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Analytic-synthetic vs priori/ posteriori

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Analytic-synthetic vs priori/ posteriori

jtoma
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Posted 07/07/09 - 10:13 AM:	#21
Are you saying hello was wrong about more than the last sentence here? hello hello wrote: I recently began studying Kant's Critique of Pure Reason. From my understanding Kant seems to distinguish between four types of propositions: analytic a priori, synthetic a priori, analytic a posteriori, and synthetic a posteriori. Kant discards analytic a posteriori as a self contradiction and does not evaluate it further. I however find synthetic a priori contradictory as well. Synthetic a priori is not contradictory, but from where Kant stands, it sometimes seems so. Synthetic knowledge defined as the complement of analytic knowledge (everything that is not analtytic), does not yield much interesting knowledge to deal with. But though, as I would have it, synthetic knowledge defined as new applications of rules can yield interesting knowledge, only at the cost of also being analytic. Such knowledge results from an application of a rule conceptually definable in a language capable of expressing a prediction thus having the potential to be known a priori via inference from selected assumptions. I find Hello hello's last line to be right when assuming Kant's definitions. Having only the power to select assumptions and rules of inference lets all knowledge (baring empirical particulars) be known by rule, without experience--other than its communication. Edited by jtoma on 07/09/09 - 10:27 AM

keda
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Posted 07/07/09 - 04:09 PM:	#22
Death Monkey wrote: I don't see how "7 + 5 = 12" could be considered to be synthetic. We have the following terms in the sentence: '7', '5', '12', '+', and '='. In order for this sentence to constitute a statement to which a truth value can be meaningfully assigned, each of those terms must be defined. Now, in principle we could assign any definitions we wanted, but in the context of this discussion it is pretty clear that what is intended is the definitions given in ordinary artithmetic. That said, the truth value of the statement directly follows from those definitions. It is not possible for the statement to be false without negating one or more of the definitions that establish the meaning of the statement in the first place. In a similar manner one could argue that all statements are analytic like Leibnitz did, but this is obviously not how language is normally used. Perhaps the dispute here is one of definitions? Maybe the proponents of the statement being synthetic are actually referring to statements about real-world examples of these abstract mathematical operations? For example, "I have 5 apples in a bag. I put 7 more apples in the bag. I then have 12 apples in the bag". This would clearly be synthetic, but it would also be a posteriori. I cannot know independantly of sense experience that my rules of artithmetic will apply to this, or to any other real-world situation. This is not either how mathetimatical language is used. Suppose you actually find 13 apples are found in the bag. The mathematician does not then conclude that 5+7=13 but that someone must have put in an extra apple without him noticing or that he had miscounted the number of apples at some point. Instead mathematics abstracts away the empirical in refering to act of counting independent of what is being counted, thereby being a priori science.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

Death Monkey
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1 of 1 people found this post helpful Posted 07/08/09 - 12:22 AM:	#23
keda, In a similar manner one could argue that all statements are analytic like Leibnitz did, but this is obviously not how language is normally used. I am unfamiliar with the argument of Leibnitz that you are referring to, but I don't see how. For example, consider the statement "My car is gray". The meaning of this statement is established by the definitions of the terms "my", "car", "is", and "gray", along with the rules of English grammar. That meaning alone is not sufficient to specify the truth value of the statement, though. It turns out that this is a true statement, but it is clearly not an analytic truth, because asserting that the statement is false does not contradict any of the definitions that establish the meaning of the statement. This is in clear contrast to the mathematical example, where one cannot assert that "5+7=12" is false without contradicting the definitions that establish what that statement means. This is not either how mathetimatical language is used. Suppose you actually find 13 apples are found in the bag. The mathematician does not then conclude that 5+7=13 but that someone must have put in an extra apple without him noticing or that he had miscounted the number of apples at some point. Instead mathematics abstracts away the empirical in refering to act of counting independent of what is being counted, thereby being a priori science. I agree. But that same abstraction makes it analytic. Here's another way to consider the issue: An analytic statement is one whose truth-value is fixed by the meaning of the statement. That is, the truth-value logically follow from the meaning of the statement. An analytic truth cannot be false because that would lead to a logical contradiction. A synthetic statement is a statement which has a truth-value (we will put aside for now statements which have no truth-value), but whose truth-value is not fixed by the meaning of the statement. As mentioned previously in this thread, though, the a priori / a posteriori distinction has to do with judgments. It is a question not of whether the truth-value is fixed by the meaning of the statement, but instead on whether or not we can decide what the truth-value is without making reference to sensory experience. Now, I think we can all agree that analytic statements must be a priori, because if the truth-value is fixed by the meaning of the statement, then that means that all of the information needed to determine what that truth-value is, can be derived from the meaning of the statement. Therefore no reference to sensory experience is needed. For synthetic a posteriori statements, the meaning of the statement alone cannot provide us with the information we need to establish whether the statement is true or false, but sensory experience provides us with the additional information we need. I think this case is also fairly clear. Where we run into problems is the case where the meaning of the statement alone is not sufficient to fix its truth value (so it is synthetic), but where we can nevertheless determine what the truth-value is without making use of information from sensory experience. If such cases do exist, they would qualify as being synthetic a priori. However, it should be clear that for such a case to exist, there would have to be some third source of information that we make use of to make the judgement. The meaning of the statement isn't enough, and we don't use information from sensory experience, so where do we get the additional information from? This question applies regardless of whether we are talking about mathematics or metaphysics. If there is some statement that one wishes to claim is synthetic a priori, then that person must show the following: 1) The statement does have a truth-value. 2) The truth-value cannot be derived from the meaning of the statement. 3) The truth-value is knowable. 4) Knowledge of the truth-value can be obtained without making use of information from sensory experience. Implicit in showing the above would be to provide some third source of information, and to demonstrate that this source of information is reliable. To my knowledge, nobody has ever done this. DM
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180 Proof
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Posted 07/08/09 - 02:30 AM:	#24
Death Monkey wrote: Here's another way to consider the issue: An analytic statement is one whose truth-value is fixed by the meaning of the statement. That is, the truth-value logically follow from the meaning of the statement. An analytic truth cannot be false because that would lead to a logical contradiction. A synthetic statement is a statement which has a truth-value (we will put aside for now statements which have no truth-value), but whose truth-value is not fixed by the meaning of the statement. As mentioned previously in this thread, though, *the a priori / a posteriori distinction has to do with judgments.* It is a question not of whether the truth-value is fixed by the meaning of the statement, but instead on whether or not we can decide what the truth-value is without making reference to sensory experience. Now, I think we can all agree that *analytic statements must* be a priori, because if the truth-value is fixed by the meaning of the statement, then that means that all of the information needed to determine what that truth-value is, can be derived from the meaning of the statement. Therefore no reference to sensory experience is needed. For synthetic a posteriori statements, the meaning of the statement alone cannot provide us with the information we need to establish whether the statement is true or false, but sensory experience provides us with the additional information we need. I think this case is also fairly clear. Where we run into problems is the case where the meaning of the statement alone is not sufficient to fix its truth value (so it is synthetic), but where we can nevertheless determine what the truth-value is without making use of information from sensory experience. If such cases do exist, they would qualify as being synthetic a priori. However, it should be clear that for such a case to exist, there would have to be some third source of information that we make use of to make the judgement. The meaning of the statement isn't enough, and we don't use information from sensory experience, so where do we get the additional information from?** This question applies regardless of whether we are talking about mathematics or metaphysics. If there is some statement that one wishes to claim is synthetic a priori, then that person must show the following: 1) The statement does have a truth-value. 2) The truth-value cannot be derived from the meaning of the statement. 3) The truth-value is knowable. 4) Knowledge of the truth-value can be obtained without making use of information from sensory experience. Implicit in showing the above would be to provide some third source of information, and to demonstrate that this source of information is reliable. To my knowledge, nobody has ever done this. I suspect Kantians will point out that this third source (no. 4) is provided by "transcendental deduction" of the conditions (i.e. "categories of reason") that allegedly make "synthetic a priori judgments" possible. However, non-Euclidean geometries, post-Newtonian physics, and other logico-empirical developments have invalidated this justificatory move. Edited by 180 Proof on 07/08/09 - 02:38 AM. Reason: ...
The question isn't "Which explanations do I believe?" but rather "Which explanations do I least disbelieve?" Absence of evidence THAT MUST BE THERE (i.e. implied by any claim, concept, or (its) predicates, that affects changes in/to the world) entails evidence of absence. [What cannot be done?[What cannot be hoped?[What cannot be known?]]]

keda
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Posted 07/08/09 - 04:59 AM:	#25
Death Monkey wrote: I am unfamiliar with the argument of Leibnitz that you are referring to, but I don't see how. According to Leibnitz the predicate is always contained in the subject so that "my car" contains "is grey" making "my car is grey" analytic. If you paint your car green, Leibnitz would say the car has two consistent predicates, namely that it is first grey and then later on green. This is in clear contrast to the mathematical example, where one cannot assert that "5+7=12" is false without contradicting the definitions that establish what that statement means. Which definitions and how? I agree. But that same abstraction makes it analytic. How so? However, it should be clear that for such a case to exist, there would have to be some third source of information that we make use of to make the judgement. The meaning of the statement isn't enough, and we don't use information from sensory experience, so where do we get the additional information from? This is correct and according to Kant, this third source is the form of sensibility. 180 Proof wrote: However, non-Euclidean geometries, post-Newtonian physics, and other logico-empirical developments have invalidated this justificatory move. This is quite a common objection leveled against Kant however Kant never denied the existence of non euclidean geometries, but on the contrary his theory anticipated non euclidean geometries and contrary to the charge does their discovery only confirm the syntheticity of Kant's claim that euclidean geometry is synthetic, for this entails that non euclidean geometries can be concieved. The denial of the parallell postulate does not lead to a contradiction. Regarding Einstein's theory if relativity, if this is what you refer to by post-Newtonian physics, it is merely a semantic disagreement of what is to be called spatial and temporal measurement. Einstein's counterintuitive notions of space and time and consequently the use of non euclidean geometry is a consequence from the adoption of the notion of mutable measurements, which is evident in that Lorentz Ether Theory makes the same predictions as Special Relativity, yet does not adopt mutable measurements, resulting in an euclidean geometry. Space is still euclidean in the intuitive sense it is normally used despite what those who may be under the semantic trance that Einstein's non standard use of language have induced, confusedly suggests.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

Death Monkey
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Posted 07/08/09 - 05:37 AM:	#26
keda, According to Leibnitz the predicate is always contained in the subject so that "my car" contains "is grey" making "my car is grey" analytic. If you paint your car green, Leibnitz would say the car has two consistent predicates, namely that it is first grey and then later on green. This seems to me to be using too liberal an interpretation of "contains". Of course, just saying that the subject is "contained" in the predicate is too vague to be useful as a formal definition in the first place. So all this really demonstrates is that interpreting "contains" here in this way just renders the term "analytic" trivial and useless. Anyway, this is why I made the specific point about the truth-value of the statement being fixed by the meaning of the statement. Certainly this is not the case for all statements, but is the case for mathematical statements. This is in clear contrast to the mathematical example, where one cannot assert that "5+7=12" is false without contradicting the definitions that establish what that statement means. Which definitions and how? As I said before, the definitions of each of the symbols used, which in this case is just the definitions and rules of ordinary arithmetic. This is not either how mathetimatical language is used. Suppose you actually find 13 apples are found in the bag. The mathematician does not then conclude that 5+7=13 but that someone must have put in an extra apple without him noticing or that he had miscounted the number of apples at some point. Instead mathematics abstracts away the empirical in refering to act of counting independent of what is being counted, thereby being a priori science I agree. But that same abstraction makes it analytic. How so? Because this abstracted mathematics defines the rules of the operations that are serving as abstractions of the empirical act of counting real objects. Those definitions are what establishes the meaning of the mathematical statement, and those same rules fix the truth-value of the statement. Put simply, the truth-value of the statement "5+7=12" cannot be false without redefining the meaning of the symbols used in that sentence to be different from what they are ordinarily defined to mean. And if we do redefine them in such a way that the statement would be false, then the meaning of the statement would be different. It would, in fact, be an entirely different statement. However, it should be clear that for such a case to exist, there would have to be some third source of information that we make use of to make the judgement. The meaning of the statement isn't enough, and we don't use information from sensory experience, so where do we get the additional information from? This is correct and according to Kant, this third source is the form of sensibility. I figured as much. The problem is that one cannot support the claim that this source is reliable. Nor can they reasonably claim that it is, itself, not making use of sensory information. keep in mind that we are talking about how we know the truth-value of a statement. One can assert all he likes that his "sensibility" or "intuition" indicates to him that the statement is true. But without justification that assertion does not constitute knowledge. This is why I made explicit #4: 4) Knowledge of the truth-value can be obtained without making use of information from sensory experience. Implicit in showing the above would be to provide some third source of information, and to demonstrate that this source of information is reliable. It is not sufficient for Kant (or anybody else) to simply assert that "sensibility" is a third source of information that does not make use of sensory experience. One must demonstrate both that this source of information is reliable, and that the information is not coming from sensory experience. That latter part is really the clincher for this. How can anybody claim that their "sensibility" or "intuition" is not informed by sensory experience? Indeed, all of the evidence pretty clearly indicates quite the opposite. DM
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keda
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Posted 07/08/09 - 06:21 AM:	#27
Death Monkey wrote: This seems to me to be using too liberal an interpretation of "contains". Of course, just saying that the subject is "contained" in the predicate is too vague to be useful as a formal definition in the first place. So all this really demonstrates is that interpreting "contains" here in this way just renders the term "analytic" trivial and useless. Maybe you've misundestood me. Containment is here used in the sense that a concept such as "apple" containing the predicate being a fruit, or bachelor containing the predicate being unmarried. As I said before, the definitions of each of the symbols used, which in this case is just the definitions and rules of ordinary arithmetic. How would you define them and prove that say 5+7=12 based on these definitions? Because this abstracted mathematics defines the rules of the operations that are serving as abstractions of the empirical act of counting real objects No, this is a misunderstanding. It is not abstracted from empirical acts, but refers to acts abstracted from its empirical objects. That latter part is really the clincher for this. How can anybody claim that their "sensibility" or "intuition" is not informed by sensory experience? Indeed, all of the evidence pretty clearly indicates quite the opposite. What evidence are you talking about? I suspect you have misunderstood something again but without further substantiation I cannot identify how.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

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Posted 07/08/09 - 07:22 AM:	#28
keda, Maybe you've misundestood me. Containment is here used in the sense that a concept such as "apple" containing the predicate being a fruit, or bachelor containing the predicate being unmarried. This sense of "containment" seems to me to coincide exactly with what I cited as the meaning of "analytic", namely that the truth-value of the statement is fixed by the meaning of the statement. The statement "the bachelor is unmarried" cannot possibly be false without redefining the terms used in that statement so that the statement no longer means what I think it means. This is not the case in my car example. The meaning of the term "my car" in no way specifies that the color of the car is gray. The statement "my car is gray" means exactly the same thing regardless of whether my car actually is gray or not. So while there may very well be a sense in which you could say that the color of my car is somehow "contained" in the predicate "my car", (and for that matter, any other property of the car), this is clearly not the same sense as the sense in which "unmarried" is contained in "bachelor". As I said before, the definitions of each of the symbols used, which in this case is just the definitions and rules of ordinary arithmetic. How would you define them and prove that say 5+7=12 based on these definitions? Didn't I just answer that question? Or are you asking me to provide here the formal definitions and axioms of arithmetic? Because this abstracted mathematics defines the rules of the operations that are serving as abstractions of the empirical act of counting real objects No, this is a misunderstanding. It is not abstracted from empirical acts, but refers to acts abstracted from its empirical objects. What difference does it make? Either way, the point is that mathematics defines the rules specify the meaning of the mathematical statement. Change those rules, and you change the meaning of the statement. Therefore the truth-value of the statement is fixed by its meaning. That latter part is really the clincher for this. How can anybody claim that their "sensibility" or "intuition" is not informed by sensory experience? Indeed, all of the evidence pretty clearly indicates quite the opposite. What evidence are you talking about? I suspect you have misunderstood something again but without further substantiation I cannot identify how. I am talking about our knowledge of how things like learning, intuition, and thinking, work. I am talking about the fact that different people's intuition can tell them very different things, and that even your own intuition can change over time, as you gain new experiences. But again, this is beside the point, because regardless the fact remains that certainly nobody has justified the claim that "sensibility" is a reliable source of information that does not come from sensory experience. 1) Unless you can justify the claim that you know the truth-value of the statement, you cannot cite the statement as an example of a true statement at all. 2) Until such time as you demonstrate that your justification for claiming that the statement is true requires information that is not implied by the meaning of the statement, you cannot cite the statement as an example of a synthetic statement. 3) Until such time as you demonstrate that your justification for claiming that the statement is true does not make use of information from sensory experience, you cannot cite the statement as an example of an a priori statement. In the case of mathematical statements, number 1 is no problem, but number 2 gets you. In the case of metaphysical statements (including claims about some sort of "objective mathematics" that is true independently of axioms and definitions), you can't even get past number 1. DM
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keda
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Posted 07/08/09 - 04:43 PM:	#29
Death Monkey wrote: This is not the case in my car example. The meaning of the term "my car" in no way specifies that the color of the car is gray. The statement "my car is gray" means exactly the same thing regardless of whether my car actually is gray or not. I'm just saying that this is how Leibnitz would interpret your sentence, namely that the greyness of the car is already contained in the subject "my car". He takes any sentence to be analytically true, simply because of a particular interpretation he makes of sentences. I'm just bringing this up to show you the absurdity in saying that mathematics is analytical, in the same way making a claim about your car analytic, because the subject no longer refers your car. Didn't I just answer that question? Or are you asking me to provide here the formal definitions and axioms of arithmetic? Of course, since I cannot see how you can prove 5+7=12 from the definitions I have in mind. I'm asking you to prove 5+7=12 from the definitions you have in mind. Without any examples, I have no idea what you are talking about. What difference does it make? You were saying that mathematics was an abstraction itself, especially of something empirical. I made no such implication. What I said was abstracted from was the empirical objects being counted e.g. 5 is not a property of a thing, but is a quantity which can be the size of any set of imaginary objects. Either way, the point is that mathematics defines the rules specify the meaning of the mathematical statement. Change those rules, and you change the meaning of the statement. Therefore the truth-value of the statement is fixed by its meaning. Does not seem to follow. The meaning of a statement "this flower is blue" is specified by the english language. If we change the rules of the english language, the meaning of the statement changes. Therefore the truth value is fixed by its meaning. Not... I am talking about our knowledge of how things like learning, intuition, and thinking, work. I am talking about the fact that different people's intuition can tell them very different things, and that even your own intuition can change over time, as you gain new experiences. An intuition doesn't tell anything, if you're refering to Kant's sense of intuition. As cognitions they only serve as immediate reference to objects of which judgments can be made about. I suggest you read more about it rather than assume "intuition" has the same meaning as in our every day usage. It certainly is a very bad translation, but unfortunately there does not exist a proper word for anchauung in English.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

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Posted 07/09/09 - 05:40 AM:	#30
keda, I'm just saying that this is how Leibnitz would interpret your sentence, namely that the greyness of the car is already contained in the subject "my car". He takes any sentence to be analytically true, simply because of a particular interpretation he makes of sentences. I understand this. If you also held all statements to be analytic, it would not make sense for you to argue that methematical statements are synthetic. I was just explaining why I do not agree with him. I'm just bringing this up to show you the absurdity in saying that mathematics is analytical, in the same way making a claim about your car analytic, because the subject no longer refers your car. But as I pointed out, the statement about my car does not qualify as being analytic in the way that statements like "that bachelor is unmarried" does, while mathematical statements do. Didn't I just answer that question? Or are you asking me to provide here the formal definitions and axioms of arithmetic? Of course, since I cannot see how you can prove 5+7=12 from the definitions I have in mind. I'm asking you to prove 5+7=12 from the definitions you have in mind. Without any examples, I have no idea what you are talking about. Well, I suppose this is as good a place to start as any: http://en.wikipedia.org/wiki/Peano_axioms In a nutshell, the symbols "5", "7", and "12" are defined as natural numbers, which are in turn defined in terms of succession. The statement "5+7=12" is a statement about a particular relationship between those natural numbers. That this particular relationship between these numbers obtains, logically follows from the definition of the natural numbers. Therefore it is not possible for the statement "5+7=12" to be false without either claiming that you are not referring to the natural numbers, or that you are not intending the "+" and "=" operators to refer to the relationships between natural numbers that I think you intend them to mean. Either way, the only way the statement could possibly be false would be if you mean something very different by the sentence than I think it means. Now, maybe this is the case. Maybe what you mean by the sentence is such that the statement is synthetic. But if this is the case, then I don't actually have any idea what you do mean by the sentence. Feel free to tell me. However, I am quite confident that when you do tell me, one of the following things will become apperant: 1) The statement is analytic, or 2) The statement is synthetic and a posterior, or 3) The statement is synthetic, but you don't actually know what the truth value of the statement is. You were saying that mathematics was an abstraction itself, especially of something empirical. I made no such implication. What I said was abstracted from was the empirical objects being counted e.g. 5 is not a property of a thing, but is a quantity which can be the size of any set of imaginary objects. That's pretty much what I meant too. The point is that in order to make such an abstraction, you have to formally define what you mean by, for example, the size of a set of imaginary objects. The truth-value of statements like "5+7=12" directly follow from those definitions. They are therefore analytic statements, because the predicate (which is that the particular relationship between the 'objects' holds), is "contained" in the subject (which is the collection of those three numbers). More to the point, it is an analytic truth because it is true by definition. Either way, the point is that mathematics defines the rules specify the meaning of the mathematical statement. Change those rules, and you change the meaning of the statement. Therefore the truth-value of the statement is fixed by its meaning. Does not seem to follow. The meaning of a statement "this flower is blue" is specified by the english language. No, the meaning of the sentence "this flower is blue" is specified by the English language. A series of symbols is not a statement until such time as its meaning is specified. Change the meaning of any of the words in that sentence, and it is still the same sentence, but is no longer the same statement. Indeed, it may no longer be a statement at all. As an anology, in base 10 the sentence "5+7=12" is a true statement. In base 8, it is a false statement. In base 2, it is nonsense, and therefore not a statement at all. That is because in the first two "languages", the symbol "12" means two different things. And in the third "language", the symbols "5", "7", and "12" are not defined at all. If we change the rules of the english language, the meaning of the statement changes. Therefore the truth value is fixed by its meaning. Not... If we change the rules of the English language, the meaning of the sentence changes. You then have a different statement, if it is still a statement at all. DM
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