It's really a bit deeper than that. The apriori is the deep underlying rules of the "grammar" of language and thought. Apriori statements look like they're saying something empirical (like they're a discovery about the world), but actually they aren't.

For example "a thing can't be red and green all over" looks like it's saying something empirical about the world, as if it's a fact - and then it seems really weird that something we think up in our heads can reach out from our minds and limit what's possible in the world out there (a discovery that's also a prescription, which seems contradictory). But actually all the statement is doing is tracing a finger around the way those concepts are used, reminding us of the basic stipulative rules of those concepts. What it's really saying is something more like, "If we find two surface properties that a thing has simultaneously, then they sure as hell ain't colours." Or perhaps, "Here's a possible property of things in the world, let's call it "colour"; one property of colour is that a thing can't have two colours at the same time."

The aposteriori, on the other hand, is existential statements (such-and-such is the case, exists, etc., or not).

The mind works, as vast swathes of Nature work (e.g. evolution, the immune system), by means of generate-and-test. We dream up possible-ways-the-world-could-be, and then we test the world to see which of those ways it actually is. The apriori is the "template" so to speak, the possible-way-the-world-could-be that we're considering, or the shape of the holes in the net we're casting over the world. We don't yet know whether the world is that way (and because of fallibility we may never be absolutely certain, no matter how much we test), but if it is that way, then it necessarily is that way.

(A further, Kantian refinement might be that it's not actually possible-ways-the-world-is that we're testing for, but possible experiences. But it's a moot point whether that really makes any difference, it depends on what earlier premises you accept, e.g. re. perception and reality, etc.)

Thank for your response @gurugeorge

I think that the "a priori" moves from language to reality, about stating the possibility of a fact; and the "a posterior" moves from reality to language. For example, in empirical science, all hypotheses tested in an experimental research are said to be possible "a priori", but not "a posteriori". If the study is informative or well-designed then results must be discard at least one of the hypotheses.

I think you are equivocating.

A priori is about having knowledge prior to experience. That is different than guessing outcomes.

You can determine that the angles on a four sided foundation will add up to 360 degrees before laying one foundation, as it can be determined strictly through rationalistic means.

I think that "prior to experience" means "predictive" insofar it introduces an temporal dimension in the truth. In mi opinion, your example of the degrees of a four sided foundation is not relevant in the a priori/posteriori distinction due to time is not included in the mathematical theories.

"prior to experience" means prior to experience, ie knowledge can be deduced theoretically without empirical evidence. You are not predicting. You already have the knowledge.

knowledge can be deduced theoretically without empirical evidence — Kamikaze Butter

In my opinion, "without empirical evidence" only can be understood as referred to certain formal knowledge (logic and mathematics, when the true can be predicted, such as the value of "x" in the equation "2+2=x"). I would accept that this is the "analytic" knowledge. But the temporal dimension of "a priori/posteriori" is neglected in my view.

You are right.

I think, therefore I am.

That is an a priori conclusion that Descartes drew up. It requires no inference from experimentation, that is no sensory evidence is needed.

Descartes believed that only certain knowledge could be known a priori, like mathematics.

He believed some knowledge, such as physics, required empirical study.

A priori vs posteri Is differentiated by how you can know the truth, either through rationality alone or through sensory input.

As if rationality alone is possible without sensory input...

Quine is relevant here...

I think that the "a priori" moves from language to reality, about stating the possibility of a fact; and the "a posterior" moves from reality to language. — Belter

I think that's sort of right, but I'm not sure what "moves from" means, and I'm not sure that "language" is the right term - "from concepts to reality" would be better. Language is a social habit pattern that we partake in, concepts are linguistic structures (I suppose that's what you might mean?)

I think it's simplest just to understand the apriori as "possible ways the world could be", IOW we dream up, think up, possibilities and then we test them. The possibilities have their own logic, but they're not really knowledge as such (that's the mistake). All knowledge is aposteriori only (IOW we've found that one of the possibilities we've dreamed up is - so far as we can tell - the way the world actually is).

That's the great philosophical confusion, and the real philosophical puzzle and it's really hard to fix in the mind: that the apriori looks like knowledge, but actually is just the description of a model or template that might represent the world or might not (and in ordinary language use, it's tried and tested models that are fairly settled).

Apriori statements look like they're saying something empirical (like they're a discovery about the world), but actually they aren't. — gurugeorge

What about 'synthetic a priori' statements? I think they're central to science.

What about 'synthetic a priori' statements? I think they're central to science. — Wayfarer

In my view, 'synthetic' means 'empirical' as opposite to 'formal' (content level) and 'a priori' means 'predictive' (formal level) as I said. Hence, 'synthetic a priori' is central to science insofar it fits its "hypothetical" side of the hypothetical-deductive method.

the apriori looks like knowledge, but actually is just the description of a model — gurugeorge

You are assuming that 'a priori' refers to models, that is, formal knowledge. It is 'analytic' in my opinion, but not 'a priori'. Only 'synthetic' knowledge can be 'a priori/posteriori' given that it need to experimentation or experience. Can you gave us other non-mathematical example of 'a priori'?

What about 'synthetic a priori' statements? I think they're central to science. — Wayfarer

Yikes, this is really difficult! :) I don't agree with the a-s dichotomy. All truths are analytic, it just depends on how well acquainted we are with the things subsumed by the concept. For example, "Man" refers to all the men that are or ever will be, all the things we know about man, and also, implicitly, to all the things we don't yet know. (Concepts are so to speak algebraic, they allow for new discoveries.)

Now, until a discovery is confirmed, it has the logical status of being a necessary component of a template or a possible model - it's a projection, or a punt, as to how things are (i.e. it's apriori, not yet a discovery in and of itself, but a self-consistent projection of possibility). Within the model, the new feature is analytic, a logically necessary and explanatory aspect of the object; but even once it's confirmed, it's still analytic, because we're now accepting a new model - it's the total new model that's confirmed, not just the new feature as a dangling addition.

IOW, what makes a discovery of a new thing a discovery, is that it's the confirmation of a priorly projected self-consistent (but new) model that necessarily includes that new feature; the discovery is not a new predicate that's arbitrarily or contingently glued on to a previously established necessary truth.

Whew! Hope that makes some sense, it's horrifically abstract.

All truths are analytic, it just depends on how well acquainted we are with the things subsumed by the concept — gurugeorge

No, I don't think so. The point about the synthetic a priori, is that allows you to make predictions based on logic and mathematical reasoning, which are not themselves strictly entailed in their postulates. And that has many applications in science. I think one outstanding example is the prediction of antimatter:

Dirac's research marked the first time something never before seen in nature was predicte – that is, postulated to exist based on theoretical rather than experimental evidence. His discovery was guided by the human imagination, and arcane mathematics.

For his achievement Dirac was awarded the Nobel prize for physics in 1933 at the age of 31.

New Scientist.

So actually I think Belter's very terse response above is pretty well on the mark.

which are not themselves strictly entailed in their postulates — Wayfarer

But isn't mathematics paradigmatically a practice that draws out what's implicit in postulates? It's just that what's implicit in physics models often isn't obvious.

One way I sometimes think about the difference between apriori and aposteriori is to think about the difference between the dictionary and the encyclopedia. The nouns in the dictionary are words that represent concepts, ideas of possible things; the encyclopedia tells you that the possible thing exists, and how it exists, gives you more information about them.

For exampe, a dictionary definition of "tiger" - "a very large solitary cat with a yellow-brown coat striped with black, native to the forests of Asia but becoming increasingly rare."

Encyclopedia entry for tiger: https://www.encyclopedia.com/plants-and-animals/animals/vertebrate-zoology/tiger (a whole bunch of detailed info about tigers, that includes what the dictionary says, but expands on it tremendously).

Now when a biologist hears the word "tiger," what comes to their mind will be something more like the encyclopedia entry, they'll have a fairly complex, detailed sense of what a tiger is. If they were responsible for writing the dictionary entry, they might fill it out just as fully as the encyclopedia entry (and then someone would tell them KISS :) )

IOW, for the biologist, everything about the tiger as they understand it is "analytic." For the layman, the encyclopedia entry is "synthetic", because it goes beyond the mere initial identificatory concept that's in the dictionary.

It's the same, at a much higher level, for a physicist and their model of reality, except there's a lot more room for error there, because the properties being thought of are recondite, not normally experienced, and the kind of testing required to check whether the model or concept actually represents something in reality (equivalently: whether such a thing as a tiger exists) is much more complex and requires particular apparatus, etc.

The physicist's model of reality is more like the full idea of a tiger that the biologist has, but it's as if the biologist were to play around at a theoretical level with the fuller kind of dictionary entry for "tiger" that he (the biologist) would make.

But isn't mathematics paradigmatically a practice that draws out what's implicit in postulates? It's just that what's implicit in physics models often isn't obvious. — gurugeorge

I don't think it's that simple; actually I think it's an overlooked question. After all, Kant himself said that one main impetus behind the Critique of Pure Reason was to understand better how such judgements are possible. Recall, he was responding to Hume's scepticism, which clearly differentiated a priori from inductive reasoning - as if these were two entirely different domains.

So I think the significance of the 'synthetic a priori' is that it provides a kind of deductive certainty with respect to some actual state of affairs - so it applies to the empirical domain, rather than for example the domain of pure mathematics. 'We can predict when and where an solar eclipse will be visible with a high degree of accuracy. Our ability to predict, however, obviously does not fall into the category of an analytic a priori judgment. An eclipse is not defined essentially by its being visible then and there. It might be visible somewhen and somewhere else, but that doesn’t negate the fact that it still is an eclipse. Our ability to predict also does not fit into the category of a synthetic a posteriori judgment. What is at stake is our ability to predict. We don’t need to wait for it to happen to see if it actually does. We already know it is going to happen before it does. Our calculations are good enough to predict these things. But how do we know it is going to happen? How can we be certain?' ¹

So again, bear in mind that Kant was critiquing both the rationalists - who said that we can arrive at knowledge solely by logical deduction - and the empiricists, who argued that we only acquire knowledge by experience. Typically, he shows that elements of both views are correct, but that neither side is correct by itself. 'For example, the law of causation ‘every event has a cause’ is necessary so therefore must be a priori, yet it is not analytic as the concept of an event does not contain within it the concept of being an effect. To Kant, metaphysical judgements such as this are therefore a priori and synthetic; they cannot be derived purely from either logic or experience.' ²

This is why it has to be recalled that in this case, these philosophers are asking fundamental questions about how knowledge is possible. It's not an exercise in cognitive science (although I think Kant still rightly has considerable influence in such disciplines, and it's an interesting topic in its own right.) But it obviously has clear implications for modern scientific method; which is why, I think, Dirac's discovery makes such a good example of a 'synthetic a priori' that was true; it wasn't until some time after he said that such particles must exist, that they were discovered. But then, we still read to this day of new observations that prove once again, that 'Einstein was right' ;-)

Have you guys paid Kant even a passing glance? If you did I missed it. My old philosophy prof. offered a definition of a priori as follows: that which is universally and necessarily so. As opposed to a posteriori, that which is known empirically, from experience. I assume this is all well-known here.

Kant matched a priori with analytical in that an example of an analytical a priori proposition is "bachelors are unmarried men," or "gold is a yellow metal."

Synthetic a posteriori propositions were those that are contingent. They may be true, but they don't have to be. "The Amazon is the longest river," and, "The Danube is the longest river," are both synthetic a posteriori propositions. One happens to be false, the other true.

Kant considered a 2x2 combining of synthetic/analytic and a priori/a posteriori propositions.

1) synthetic a posteriori. Covered.

2) analytic a priori. Covered.

3) analytic a posteriori. Nonsense.

4) synthetic a priori. Interesting!

The idea is that we have some knowledge simply because that's how our minds work; how our minds make sense of what we perceive: synthetic: a putting together, a priori, universally and necessarily. All this within the first couple of hours reading A Critique of Pure Knowledge or any decent secondary literature on that book or Kant's thinking on knowledge.

It remains to note that this is not physics. Kant in his day was a world class physicist. He's credited with being the first to conjecture the existence of galaxies! He argued that both space and time were categories of thinking, that is, created by our minds. But that in no way suggests that he supposed that his mind created the universe in which those phenomena occurred. His is a metaphysical accounting of how knowledge is possible, not a theory of physics.

If you did I missed it. — tim wood

Q: What do you call a Greek Skydiver?
A: Con Descending ;-)

My old philosophy prof. offered a definition of a priori as follows: that which is universally and necessarily so. As opposed to a posteriori, that which is known empirically, from experience. I assume this is all well-known here. — tim wood

In my view (which can becomes different through the discussions insofar I clear my ideas), there are several interesting ways of define the "analytic/synthetic", "a priori/posteriori", "contingent/necessary","formal/empirical" and so.

Another way could be the following:
The mathematical and logical theorems, such as Pythagoras's Theorem, are contingent a priori, that is, before demonstrating it, it by definition could be "false". To prove its necessity (it is a logical consequence of premises), which always is a posteriori (step by step to the last one), we state its contingency a priori, and then we conclude -necessarily and a posterior- that it is impossible, so it is a theorem. If we are sure that it is necessary a priori, its demonstration is irrelevant, such as religious dogmas. The knowledge such as mathematical theorems is also "formal" (about abstract objects) as opposite to "empirical" (about concrete objects).
Thus, the empirical knowledge, such as physics equations like "E=mc^2", is also contingent a priori. That is, null hypothesis rejecting in experimental studies is the way of inferring the true of the law (a posteriori). However, contrary to mathematics theorems, physics laws do not become necessarily true when they are empirically confirmed a posteriori.
In the two cases, the theorems and physics laws are "synthetic" (the "integration" or final product of knowledge procedure; from the components to the totality), meanwhile definitions, rules, axioms, etc., are the "analytic" side (the inverse procedure; from the totality to its components).

To sum, the knowledge can be divided at least into these four dimensions:
Temporal: "A priori/posteriori" (before/after)
Structural: "Analytic/synthetic" (components/totality)
Semantic: "Formal/empirical" (abstract/concrete)
Epistemic: "Necessary/contingent" (reason/evidence)

I thought a priori meant arrived at by reason alone and a posteriori meant requiring, in addition to reason, observation.

The issue I find relevant in this dichotomy is whether ALL knowledge is one of the two or a combination of both.

A priori knowledge seems to be most clearly present in the conceptual models we create (math, logic) but it's debatable whether this form of implication by necessity of rules and definitions counts as real knowledge.

A posteriori knowledge seems not so clearly delineated as all knowledge requires mapping of concepts onto the world external and finding relations between them.

Also there seems to be some form of deep connection between our conceptual models, especially math, and the real world outside. Does this count for anything?

The issue I find relevant in this dichotomy is whether ALL knowledge is one of the two or a combination of both. — TheMadFool

I think that all knowledge can be:
1) Necessary a priori (axioms, A ->B = not-A v B, etc.): they do not require any more logical or empirical evidence.
2) Contingent a priori and necessary a posteriori (Pythagoras's Theorem): they require logical evidence (logical consequence)
3) Contingent a priori and a posteriori (E=mc2): they require empirical evidence (physical effect).
Then, all knowledge is a priori either contigent or necessary, and if contingent, then it can be shown either also contingent or necessary a posteriori.

Also there seems to be some form of deep connection between our conceptual models, especially math, and the real world outside. Does this count for anything? — TheMadFool

This connection is a modeling one: that is, math are used to quantify reality and then make our possible to infer testable predictions.

So I think the significance of the 'synthetic a priori' is that it provides a kind of deductive certainty with respect to some actual state of affairs - so it applies to the empirical domain, rather than for example the domain of pure mathematics. — Wayfarer

Yes, but we're never at any point departing from a distinction between possibility (or in Kant's terms possible experience) and actuality. The apriori is simply the delineation of possible ways of being, possible existents, possible laws, etc., that's why it's not actually knowledge, even though it's cast in terms of what look like existential statements (it's the dictionary, not the encyclopedia). The actuality is what is found, discovered in experience aposteriori; experienced reality either conforms to the posited, punted, projected model or not (and if it does, it goes in the encyclopedia, not the dictionary).

Again, just think about a dictionary entry, and about all those puzzles about the existential status of things like characters in stories, etc. Unicorns are what's called "logically possible," but so far as we know no such thing exists. Yet we can talk about them in stories. What is the "thing" we're talking about? We're not talking about a thing, but just a possible thing. Likewise, a mathematical model per se is just a bunch of lego pieces that fit together; and it remains so even when it's interpreted. But the interpretation just means we've found something in reality that has that particular lego-fitting quality, that particular logic.

Perhaps we could put it this way: apriori and aposteriori are modes of discourse that can even use exactly the same words, but in the former case, we're not making an existential commitment, we're only tracing the outline of the possibility, its internal logic; whereas in the latter case we are making the existential commitment, we are saying that the world is in fact the way we were only positing as a possibility when we were in apriori mode.

But that's why the distinction analytic/synthetic, as it intersects with the apriori/aposteriori distinction, is a movable and somewhat arbitrary line that simply depends on depth of acquaintance. (As above, to someone deeply familiar with a topic, everything is analytic, because everything about the topic has been found to be interconnected in reality.)

A priori knowledge seems to be most clearly present in the conceptual models we create (math, logic) but it's debatable whether this form of implication by necessity of rules and definitions counts as real knowledge.

A posteriori knowledge seems not so clearly delineated as all knowledge requires mapping of concepts onto the world external and finding relations between them.

Also there seems to be some form of deep connection between our conceptual models, especially math, and the real world outside. Does this count for anything? — TheMadFool

As above, if we find that reality works in the way our mathematical model projected, then reality has that logical structure, we're not fooling ourselves.

It was a great mistake of modern philosophy (although it was probably necessary as a phase) to divorce logic from being fundamentally about reality, and making it only about the way the lego pieces of our language and models logically click together in vacuo. While we are still concerning ourselves with models of possible things (i.e. before we're making the existential commitment that we've found something in reality that conforms to the model), and so long as we're in the mode of conceiving the models as just possible ways reality could be, and we're just playing around with them in the abstract, then yes, all we're doing is shuffling symbols around in coherent ways according to a self-chosen game; but once something has been found in experience that conforms to the model, then it's reality itself that has that logical structure, and it just so happens that the "rules" or "laws" of how reality fits together is the same as the rules and laws of our self-chosen game.

The mere fact that our game was a game, that the rules of the game were invented from our side, never at any point meant that reality itself couldn't just happen to have those same rules. (I think this is the massive howler at the root of subjectivism and relativism.)

if we find that reality works in the way our mathematical model projected, then reality has that logical structure — gurugeorge

Reality has a "real" structure (for example, elementary waves-particles and forces) like logic has a "logical" one. You are mixing the structure of the model with the structure of the modeled. In your rationalist view, it would be possible, for example, to demonstrate that reality is or not consistent, complete, etc., which in my view is a nonsense.

Reality has a "real" structure (for example, elementary waves-particles and forces) like logic has a "logical" one. You are mixing the structure of the model with the structure of the modeled. In your rationalist view, it would be possible, for example, to demonstrate that reality is or not consistent, complete, etc., which in my view is a nonsense. — Belter

Reality's "real structure" can be consistent, complete, etc., reality itself just is what it is, whatever it happens to be, whatever we find it as. But it wouldn't be possible to demonstrate that consistency analytically without omniscience, or a "view from everywhere" as Janus recently neatly put it.

Reality's "real structure" can be consistent, complete, etc., — gurugeorge

How it is possible to prove logical properties in something that is not a logic (a formal language constructed by humans)? I view it as a complete nonsense.

How it is possible to prove logical properties in something that is not a logic (a formal language constructed by humans)? — Belter

Again, this just presupposes that logic is a property only of formal systems we construct. I can only repeat what I said above: if and when a logical schema or structure is in fact applicable to reality, then what are we to think other than that reality itself just happens to have that logical structure?

Again, this just presupposes that logic is a property only of formal systems we construct — gurugeorge

"Logic" are these systems and by definition no other thing. "Reality" has not a logical structure but physical. You seem to be a rationalist like Hegel.