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[Logic_ill]

I would ask the philosophers who believe that some a priori knowledge is possible to give examples, Perhaps I'm missing something, but I think it's all a posteriori...

I'm not a philosopher as my background is in sciences but how about our knowlege in language? Consider the 'Poverty of stimulus' argument with respect the way children attain lingustic knowledge:

A central goal of modern generative grammar has been to discover invariant properties of human languages that reflect ‘‘the innate schematism of mind that is applied to the data of experience’’ and that ‘‘might reasonably be attributed to the organism itself as its contribution to the task of the acquisition of knowledge’’ (Chomsky, 1971). Candidates for such invariances include the structure dependenceof grammatical rules, and in particular, certain constraints on question formation. Various ‘‘poverty of stimulus’’ (POS) arguments suggest that these invariances reflect an innate human endowment, as opposed to common experience: Various ‘‘poverty of stimulus’’ (POS) arguments suggest that these invariances reflect an innate human endowment, as opposed to common experience...In our view, the way forward begins with the recognition that environmental stimuli radically underdetermine developmental outcomes, and that grammar acquisition is a case in point. Then one can try to describe the gap between experience and linguistic knowledge attained, reduce that gap to basic principles that reflect the least language-specific innate endowment that does justice to the attained knowledge, and thereby help characterize the true role of experience in a manner that illuminates cognition.

Poverty of the Stimulus Revisited
http://isites.harvard.edu/fs/docs/icb.t ... isited.pdf

A central goal of modern generative grammar has been to discover invariant properties of human languages that reflect ‘‘the innate schematism of mind that is applied to the data of experience’’ and that ‘‘might reasonably be attributed to the organism itself as its contribution to the task of the acquisition of knowledge’’ (Chomsky, 1971).

I do not grasp this argument in its entirety, but there may be such a thing as an innate schematism of mind that might be attributed to the organism itself, if such schematism is found. I do not know enough about languages to detect it or point it out, but in such a case it may be attributed to the organism itself. If we want to call that a priori knowledge that´s fine, but I might label it a posterirori because it takes a human being, who is a product of evolution and generations before him-her to have acquired the attribute, just as much as we acquired our ability to walk erect and on two legs. It´s wired into us or part of the organism itself, but it has to be triggered by example. I mean that walking has to be socially learned, just as much as language. A modern human may not learn how to walk, if it has no examples.

Would a blind infant eventually learn how to walk, if no one encouraged or taught him how? I´m really asking becasue I don´t know the answer.

One important question and probably unanswerable is how did these "abilities" spring in the first place? What triggered those first few primates to walk on two legs, and why did the others imitate the behavior? That might be an example of a priori knowledge because these primates were not necessarily wired to walk on two legs.

-- Updated May 2nd, 2014, 9:27 pm to add the following --

But still that is not entirely a priori. Walking on two legs might have happened gradually. Maybe some were able to stand erect longer than others, until their four legs or using all fours was a thing of the past. Generation after generation of primates got better and better at the task. I wonder if they too discriminated on the basis of walking style? A priori knowledge may be possible but there is always a foundation of other experiences to lean on. Perhaps that is how it works. Some individuals within a "apparently uniform" population acquire some knowledge, abilty, or present a distinguishing trait, until it is becomes widespread because it gives them an advantage.

[Belinda]

Chomsky is separating nature from nurture. Nature, and one may like to think of nature as the synthetic a priori, is deep generative grammar(Chomsky). Nurture is, in the instance of acquisition of language, how any child, including a blind from birth child is inducted into its native culture. Socialisation(nurture) and the innate deep grammatical ability(nature) are both of them necessary for the child to learn language. Moreover there is a stage in its maturation at which the child will most readily learn its native language .

[Londoner]

Fafner

I said it a hundred times already, what's known a priori is not whether some object is circular, but some other empirical properties of the object given that it was correctly recognized as a circle and not a triangle or whatever (that you can't know a priori).

OK. Then since that 'a priori' knowledge about a geometric shape is not sufficient in itself to provide knowledge of the object, it is not 'synthetic a priori' knowledge of the object.

The examination of the object is conducted through the senses; the whole object of seeking some 'synthetic a priori' knowledge was to find something that we can know is true of the world that doesn't depend on our senses (as we cannot be sure they are reliable).

Now suppose that you see a very symmetrical circle (like in the picture I linked), can you know that its ratio is going to be very close to pi without calculating? The obvious answer is yes, this is something that every kid learns at school. If knowing accurately without calculation a ratio of a physical object is not synthetic a priori knowledge then I don't know what 'synthetic a priori' is supposed to mean on your view.

But the kid doesn't need a drawn 'very symmetrical circle' to know this. If they did, if the information was taken from the drawn object and not the idea, then their knowledge would be empirical. (And different; they could only know pi as an indication, not an exact figure).

You will not take the point that ideas in geometry are fundamentally different to actual objects; their definitions are incompatible and the difference is further clearly signaled by the fact that what is always true of geometric shapes is never true of real ones. You only know 'accurately' ratios in a circular object if you pretend it is something that never exists in fact; a perfect geometric shape.

This is taking too far the process involved in all description. Every object is unique, but I cannot communicate uniqueness. I can only communicate similarities; 'Fido is a dog' tells others that part of Fido's character is one shared by all dogs. Yet 'Fido' is not 'dog'. 'Dog' is not the name of an individual animal. Dog as a category is not a thing, it doesn't exist. The same is more obviously true of adjectives; 'snow is white' but 'white' does not mean the same as 'snow'. Similarly, if we say 'a plate is circular', we don't really mean the plate is a circle, it remains a plate.

We have no other way of understanding or communicating except this; if we didn't have this ability we could not begin to make sense of our thoughts or sense impressions. So if you say that our thinking this way, abstracting ideas like colour and shape from specific examples, constitutes 'synthetic a priori' knowledge, then you are describing not the world but our cognitive processes.

To put it another way, isn't the interesting point that that kid in school can look at a round object and yet think about it as if it were a two dimensional abstract shape that they have never, and can never, encounter empirically?

The argument of Quine that I've presented earlier was aimed specifically against the doctrine of A.J Ayer, and it was one of the arguments that persuaded philosophers that logical positivism is false. So it is recommended to read some criticisms of Ayer's views because almost no philosopher accepts them today. One good book will be "Philosophical Analysis in the Twentieth Century" by Scott Soames (the chapters about Ayer and Quine).

I mentioned Ayer because I think it explains the terms used here (and by Kant etc.) nice and clearly. I agree that Quine responds to Ayer, but his argument is not yours; I don't think either would agree with you (although for slightly different reasons!) I would not say Quine disagrees with Ayer; surely he is even more categorical than Ayer. Ayer, the empiricist, thinks propositions are either empirical or analytic (or meaningless). Quine doubts that there are any analytic propositions at all! He thinks all propositions are subject to empirical revision (so are not true 'a priori'). I don't see how you can work that round to supporting your claim to have a priori knowledge of objects known empirically.

[Fafner88]

OK. Then since that 'a priori' knowledge about a geometric shape is not sufficient in itself to provide knowledge of the object, it is not 'synthetic a priori' knowledge of the object.

That's not very accurate. Consider the claim that we can know a priori that all triangles satisfy Pythagoras's theorem, and therefore one can know a priori that a particular triangle satisfies Pythagoras's theorem via an argument like this:

(1) If x is a triangle* then Pythagoras's theorem is true about x. (2) P is a triangle. (3) Therefore Pythagoras's theorem is true about P.

(*of course it's supposed to be true only of right angle triangles)

Now, what is known a priori is premise (1), whereas premise (2) of course is not a priori. But premise (1) doesn't follow from premise (2) by itself, so one can't know the conclusion just from the empirical knowledge that P is a triangle, one has to supplement it with an additional premise, and I claim that this premise can be known a priori because (1) can be arrived at by reasoning alone, without the examination of actual triangles. So the point is this: even though the argument requires a premise which is not a priori, this doesn't mean that premise (1) isn't known a priori. Is the conclusion known a priori? In a sense it is: I gain new empirical knowledge about the triangle by using an a priori premise. This is something you can't do for example in knowing that Felix the cat has a tail because you can't ever be warranted a priori to believe that all cats have tails.

And then philosophers do call argument that contain at least one a priori premise 'a priori arguments', but its doesn't matter how you call it, the crucial claim is that one can know empirical facts (partly) on the basis of non-empirical premises, and I don't see how you can explain this fact without assuming synthetic a priori knowledge.

If they did, if the information was taken from the drawn object and not the idea, then their knowledge would be empirical. (And different; they could only know pi as an indication, not an exact figure).

I hope that I made it clear already that one can't know all kinds of complex geometrical properties of objects solely on the basis of knowing that something is a shape of a certain kind, without supplementing it with further premises that I claim can be known a priori. To go back to the previous example, one can't know empirically that a certain triangle satisfies Pythagoras's theorem without doing some elaborate measuring and calculations, you can't know this if all that you know that it's a triangle, unless you are already warranted independently of experience to believe something like premise (1) (and by the way, no amount of observational evidence can suffice to know the truth of premise (1) ).

You only know 'accurately' ratios in a circular object if you pretend it is something that never exists in fact; a perfect geometric shape.

This is an odd thing to say, you don't have to know anything about 'perfect geometric shapes' to know the ratios of the object that you've drawn on the page. Even if you don't agree that actual circles approximate abstract circles, it's still an empirical fact that physical objects have objective spatial properties that can be measured and calculated. Is it a fact that if you measure the circumference of the London Eye its going to be around 3.14 to its diameter, and not 1 or 367? One doesn't have to take a position on the reality of "abstract circles" to acknowledge the fact that objects in reality exhibit objective spatial properties, or do you also deny that physical objects have size and shape?

Quine doubts that there are any analytic propositions at all! He thinks all propositions are subject to empirical revision (so are not true 'a priori'). I don't see how you can work that round to supporting your claim to have a priori knowledge of objects known empirically.

Well, I'm not subscribing to Quine's position. You don't have to agree with Quine on the analytic/synthetic distinction to accept that particular argument, because it's a self standing argument.

[Daviddunn]

(1) If x is a triangle then Pythagoras's theorem is true about x. (2) P is a triangle. (3) Therefore Pythagoras's theorem is true about P.

Premise 1 of this argument is false. Not every triangle satisfies the pythagoras theorem, only right angle triangles satisfy the pythagoras theorem.

[Fafner88]

(1) If x is a triangle then Pythagoras's theorem is true about x. (2) P is a triangle. (3) Therefore Pythagoras's theorem is true about P.

Premise 1 of this argument is false. Not every triangle satisfies the pythagoras theorem, only right angle triangles satisfy the pythagoras theorem.

Oh, yes you are right, I forgot about that. I did very badly at school on this subject...

[Logic_ill]

knowledge doesn´t exist for me...

[Bohm2]

I do not grasp this argument in its entirety, but there may be such a thing as an innate schematism of mind that might be attributed to the organism itself, if such schematism is found.

I don't see how it can be any other way. Here's their basic argument using a simple example: a human embryo. It grows to develop hands and not wings in comparison to a bird embryo. One assumes that's due to heredity (nativist position). Environmental factors are considered to play a minor role. I don't think many would disagree with this. Okay, given that the vast majority agree with this and we don't know the details how that occurs, why do some still believe that such an internal genetic programming that does apply with respect to physical organs does not apply to mental organs like the human language case? As Chomsky writes:

In fact, if someone came along and said that a bird embryo is somehow "trained" to grow wings, people would just laugh, even though embryologists lack anything like a detailed understanding of how genes regulate embryological development...The gene-control problem is conceptually similar to the problem of accounting for language growth. In fact, language development really ought to be called language growth because the language organ grows like any other body organ.

These authors further write:

Poverty of stimulus problems are ubiquitous. Every aspect of growth and development poses huge poverty of stimulus problems. Now the term isn't used in biology and the reason is it's taken to be so obvious that there is no need for a term, so it's obvious that there is a poverty of stimulus problem when humans develop arms instead of wings or a mammalian visual system but not an insect visual system. There is a stimulus. There's external data like nutrition but there's – no one even bothers to argue about it - there is no way for nutrition to determine that you have a mammalian visual system so that's got to be accounted for by something internal, some genetic property. And then you go on to try to find out what it is and ask why it's that way and not some other way. In the case of language, there is a term, poverty of stimulus, and it's considered highly controversial, but just about everything about language is considered highly controversial, even if it is perfectly obvious, a total truism.

‘Poverty of the Stimulus’ Revisited: Recent Challenges Reconsidered
http://csjarchive.cogsci.rpi.edu/procee ... s/p383.pdf

Poverty of Stimulus: Unfinished Business
http://www.stiftung-jgsp.uni-mainz.de/B ... edited.pdf

The implication here (a sound one, in my opinion), is that our biologically-determined properties of the mind/brain play a crucial role in determining what and how we perceive the “external” world, since the perceptual knowledge we attain vastly transcends any environmental input. Thus, like physical growth and development (i.e. humans are designed to grow arms and legs, not wings-to use one of above author's well-known examples), human development (including our systems of belief and knowledge) largely reflects our particular, biological endowment (i.e. a consequence of the organizing activity of the mind) and not the properties of our physical environment. Thus, environmental input may act only as a trigger to set off a rich and highly articulated system of beliefs that, to a large extent, is intrinsically determined, following a predetermined course (in the same way that oxygen and nutrition are required for cellular growth to take place). Thus, our various systems of knowledge and belief do not resemble the “real” properties of the world, in any sense of the word, any more than our physical organs reflect our environment. It then follows that,

Our knowledge...even in science and mathematics is not derived by induction, by applying reliable procedures, and so on; it is not grounded or based on ‘good reasons’ in any sense of these notions. Rather, it grows in the mind, on the basis of our biological nature, triggered by appropriate experience, and in a limited way shaped by experience that settles options left open by the innate structure of mind.

So that,

If I had been differently constituted, with a different structure of mind-brain...I would come to know and follow different rules (or none) on the basis of the same experience, or I might have constructed different experience from the same physical events in my environment.

[Daviddunn]

Oh, yes you are right, I forgot about that. I did very badly at school on this subject...

I am here to learn too. I am glad something positive came out of this. . Some more may be?!

Assuming flat euclidean geometry throughout.
Proposition A: The ratio of the circumference to the diameter of any circle is a constant. The previous proposition can be said to be an apriori judgement. Its PROOF shows that it is necessarily true, based on our laws of reasoning alone. That this constant is pi is contingently true. Pi is a calculated value, experience is required to determine its value. Hence that the ratio of circumference to diameter of a circle is pi, is not apriorily true.

Proposition B: All right angle triangles satisfy the pythagoras theorem. The previous proposition is PROVED to be true for all right angle triangles by the laws of our reasoning. Hence it can be said to be apriorily true. That a particular triangle is a right angle triangle requires experience (measurement of its angles). That a triangle satisfy the pythagoras theorem cannot be known a priori, as one has to determine by experience whether it is a right angle triangle.  


One has to know how proposition A and B are proved by reasoning alone.

Proof of proposition A: http://m.youtube.com/watch?v=0iU60fFbqNY(an 'intuitive' proof, that is, it is not rigorous, it will do very well here) I hope this link works, I am writing from a mobile device. If it does not work, remove the 'm.' before 'youtube' in the url.
Proof of proposition B:http://www.cut-the-knot.org/pythagoras/index.shtml(several proofs here, but any single one will do. I like Euclide's proof, I find it neat!)

Interesting it is to note what Kant has to say.
In the Norman Kemp Smith translation of CPR we can read:

P 043
The expression 'a priori' does not, however, indicate with sufficient precision the full meaning of our question. For it has been customary to say, even of much knowledge that is derived from empirical sources, that we have it or are capable of having it a priori, meaning thereby that we do not derive it immediately from experience, but from a universal rule -- a rule which is itself, however, borrowed by us from experience. Thus we would say of a man who undermined the foundations of his house, that he might have known a priori that it would fall, that is, that he need not have waited for the experience of its actual falling. But still he could not know this completely a priori. For he had first to learn through experience that bodies are heavy, and therefore fall when their supports are withdrawn.
In what follows, therefore, we shall understand by a priori knowledge, not knowledge independent of this or that experi- ence, but knowledge absolutely independent of all experience. Opposed to it is empirical knowledge, which is knowledge possible only a posteriori, that is, through experience. A - priori modes of knowledge are entitled pure when there is no admixture of anything empirical. Thus, for instance, the proposition, 'every alteration has its cause', while an a priori proposition, is not a pure proposition, because alteration is a concept which can be derived only from experience.

From this we can see that neither proposition A nor B is a pure a priori proposition, although they are apriori propositions( one make use of the concept of circle, and the other triangle.) This distinction, however, does not concern us here. Suffice that they are apriori propositions. They are not arrived at by induction, from that which holds for most cases to that which holds in all cases. The proofs shows that it is necessarily true for all circles and triangles.

Now all through, we have been assuming flat euclidean geometry. Whether our intuition of objects, be it a circle or a triangle, has an underlying euclidean structure cannot be proved. One can go on to calculate ratios of circumference to diameter or measure the hypotenuses, adjacents and opposites, to confirms these geometric laws to investigate the euclidean nature of our spatial intuition, but no matter how many times one tries, the conclusion will be an apostiori judgement. We are not in possesion of a deduction concerning this matter. It will be an induction from that which holds in most cases to that which holds in all cases. The latter method is indispensable for us, but it is not a priori. We have to trust the reasonable, no way out of this!

[Spectrum]

Interesting it is to note what Kant has to say.
In the Norman Kemp Smith translation of CPR we can read:

I prefer to explore the a priori concept of Kant which is a specific keystone to his architectonic/systematic arches of knowledge.

Nevertheless, I mentioned earlier the question of a priori is a matter of consensus, i.e.
Post #26
Thus it is not a question of whether a priori knowledge is possible, it is a question of whether one or a group agreeing to the above definitions and categories prior to any philosophical discussion. If one agree, then go ahead with the discussion, if not, then avoid discussing the concept of the 'a priori' altogether.

Prior to Kant, a priori refers to modes of logical demonstration, i.e.
'When the mind reasons from Causes to Effects, the demonstration is called a priori;
when from Effects to Causes, the demonstration is called a posteriori'

[Londoner]

Fafner

You describe '(1) If x is a (right angled) triangle then Pythagoras's theorem is true about x' as a premise. Yet you also say '(1) can be arrived at by reasoning alone, without the examination of actual triangles'. So why did you include the term 'x'? Why not instead write 'Pythagoras's theorem describes right angles triangles'?

The conclusion is then shown to be the same as the premise: Your argument goes: 'If Pythagoras's theorem describes right angles triangles, then a right angled triangle will be described by Pythagoras's theorum'.

The reason for the circularity is that the first premise was true 'a priori' because it was analytic; the theorem describes something already inherent in the notion of right angled triangles. An analytic premise can only lead to an analytic conclusion.

Quite independent from the rest is:

(2) P is a triangle.

Accepting that clause that begs the whole argument. That P exits and, if it exists, that it is a triangle, is information that can only gained through our senses. (Leaving aside the impossibility of a geometrical form existing as an object). The search for a synthetic a priori is for the search for such a premise about the real world which isn't an assumption, which isn't something that may be true or false, but a certainty. If we could know - without doubt - that 'P is a triangle' then that would be the 'synthetic a priori' on its own!

This is something you can't do for example in knowing that Felix the cat has a tail because you can't ever be warranted a priori to believe that all cats have tails.

That belief would also be analytic, it would depend on whether the meaning of 'cat' included specifics about tail possession. If the description 'cat' requires having a tail, then I can know all cats must have tails - just as you can know the attributes of the term 'right angled triangles'. But again, information about the meaning of a word isn't evidence about the existence or attributes of Felix, or any other object, which must be obtained empirically.

One doesn't have to take a position on the reality of "abstract circles" to acknowledge the fact that objects in reality exhibit objective spatial properties, or do you also deny that physical objects have size and shape?

We have addressed this before. The problem is with that 'have'. I think we are the ones who have general ideas, like those of shape, colour etc. and we apply them to objects - and not just objects. I claim to see an enormous green dragon holding a right angled triangle. That I think something has size and colour and involves a shape about which there are theorems doesn't prove it is real. So how can we tell which objects are 'real' and 'really' have properties like size and shape? That is the philosophical problem we are working on.

[Fafner88]

You describe '(1) If x is a (right angled) triangle then Pythagoras's theorem is true about x' as a premise. Yet you also say '(1) can be arrived at by reasoning alone, without the examination of actual triangles'. So why did you include the term 'x'? Why not instead write 'Pythagoras's theorem describes right angles triangles'?

Premise (1) is a universal generalization, that's how you formulate it in predicate logic by using variables. I just wanted to make the structure of the argument clear.

The conclusion is then shown to be the same as the premise: Your argument goes: 'If Pythagoras's theorem describes right angles triangles, then a right angled triangle will be described by Pythagoras's theorum'.

No, that's not my argument, premise (1) doesn't mention triangle P (which is a particular triangle) so the argument is not circular. Don't you know how modus ponens arguments work? (circularity is when you write the conclusion as a premise and I didn't do that. of course the conclusion follows logically from the premises, but being a valid argument is not the same as being a circular argument).

The reason for the circularity is that the first premise was true 'a priori' because it was analytic; the theorem describes something already inherent in the notion of right angled triangles. An analytic premise can only lead to an analytic conclusion.

Now this is what bagging the question looks like. You can just assume that geometry is analytic and I can assume that it's synthetic and we can go home and do something else, so what's the point? Do you have any arguments to show that the theorem is inherent in the notion of right angled triangles?

(2) P is a triangle.

Accepting that clause that begs the whole argument. That P exits and, if it exists, that it is a triangle, is information that can only gained through our senses. (Leaving aside the impossibility of a geometrical form existing as an object). The search for a synthetic a priori is for the search for such a premise about the real world which isn't an assumption, which isn't something that may be true or false, but a certainty.

Why assuming that one can know that something is a triangle is begging the question? The topic is not scepticism about knowledge in general, the topic is whether someone can gain knowledge about the world by virtue of a priori reasoning, and premise (2) is not supposed to be a priori under any account, nor accepting it by itself proves a priori knowledge, so I don't see what's the problem. You fail to distinguish between

(a) knowing that P is a triangle, and (b) knowing that P has some property x.

It's true that you must know (a) before you know (b), but the question is can you deduce (b) from (a) by a priori reasoning or not, so granting that one can know (a) doesn't beg the question in any way, because even if one doesn't agree that a priori knowledge exists or that all a priori knowledge is analytic, he can nevertheless accept the truth of (a), because everybody accepts it except the radical sceptic, and I'm not interesting in persuading the radical sceptic. My goal is to persuade someone who agrees that at least that empirical knowledge is possible.

If we could know - without doubt - that 'P is a triangle' then that would be the 'synthetic a priori' on its own!

How does it suppose to follow? If I know that something is a cow, it doesn't make my knowledge a priori.

That belief would also be analytic, it would depend on whether the meaning of 'cat' included specifics about tail possession. If the description 'cat' requires having a tail, then I can know all cats must have tails - just as you can know the attributes of the term 'right angled triangles'. But again, information about the meaning of a word isn't evidence about the existence or attributes of Felix, or any other object, which must be obtained empirically.

How is it even relevant? (and it's not analytic that cats have tails, something can still be a cat even without having a tail)

We have addressed this before. The problem is with that 'have'. I think we are the ones who have general ideas, like those of shape, colour etc. and we apply them to objects - and not just objects. I claim to see an enormous green dragon holding a right angled triangle. That I think something has size and colour and involves a shape about which there are theorems doesn't prove it is real. So how can we tell which objects are 'real' and 'really' have properties like size and shape? That is the philosophical problem we are working on.

As I said, the topic is not scepticism, let's just assume for the sake of the argument that we can know whether some object has a triangular shape, because this assumption doesn't say anything about a priori knowledge.

[Daviddunn]

No, that's not my argument, premise (1) doesn't mention triangle P (which is a particular triangle) so the argument is not circular. Don't you know how modus ponens arguments work? (circularity is when you write the conclusion as a premise and I didn't do that. of course the conclusion follows logically from the premises, but being a valid argument is not the same as being a circular argument).

Your argument is a valid modus ponens, that is not being denied. But then consider the following modus ponens. Let be domain of discourse be man.

1. If x is unmarried then x is a bachelor. 2. Liam Neeson is unmarried. 3. Therefore, Liam Neeson is a bachelor!

The conditional of premise 1, does not make it a synthetic proposition. It is still the same as the categorical proposition, 'All bachelors are unmarried'.

Now this is what bagging the question looks like. You can just assume that geometry is analytic and I can assume that it's synthetic and we can go home and do something else, because what's the point of just making assertions? Do you have any arguments to show that the theorem is inherent in the notion of right angled triangles?

Of course! Pythagoras theorem is theorem specifically about right angled triangles. Right angled triangles is already contained in the notion of pythagoras theorem. So it is really an analytic proposition.

Note: I apologize to you for inviting myself in your interesting exchange.

[Fafner88]

Your argument is a valid modus ponens, that is not being denied. But then consider the following modus ponens. Let be domain of discourse be man.

1. If x is unmarried then x is a bachelor. 2. Liam Neeson is unmarried. 3. Therefore, Liam Neeson is a bachelor!

The conditional of premise 1, does not make it a synthetic proposition. It is still the same as the categorical proposition, 'All bachelors are unmarried'.

Yes you are right, it doesn't prove that the conclusion is synthetic, nor does my original argument. I presented the argument to show that geometry can give us empirical knowledge about actual objects because Londoner disputes even this.

But the difference between the two is this: the proposition "all bachelors are unmarried men" doesn't say anything interesting about reality (it's just a linguistic convention), but there's a big leap from knowing that something is a right angle triangle to knowing that it satisfies the theorem. Someone can know that something is a right angle triangle without knowing the theorem while understanding perfectly well the meaning of 'right angle triangle'. We need mathematicians to know the theorem, so understanding language can't be enough (therefore the Pythagoras theorem is considered a discovery while "all bachelors are unmarried men" is not).

Of course! Pythagoras theorem is theorem specifically about right angled triangles. Right angled triangles is already contained in the notion of pythagoras theorem. So it is really an analytic proposition.

It doesn't quite follow, because the reverse also must be true, but is Pythagoras' theorem contained in the notion of a right angle triangle? I think not because as I said, someone can know what a right angle triangle is without knowing anything about the theorem, while one can't know what 'bachelor' means without also knowing that all bachelors are unmarried man (because then he simply doesn't understand the word). Perhaps it doesn't show that Pythagoras' theorem isn't analytic, but frankly it's quite difficult do see how it can be analytical while competent users of language can still fail to see the truth of the theorem. If indeed Phytagoras' theorem is true by virtue of the meaning of the words, how can someone understand the words while not knowing the theorem?

Note: I apologize to you for inviting myself in your interesting exchange.

It's OK.

[Londoner]

Fafner

Don't you know how modus ponens arguments work? (circularity is when you write the conclusion as a premise and I didn't do that. of course the conclusion follows logically from the premises, but being a valid argument is not the same as being a circular argument).

It is a valid argument, but however valid, for the conclusion of an argument to be true the premises must be true. One of your premises is an assumption; if the conclusion was meant to be a truth 'synthetic a priori' then it can't depend on an assumption.

Now this is what bagging the question looks like. You can just assume that geometry is analytic and I can assume that it's synthetic and we can go home and do something else, so what's the point? Do you have any arguments to show that the theorem is inherent in the notion of right angled triangles?

So do you argue that geometry is synthetic? In that case I would again point out that its claims aren't true, since no geometric shapes can exist as objects. And even if it were true, then it would only be a truth gained through experience and inductive reasoning, like any scientific proposition, and it would fail to qualify as being true 'a priori' for the same reasons as anything derived that way.

By contrast, my reasons for thinking that theorems in geometry are analytic is that they can be proved by reason alone - a theorem is that which has been proved to be true using facts that were already known. We can use algebra to prove Pythagoras in which there is no mention of specific measurements.

(The proof of a complicated theorem, like the answer to a complicated sum, ultimately rests on a combination of very simple mathematical postulates, like 'if A equals B, B equals A'. The eventual result may surprise those of us who are not good at the spatial manipulation questions in IQ tests, we therefore feel a new truth has been discovered, but the reality is that it is only a construction made from those very simple cognitive building blocks.)

Why assuming that one can know that something is a triangle is begging the question? The topic is not scepticism about knowledge in general, the topic is whether someone can gain knowledge about the world by virtue of a priori reasoning, and premise (2) is not supposed to be a priori under any account, nor accepting it by itself proves a priori knowledge, so I don't see what's the problem.

The problem is skepticism about knowledge in general. The search for a synthetic a priori is the search for an answer to the question: How can we know anything for sure about any thing? If you assume that X both is a thing and it is this sort of a thing you have assumed what you need to prove.

(a) knowing that P is a triangle, and (b) knowing that P has some property x.

This is the same argument as before, except that you have run two things together in the first part. There is 'if P' and separately 'if P is a triangle'. Yes; if P is a triangle then it will have the properties of a triangle. But that doesn't deal with the 'if' in 'if P exists'.

How is it even relevant? (and it's not analytic that cats have tails, something can still be a cat even without having a tail)

You have just made that declaration about cats based on your understanding of the word 'cat'; you declare that cats need not have tails. Suppose everyone else thought that cats must have tails; that the meaning of 'cat' included 'has a tail'? How would you go about proving they were wrong and you were right? It would be no good being empirical and going out and looking at tail-less Felix. You would say; 'It's a cat!' and everyone else would say;'No it isn't!'

You would just have to accept that when anyone else refers to a 'cat', they refer to something with a tail. The proposition 'Cats have tails' would be analytic; the predicate is contained in the subject.

For the relevance, you will have to chase this strand of the argument back to your earlier post.