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No. Even if there are no blue dogs, no blue things and no dogs, still all the blue dogs are blue dogs, just as all unicorns are unicorns.

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Exactly. It's called pure reason, and it has limitations. That's what it means for a proposition to be analytic, that it says nothing about the world. And that is why the number of beers in the fridge, or rabbits in the hutch is synthetic, whereas that 6 - 3 = 3 is analytic. The price of necessity is vacuity.

Where philosophers start to use both together is where they can say interesting and meaningful things.

Of course not. Logical truths depend on logic.

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Logical truths are known by studying logic. Logical truths' truths depend on purely logic. You should distinguish what makes statements true from how we know statements.

There are two somewhat distinct questions that can be discussed in connection with the OP. A historical, philological question concerns Kant's own notoriously ambiguous treatment of the concepts that he coined. What Kant thought depends on who you ask, and different commentators will typically massage and harmonize the text to favor their own views on the matter. It seems to me that StreetlightX's referenced summary is rather too neat. It, for instance, glosses the issue of "containment," also raised by the OP, that Kant seemed to take seriously enough that he would not recognize even simple mathematical statements such as 2+2=4 as analytic (because, the argument goes, "4" is not contained in either "2" or "+").

Apart from specifically Kantian scholarship, modern discussion of the analytical/synthetic owes more to the way these concepts were framed later, when Western analytical philosophy took a logical and linguistic turn. And here the debate is not dead, despite Quine's valiant efforts. The reason, I think, the idea of analytic/synthetic distinction will not go away is that we intuitively feel a categorical difference between groups of statements such as the following:

I.

(1) Some doctors that specialize on eyes are rich.
(2) Some ophthalmologists are rich.
(3) Many bachelors are ophthalmologists.
(4) People who run damage their bodies.
(5) If Holmes killed Sikes, then Watson must be dead.

II.

(6) All doctors that specialize on eyes are doctors.
(7) All ophthalmologists are doctors.
(8) All bachelors are unmarried.
(9) People who run move their bodies.
(10) If Holmes killed Sikes, then Sikes is dead.
(11) If Bob is married to Sue, then Sue is married to Bob.
(12) Anyone who's an ancestor of an ancestor of Bob is an ancestor of Bob.
(13) If x is bigger than y, and y is bigger than z, then x is bigger than z.
(14) If something is red, then it's colored.

Until this distinction is not at least explained away, the work is not done.

The above examples are taken from the SEP article The Analytic/Synthetic Distinction, which offers a comprehensive introduction to the issue.

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I agree with you.

I wouldn't completely dismiss the distinction--there's a difference between experience gained via interacting with the world and the way that your brain works because of its structure, function and processing, but it's not at all a black and white distinction, and many people make a mistake of taking the analytic/a priori side to somehow transcend individuals and their brains.

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It seems to me that everything turns out to be a combination of a priori and a posteriori, really, with nothing purely one or the other.

Your brain doesn't actually arrive with anything like propositions fully formed. You need to have experiences and to think about them, while your brain physically develops, to arrive at beliefs, propositions, etc.

And you can't just experience the world. Your brain has to process that experience, and it's going to process it a particular way because of its structure and internal interactions.

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If we find that everything that has a heart has kidneys, and everything that has kidneys has a heart, then the two sets, things that have a heart and things that have kidneys, are the same. But is saying something has a heart the same as saying it has kidneys? It doesn't seem like it is. It seems like two different concepts, even though they apply to exactly the same things.

On the other hand, would we have to do research to find that everything that has a heart has a body? How would you search for something that has a heart but no body? (Leaving aside Roy Orbison.) It's already built into the concept "has a heart" that it can only apply to things that have bodies. That's the analytic part. So you can know that the one set is a subset of the other without looking. That's the a priori part.

A couple of comments:

(a) When reading Kant, it is often useful to take a look at his historical predecessors in order to understand how some of his distinctions are actually tactical maneuvers deployed against positions which he rejected (two books that, incidentally, show the power of such an account for Kant is the classic Kant and the Capacity to Judge, by Longuenesse, and, more relevant to this thread, The poverty of conceptual truth, by R. Lanier Anderson, which is entirely devoted to an elucidation of Kant's distinction between analytic/synthetic). Case in point, Kant considered his analytic/synthetic distinction as a weapon against Wolffian metaphysics. Very roughly, this is the idea:

Wolff and his followers apparently thought that every concept could be positioned in a logical hierarchy, in such a way that immediately above it would be its genus and right below it its species. Thus, we could picture this hierarchy as an upside-down tree, with the most general concept at its root (say, the concept of <object in general>), infinitely branching downward in subdivisions that would systematize the rational order of the world. Importantly, the branches below a given node of this tree had two main characteristics: (i) exhaustiveness, that is, the branches below a node collectively exhaust its species and (ii) exclusiveness, that is, there are no intersections in the path below the branches. This gives us a neat picture of conceptual containment: a concept A is contained in concept B iff A is a node in the path above B.

Wolff wanted to put this picture to use in order to establish a metaphysical system which would mirror the rational structure of the world. The philosopher's task, according to Wolff, was essentially to distill each concept in its component parts (an activity he called "analysis") in order to locate in this hierarchy. This would allow one to ground every (conceptual) truth in this hierarchy. Remarkably, since Wolff actually thought that even our empirical judgments were reducible to conceptual ones, this would mean that every truth would be ultimately grounded in this logical hierarchy. One can thus understand (i) why he gave pride of place to categorical syllogistic inference (because such inferences made explicit precisely relations of conceptual containment, as Arnaul and Nicole had long argued in their logic) and (ii) why he considered the principle of non-contradiction to be a sufficient (!) ground for every truth, including empirical ones (because conceptual containment claims are settled by appeals to the pnc).

Of course, appealing as it is in its simplicity and elegance, Wolff's system is woefully inadequate for the task of describing the structure of the world. In particular, conceptual containment is too crude an instrument to capture even mundane truths such as "The sun is warming this stone" or, to give a more interesting example, "7+5=12". That is the point of Kant's analytic/synthetic distinction: Kant grants to Wolff that conceptual containment does capture some interesting class of truths (namely, the analytic ones); but that class is too meager. In particular, he claims, the B Introduction to the Critique, that (i) Judgments of experience are synthetic; (ii) Mathematical judgments are synthetic (iii) Natural science's judgments are synthetic; (iv) Metaphysical judgments are synthetic. As Anderson makes clear, these four theses are essentially a blunt polemic against the Wolffian paradigm: it is essentially saying that every interesting judgment is outside its scope (hence the title of Anderson's book: conceptual containment has scarce resources to express anything of interest).

Notice that, according to Anderson's analysis, even some logical judgments are synthetic. Kant admitted as distinctive types of inference both hypothetical and disjunctive. But these inferences are grounded in relations between judgments, not concepts, and so a fortiori can't be grounded in conceptual containment relations! So it may be that, perhaps unwittingly, Kant excluded from the class of analytic judgments a whole swath of logical judgments (that this probably wouldn't bother Kant may be gleaned from the fact that, when he announces the pnc as the supreme principle of analytic judgments, he merely says that every analytic judgment is grounded in the pnc, not that every judgment grounded in the pnc is analytic).

In summary, you are right that analytic judgments are a rather impoverished class of judgments. But it was precisely in order to show this that Kant introduced the distinction in the first place.

(b) Note that this way of carving out the distinction between analytic/synthetic is completely different from the current way of appealing to the meaning of the terms involved. For a contemporary treatment of the distinction, you should read David Lewis's outstanding work, especially Conventions andLanguage and Languages.

(c) Finally, regarding the a priori/a posterior distinction, note that this is an epistemic distinction, not a psychological one. In other words, it is a distinction between the grounds of justification for a given proposition, not a distinction about the particular sources for the proposition in question. As Kant himself remarks, "There is no doubt that all our cognition begins with experience (...). But although all our cognition commences with experience, yet it does not on that account all arise from experience". So even though experience may be a necessary condition for me to acquire certain concepts, and hence for me to even formulate certain propositions, that does not mean that experience is a necessary condition for me to justify the propositions in question.

Take, for instance, Fermat's Last Theorem (FLT). Clearly Andrew Wiles needed to go through certain experiences in order even to formulate FLT (in particular, he had to read a certain mathematical textbook when he was a child which explained it to him). But these experiences do not need to be mentioned, and in fact are not mentioned, in his proof of FLT. Only axioms, mathematical definitions, etc., are used in this proof. So it is an a priori, not an a posteriori truth.

Your version of radical empiricsm simply cannot work (i.e., that all cognition is explainable by experience), and even the logical positivsts understood this, because they saw that one must presuppose the analytic/synthetic distinction for the viability of their empiricst project.

The reason is that syntetic sentences, that is, sentences which derive their meaning from experience, must presuppose some analytic definitions to function as 'synthetic' in the first place. Because think in what sense can a sentence said to be 'derivable' from experience: it must mean that there is some general a-priori rule which justifies you in inferring the sentence from a certain experience. Thus the positivists thought that all empirical terms have a fixed meaning or definition (or at least we can construct such definitions for a 'perfect' scientific language). So for example 'cat' is defined via the experience of such and such shapes and colors occurring in a certain configuration; and therefore in the present of such an experience it follows analytically that you are seeing a cat (because this is what the term means). And this is where their famous verification criteria of meaningfulness comes from: every empirical sentences must be connected to experience (empiricism); to be connected to experience is to be verifiable by experience; and to be verifiable is to have an analytic definition - prior to any experience - that tells you which experience should count as the verification of the sentence. Otherwise, if you don't have such definitions that fix the meaning of your empirical terms, then nothing could follow from your experience, no sentence could be ever verified by experience (and hence you wouldn't have empiricsm).

The conclusion is therefore that there cannot be only synthetic sentences, but also analytic which are true independently of experience. And this is also how they thought that science works: you have a theory from which (by virtue of the meaning of the terms it contains, as they have been defined) a set of certain possible observations follow. Then when you go on testing the theory, if the predicted observations obtain, then the theory is verified; if not then it is disproved. And the idea here is the dame: unless theory and observation are connected by definitions, no observations could follow (deductively or inductively) from any theory, and there will be no way to test it experimentally, in which case science would be impossible.

And as a historical sidenote, it is important to note that though Quine famously attacked the analytic/synthetic distinction, what he was primarily concerned with is the positivists' conception of the 'meaning' of sentences (this is why he talks so much about synonimity in "two dogmas"); however, he himself didn't reject the distinction (which was central to the positivsts as well) between theory and observation, but only claimed that they are mutually interdependent, and that observation cannot verify or disprove individual sentences, but whole theories (and maybe the whole body of science). And so according to Quine, the way experience is connected to our body of beliefs is not via "definitions" but whole theories or world views ("conceptual schemes"), but it still retains the same idea that our experience is mediated by logical connections which themselves are prior to experience.

I don't really understand why Kant made this distinction, was it important for something? — Hallucinogen

Analytic statements are true by definition, synthetic a priori statements are true not by definition, but because they are self-evident; we do not need to check empirical conditions in order to confirm either, but the latter are not true merely by definition, that is the main point. This distinction is a perfectly valid and useful one, as far as I can tell.

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Thank you for taking the time to write that fascinating reply. I see that the a priori / a posteriori distinction is based on the justification of the terms now, - it is an epistemological, not ontological distinction. — Hallucinogen

I'm glad you found my reply useful.

I can't see how natural scientific statements or mathematical statements fall exlusively into synthetic or analytic statements. In order to make scientific explanations, causation has to be involved, ways of referring to time and space, or movement. An example would be "the sun is warming this rock". I don't think synthetic statements are able to do this, they're purely associative right? All they can do is refer to how sets of objects overlap. — Hallucinogen

Well, the whole point of the first Critique is to argue against this idea, that is, to argue that there re synthetic claims which are not grounded merely on empirical association. Kant's argument here (which comprises the entire Transcendental Analytic) is notoriously complex (I myself don't fully understand it---I don't know if anybody does), but the gist of it is that our consciousness of ourselves as abiding (i.e. our consciousness of our own identity throughout time) requires certain conditions which allow us to distinguish between (to use Strawson's turn of phrase) the subjective route of our experiences and the objective world through which it is a route. This distinction in its turn is grounded on certain principles which allows us to distinguish our subjective spatio-temporal order and the objective order of the world (for instance, to use Kant's own example, when I successively experience the different aspects of a house, I hold these to be successive apprehensions of a single object which does not change, whereas if I successively experience the different aspects of a boat going downstream, I hold these to be successive apprehensions of an object in the midst of change, so to speak).

Of course, that does not guarantee that when I judge that something belongs to the objective spatio-temporal order it does in fact belong to that order (unity is never given for Kant, but always produced). Nevertheless, the mere fact that there are such principles which allow me to make this distinction opens up the possibility that some judgments are grounded in these principles themselves, instead of being grounded in my experience of something as this or that. Mathematical judgments, for example, are grounded for Kant in the way we apprehend things as being conceptually identical yet still distinct. And some scientific judgments (Kant thought of his own rather Newtonian Metaphysics of Nature) are grounded in way an objective spatio-temporal order is structured in terms of a community of substances reciprocally acting one upon the other. So for Kant there is a middle term being a judgment being grounded purely in terms of conceptual containment (analytic judgments) and a judgment being grounded merely on empirical association (synthetic a posteriori judgments): some judgments are grounded in principles that make an objective spatio-temporal order possible in the first place (synthetic a priori judgments).

That would be Kant's reply, anyway. For better or worse, it is almost universally rejected today, in part because the supposed principles identified by Kant turned out not to be so necessary---Newtonian physics, for instance, was famously displaced by relativity and quantum mechanics. Some (e.g. Michael Friedman) have attempted to salvage something of the Kantian program by blending it with some variety of positivism: the task of the philosopher would be to identify the conceptual structure that underlies our best scientific theories. Myself, I personally think that the increased power in our logic has given us a much better picture of mathematics, which allows us to defend a variety of (structural) platonism: mathematics describes certain structural features of reality, to which we have epistemic access via proof. On the other hand, I think scientific theories (such as relativity) are not in the business of aiming at truth, but merely of empirical adequacy, of providing nice (generally mathematical) models which save the phenomena, so to speak. These are not merely associations because they turn on certain mathematical or structural features of reality.

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