There was never a statement that did not depend on prior experience or sensation. — Hallucinogen

So you put six beers in the fridge and see someone take three of them out. You don't know there's three left until you open the fridge door and verify it by looking.

So you put six beers in the fridge and see someone take three of them out. You don't know there's three left until you open the fridge door and verify it by looking. — Wayfarer

One of the beers might have been pregnant.

Analyticity is related to necessity and a priority. If analytic statements are necessarily true, they are automatically true come what may. If analytic statements are true a priori, they are known without experiential knowledge.

Well let's have a little go at this. Language works by means of distinctions: beer is distinguished from not-beer, otherwise we don't know what we are talking about when we say "beer". and that would be a tragedy, because it is a useful and important distinction, even if it becomes blurry at the barleywine edges.
One of the distinctions that philosophers find useful and important is between the word and the thing, sometimes called the signifier and the signified. The convention is that when one wants to talk about the word "beer", one puts it in quotes, and when one wants to talk about the drink beer, one does not. Thus there is a clear difference, beer is a nourishing drink, whereas "beer" is a word.

So now, philosophers of beer can discuss the defining (necessary and sufficient) features of beer, What makes something beer and not a rabbit? Cue much talk of hops, barley malt, fermentation, and the amount of froth on top. Does it have to be liquid, or is a frozen beer still a beer? Are the yeasty dregs at the bottom of the barrel beer? All this talk is talk about what the word "beer" means, about what counts as beer, we are trying to sharpen up those blurry edges of the distinction between beer and not-beer.

So it is the case that(P1.) Fosters is not beer, but the recycled piss of inebriated Australians, and this is a matter of fact, given our shared understanding of what "beer" means. And this is what we call synthetic proposition, because it turns out that Australians also make proper beer that they do not export, but wisely drink themselves.

However, it is based on not only the facts of the case, but also the analytic proposition that (P2.) the recycled piss of inebriated Australians is not beer. This is analytic, because it is not about beer, but about "beer". The facts of the case - that Fosters call their drink "beer", are not decisive, because to most philosophers of beer that is simply an abuse of language.

Perhaps the Hegelian perspective is useful to you. Analytic thought produces a thesis. The existence of the thesis means that there is an antithesis. If the antithesis is also meaningful, then one can use synthesis to combine the thesis and antithesis and generate a more complete system of explanation. I should mention, some feel that Kant does not use the thesis/antithesis/synthesis method in this manner, although he used those words, and they state that was something Frege first did instead. However I believe they are wrong in tat. The difference, if there is one, was that Frege was the first to indicate that a synthesis is itself another thesis, leading to a dialectic; whereas Kant simply was interested in stating triads of ideas with that relationship, rather than pursuing dialectical investigation.

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you know there's 3 beers left prior to opening the fridge door. That's all 'a priori' refers to and it's what is called an apodictic truth, i.e. cannot plausibly be denied. The rest is blather.

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So, basically, I can't see how dividing all statements into either analytic or synthetic is correct. — Hallucinogen

It is a distinction that seemed to made sense at the time it was hotly discussed, which was the 16th-17th centuries. That was before a modern understanding of logic was developed, which arose in the late 19th to early 20th centuries. That understanding has revealed that the distinction is an illusion - for instance that the statement '7+5=12', which Kant thought was synthetic, is not different in kind from 'all bachelors are unmarried', which Kant thought was analytic. I presume that is one of the reasons why hardly any professional philosophers discuss it any more, other than as a historical phenomenon.

However the distinction is very important historically. It is important because the controversy about it awakened Immanuel Kant from his 'dogmatic slumber' and goaded him to write the Critique of Pure Reason, which is still very relevant, meaningful, and much discussed today. Many see it as one of the most important philosophical works ever written. The bits of CPR about the analytic/synthetic distinction are obsolete and can be skimmed over in favour of those that give real insights, like the Transcendental Aesthetic, the Transcendental Deduction of the Categories and the Synthetic Unity of Apperception. These give difficult, but highly significant, insights into how we think, how we approach the world.

but memory doesn't explain mathematics. You're not struggling to understand something, you're struggling to make any kind of argument.

Analytic thought produces a thesis. The existence of the thesis means that there is an antithesis. If the antithesis is also meaningful, then one can use synthesis to combine the thesis and antithesis and generate a more complete system of explanation."
I'm not sure what this means, could you give an example? — Hallucinogen

Sure, here is a simple example
analytic: what noise do animals make?
analytic thesis: dogs bark
analytic antithesis: but cats meow
synthesis: different species make different noises.

Analytic things are true by definition, or tautological. Reason is a process of truth retention, and is deleterious by nature. You're to find the necessary by deleting the coincidental.

The precise example of an analytic claim that Kant gave, I believe was "all bodies are extended in space" or something like that. Married bachelors were too simple and obvious for him.

"All blue dogs are dogs." This statement is true in virtue of logical forms. I think this is true a priori.

you know there's 3 beers left prior to opening the fridge door. That's all 'a priori' refers to and it's what is called an apodictic truth, i.e. cannot plausibly be denied. The rest is blather. — Wayfarer

You're just wrong about this. How many beers are in the fridge is settled by opening the fridge and counting the beers, and the result of this experiment trumps any amount of mathematical and logical reasoning. If there turn out to be 4 beers, then one thought one knew but was mistaken. Perhaps Jesus passed by and turned the milk into beer, perhaps beers can breed, perhaps there was already a beer in the fridge, perhaps a wormhole opened and a beer fell through, or perhaps you miscounted the beers you put in, or the ones taken out, but anyway it is not analytic that there are 3 beers, nor a priori. It is a matter of fact, that might be otherwise.

Perhaps Jesus and passed by and turned the milk into beer, perhaps beers can breed, perhaps there was already a beer in the fridge, perhaps a wormhole opened and a beer fell through, or perhaps you miscounted the beers you put in, or the ones taken out, but anyway it is not analytic that there are 3 beers, nor a priori. — unenlightened

If you think that amounts to an argument then please, go and have another beer.

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If you think that amounts to 'an argument', then please, go and have another beer. — Wayfarer

These are not arguments, they are examples of explanations one might make if there turn out not to be 3 beers in the fridge. The point is that how many beers are in the fridge is not a priori, it is a contingent fact. What one would not do, however, unless one had had several too many beers, is claim that 6 - 3 = 4, because that is analytic.

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But if you forget how many beers were placed in the fridge, — Hallucinogen

Yes, you could be shot dead en route, or have your brain consumed by a parasitic worm, or some of the beers could be turned into beetroot juice by some as yet unknown process. All this is true, and all beside the point.

If you were not capable of knowing basic maths, then you probably couldn't type anything at all, the fact that you can think of anything to say, even if what you're saying is completely meaningless, relies on the fact that you're able to grasp basic truths of grammar, mathematics and the like.

ultimately it is those experiences of objects in the world that gets plugged into our mathematical thinking when we're learning to count as children — Hallucinogen

Kant argued that the structures of logic which organize, interpret and abstract observations were built into the human mind and were true and valid a priori. Mill, on the contrary, said that we believe them to be true because we have enough individual instances of their truth to generalize: in his words, "From instances we have observed, we feel warranted in concluding that what we found true in those instances holds in all similar ones, past, present and future, however numerous they may be". Although the psychological or epistemological specifics given by Mill through which we build our logical apparatus may not be completely warranted, his explanation still nonetheless manages to demonstrate that there is no way around Kant's a priori logic. Mill argues: "Indeed, the very principles of logical deduction are true because we observe that using them leads to true conclusions" - which is itself an a priori presupposition.

Wikipedia.

In other words, to make a judgement about whether mathematics corresponds with experience, we must first make a mathematical judgement. Otherwise, how would we know that it corresponds? The knowledge of mathematical truths can't rely on, or be explained in terms of, anything else; it is the source of explanations, not the target of explanations.

The point is that how many beers are in the fridge is not a priori, it is a contingent fact. — unenlightened

The idea that they're 'beers in fridge' is only a rhetorical device to illustrate the fact that 6-3=3 in a rather less boring manner. And that is something that is obviously know a priori - it's simply an example of a tautological truth.

The idea that they're 'beers in fridge' is only a rhetorical device to illustrate the fact that 6-3=3 in a rather less boring manner. And that is something that is obviously know a priori - it's simply an example of a tautological truth. — Wayfarer

Then we agree. But your rhetoric serves to blur the distinction rather than clarify it. Change the example:

I put 2 rabbits in an empty hutch with some lettuce, and then see my friend take out 5 rabbits. I do not know that there are - 3 rabbits in the hutch, nor have I proved arithmetic wrong, I know they've been breeding. Experience tells me that rabbits multiply and beers only add and subtract.

I put 2 rabbits in an empty hutch with some lettuce, and then see my friend take out 5 rabbits. I do not know that there are - 3 rabbits in the hutch, nor have I proved arithmetic wrong, I know they've been breeding — unenlightened

You only know that because you can count! So you say - ah, five rabbits, they must have multiplied. And you know that, even if you didn't actually see them at it. ;-)

] The first sentence is the SUBJECT of the analysis which directs the investigation to the thesis.

The Kantian method and Hegelian dialectic are considered methods of investigation, so they doesn't really need a justification. Here is the wikipedia reference

https://en.wikipedia.org/wiki/Dialectic#Hegelian_dialectic

The difference was, Kant kept exploring the same subject until he was sure he had the right assertions, whereas Hegel's method was iterative.

Two things you need to remember to properly understand the analytic/synthetic distinction (along with the a priori/a posteriori distinction). First, is that Kant mobilises these terms in the context of the problem of causality, and they cannot be understood apart from that context. Specifically, remember that Kant is responding to Humeian skepticism about causality: qua Hume, the constant conjunction of events does not guarantee the universality of causal connection. In other words, causality does not admit of the order of logical necessity. And connection via the force of logical necessity is just what Kant refers to as an analytic connection. By contrast, synthetic connections - in this case causality - are those that admit of extralogical reasons to explain their connection.

In Christian Kerslake’s terms: “Whereas an analytic connection contains its reason solely in the logical explication of the presupposed meaning of a concept, a synthetic connection must involve an extralogical reason. [For Kant,] the concept of a causal relation must be synthetic… [Hence,] Kant’s notion of the synthetic a priori simply names a problem faced by eighteenth-century philosophy – that of how to account for any possible nonlogical a priori connections”.

The second thing to understand then is that the above means that Kant here is dealing with the problem of how to move from the sphere of logic to the sphere of existence. Insofar as analytic statements are those driven by logical necessity, synthetic statements by contrast involve a measure of reality. It is this move from logic to existence that in turn - for example - grounds Kant’s famous response to the cosmological argument: it is not enough to argue - as the cosmological argument more or less does - that God is perfect and that because existence is a perfection He must exist: for this simply begs the question of God’s existence to begin with. Essence - or analyticity - cannot ground existence - which belongs to the order of the synthetic.

So just remember: at stake in the analytic/synthetic distinction is the question of both causality on the one hand, and the move from logic to reality on the other: both of which turn upon the question of logical and extra logical necessity respectively.

"All blue dogs are dogs." This statement is true in virtue of logical forms. I think this is true a priori. — quine

What about "all former students are students"?

What about "all former students are students"?

Shouldn't it read: " all former students were students" the other way round mixes up tenses.

Shouldn't it read: " all former students were students" the other way round mixes up tenses. — Cavacava

That's the problem with quine's claim that "All blue dogs are dogs" is true by virtue of its logical form, i.e. all X-type Ys are Ys.

Meaning matters.

Well that's a much more detailed answer than I have seen anyone else write here. I look forward to seeing more of your posts :)