bd-from-kg

Longbow:

This is in response to the latter part of your next-to last post.

Look. Every time I make a declarative statement about morality and moral theories, are you going to say that it must be expressing a proposition, and that therefore I’m contradicting myself? This is the same issue over and over again.

This isn’t much of an argument against noncognitivist theories in general, since it can be argued that an theory that does not involve impartiality isn’t really a “moral” theory at all; that the essence of the “moral” point of view is precisely that it is independent of the agent, the “evaluator”, or anyone else. In other words, that my statement above really does express a proposition, but an analytic one because it follows from the meaning of “theory of morality”. However, that’s not the direction my theory takes.

Let’s review what I mean by “valid principles of action”. I tried to describe this earlier as it applies to Occam’s Razor, but it applies equally to any VPOA, so I’ll rephrase it accordingly. Any POA can be phrased as “Act in accordance with such-and-such a rule or principle.” Moreover, “One should always act in accordance with such-and-such a rule or principle” means that the POA in question is valid. Another way of saying the same thing is that violating this POA is irrational. But all of these modes of expression involve an undefined term: “should” or “rational” or “valid”. As I said earlier:

I further contend that any fully rational person who fully understands a VPOA will “just see” that adhering to it is “the thing to do”; that it is (for him as a rational person) a partial answer to what Aristotle identifies as the fundamental question dealt with by morality: “How shall I live?” And that therefore such a person will adhere to it.

Note: Not all VPOA’s are moral principles; the division between moral principles and other VPOA’s is actually somewhat arbitrary. But the paragraphs above apply to all VPOA’s, so the distinction has no practical import.

But POA’s themselves are not even in the form of declarative statements, and saying that a POA is valid does not express a proposition. (I’ll get into this again in my next post.)

Now to say that any reasonable moral theory must involve impartiality is to say (in very general terms) that certain POA’s are clearly not VPOA’s. And of course if a statement to the effect that a particular POA is or is not a VPOA does not express a proposition, the preceding sentence does not express a proposition either, for the same reasons.

First off, the Golden Rule really does say that masochists should go around hurting people. But then, everyone understands that it is not really meant to be taken literally as a statement of a valid moral principle, any more than people (other than madmen) take “let justice be done thought the heavens fall” literally. Both of these are loose expressions of sentiments that can be spelled out more precisely, but in doing so the poetry is lost. But we are doing philosophy here, not poetry, so it is reasonable to ask you to be as literal and precise as possible.

If I were to adopt “Give pain to everyone” as a moral rule because I was a masochist, I would not be acting impartially, since I’d be giving myself special preference. But the rule itself is perfectly impartial; it applies the same to everyone. Conversely, if I were to adopt “Give pleasure to everyone” as a moral rule because I desire pleasure, I wouldn’t be acting any more impartially than the masochist. But again, the rule itself is perfectly impartial.

Conversely, “Other things being equal, give people what they prefer” seems to me to be a valid moral principle, but it doesn’t seem to be entailed by the principle of impartiality. (Nor can it be derived from empirical facts.)

Thus the question is: on what basis (other than self-interest) do we distinguish “the “valid” impartial rules from the “invalid” ones? In the case of the rules “Give pain to everyone” and “Give pleasure to everyone” the answer would seem to be either that pleasure is intrinsically more desirable than pain (regardless of how many people prefer the latter) or that more people prefer pleasure to pain than the reverse. In any case, there is some other principle in play besides rationality.

Of course, it might be argued that both principles are invalid; that the “correct” principle is “Give everyone what he most desires – pleasure to those who most desire pleasure, pain to those who most desire pain, knowledge to those who most desire knowledge, etc.” But again, this rule is based on something more than impartiality, because the rule “Give everyone what he least desires (or most desires not to have)” is just as impartial as the first.

Here you lose me once again. What is it that “does not adequately correspond ...”? If virtue is “partially subjective” but is involved in a “positive (moral) principle)” doesn’t that make morality subjective?

You’ve got to be kidding. This is obviously a synthetic (i.e., an empirical) proposition.

But even “justice” does not equate to “impartiality”, unless you simply define “impartiality” to mean “justice”. It’s quite possible to have a “code of justice” (not to be confused with a body of positive law) which is perfectly impartial but quite unjust.

This is a really fundamental difference between us. I argued this point at some length with Alonzo earlier on this very thread. In fact, this is an important reason why I think moral statements cannot be treated merely as propositions.

But your position does not “square up” at all with common usage. Common usage presumes that the belief that an act would be morally wrong will, for a rational agent, be a reason or motive for not doing it, and conversely, that a belief that he should (in the moral sense) do something will be a reason or motive for doing it. (To be sure, it is not expected to necessarily be a decisive or controlling reason or motive.) But a mere fact cannot, in itself, be a reason or motive for anything. Something very important is missing from any theory that purports to be an account, or interpretation, or explanation of common usage which fails to come to grips with this aspect of morality. After all, it’s the very thing that makes morality of practical interest: the purpose or function of morality is obviously to influence people’s behavior. If it does not, and cannot be expected to, influence anyone’s behavior, what’s the point? As someone pointed out on another thread, we could easily set up a system of rewards and punishments to encourage socially desirable behavior, and we can (and often do) hold people responsible for their actions without holding them morally responsible. The whole point of identifying some acts or types of behavior as morally wrong is that knowledge of this fact could reasonably be expected to influence the behavior of a moral agent independently of any associated rewards or punishments.

This is a side issue, so I’ll content myself with quoting the Britannica:

Now we go back to Occam's razor:

Well, in the case of stuff like religion, emotions get in the way to the point that otherwise rational people do form beliefs in clear violation of Occam’s Razor. But in everyday life everyone applies it all the time (thousands of times a day) unless they are stark raving mad.

But that’s just what I mean when I say that it’s a valid principle of rational action! Thus, when you say that if you didn’t apply Occam’s Razor in arriving at a belief you don’t really “know” it, you’re really saying that it fails to satisfy the JTB (justified true belief) condition because it isn’t justified, which is to say that it isn’t rationally justified, which is to say that a rational person would not form this belief on the basis of the evidence in question.

I’d say that it’s more “objectively valid” rather than more “real”, but at this point we’re just quibbling about words. And as I said, things like Occam’s Razor are more in the nature of a recommendation or injunction, not that they literally are recommendations or injunctions.

In my terminology, something is a “principle of rational action” if and only if any perfectly rational person who fully understands it will adhere to it. And a part of my theory is that any partially rational person desires to be a fully rational person. Moreover, this is a “meta-desire” or “second-order” desire in the sense that it automatically takes priority over all ordinary or “first-order” desires.

However, what it means to be a “fully rational person” cannot really be defined. Any ordinarily rational person has a pretty good understanding of what it means to be fully rational. And principles like Occam’s Razor are not propositions, but attempts to articulate certain aspects or properties of rational action.

Or, it’s your stubborn refusal to accept that moral statements do not express propositions that leads you to invent the fictitious category of “synthetic a priori” propositions.

Anyway, while my interpretation may be unusual by historical standards, I think it’s fairly mainstream by modern ones. Which is not to say that it’s universally accepted by any means.

On the contrary. The fact that the principles of epistemology, for example, are not propositions is that they are not about reality, but about how to go about creating and modifying a conceptual framework for understanding, interpreting, and predicting our experiences. In fact, it would completely inappropriate for epistemology to make any substantive statements, because it has to be in place before we have any basis for drawing any conclusions whatsoever about reality.

What a strange question! What’s the similarity between these subjects?

Not all “noncognitivist” theories are alike. I didn’t use the word “flaky”. I used the word “nonsense”, which was a word actually used by Ayer and others, and I said that positivists generally characterized moral statements as merely expressions of an emotion or attitude. I think the word “mere” here is quite appropriate. The negative attitude toward doing “morally wrong” things per se is essentially an intellectual attitude, just as our negative attitude toward someone who chooses to believe a hypothesis far more complicated than simpler ones the fits that facts (in violation of Occam’s razor) is essentially an intellectual attitude – specifically, the attitude we adopt toward someone who acts irrationally. (This is often hidden by the fact that we often have other attitudes and emotions about the very same act.)

I wasn’t saying that, but of course it’s true. Science looks for the best (i.e., simplest, most elegant) conceptual scheme that comprehends as many known natural phenomena as possible. It’s meaningless to ask whether this conceptual scheme is “true” as opposed to whether it’s the simplest or most elegant. Gravity, for example, is not a feature of the “real world”; it’s a concept which helps us to understand to interpret our experiences.

Anyway, the sense and extent to which science consists of propositions, and the sense in which these propositions are meaningful, is the heart of logical positivism. This is a very, very complex question. I thought we were talking about morality. Are we now going to get into a detailed discussion of the philosophy of science?

I’m not sure what exactly is included “other philosophical statements”. But many such statements are either in the nature of definitions, which are analytic, or analyses of how language is actually used, which are descriptive, or proposals for using language differently from the way it is currently used (at least for philosophical purposes), which are prescriptive. The remainder are indeed essentially principles of rational action. Philosophy is not in the business of determining the truth or falsehood of factual questions, so any philosophical statement that purports to be a substantive statement (i.e., to be expressing a non-analytic proposition) should be regarded with extreme suspicion.

It’s not an accident that the kinds of statements you have in mind are “informal”. “Formalizing” them would consist essentially of finding a precise way of expressing the propositions that they express roughly or approximately. Since they do not express propositions, not even roughly or approximately, this cannot be done.

Again, Occam’s Razor illustrates the situation as well as anything. What does it say? Well, it seems to say that the simplest explanation that fits the facts is the best one. But what does “best” mean? Perhaps it means it’s most likely to be correct? But this is by no means clear. For example, suppose that event A is followed by event B thousands of times in a row. The simplest explanation would seem to be that A causes B. But is this explanation “most likely” to be correct? This seems implausible. After all, by any ordinary definition of “probability”, this regularity makes it more likely that A will be followed by B on the next occurrence, but the probability that it will do so every time is still infinitesimal, even after thousands of repetitions. Perhaps there is some understanding of probability which does not give this result, but I don’t know of it. And in any case, there’s no obvious way to decide objectively which definition or analysis of probability is “objectively correct”. In fact, the proper analysis of probability seems to be far more problematic than Occam’s Razor itself. Something appears to be wrong here.

Also, is this really the simplest explanation? For example, isn’t the explanation that B causes A just as simple? And what about the explanation that it was just coincidence; that the fact that B followed A thousands of times in a row has no significance whatever; it “just happened”, and there is no reason to expect that it will happen again next time? Isn’t this the simplest explanation of all? Why not just always go with the “pure chance” explanation? That seems to really be the one always favored by Occam’s Razor.

But in reality we are looking for explanations that allow us to “make sense” of[/i] reality; to summarize a massive amount of sensory experience in a concise way, to predict and thus to some extent control the future. We look for causal explanations, not because they’re “simplest” (actually the whole concept of causation is quite puzzling and problematic) but because they help us to anticipate and predict the future. And of course the causality has to be in the forward direction because we normally want to predict the future, not “retrodict” the past.

This gives us a valuable clue to understanding the true nature of Occam’s Razor. It’s a strategy for coping with the real world; for functioning effectively as a rational agent. The question is not whether it’s “true” or “false”, but whether it’s a rational strategy. The reason it’s impossible to say under what conditions it would be true or false is that it isn’t a statement about the world, or even about ourselves – i.e., it doesn’t express a synthetic or empirical proposition. It’s a statement about how to proceed.

Not all “principles of rational action” are strategies for coping with the “real world”, but all of them are statements about “how to proceed” which can be justified rationally. But they can’t be proven, because they don’t assert anything. They are recommendations or prescriptions. And their nature is such that any fully rational person who understands them will adhere to them.

I’m not sure what you mean by this. Many of the statements in the theory are true; but the moral statements themselves are not true (or false). As for being “compelling”, I think that, properly understood, moral principles and moral statements generally consist of advice (or comments) as to what one would do if one were fully rational and had a full understanding of the situation. As such, they are just as “compelling” as the statement “move the queen’s pawn forward one square” would be if it were made by Kasparov to a close friend who trusts him completely (and wants to win). He may not understand why Kasparov recommended this move, but knows that it would be very wise to take his advice.

[ October 05, 2002: Message edited by: bd-from-kg ]</p>

bd-from-kg

Longbow:

I had pretty much finished a reply to your latest post when I came across this fascinationg <a href="http://home.ican.net/~arandall/Kant/Geometry/" target="_blank">article about Kant’s view of the nature of geometry</a>, which also sheds a good bit of light on what he meant by a “synthetic a priori truth”. I think I understand better what you were getting at with some of your comments. I’m revising my response accordingly. Stay tuned.

[ October 09, 2002: Message edited by: bd-from-kg ]</p>

Longbow

I am suddenly very short on time, so I have more or less dropped the threads I was in. But, I am reading the replies. I don't want you to wonder if your replies are falling on deaf ears -- I am definitely reading them. But, it looks you will get the last word for now...

bd-from-kg

Longbow:

Here finally is a response to your last post (not counting the very short one.)

Unfortunately these posts seem to be getting longer and to be covering more and more material. Pretty soon we’ll be covering every area of philosophy at this rate. Please try to start focusing on the issues that seem to you to be most relevant to morality. Otherwise I’ll have to decide more or less arbitrarily to ignore whole areas just to keep the length manageable.

Note that all of the sections but the last were written before I read the Randall article. I cut and trimmed a good bit of stuff in view of it but didn’t revise much. So the last section may not mesh well with what comes before.

1. On truth

Well, I suppose that there is a sense in which saying that something is “true” brings up the “issue” of truth. It’s just that I don’t see that there’s really much of an “issue” there.

I suggest that we avoid using the word “truth” as if it were a “thing” that needs to be “defined” or in terms of which other things can be defined. In fact, I think this whole discussion is rather pointless. You seem to think that what it means to say that a proposition is “true” is rather mysterious and in need of some sort of profound metaphysical analysis, and that only when we understand the results of this analysis will we really understand what it means to say that a statement or proposition is true. I say that what it means to say that something is true is so ridiculously simple that it can’t really be defined except in a trivial sense. Thus, to take Tarski’s famous example, “Snow is white” is true if and only if snow is white. In fact, the only point of the grammatical form “X is true” is that sometimes it improves the flow of a definition, or a logical statement involving undefined parameters. For example, if I want to explain what “modus ponens” means, I could say, “If A and A implies B, then B.” But this is awkward, confusing, and ambiguous, so it’s better to say, “If A is true and A implies B, then one may validly infer that B is true.” This means the same thing, but it reads more smoothly and is clearer.

As for “truth values”, again this is getting needlessly formal. In formal logic one defines a set of “initial strings” with “values” assigned to them. (Typically the permitted values are “t” and “f” or “0” and “1”, or “true” and “false”.) In addition we have “rules” specifying that one can assign a specified value to a string if it is formally related in specified ways to strings with already assigned values. We now have a kind of game, which may or may not be interesting depending on what initial strings and rules we chose and what values were assigned to the initial strings. For some choices it is possible to interpret the strings (or some of them anyway) as propositions (or proposition schema – i.e., they become propositions if we substitute actual propositions for the “constants” A, B, etc.), the values as representing the “truth” or “falsehood” of these propositions, and the rules as representing valid inference rules. But of course such an interpretation is only possible because we already know what a proposition is, what it means to say that a proposition is “true”, etc. So this sort of thing cannot be used to define truth, or what it means for an inference to be valid, etc.

2. On skeptical paradoxes and definitions

In the first place, there is no such thing as “correctly defining” something. Defining a term does not consist of asserting that the definition is “correct”; it just consists of stipulating that this is how you intend to use it. (Of course, when a dictionary gives a definition it is implicitly making the factual claim that the term is, or has been, actually used that way by a significant number of people. But we are not compiling a dictionary here; we’re doing philosophy.)

Anyway, it would be more profitable to discuss any issues that you think come under the heading of “skeptic’s paradox” directly rather than trying to deal with problems that are already quite abstract enough in terms of generalization – that is, abstractions about abstractions.

3. On analytic and a priori truths

As to the rather pointless dispute about whether saying that a proposition is analytic means that its predicate is “contained” in its subject, I suggest that we drop it. I don’t understand 80% of what you’re saying on this point anyway. Hopefully we can agree that a proposition is analytic if and only if it is true by virtue of the meanings of the terms involved. If you mean something else by “analytic”, we are definitely not on the same page.

Well, actually it does. One can refer to an “a priori proposition”, but what this means is that it can be known independently of any evidence.

Since we’re being pedantic today, there are two problems here:

(1) “Knowledge” does not consist of a proposition; it consists of a complicated relationship between a proposition, the knower, the evidence, and reality.

(2) You’re essentially stating the JTB (justified true belief) definition of knowledge, which is well-known to be defective because of Gettier-type problems.

I don’t see any sense in which this can be right. “A priori knowledge” is by definition knowledge, which of course entails that it’s true. An “a priori proposition” is one that can be known to be true (under certain conditions, the exact meaning of which need not concern us here), which again entails that it’s true.

As for what sorts of things the adjectives “a priori” and “analytic” can properly be applied to, I note that the Britannica has entries for “a priori knowledge” and “analytic proposition”; it does not have entries for “a priori propositions” of “analytic sentences”. That doesn’t prove that the latter terms are incorrect, but it certainly shows that in saying that “a priori” applies to knowledge and “analytic” applies to propositions I am following standard usage.

4. On mathematics

No, it’s not. But it entails that they are a priori. Even Kant agreed that there are no analytic a posteriori propositions. Am I missing something here?

More precisely, from the meanings of the terms.

Well, of course that’s what you’re doing. Nowadays formalism is all the rage, so the process is often formalized as follows: First you define a set of axioms which are “self-evidently” true in view of the meanings of the terms. Second, you produce a formal proof (or more commonly, a rather hand-waving outline of how to produce such a proof, from which any competent mathematician can see how he could produce one if he cared to). In the old days the approach was more informal; one simply started with any premises that were convenient to the purpose at hand and which were obviously true in view of the meanings of the terms involved, and then proceeded to a series of conclusions, each of which obviously followed from what came before, the final one being the theorem in question. Either way one is deriving the truth of the theorem from the meanings of the terms involved. Do you know of some other, magical way to prove mathematical theorems that we mathematicians are not familiar with?

What are you questioning here?

(1) Whether R&W demonstrated that mathematics is a branch of logic? If so, I’m not interested in discussing it.

(2) Whether the whole of mathematics can be put on an axiomatic basis? If so, I’m not interested in discussing it, beyond noting that (i) notwithstanding what the author of the article you cited seems to think, R&W had no interest in doing this, or in the question of whether it could be done. As you note, the Principia is not an example of the formalization of mathematics (in the sense of putting it on an “axiomatic basis”. They believed that mathematics was a branch of logic and that therefore the only relevant axioms were those of logic. As for mathematics proper, their contention was that all of its results could be derived from the axioms of logic and the relevant definitions. (ii) Godel himself did not believe that his results showed that this is false, and indeed seems to have believed that it is true notwithstanding the Incompleteness Theorem.

(3) Whether all of mathematics is analytic? If so, I don’t see what the Incompleteness Theorem has to do with it. This theorem has to do with what can be proven formally from a formal (finitely definable) set of axioms. R&W’s project was to derive mathematical theorems from the definitions alone. As you may know, the IT applies only to formal axiomatic systems in which the axioms are “strong” enough to allow one to formally define the natural numbers from them. The axioms of logic do not qualify. In fact, it can be shown that the axioms of logic, taken by themselves, are complete.

5. On alleged synthetic a priori truths

(A) “I exist” (with comments on justified belief and a priori truths)

To refute your claim that “I exist is a synthetic a priori truth, I showed that it is not a priori at all, but that the belief that I exist is based on evidence. but the evidence on which this belief was based was obtained, of course, when I was very young. You pointed out:

Certainly. But once again, you seem to have missed the point of my “historical narration” of how I came to believe that I exist, which was not that this history justifies my belief, but that it explains how the knowledge that one exists might seem to be a priori. In other words, I was “explaining away” the fact that we have a “gut feeling” that this is something that we “just know” independently of any evidence. This “gut feeling” is just plain wrong; actually this belief was formed on the basis of evidence, just like all other beliefs about the “real world”.

As to whether the belief is rationally justified, there are two slightly different ways to interpret this. One is whether it is possible to rationally justify it. To answer this question we don’t need to consider how the belief was formed. Suffice it to say that the kinds of considerations I cited are quite sufficient to justify it; it is at least as justified as any other belief about the real world; in fact, far more justified than most. The second interpretation of the question is that it is asking whether my belief that I exist (assuming that I am not an especially reflective person and have never gone back and “reconstructed” a rational basis for this belief in terms of inferences from empirical evidence) is rationally justified. My answer is again yes. I say that that the process that led to this belief in the first place is in itself a rational justification for the belief, even though it wasn’t a rational process.

No, you don’t. Say that I’m at the mall and my wife and I have split up temporarily. I see her a few yards distance away, and she waves and says “Hi, honey! Here I am!” I contend that I’m rationally justified in my belief that this really is my wife. But this belief is based almost entirely on “pattern recognition” circuits in my brain that are able to distinguish my wife’s appearance and voice from other people’s with astonishing accuracy. And all this occurs at a subconscious or preconscious level; I’m not aware of it. I couldn’t tell you how I know that it’s my wife to save my life. Similarly, my coworker who works in the office with me is behaving abnormally; she seems preoccupied, unable to concentrate, distracted. I conclude that she has something on her mind; something is bothering her. Is this a rationally justified belief? I’d say so. I know her pretty well; we’ve shared an office for over twenty years. But if I were challenged to give an argument for this belief I’d be stumped.

But beyond this, how can the claim that one must have an argument for a belief in order for it to be justified be squared with the claim that there are synthetic a priori truths? What kind of argument could possible be given for such a truth? Certainly not an argument from logic: if such an argument were available the proposition in question would be analytic. But not an evidential one either: if evidence were needed to justify the belief the proposition in question would not be an a priori truth. But what other kind of “belief-justifying” argument is there?

Of course it is! And the whole point is that we are not created with this belief. It’s based on evidence. And it must necessarily be based on evidence.

Unless, of course, you can offer an alternative account of how I could have arrived at this belief (and been rationally justified in believing it) through some process that was completely independent of all evidence. It seems pretty obvious to me that this is impossible.

(B) Ockham’s Razor

[Note: While “Occam’s Razor” is a correct alternative spelling, it gets jarring to see the same thing repeatedly spelled different ways, so I’ll use your more standard spelling.]

Earlier we had this exchange:

No, I’m not saying that at all. But presumably you must mean something by the claim that OR is true in all logically possible worlds. And it can hardly mean that you have examined all logically possible worlds and found empirically that OR holds in all of them! The only reasonable meaning of a claim that something is true in all logically possible worlds is that its negation is self-contradictory. If the negation isn’t self-contradictory, it’s logically possible for it to be true. (This is a tautology.) And if it’s logically possible for it to be true, there is a logically possible world in which it’s true. (Another tautology.) But that would mean that the original proposition is not true in all logically possible worlds. [See my next response for more on this.]

Ah, but in that case the claim that it is an a priori truth is meaningless. Let me explain.

To say that we can know that OR is an a priori truth is to say that we can know it independently of any evidence. But what makes something “evidence” is that it tells us something about the world we live in – that is, it rules out the possibility that we live in certain logically possible worlds. Thus to say that something is an a priori truth is to say that it is true in all logically possible worlds, or equivalently that its negation is false in all logically possible worlds. But to say that a proposition is false in all logically possible worlds is to say that it is logically incompatible with the laws of logic – i.e., that it entails a contradiction.

This leads to the question of what it means in general (i.e., outside of formal systems) to say that A entails B. So far as I can see, the only intelligible interpretation that “works” in general is that it is possible to express in formal terms at least a part of what A and ~B mean, and to show that it is possible to derive a contradiction from these formal expressions. For example, say that A and B are “informal” statements whose meaning cannot be “spelled out” or “explained” in full, and that it is asserted that A entails B. The only way to construe this claim that makes sense is that there are propositions C and D such that C “captures” at least part of the meaning of A and D “captures” part of the meaning of ~B, and that C and D taken together formally entail a contradiction. Obviously it is up to the person making the claim that A entails B to supply us with C and D. If he cannot specify any clear, unambiguous propositions that are entailed by A and B and their negations, we are justified in concluding that he has no idea what he actually means by them. And if he cannot supply us with C and D, we are justified in concluding that he has no idea why A might entail B, and therefore no justification for thinking that it does.

Where the Completeness Theorem enters into all this, of course, is that once you have C and D, it says that C and D entail a contradiction if and only if a contradiction can be rigorously proved from them. So any claim that C and D are really contradictory, but that there is no proof that they are, is out of court.

In our case the situation is simpler: the claim is simply that ~OR entails a contradiction. So the person making this assertion need only express formally enough of what he means by ~OR to yield a contradiction. If you cannot do this, I conclude that either you don’t have any kind of remotely clear idea of what proposition OR supposedly expresses or you have no real idea of why its negation might entail a contradiction. Either way I conclude that you have no basis for your claim that OR is an a priori truth – in fact, that you don’t really know what you mean by this claim.

Now let’s take another look at your statement:

On the basis of the analysis above, I say that all of this is literally nonsensical. The concept of a proposition that can be known a priori, but that cannot be shown to be true through appropriate application of predicate calculus, is untenable. Either you understand what you’re talking about well enough to pin down precisely why the negation of the proposition leads to a contradiction, or you do not know that it’s true.

(C) Propositions about the physical world

If there is no logically possible world in which a proposition is false it is analytic. And I don’t see what “possible” might mean in this context other than “logically possible”. It appears that you want to say that some logically possible worlds are not “really” possible in some undefined sense. But it’s very difficult to make any sense of this. It would require the existence of some mysterious principle that excludes certain logically possible worlds from existing. But what would be the nature of such a principle? It couldn’t be contingent , because that would mean that it is true in some possible worlds but not others, which makes no sense for a principle that supposedly applies to all possible worlds. And what would it mean to say that it’s “necessary”? It certainly cannot be logically necessary, because by definition any logically possible world conforms to any logically necessary principle.

To illustrate the problem, let’s imagine a logically possible world which violates your hypothetical principle (i.e., one that you say is “impossible”). You say that this world cannot “possibly” exist. I ask “why not”? You reply that it violates your principle. I reply “Yes, that’s a fact about this world. So what?” At this point I can’t imagine any intelligible reply.

Name one.

It seems to me that it would be the presence of such statements that would make the theory unintelligible. I know how to interpret the meaning of an analytic a priori proposition: I analyze the meanings of the terms involved. I know how to interpret the meaning of a synthetic a posteriori proposition: I consider how possible worlds in which it is true differ from possible worlds in which it is false. But I’m at a loss as to how to understand the meaning of a “synthetic a priori” proposition. It doesn’t follow from the meanings of the terms involved. And I can’t look at how possible worlds in which it is true are different from possible worlds in which it is false, since (according to you) it is true in all possible worlds.

Besides, how can a statement which is true in all possible worlds be a legitimate part of a scientific theory? Scientific theories are supposed to say something about the world. How can a statement that fails to differentiate this world from any other possible world be “saying something” about this world?

Ah, so that’s the sort of thing you have in mind. Fine, but these aren’t propositions, nor are they assumptions. They are part of the definition of what constitutes science.

Also, for future reference, what it means to say that a statement is analytic is not that it can be derived formally from first-order predicate logic. It means that it is true by virtue of the meanings of the terms involved. First-order predicate logic is a subset of logic.

6. Kant, synthetic a priori truths, and “objective morality”

According to Allan Randall’s article “A Critique of the Kantian View of Geometry” (which I gave a link to earlier) Kant had something quite different in mind when he wrote about synthetic a priori truths than you seem to. Randall says:

Thus for Kant “synthetic a priori truths” are not truths about the “external world” at all, but rather truths about how we necessarily perceive the world. They represent “intuition filters” through which all of our sensory input passes before it ever reaches our consciousness. Thus we do not have access to “things in themselves”, but only to the representations of these things presented to our conscious minds after extensive cognitive processing of the sensory inputs.

Under the circumstances it is legitimate to ask why Kant considered such things to be “truths” at all. The answer is that they are just as true as anything we “know” about the external world. Since all of our conceptions about the world necessarily conform to these “sensory intuitions”, none of them can possibly be any more true than the “synthetic a priori truths” that shaped them.

Kant believed that one of these “intuition filters” was the fact that we necessarily interpret the world in terms of objects embedded in three-dimensional Euclidean space. According to Kant,

The fact that it is possible to construct a scientific theory involving a non-Euclidean space is irrelevant to the point Kant is making. The crucial point is that even though we “know” that space is “really” non-Euclidean, we cannot help but visualize it as Euclidean. As we all know, when we try to understand the relativistic concept of space, we do so by imagining things like a plane with “bumps” (a 2-D version of warped space embedded in 3D Euclidean space) and other “visualization aids”, all of which render the concept intelligible to us through simplified “models” that can be envisioned in our “conceptual” 3D Euclidean space.

The main point here is that:

Another way of putting this is that “synthetic a priori truths” represent how human minds necessarily structures reality, they are part of our innate conceptual framework for interpreting the world.

Now while your specific examples of “synthetic a priori truths” are misguided (at least from a Kantian point of view), the notion of an innate conceptual framework for structuring or interpreting reality has obvious application to morality. Perhaps what we think of as “objective moral truths” are actually also part of the innate conceptual framework for interpreting reality (though in a somewhat different sense) that all humans have in common. This is getting very close to my own viewpoint.

[ October 11, 2002: Message edited by: bd-from-kg ]</p>

Longbow

Okay, here is a quick reply, then, to the main issues. For starters, I am not taking knowledge to be "justified, true belief". In fact, that is specifically why I used the term "warrant" in that statement. Secondly, what makes knowledge knowledge and not something else certainly is all the characteristics that you mention. But, what the actual thing is that has those characteristics is a proposition. It is plainly possible to speak, for instance, of a priori propositions that are false. For instance, "I do not exist," is a priori. And thirdly, I am using the terms in the more precise usage that has given rise to the imprecise common usage. Specifically, that a term like "analytic" refers to the sentence and strictly speaking not the proposition it contains, I believe (where such a distinction is recognized), is indisputably congruent with how the term is used.

The point in bringing these things up is to draw a distinction between the sentence and the meaning of the sentence (the proposition). And, this is done largely to make the rather profound statement that whenever a sentence has a meaning that corresponds to an a priori proposition, it turns out that such a sentence is analytic. What this would do for us is say that to analyze any a priori proposition, we merely need to scrutinize the form such a proposition appears in rather than to analyze the concepts that make up such a proposition. The idea is precisely, then, to be able to derive the truth of all a priori propositions through an anlysis of the terms they are stated with.

And this brings us back to Godel. Actually there is an article he wrote in 1961 (granted quite awile after the "On formally undecidable..." in 1931), "The modern development of the foundations of mathematics in the light of philosophy". Here is a quote:

"What I have said so far are really only obvious things, which I wanted to recall merely because they are important for what follows. But the next step in the development is now this: it turns out that it is impossible to rescue the old rightward aspects of mathematics in such a manner as to be more or less in accord with the spirit of the time. Even if we restrict ourselves to the theory of natural numbers, it is impossible to find a system of axioms and formal rules from which, for every number-theoretic proposition A, either A or ~A would always be derivable. And furthermore, for reasonably comprehensive axioms of mathematics, it is impossible to carry out a proof of consistency merely by reflecting on the concrete combinations of symbols, without introducing more abstract elements. The Hilbertian combination of materialism and aspects of classical mathematics thus proves to be impossible."

He basically rejects Hilbert's formalism as well as the Principia. In fact, he has a lot to say about Kant. You might find it interesting...http://www.marxists.org/reference/subject/philosophy/works/at/godel.htm. In any case, I believe something like this cuts close to the heart of the matter.

It sounds like you are rejecting that which is informal as not being "true" or "false" and not really being "knowledge". Instead these sorts of things rank as "compelling" recommendations. On the other hand, that which could be considered "true" or "false" or "knowledge" is, at least in principle, formal and/or scientific. You are treating math and science like it can stand on its own without informal philosophical underpinnings. Actually, I don't know which one to emphasize more -- the word "informal" or "philosophical" -- the two seem largely equivalent with respect to this issue.

The reason we disgree about morality is because we disagree about what is even possibly something like "knowledge". I do not think that an informal subject or even an inherently informal subject cannot contain knowledge. I think you implicitly (or perhaps even explicitly) do. You might not say this exactly, or you might imagine that it could be technically not true, but this is for the most part where you seem to fall. For the most part, if a given subject matter is informal, it is about "recommendations" not "knowledge". And, those subjects that are about kowledge are formal like science and math. I think that moral philosophy, in fact all of philosophy, is a counter-example to this.

And, I think that a big step in my direction is the idea that math rests on a partially informal foundation. (At least it undermines the dichotomy of formal vs informal as being a relevant epistemological one.) I think you would tend to disagree with this, and I think that is a key aspect of all of our disagreements. In short, I think it really does come down to the Sorites Paradox, philosophical vagueness, et al and how we each deal with these issues differently that drives our different views.

(Just make a short reply like this one that praphrases rather than breaking my post up an responding to each section. It will keep it shorter and more to the point for us, I think. I sincerely do want to be able to adequately state your position in my own words, so I think there is sufficiently less danger of misrepresentation for us to do it this way.)

[ October 11, 2002: Message edited by: Longbow ]</p>

Longbow

I almost missed this one. For starters, I do not say that synthetic a priori truths are about the "external world". Obviously the whole point is that they aren't. I did say that a proposition such as "I think" is not about the external world. And finally, the more elaborate description of Kantian philosophy as being "how we think" is a bit more of a particular interpretation of what he said than it is what he really did, in fact, say.

In any case, I think this is perhaps less to the issue. I think the real issue that dirves most of our disputes is a metaphilosophical one about the place philosophy has in intellectual pursuits. And what drives this is how we each react to the issue of philosophical vagueness differently from one another. From your perspective, I have to redefine "reality" to achieve my world view. From my perspective you have to redefine something even more profound, namely "knowledge", to arrive at yours.

By the way, I'm not a professional philosopher, so it wouldn't offend me if you were to say, for instance, that philosophy as an academic field is in some sense bogus. That is not to say that you do contend this or that I would agree with it if you did. It is just to say that you need not restrain yourself out of politeness from saying something like that...

[ October 11, 2002: Message edited by: Longbow ]</p>

bd-from-kg

Longbow:

Well, your original statement was:

This is practically an exact statement of the JTB definition of knowledge except that “justified” is replaced by “warranted”. So are you merely objecting that you do not define knowledge as justified true belief, but prefer to say that it is warranted true belief? If so, this is a distinction without a difference. In particular, a “warranted true belief” definition has the same Gettier problems as the JTB definition.

If “I exist” were an a priori propositions, this would be true. But given that I have argued extensively that it isn’t (an argument which, IMHO, you have not come remotely close to refuting) I am somewhat taken aback that you should choose this example as if it were completely unproblematic.

I think that your notion that “I exist” is a priori is based on a confused reading of Descartes and Kant. Descartes claimed that one could be certain that “I think[/i] is true. but he did not claim that this is an a priori truth; he simply believed that the evidence, being direct and not subject to misinterpretation, was such as to make the conclusion certain. Kant, on the other hand, so far as I have been able to determine, had nothing whatever to say about the proposition “I exist”, and certainly did not include it among his “synthetic a priori truths”.

Here you’ve lost me. In what sense is this usage more “precise”, as opposed to merely being different? In what sense is it “indisputably congruent with how the term is used” when (as can easily be shown) references to “analytic propositions” are quite common, and probably more common than references to “analytic statements” in the literature? And why do you think any of this is at all important? If you want to refer to “analytic statements” I have no real objection; I just don’t see why you keep harping on this as though how a word is used has some deep philosophical significance.

I’m still baffled. What point are you trying to make here? Is this intended to a be a statement of my position?

Yes, I’m familiar with the article you cite. As he says, the Incompleteness Theorem shows conclusively that the Hilbert program is impossible. But the article says nothing about logicism or the Principia. I discussed this in my last post, so I see no point in going through it again. The last thing R&W were interested in doing was showing that mathematics can be put on an axiomatic basis. There are serious arguments against logicism, but they have nothing to do with the Incompleteness Theorem.

I’m not sure that I understand what you mean by this. If you mean that I insist that it must be possible to “formalize” a statement by expressing it in first-order predicate logic, that just doesn’t mean anything. In one sense any proposition can be expressed in first-order predicate logic. I need only choose a letter (say P) to denote it and viola! – I’ve expressed it in first-order predicate logic.

But in a deeper sense, it isn’t possible to express any proposition in first-order predicate logic. FOPL is just a system of manipulating formal symbols; it doesn’t mean anything at all in itself. A system of axioms, etc., must be interpreted before it can be said to mean anything. One cannot look to a formal system for meaning; the only real point of such a system is to allow one to check the validity of a proof. If an argument can be expressed in such a system according to a particular interpretation, and if one accepts the transformation rules as corresponding to valid inference rules, and if the final string of symbols corresponds (according to the chosen interpretation) to the desired conclusion, then one can safely conclude that the proof is valid.

But if you mean that before accepting that an “informal” statement S actually expresses a substantive (i.e., non-analytic) proposition I insist that you give some reasonable indication of what logically possible worlds it is true in, you’re right: I do insist on this. Ordinarily this is expressed in terms of evidence. If you tell me what counts as evidence for and against S, you are in effect telling me what logically possible worlds it is true in. And in so doing, you are telling me what S means. If you insist that S really expresses a substantive proposition but you can’t give any indication of what would count (at least in principle) as evidence for or against it, or if you insist that it’s true in all logically possible worlds, I can only conclude that you’re talking gibberish. Thus, if you say that there are black swans in Australia, I understand what would count as evidence for or against your statement, so I understand what you mean. But if you say that the Absolute is ineffable, I have no idea what would count as evidence one way or the other for this statement, so I have no idea what you mean. If you go on to claim that this is necessarily true – that it’s true in all logically possible worlds – I’ll be certain that you’re talking gibberish. (Ayer gives another good example of this: the monist assertion that Reality is One is nonsensical, since no empirical situation could have any bearing on its truth.)

So if your objection is that I don’t consider anything to be a “real” proposition until it has been expressed in a formal axiomatic system, I plead not guilty. But if it’s that I insist that you have to put some “bones” on an assertion by specifying what would count as evidence for and against before I consider it to be expressing a proposition, I’m guilty as charged.

A misunderstanding. I think that some things that appear at first sight to be expressing propositions are actually expressing principles of action. But this is hardly a general strategy for interpreting statements that turn out not to be expressing propositions. A great many of them are simply nonsensical, others are expressions of approval or of personal preference, etc.

Perhaps what you’re trying to say is that we disagree about what statements express propositions, in which case you’re right. But if you disagree – if you think that certain statements that I think don’t express propositions really do – you need only explain (informally, if you like) what propositions they express. That is, explain what would count as evidence for or against them. You might start with Ockham’s Razor.

I think you must have a technical meaning of “informal” in mind. I count all sorts of “informal” things as knowledge. for example I think I know that my wife loves me. I know that I live pretty close to a certain town, that a certain paper has the best coverage of local news, that so-and-so is the most reliable plumber in the area, etc., etc. The problem seems to have little to do with “informality” as such.

I think that you’re saying that some math is about concepts that cannot be fully expressed in an axiomatic system. And this is certainly true. Strictly speaking, no concepts – even very simple ones – can be expressed in an axiomatic system. But once we reach the level of the natural numbers it gets worse: we cannot even define, in any finite way, a system of axioms from which all true statements about natural numbers can be derived. But so far as I can see, this has nothing to do with whether math is analytic. All true statements about the natural numbers are entailed by the meaning of the concept of a natural number, and therefore they are analytic. The fact that this meaning cannot be “captured” fully by a finite system of axioms is irrelevant. What you seem to want to say is that the concept of a natural number is not analytic. But as I pointed out earlier, this is trivially true. The term “analytic” doesn’t even apply to concepts. The concept of natural numbers isn’t analytic for the same reason that a triangle isn’t green.

First off, I’ve made it quite clear that I accept that any statement that is “about reality” is meaningful. To be more precise, if it is logically possible that a statement could be true (and therefore is true in some logically possible world) and logically possible that it could be false (and therefore is false in some logically possible world), it expresses a synthetic proposition. But what you apparently insist on is that there are meaningful statements (i.e., statements that can meaningfully be said to be true or false) such that it is not logically possible that they are false, yet are not analytic.

But it is about this world. That is, there are logically possible worlds in which you do not think, and it asserts that this is not one of those worlds. That makes it a synthetic proposition. But it isn’t a priori because your knowledge that you think is based on evidence. (Note that “I think” entails “I exist”, so if the latter is a posteriori, the former must be as well.) Ockham’s Razor is a very different cup of tea. It does not appear to be saying that there are logically possible worlds in which something-or-other is false, and that this is not one of those worlds. On the contrary, it appears to be an expression of a principle which, if valid at all, is valid for any logically possible world.

OK. But as far as I’m concerned, this is the only interpretation of what he said that represents a tenable position.

As usual, I find this kind of statement pretty much useless. I don’t really know what you mean by “philosophy” or what the “issue” of “philosophical vagueness” is. I don’t know what you mean by “redefining ‘reality’”. If it just means redefining the word “reality”, why should this be necessary to “achieve” a worldview? Similarly for “knowledge”. It seems to me that we’re not at all far apart on the question of what it means to “know” something; the question seems to be what things there are to know. For example, is it possible to “know” that Ockham’s Razor is “true”? That depends on whether it expresses a proposition. It seems to me that it would be far more productive to discuss questions like this than to talk vaguely about what “drives” our disputes.

At this point I’m getting very impatient. You’ve ignored my long post of Oct. 5, which was actually about morality. (This is the “Moral Foundations” forum, after all.) Apparently you prefer to go on endlessly about these abstruse metaphysical issues which seem to be a bottomless pit. If you don’t want to discuss morality, let’s just call it a day. If you do, let’s get on with it.