a concept and reasoning from a concept, one is reasoning purely on the basis of the definition.
Virtue of meaning: The same is not true of the “red and green all over at once” example. One can speak of the virtue of meaning or at least convention - which is basically the conventional meaning of the terms red and green.
Conventionalism ground truths of reason, especially definitional conventions, regarding meaning. It also conceives our knowledge of them based on knowing these conventions, since knowledge of such convictions should be taken as empirical knowledge, based on observations of linguistic behavior. Conventionalism is a kind of empiricism regarding the truths of reason. The claim is not that these truths are *about words, but knowledge of them is based on empirical knowledge of linguistic usage.
A classical view might say that definitions are our way of understanding the concepts. There’s also the point that one can grasp the truth analytically by grasping the relevant conceptual relations without understanding the definitions.
Example: The spruce is taller than the maple, therefore the maple is shorter than the spruce. One can know this truth without a definition of spruce or maple. You don’t even need to know they’re trees.
Analytic truths would be essential in one’s route to the definitional knowledge, not the other way around.
Difficulties & Strengths of the Classical View
Consider being ‘red’. What does it mean to ‘be red’ and how would that differ from a shade off of red (like orange)? Well even though the concept of red could be vague, the concept of red is not. The more vague the terms, the harder it becomes to discern what the proposition is expressing.
Meaning Change & Falsification
There is a problem that can emerge with what the term means and the concept being expressed. There is also a definition creep that can occur to a word over time. In the proposition “all vixens are female”, the term vixen may not simply represent the concept being expressed. Also scientists could find the vixens contain enough to make qualities to no longer be female (meaning change). There’s also the possibility that scientists discover that vixens are actually totally different than was thought - completely changing the way the term is used (falsification).
Classical view takes the falsification as something possible. They see the difference between meaning change and falsification as clear enough to conclude that what seems like a falsification of an analytic proposition is really only a change of meaning that leads us to substitute (for an analytic truth), what looks like a proposition inconsistent with it, yet is actually compatible with it.
The Possibility of Empirical Necessary Truth
The classical view holds every a priori truth as necessary. It is less plausible that every necessary truth is a priori. Take “sugar is solvable in water”. This is a law of nature and something that is necessary. The difference is that it is not self-evident. It appears to be a truth that is empirically discovered.
Proponents of the classical view hold that this particular necessity is not from logical, but from nomic (Greek nomos, for law) in roughly the sense of characterizing laws of the natural world - as opposed to every possible world or situation. We can clearly conceive of sugar not dissolving in water, but one cannot clearly conceive something both round and square.
Essential and Necessary Truths
The necessary truths discussed so far concern the general and non-existential - such as not being both round and square. Consider that human beings have parents. The empirical proposition “I’m the son of C and C” is necessary. The parent example is existential, implying the existence of the particular thing it concerns (me).
The classical view says that the proposition that I have parents is an essential truth. One attributing to a thing a property absolutely essential to it. It would not exist without it, but a necessary truth. Necessary truth holds in any possible world or situations. Essential truth holds in those possible worlds or situations in which what it is about exists.
With that said, even if water doesn’t exist in a specific world, we can still speak of water as H2O. The classical view must distinguish between necessary truth - those applicable to entities that must exist (she as numbers) and those applicable entities that don’t exist.
Another objection to apriority of all necessary truth. If you looked at a theorem, it follows from true propositions and is therefore a necessary truth - since what follows from a necessary truth is itself necessarily true. But this is not a priori since it is not self-evident. We cannot simply assume every theorem must proceed by self-evident steps from a self-evident proposition.
Not all provable propositions need to be self-evident - a self-evident proposition may be provable. Self-evident propositions are knowable without proof, on the basis of understanding them. Unlike theorems, they are not premise dependent.
So it appears that there are necessary truths knowable only through empirical investigation or laborious mathematical proof.
A Priori Beliefs
A priori is not normally applied to beliefs, but the apriority of a belief tends to indicate some degree of justification. The principle of justification for a priori belief is normally, if a rational person believes a proposition solely on the basis of (adequately) understanding it - believes it in a strict a priori way - this belief is prima facie justified.
There is a counter part plausible epistemic principle called a principle of knowledge for correct a priori beliefs. If a rational person believes a true proposition in the a priori way described above, this belief constitutes knowledge.
Loose & Strinct Senses of A Priori Justification and Knowledge
When knowledge or justification that arises from believing in an a priori way is not strictly speaking it is regarded as the loose sense.
Outline of Four Dimensional Conception of the A Priori
A priori in the narrow sense: Self-evident with understanding sufficient to ground for justification, belief based on such understanding constitutes knowledge.
A priori in the broad sense: Not directly self-evident (a) indirectly self-evident - self-evidently entailed. (b) ultimate a priori - not self-evident, but provable by self-evident steps.
A priori in the strict sense: (a) based on understanding of a directly self-evident proposition or (b) indirectly based on such an understanding via a self-evident entailment of the proposition by a self-evident proposition.
A priori in the loose sense: Not a priori in the strict sense, but based on an understanding of the proposition (proposition itself need not be true or a priori).
A priori in the strict sense: Knowledge (a) of an a priori proposition that is directly or indirectly self-evident, and (b) constituted by a belief that is a priori justified in the strict sense.
A priori in the loose sense: Knowledge (a) of a proposition that is not directly or indirectly self-evident, but is provable by self-evident steps from some self-evident proposition and (b) constitute by believe based on understanding such a proof.
A priori in the narrow sense: (a) held in an a priori way; roughly based on an understanding of the proposition, and (b) of a proposition that is a priori (in the narrow or broad sense).
A priori in the broad sense: (a) held in an a priori way but (b) of an empirical proposition.