Philosophy Discussion Forums

Londoner wrote:The word 'bachelor' refers to a concept. True, we might have instead chosen the word 'caelibem' (Latin) to refer to that concept, and provided everyone understood it then that would do, but what we are interested in is the concept, not the sign for that concept.'A bachelor is an unmarried man' is an accurate description of the concept 'bachelor'.

Belinda describes 'analytic' as a glorified tautology, but that doesn't imply it is so obvious. When you ask me what I mean by 'analytic'; you are asking me to give the equivalent of 'a bachelor is an unmarried man' answer...although in this case it is far more difficult. Far from being arbitrary, the meaning of 'analytic' is a subject of contention amongst philosophers.

So, although a dictionary of philosophy would say 'Analytic propositions are propostions that are true by virtue of their meaning', that doesn't tell us whether a proposition falls into that category - and if it does, exactly what that meaning would be. Indeed, isn't this expressed by the reflex response of philosophers to any question i.e. 'It depends what you mean by...'?

I wasn't really asking how can one tell between analytic and synthetic propositions, I rather meant to ask where the truth of analytic propositions derives from? It seems that we agreed that synthetic propositions, if they are true, then they are true by virtue of the way the world is. But what makes analytic propositions true on your view? Do you accept the common view that analytic propositions are true by virtue of the meaning of the terms?

I disagree. If we were to try to create a form of geometry that was true of real objects in the same way that Euclidean geometry is true of two dimensional ones, we would first have to have the equivalent of his postulates. These postulates would have to be (necessarily true) about real objects.

What do you mean by 'to have the equivalent of his postulates'?

And we don't have to create a new form of geometry because we already have it, both euclidean and non euclidean types of geometry function perfectly well as parts of Newtonian and relativistic physics to describe and predict physical phenomena. How did Einstein manged to make all his predictions with such astonishing accuracy if the mathematical and geometrical apparatus that he was using wasn't describing correctly how space-time behaves?

But the discovery that E=MC^2 isn't like that either, but it doesn't mean that relativity wasn't a discovery of new facts about the world.

Unlike geometry, that formula is ultimately based on science; unlike a theorem its truth or otherwise would be determined by empirical evidence.

This is beside the point. The example of E=MC^2 just illustrates that your objection that Pythagoras's theorem can't be a discovery of a fact about the world because it's not of the form "there exists a continent in the west..." is not very sound, because there are tones of true statements about the world which are not of the form "there exists an x...", and physical laws is one obvious example. On my view, Pythagoras has discovered some necessary properties of all possible euclidean spaces, and this seems to me like a significant discovery about reality (which feature of reality the theorem is supposed to be about is of course another question, I'm not saying that it's the same kind of thing as E=MC^2).

This isn't to claim that empirical evidence is conclusive, however the conventions of science accept as an assumption that they are, and also that certain sorts of reasoning are valid.

Earlier in this exchange I mentioned the idea that the truth of statements are understood in their context. What verifies a proposition in science isn't the same as what verifies one in geometry (or psychology etc.) Similarly, the assumptions required for one are different to those required for another. That is why you cannot mix them. For example 'I dream of monsters' is evidence of a psychological state, it may be a fact, but it cannot be used as evidence of a scientific proposition.

In other words, there isn't 'the world' about which there are discoveries. There are various worlds of understanding about which we seek to better comprehend.

I don't understand what the epistemic uncertainty of scientific discoveries has to do with my claims about geometry?

Fafner

I wasn't really asking how can one tell between analytic and synthetic propositions, I rather meant to ask where the truth of analytic propositions derives from? It seems that we agreed that synthetic propositions, if they are true, then they are true by virtue of the way the world is. But what makes analytic propositions true on your view? Do you accept the common view that analytic propositions are true by virtue of the meaning of the terms?

To be precise I'd say that synthetic propositions are true both by virtue of the meaning of terms used and by their reference to something other than those words. But what constitutes 'truth' in any proposition is a subject in itself, being obviously linked to the topics of 'meaning' and 'linguistics'. I cannot face writing an essay on it and explaining my own thoughts, but I would refer you the Stanford Encyclopedia essay on the Analytic/Synthetic distinction.

And we don't have to create a new form of geometry because we already have it, both euclidean and non euclidean types of geometry function perfectly well as parts of Newtonian and relativistic physics to describe and predict physical phenomena. How did Einstein manged to make all his predictions with such astonishing accuracy if the mathematical and geometrical apparatus that he was using wasn't describing correctly how space-time behaves?

Einstein did it by moving beyond that Newtonian physics that you said functions perfectly well! But I do not think you have followed my point, which was that Einstein did not come up with his formula first, then use it to show things about the universe (in the same way that the postulates in geometrical systems enable us to build theorems). Rather, his formula were the result of his analysis - an analysis of the external world.

...there are tones of true statements about the world which are not of the form "there exists an x...", and physical laws is one obvious example.

I disagree. I do not see how you can assert a physical law without also asserting 'and this exists in the world'. Not in itself; obviously 'gravity' is not an object. But unless there existed objects being affected by gravity, then what would the term signify? How could gravity be 'true'?

Again, this isn't special to science. If I just say 'green' that doesn't assert anything. For 'green' to be true or false it has to be attached to an object (that exists).

I don't understand what the epistemic uncertainty of scientific discoveries has to do with my claims about geometry?

It was part of an argument showing that what is sufficient to constitute evidence of truth in one area; science say, was different to what constituted evidence in another; geometry say. Scientific laws are not considered true simply because they are internally consistent, they must also match empirical evidence.

Londoner wrote:To be precise I'd say that synthetic propositions are true both by virtue of the meaning of terms used and by their reference to something other than those words. But what constitutes 'truth' in any proposition is a subject in itself, being obviously linked to the topics of 'meaning' and 'linguistics'. I cannot face writing an essay on it and explaining my own thoughts, but I would refer you the Stanford Encyclopedia essay on the Analytic/Synthetic distinction.

So if you don't have a clear idea of what analytic propositions are supposed to be, then how do you know that mathematics and geometry must be analytic and not synthetic? You can't argue that they are analytic, without taking a position on what analiticity supposed to mean, because this way it's impossible to asses your claim. I gave an argument why geometry can't be analytic at least on one account of analiticty but you said that it's not your view, so if you want to continue this discussion any further, then you must say something more on this.

Einstein did it by moving beyond that Newtonian physics that you said functions perfectly well! But I do not think you have followed my point, which was that Einstein did not come up with his formula first, then use it to show things about the universe (in the same way that the postulates in geometrical systems enable us to build theorems). Rather, his formula were the result of his analysis - an analysis of the external world.

Yes Einstein did use empirical data as a basis for his theory, I never said that relativity is an a priori discovery, but I don't see what it has to do with my argument. What I said is that your claim that we can't use geometry to describe the physical reality is extremely implausible, because that's precisely what people like Einstein and Newton did with undeniable success. If equations in physics don't describe reality then why the hell they need them and why they work so well to predict observable phenomena?

I disagree. I do not see how you can assert a physical law without also asserting 'and this exists in the world'. Not in itself; obviously 'gravity' is not an object. But unless there existed objects being affected by gravity, then what would the term signify? How could gravity be 'true'?

Yes there's a problem about the truth-makers for mathematics and geometry as I indicated earlier, but I don't think that this problem is enough to deny that they are genuine facts about reality, because the same problem applies to many other things (and of course an inability to explain something doesn't mean that this thing is false). For example what makes the proposition "a 6 meter tall man is taller then Barack Obama" true? Certainly there's no object in the world which is a 6 meter tall man, so how this proposition can be true? And such examples can be multiplied indefinitely, so it can't be the case that there must always be an object corresponding to any true statement about reality as you claimed (and this is only the tip of the iceberg, what about modal facts, properties, relations?)

It was part of an argument showing that what is sufficient to constitute evidence of truth in one area; science say, was different to what constituted evidence in another; geometry say. Scientific laws are not considered true simply because they are internally consistent, they must also match empirical evidence.

So what's the point?

Daviddunn wrote:Ten years ago I read Kant, painful but worthy reading. He has had good points, some still stand but after the scientific discoveries that came after the Critique, the Critique itself stand in need of a critique.

Anyway, this exchange has been very enriching for me.

I am reading Kant's work at present. As far as the structure and framework for theoretical knowledge is concern, imo, Kant's Critique of Pure Reason is quite complete. What is needed is to connect some small gaps and fill in the 'flesh'.

Kant wrote:In this enquiry [1st Critique] I have made Completeness my chief aim, and I venture to assert that there is not a single metaphysical problem which has not been solved, or for the solution of which the key at least has not been supplied.

As for Science [philosophical perspective], it was Kant (via his a priori principle) who saved Science's 'sanity' after Hume threw in a big spanner (Problem of Induction).
Subsequent scientific discoveries has and will validate Kant's theory of knowledge.

If you recall, as far as Kant is concern, there is no issue with synthetic a priori judgments in Science, Mathematics or Geometry.
The serious philosophical issue with synthetic a priori judgment is that of Metaphysics.

Fafner

So if you don't have a clear idea of what analytic propositions are supposed to be, then how do you know that mathematics and geometry must be analytic and not synthetic?

What I actually wrote, as you can see from the extract you chose to quote, was that the idea of 'truth' was a topic in itself.

If equations in physics don't describe reality then why the hell they need them and why they work so well to predict observable phenomena?

Because they deal in observable phenomena. In so doing, they take certain things, like the reliability of sense impressions and the validity of inductive reasoning as given. A synthetic a priori (in your sense) should be self-supporting; it should not take anything as given.

For example what makes the proposition "a 6 meter tall man is taller then Barack Obama" true? Certainly there's no object in the world which is a 6 meter tall man, so how this proposition can be true?

That isn't a proposition, it is an argument in logic; If there are no 6 meter men, and if Obama is a man, then Obama is under 6 meters.

The truth of the assumptions in that argument would be determined empirically. By doing measurements we could verify 'there are no men 6 meters tall'. In practice, if we measured sufficient people we would feel justified in reasoning that our proposition was always true, but we wouldn't have proved it absolutely. We could also confirm Obama qualified as a man (or not! Google 'reptilians')

Or, if you only meant to assert something about Obama's height, similarly we would verify it empirically in that we could take a ruler and measure Barack Obama.

(In either case, there would still be scope for the doubts that always go with empirical evidence.)

..so it can't be the case that there must always be an object corresponding to any true statement about reality

In your example, the objects were 'men' and 'Obama'. When you say 'there are no 6 meter men' you imply you are referring to 'nothing', but in fact you are referring to men. You are asserting; 'a characteristic of all men is that they are less than 6 meters'.

'No-men' are not a sort of object. They cannot have attributes like height. Otherwise I might argue; 'But suppose Obama is a no-man; I'm told they can be 6 meters tall!

Londoner wrote:What I actually wrote, as you can see from the extract you chose to quote, was that the idea of 'truth' was a topic in itself.

It's not a different question, the analytic/synthetic distinction is largely about what makes each class of propositions true. The definition of analytic is after all "true by virtue of meaning", so you can't ignore it.

Because they deal in observable phenomena. In so doing, they take certain things, like the reliability of sense impressions and the validity of inductive reasoning as given. A synthetic a priori (in your sense) should be self-supporting; it should not take anything as given.

This doesn't answer my question. If the abstract geometrical concept of a Minkowski space-time is not about the actual space time, then why Einstein put it there? This is even to ridiculous to argue about, everybody who knows anything about relativity knows that the whole point of adopting a non euclidean geometry is because it describes correctly the actual physical space. So do you claim that Einstein was wrong thinking that he discovered some important properties of space time because "geometry can't describe reality"?

That isn't a proposition, it is an argument in logic; If there are no 6 meter men, and if Obama is a man, then Obama is under 6 meters.

It is a perfectly fine proposition, and logic does deal with propositions.

The truth of the assumptions in that argument would be determined empirically. By doing measurements we could verify 'there are no men 6 meters tall'. In practice, if we measured sufficient people we would feel justified in reasoning that our proposition was always true, but we wouldn't have proved it absolutely. We could also confirm Obama qualified as a man (or not! Google 'reptilians')

I wasn't asking about confirmation. The sentence doesn't say "there's no men 6 meters tall", it says that if there was such a man then he would be taller then Obama.

Or, if you only meant to assert something about Obama's height, similarly we would verify it empirically in that we could take a ruler and measure Barack Obama.

It's not only an assertion about Obama's height, it's also an assertion about a hypothetical non existent person.

(In either case, there would still be scope for the doubts that always go with empirical evidence.)

The question has nothing to do with evidence or knowledge.

You are asserting; 'a characteristic of all men is that they are less than 6 meters'.

No, I'm not asserting that. The proposition would still be true if in fact there were 6 meter tall men.

'No-men' are not a sort of object. They cannot have attributes like height. Otherwise I might argue; 'But suppose Obama is a no-man; I'm told they can be 6 meters tall!

So now you are saying that it's not a true proposition?

Fafner

It's not a different question, the analytic/synthetic distinction is largely about what makes each class of propositions true. The definition of analytic is after all "true by virtue of meaning", so you can't ignore it.

I don't ignore it; crudely that is true, but earlier I pointed out that that definition also just shifts the question to your understanding of 'meaning'. I'm sorry, but philosophy is like that!

You seem to be asking me to write a general essay on the philosophy of the synthetic/analytic labels; I have pointed you to a website that does that. I have already tried several times to link our discussion to the works of other philosophers, but you will not go there! So, if you don't mind, I will stick to discussing your own understanding of a synthetic a priori.

This is even to ridiculous to argue about, everybody who knows anything about relativity knows that the whole point of adopting a non euclidean geometry is because it describes correctly the actual physical space.

Yes. Except I would have emphasised the word 'because'. The nature of the physical determines the geometry adopted, the geometry we adopt does not determine the nature of the physical.

No, I'm not asserting that. The proposition would still be true if in fact there were 6 meter tall men.

So what made you choose to include in your proposition the sentence: 'Certainly there's no object in the world which is a 6 meter tall man'? If that bit wasn't relevant then you are left with: 'A 6 meter tall man is taller then Barack Obama', or if we remove the man altogether: '6 meters is taller than Barack Obama'. You ask 'What makes that proposition true?' I reply; 'The height of Barack Obama being observed to be less than a 6 meters is what makes that true.'

What exactly is your alternative? That the height of Obama can be known in a non-empirical way because you have linked a statement about his height to another statement about some hypothetical giants who may, or may not, exist?

So now you are saying that it's not a true proposition?

I'm running out of ways to explain that things that don't exist can't have qualities. Theories in physics are only meaningful if they describe objects in physics, only men who exist can have the property of height etc. A proposition which doesn't assert the existence of its subject isn't true or false - it isn't a proposition because it doesn't propose anything!

Londoner wrote:I don't ignore it; crudely that is true, but earlier I pointed out that that definition also just shifts the question to your understanding of 'meaning'. I'm sorry, but philosophy is like that!

You seem to be asking me to write a general essay on the philosophy of the synthetic/analytic labels; I have pointed you to a website that does that. I have already tried several times to link our discussion to the works of other philosophers, but you will not go there! So, if you don't mind, I will stick to discussing your own understanding of a synthetic a priori.

I'm not asking you to write an essay, but if you claim that it's not accurate to say that analytic propositions are true by virtue of arbitrary linguistic convention then I would like to know why. Because if they are, then on my view it would be very implausible to say that mathematics or geometry are analytical since they don't seem to be true just by virtue of linguistic conventions, but rather being significant discoveries about reality.

Yes. Except I would have emphasised the word 'because'. The nature of the physical determines the geometry adopted, the geometry we adopt does not determine the nature of the physical.

Fine, but this wasn't my claim anyway. The example simply contradicts your view that we can't describe reality through geometry, and if we indeed can then how come it's not synthetic?

So what made you choose to include in your proposition the sentence: 'Certainly there's no object in the world which is a 6 meter tall man'?

This bit isn't included in the proposition, it simply an independent fact which happens to be true.

If that bit wasn't relevant then you are left with: 'A 6 meter tall man is taller then Barack Obama', or if we remove the man altogether: '6 meters is taller than Barack Obama'. You ask 'What makes that proposition true?' I reply; 'The height of Barack Obama being observed to be less than a 6 meters is what makes that true.'

I agree that the fact that Obama is shorter then 6 meters is part of what makes the proposition true, but I don't think it's sufficient. "x is taller then y" is a relation, and for a relation to be true there must be two objects related to each other, but in my example there's no object which is a 6 meter tall man to whom Obama is related. Perhaps I should've used a different example like "a 6 meter man is taller then a 5 meter man" to make the point more clear. Surely this proposition isn't true because there are two particular man in the world whom we can locate, so it must be true because there's some other non specific fact in the world which isn't an object or a pair of objects.

What exactly is your alternative? That the height of Obama can be known in a non-empirical way because you have linked a statement about his height to another statement about some hypothetical giants who may, or may not, exist?

No, the alternative is to say that there are facts in the world which aren't constituted by particular and locatable objects as you claimed. Of course Barack Obama must exist for the sentence to be true, but as I pointed out, his existence is not sufficient to make the proposition true.

I'm running out of ways to explain that things that don't exist can't have qualities. Theories in physics are only meaningful if they describe objects in physics, only men who exist can have the property of height etc. A proposition which doesn't assert the existence of its subject isn't true or false - it isn't a proposition because it doesn't propose anything!

So do you agree or not that "a 6 meter tall man is taller then Obama" is a true proposition?

Firstly, I'm finding this a riveting read! I just had to put that out there before I continue.

I present to you this: I would like to know where the following sentence might fit into priori logic.

'Never nix that which is not no, is always yes.'

Londoner wrote:You seem to be asking me to write a general essay on the philosophy of the synthetic/analytic labels; I have pointed you to a website that does that.

This is a reasonable, rational and philosophical point.
I suggest Fafner88 have a go at it, at least for a quickie.
http://plato.stanford.edu/entries/analytic-synthetic/

IMO, it is a matter of consensus on which direction the participants want to take the discussion.

Fafner

I'm not asking you to write an essay, but if you claim that it's not accurate to say that analytic propositions are true by virtue of arbitrary linguistic convention then I would like to know why.

I have already answered this; look at the first paragraph in post 99 for example.

The example simply contradicts your view that we can't describe reality through geometry, and if we indeed can then how come it's not synthetic?

'Gravity' describes the behaviour of objects. Its truth depends on whether it does so accurately i.e it is determined empirically. Synthetic statements are ones, which like gravity, are true by how their meaning relates to the world.

But if it is to also serve as a theory, our ideas about 'gravity' must also be internally consistent. The terms it uses, like 'attraction' and 'acceleration' must relate to each other and to other terms in a consistent way. Thus statements about 'gravity' must also be true in an analytic sense.

However, it is possible to come up with endless other theories that are also internally consistent and would thus be just as true in an analytic sense as 'gravity', but which do not accurately describe reality experienced empirically. In other words, the validity of 'gravity' in a 'synthetic' sense is not proven just by its validity in an 'analytic' sense.

Perhaps I should've used a different example like "a 6 meter man is taller then a 5 meter man" to make the point more clear. Surely this proposition isn't true because there are two particular man in the world whom we can locate, so it must be true because there's some other non specific fact in the world which isn't an object or a pair of objects.

This formulation is ambiguous. Are you saying; '6 meters is more than 5 meters'? Or are you saying; 'There exists a man, who at 6 meters is taller than 5 meters'?

The truth of the first version depends on our understanding that '6' is bigger than 5 ''. The second version also requires us to understand that, but in addition there needs to exist a 6 meter tall person.

If we dropped the person or any other object entirely and stick with '6 meters is more than 5 meters', then that isn't true of the world. The world consists of things. A 6 meter 'no-thing' isn't 6 meters; a 'no-thing' cannot have qualities like dimensions.

So do you agree or not that "a 6 meter tall man is taller then Obama" is a true proposition?

Not a non-existent 6 meter tall man - non existent things are not taller (or shorter) than anything.

Otherwise we get into all sorts of problems; I could argue that although God may or may not exist, it is an undeniable fact that God owns a bigger house than any human who owns a smaller house.

Londoner wrote:I have already answered this; look at the first paragraph in post 99 for example.

If analytic statements are about concepts as you seem to suggest, then I don't see how it's different from my claim that they are true by virtue of linguistic convention.

However, it is possible to come up with endless other theories that are also internally consistent and would thus be just as true in an analytic sense as 'gravity', but which do not accurately describe reality experienced empirically. In other words, the validity of 'gravity' in a 'synthetic' sense is not proven just by its validity in an 'analytic' sense.

Fine, maybe physics is underdetermined by observations in some sense, but this is a general problem for science, and I don't think that attacking science is a very good strategy to argue against synthetic a priori. But suppose that relativity is in fact that best possible model to describe and explain the current empirical data, then what?

This formulation is ambiguous. Are you saying; '6 meters is more than 5 meters'? Or are you saying; 'There exists a man, who at 6 meters is taller than 5 meters'?

It's neither, it simply says that if there are two men of this height then one must be taller then the other.

If we dropped the person or any other object entirely and stick with '6 meters is more than 5 meters', then that isn't true of the world. The world consists of things. A 6 meter 'no-thing' isn't 6 meters; a 'no-thing' cannot have qualities like dimensions.

This is simply ridiculous, of course 6 meters is longer then 5 meters, you can't say it's false.

Otherwise we get into all sorts of problems; I could argue that although God may or may not exist, it is an undeniable fact that God owns a bigger house than any human who owns a smaller house.

No, this doesn't follow from what I said. From the truth of P->Q it doesn't follow by itself that either P or Q are true.

Fafner

If analytic statements are about concepts as you seem to suggest, then I don't see how it's different from my claim that they are true by virtue of linguistic convention.

Then perhaps you should explain what you mean by 'linguistic convention', to which you usually add 'arbitrary'? But I will expand what I wrote earlier:

The letters o-n-e are used in English to refer to the first cardinal number. That is a particular linguistic convention. In other conventions, the same idea might be expressed by '1' or 'uno'. So the symbol can vary, although it isn't 'arbitrary' in the sense that anyone can make up their own alternative symbol whenever they get the urge.

However, the concept all those symbols refer to is the same. That 'one is half of two' is going to be true in any language. That isn't a convention; it isn't true just because a group of English speakers have agreed it will be.

Fine, maybe physics is underdetermined by observations in some sense, but this is a general problem for science, and I don't think that attacking science is a very good strategy to argue against synthetic a priori.

What sort of objective do you think physics has, such that its ideas might be 'underdetermined' by observation of actual physical objects? As for my attack on science, I'm not saying anything that actual scientists won't have covered in their 'philosophy of science' module.

No, this doesn't follow from what I said. From the truth of P->Q it doesn't follow by itself that either P or Q are true.

It doesn't 'follow', it is a necessary antecedent. To argue a conclusion (Q) from any premise (If P then Q) then two things have to be assumed true; that premise - but also 'P'. 'Assume P' is the first line of your proof.

'P->Q' on its own isn't 'true'! How can it be true or false unless the terms stand for something? If they don't, then why isn't 'Not P->Q' equally true?

Logic enables us to move from one truth to another, but it neither generates those truths, nor guarantees them. Computer science has created a useful reminder of this with the term 'GIGO'.

Londoner wrote:Then perhaps you should explain what you mean by 'linguistic convention', to which you usually add 'arbitrary'?

The analytic truth "bachelors are unmarried" is true because we happen to use the term 'bachelor' to signify unmarried people, but this is an arbitrary convention about language (we may not had the term 'bachelor' for example), hence it's not an interesting fact about reality that we happened to have a special term for unmarried men. On the other hand, a synthetic proposition like "dolphins are mammals" isn't true by virtue of linguistic convention but by the way the world is, we can't know that just by analyzing the meaning of the words, we have to see how the world is like to know it.

The letters o-n-e are used in English to refer to the first cardinal number. That is a particular linguistic convention. In other conventions, the same idea might be expressed by '1' or 'uno'. So the symbol can vary, although it isn't 'arbitrary' in the sense that anyone can make up their own alternative symbol whenever they get the urge.

If anybody can make up their own word then how it isn't arbitrary?

What sort of objective do you think physics has, such that its ideas might be 'underdetermined' by observation of actual physical objects? As for my attack on science, I'm not saying anything that actual scientists won't have covered in their 'philosophy of science' module.

Actual scientists don't really care about philosophy of science. About the problem of underdetermination there's a nice paper by Larry Laudan called "Demystifying Underdetennination" which is as far as I remember argues that it's really just the problem of induction in disguise, and therefore if there's a solution to the problem of induction, then it's not the case that any scientific theory (e.g. relativity) has an infinite number of alternative interpretations that are consistent with the data, or at least it doesn't threaten the truth of relativity as it may seem.

'P->Q' on its own isn't 'true'! How can it be true or false unless the terms stand for something? If they don't, then why isn't 'Not P->Q' equally true?.

But the fact is that P->Q can be true nevertheless. For example the conditional "if it is raining, then I will bring an umbrella" can nevertheless be true even if it isn't raining (and hence I didn't bring an umbrella). This is just how the truth table of logical implication works, you can check any logic textbook and see.

The only condition under which P->Q is false is when P is true and Q is false, otherwise the conditional is true. See for example http://www.millersville.edu/~bikenaga/m ... ables.html

And by the way, being false doesn't mean "the terms don't stand for something" in the sense that a false sentence still has a meaning.

It was a long time ago, but I think it was a personal communication from one of my professors, probably Ezra Stotland, that Jean Piaget held that there are some forms of mathematics which are not reflected in any psychological processes. That leads me to consider the idea that if there is a priori knowledge, it might restrict the psychological processes we can use.

That might be part of a test. Do we find limitations in our thinking that we would expect would be the results of a priori knowledge.

On the Einstein point, I don't remember the reference, Einstein was interviewed on how he did his creative thinking. He replied that he relaxed and turned the ideas he was thinking about into vague floating meanings in his mind, and then he moved those clouds of meaning around in various ways. That was similar to what successful subjects said they did with Mednick's Remote Associates Test, a test of creative problem solving in which the subjects give a word which relates to all three words given in each test question. They said they turned the words into vague meanings and worked with them.

Sorry about being vague om references. Long ago, realizing that there are limitations in my memory, I trained myself to strongly remember ideas, and sacrificed things like names and dates. If I had worked to remember names and dates, I wouldn't have been able to remember as many ideas.