Yes, but did Kant himself think that "one simple unity" was empirical? I think to him, numbers themselves and counting were all a priori, though possibly synthetic. — schopenhauer1

I agree, I was using that as an example of why Kant's category of a priori is not very good. I think other comments on this thread have illustrated the same thing.

Again, that's where I get confused with Kant. He doesn't demarcate enough. His examples are kind of fuzzy and taken as givens of why they are a priori sometimes. — schopenhauer1

My opinion is that we must allow that there is such a thing a priori knowledge, but it would probably be impossible to give an example of it, because of the a posteriori basis of language itself. So a priori knowledge is not something which can be exemplified, but we can know from logical demonstration that there must be ( necessarily is) such a thing. That's why examples are always fuzzy and confusing.

Banno is indeed Banno in every possible world in which I exist. In no world am I a doughnut, a fruit cake or anything other than human. Although in some worlds my name is Tim Wood. I am not Banno in all possible worlds, since one can posit a possible world without me in it. — Banno

It's not at all clear to me why that would be the case, though.

And I don't know how we'd convince anyone that it's not conceivable, because we do things like this in films all the time. For example, a witch or genie might turn a person into a dog, into a biscuit--whatever. It's not at all difficult to imagine that the person is now a dog or a biscuit. So how would we argue that in all possible worlds, the person can't be a biscuit?

There might be problems with the metaphysical plausibility of such things, but no moreso than that's a problem with any counterfactual scenario, because in all of them, we have to pretend that things can be different than they are yet still be the "same thing" somehow, which doesn't really make any metaphysical sense (at least to me as a nominalist).

2+1=3 in all possible worlds. If it did not, we would not be talking about 2,3,+, or =.

Water is H₂O in all possible worlds. If it were not, we would not be discussing water. — Banno

If we can say this of numbers, and water, and even that the atomic number of Gold is 79 and to find something else would simply be to misuse the word "gold" -- why is it that color cannot function in the same way?

You say that it is a secondary characteristic. Perhaps because it just seems plausible that Gold could be some other color, perhaps if our eyes had different cones in them or something along those lines. But it seems to me that another could just insist that you're using the word "Gold" incorrectly if you're referring to something that is not yellow yet has all the other properties of Gold. They could insist that this is not Gold, clearly, because it is not yellow, but should be called Rold.

I mean why not?

Everybody seems to know what a possible world is - any world without a contradiction. — TheMadFool

I don't. If non-contradiction applies in our world, why should it apply in another "possible" world. After all, we only need to "posit" them, not account for them.

The why......
“.....Our knowledge springs from two main sources in the mind, first of which is the faculty or power of receiving representations; the second is the power of cognizing by means of these representations. Through the first an object is given to us; through the second, it is thought....”
“....We apply the term sensibility to the receptivity of the mind for impressions, we call the faculty of spontaneously producing representations, or the spontaneity of cognition, understanding....”
“....Neither of these faculties has a preference over the other. Without the sensuous faculty no object would be given to us, and without the understanding no object would be thought. Thoughts without content are void; intuitions without conceptions, blind. Hence it is as necessary for the mind to make its conceptions sensuous (that is, to join to them the object in intuition), as to make its intuitions intelligible (that is, to bring them under conceptions). Neither of these faculties can exchange its proper function. Understanding cannot intuit, and the sensuous faculty cannot think. In no other way than from the united operation of both, can knowledge arise....”
“....We therefore distinguish the science of the laws of sensibility, that is, aesthetic, from the science of the laws of the understanding, that is, logic....”
“....logic in its turn may be considered as twofold—namely, as logic of the general use, or of the particular use, of the understanding. The first contains the absolutely necessary laws of thought, without which no use whatsoever of the understanding is possible, and gives laws therefore to the understanding, without regard to the difference of objects on which it may be employed....”
“....general logic has nothing to do with the origin of our cognitions, but contemplates our representations, be they given primitively a priori in ourselves, or be they only of empirical origin, solely according to the laws which the understanding observes in employing them in the process of thought, in relation to each other. Consequently, general logic treats of the form of the understanding only, which can be applied to representations, from whatever source they may have arisen....”
“....All intuitions, as sensuous, depend on affections; conceptions, upon functions. Conceptions, then, are based on the spontaneity of thought, as sensuous intuitions are on the receptivity of impressions. Now, the understanding cannot make any other use of these conceptions than to judge by means of them...”
“....In all judgements wherein the relation of a subject to the predicate is cogitated, this relation is possible in two different ways. Either the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A; or the predicate B lies completely out of the conception A, although it stands in connection with it. In the first instance, I term the judgement analytical, in the second, synthetical...”
“....The former may be called explicative, the latter augmentative judgements; because the former add in the predicate nothing to the conception of the subject, the latter add to our conceptions of the subject a predicate which was not contained in it, and which no analysis could ever have discovered therein....”
(Insert for clarity: “...analyse the conception, that is, become conscious of the manifold properties which I think in that conception,...”)
“....Judgements of experience, as such, are always synthetical. For it would be absurd to think of grounding an analytical judgement on experience, because in forming such a judgement I need not go out of the sphere of my conceptions, and therefore recourse to the testimony of experience is quite unnecessary....”

And the wherefore.......
“....But as in all the attempts hitherto made to answer the questions which reason is prompted by its very nature to propose to itself, for example, whether the world had a beginning, or has existed from eternity, it has always met with unavoidable contradictions (...), but it must be possible to arrive at certainty in regard to the question whether we know or do not know the things of which metaphysics treats....”
“....For what of analysis, that is, mere dissection of conceptions, is contained in one or other, is not the aim of, but only a preparation for metaphysics proper, which has for its object the extension, by means of synthesis, of our a priori knowledge. And for this purpose, mere analysis is of course useless, because it only shows what is contained in these conceptions, but not how we arrive, a priori, at them; and this it is her duty to show, in order to be able afterwards to determine their valid use in regard to all objects of experience, to all knowledge in general....”
————————————————————————

Logic is the law of thought, understanding is the means of thought, therefore logic rules understanding. Understanding manifests in judgements, judgements are of two kinds, one for objects, one for thought of objects. Therefore the laws of thought in the form of logic rules judgements. One form of logic, in which thought is transposed to intelligible communication, is the proposition. A logical proposition requires a subject and a predicate, and the relationship between them determines the kind and the source of the judgement being communicated.

Analytic propositions dissect conceptions in order to determine whether the conception in the predicate belongs to the conception in the subject by the logical law of identity, re: all dogs are canines.

Propositions where the conception in the predicate does not belong to the conception in the subject by identity, but must be connected to it by synthesis from an intuition of experience, earn the title synthetic propositions, re: all dogs have four legs.

Analytic judgements a priori occur when the relations in the conceptions in the predicate belong to the conceptions in the subject from the logical principles of universality and necessity, re: all bodies are in space.

Synthetic a priori judgements, synthetic in the method of their subject/predicate relation and a priori from the nature of the conceptions in them, are shown by any pure mathematical expression.

An analytic a posteriori proposition may be merely a tautology, hence useless, re: my hat is a hat.

Synthetic a posteriori propositions are redundancy in kind, for synthetic propositions are already completely empirical judgements, the concept in the predicate supplements the concept in the subject, even if not contained in it.

The proof of the possibility and validity of synthetic a priori propositions is given in mathematics and the principles for them are given to all natural sciences. Their true value, however, lays in their employment by reason in questions of metaphysics, which is in effect, the examination of grounds for possible truth.

Thus is the division into and function of differences in the approach to the attainment of knowledge, and the justification for synthetic a priori judgements being by far the most important.

Theoretically.

Posit any world you like. This might have been where @frank was thinking with his talk of imagination. Whatever you want.

Is it possible?

PWS gives us a way of at least making sense of that question.

Posit a world that contains a square circle. Is it possible? No, because it leads to contradiction. Posit a world in which water is He₂O. Is that possible?, no, because what we call water is H₂O. SO a world in which what was called "water" was found to have the chemical structure He₂O is a world in which "water" refers to something other than water.

A world in which gold is purple is just a world in which gold is not yellow. A world in which gold has the atomic number 63 is a world in which "gold" refers to something else.

Now while PWS was somewhat controversial for a wile, it has settled in as a neat way of dealing with counterfactuals, possibilities and necessities.

I don't. If non-contradiction applies in our world, why should it apply in another "possible" world. After all, we only need to "posit" them, not account for them. — tim wood

Well, if contradictions are allowed, the word "possible" loses meaning. Possibility and impossibility imply constraints to worlds. If there are none then we might as well as get rid of the words "possible" and "impossible" as then everything would be possible.

You were saying at one point that logic is a set of rules for the way we talk. I think logic is about the limits of imagination, and we discover those limits.

Quine pointed out that even if the way we talk is entirely conventional, it doesn't appear that the ability to apply logic to new situations can be learned second hand.

Well, if contradictions are allowed, — TheMadFool

Is "contradiction" well-defined in all possible worlds? And is it "well-named?" Folks who think "all possible worlds" is a meaningful idea are content to have gold not be gold, but at the same time be gold, in various "possible worlds" (I do not know how that works...). Why should the idea of non-contradiction be any less malleable? You argue from your world, but that's not the world in question.

Posit a world that contains a square circle. Is it possible? No, because it leads to contradiction. — Banno

In this world, sure, but not in the world I "posit." But I think we're in agreement: there is nothing true in any world that is not at the same time true in all possible worlds, yes?

Is "contradiction" well-defined in all possible worlds? — tim wood

In PWS, propositional and predicate logic hold in each world.

"all possible worlds" is a meaningful idea are content to have gold not be gold, but at the same time be gold, in various "possible worlds" (I do not know how that works...). — tim wood

It doesn't because it is wrong. Gold is gold in every possible word in which it exists. Posit a world in which gold is purple, and get a possible world in which gold exists. Posit a world in which gold has the atomic number 1, and you will have to decide if in that world "gold" means hydrogen, or in which the atomic number system admits to fractions...

In this world, sure, but not in the world I "posit." — tim wood

No; no possible world may contain a contradiction.

But I think we're in agreement: there is nothing true in any world that is not at the same time true in all possible worlds, yes? — tim wood

Far from it. The whole point of the exercise is to allow for different truths in different worlds.

But it seems to me that another could just insist that you're using the word "Gold" incorrectly if you're referring to something that is not yellow yet has all the other properties of Gold. They could insist that this is not Gold, clearly, because it is not yellow, but should be called Rold. — Moliere

Of course they could. Humpy Dumpty beckons.

But what about you, Moli? What would you say?

Good to see Quine get a mention.

There are only logical tautologies as necessary truths all the rest are akin to be possible in some worlds while not possible in others.

For example, physical theories are only possible truths, they aren't necessary to hold in every world there is.

No; no possible world may contain a contradiction.

Far from it. The whole point of the exercise is to allow for different truths in different worlds. — Banno

But not just any old truth, Just the ones from the approved list. Makes a hash of the notion of "all possible" though, doesn't it. And reduces truth to a word game. If in some "possible" world we call Hydrogen "gold," then there's gold in that world. Which is to say that a rose by any name other than "rose," is maybe not a rose.

I'm still waiting for the simple and coherent statement of just what "all possible worlds" means. This from Stanford (SEP):
"That the truth values of ⌈□φ⌉ and ⌈◇φ⌉ are not in general determined by the truth value of φ at the world of evaluation is, semantically speaking, nothing more than the fact that the truth values of ‘∀xFx’ and ‘∃xFx’ are not in general determined by the truth value of ‘Fx’, for any particular value of ‘x’."

I should think for ExFx at least, for some particular value of x. Or maybe it's "world of evaluation." Is that a term of art?

Colloquially, isn't possible just self-consistent and either of non/hypothetical?
Self-consistent is at least the usual identity (ontological, propositional) and non-contradiction (propositional).
I guess that implicitly assumes our world is self-consistent, but that seems required for propositions to be meaningful anyway.

But what about you, Moli? What would you say? — Banno

I don't mean to invoke humpty dumpty, it's just that to myself it seems that the invocation of atomic number is just as arbitrary as color in designating some aspect as a necessary feature.

For myself it does strike me as odd to say that the color of gold is known a priori. There's a sense in which I can make sense of it -- like, now that I think of Gold as golden it seems that I can think of gold in a purely conceptual sense and say that its color is now part of its concept. But, then, it seems to me that gold could have been other than golden.

However, from my perspective, it seems to me that gold could have been other than having 79 protons in its nucleus too. This was something we discovered a posteriori, and is not a necessary feature of gold -- that is, there is a possible world in which gold has more or less protons than 79. Your thought experiment would apply equally well here too -- imagine that we had some substance who retained the same ductility, the same color, the same melting point and freezing point, the same ratio for certain alloys, and in all other ways was the same as the gold we know in our actual world -- save for the number of protons we find in its nucleus.

Wouldn't we call this gold? Or would we call this something else?

To me I think it's similar to your color example. initially I wouldn't call something gold that happened to be red, but I'd say, perhaps, that it is red gold. And if we found some substance with a different number of protons in it then I'd say it is Gold-78 or some such, if I were being precise.

Sort of like how we came to know there are isotopes of various elements with differeing number of neutrons. They were similar in structure, but different in certain respects (and had different properties too because of that).