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Hodge'sPodges

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  1. I guess there are people who may feel insulted to some degree by even mild jocularity on some subject or other. By the way, Obama didn't say marijuana is evil (whether he thinks that or not, I don't know) but only that he does not believe that legalizing marijuana would help the economy and the creation of jobs. (And I'm not claiming that you are or are not arguing that Obama revealed himself to think that marijuana is evil or that his answer was meant to pander to those who do.)
  2. He didn't shrug the question. Whether one likes his answer or not, he gave a direct answer. And he prefaced with some jocularity, hardly meant as "spitting in the face".
  3. Probably the first formal set theory was a proposal of Zermelo (though it was not yet fully formal). His primary purpose was to prove the well ordering theorem. ZF came later and was a refinement (closer to full formalization) and an expansion of Zermelo's proposal. If ZFC is consistent, then the consistency of ZFC cannot be proven by ZFC (nor, perforce, by a theory weaker than ZFC). However, there are theories that do prove the consistency of ZFC. Of course, the epistemological value of such proofs may not be great; nevertheless, they are formal proofs. In some treatments, the axiom of infinity is used, but this is not necessary; the existence of an empty set can be derived from the axiom schema of separation and uniqueness from the axiom of extensionality. I don't know what sense of construction you have in mind. Rather, the natural numbers and the basic operations on them are a MODEL of first order Peano arithmetic. That does not really capture the range of thinking that comes under the formalist school. Most reasonable formalists (say, of the Hilbertian bent) do not consider mathematics to be ONLY such a game, but rather that there is an ASPECT of mathematics that is as definite in its rules as those of games such as chess. There are also many other schools of thought, including finitism (related to formalism on one end and to intuitionsim on the other end), constructivism, intuitionism (a form of constructivism), predicativism (related also to constructivism), Russian constructivism, Bishop constructivism, structuralism, consequentialism, fictionalism, instrumentalism, logicism, variations on realism other than full-fledged platonism, etc. There were several other points in your posts that are correct; I did not include them but rather I only commented on those points I felt in need of comment.
  4. One could start right at the first pages of Logical Investigations. One could find lots of relations. Whatever Odden had in mind, I don't know what formal logician's views can be fairly characterized as Odden did. No, it's not. Support has not been given. But my saying that does not preclude that one might eventually provide such support. I don't go around shouting in all-caps. I just used all-caps as a text-only means of emphasis of certain words or phrases. As to 'method', it's the word used by Odden himself. I have no desire to stick you with agreeing with Odden, but when you argue that the views of a certain logician do (or may, or whatever) fit the description Odden gave, then this part about method is part of that description. It is about formal logicians, not just philosophers, and maybe it was meant by Odden to be only about some, or perhaps, as he left it unquantified, it supposed to be general in whatever sense. He or anyone may now sharpen to be more specific about that; that's fine with me. But in the meantime, I would have liked to know what formal logicians he has in mind. Who, specifically, are at least some of the people he's talking about? (Though, he may or may not wish to produce such an example, in which case it's fine with me to let the matter there rest.) We don't have to quibble about what a "formal logician" is, and we don't need to poll. All that is needed for a start is for Odden to say who, specifically, he has in mind. (Previous disclaimer again.) I don't mean to merely nitpick, but while Frege is one of the most important of his generation, he is not the only fountainhead of modern logic. Of course platonism, and realism of various kinds, and the influence of Fregean realism are very much at work in modern philosophy of logic and mathematics. But I don't recognize such views as Odden characterized them. Otherwise, it would help if he would just say who specifically he has in mind. (Previous disclaimer again.) At this moment, I'm not too interested in hectoring Odden himself on that matter; as I've already asked him, he's not yet responded, so I'm content just to let it rest there with him specifically as he may elect to say more or not to say more at any later point. But you've continued the matter, and also as to the shape of the conversation among the three of us, so I can't properly address that without mentioning Odden's role too. In closing, I have to say, this kind of getting bogged down in discussion about the conversation itself is not very interesting to me. My initial point is that I don't recognize the views of any particular formal logician in the characterization Odden made. And I've yet to see that characterization upheld. That is good enough for me to let the matter rest, unless we do find some specific writings of a formal logician that are accurately paraphrased as ""For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method."
  5. I'm not interested in sticking you with any particular claims. Rather David Odden made a claim that formal logicians hold a certain view (part of which was the bit about "symbol transformation") and I asked what formal logician holds such a view, then you offered Frege, though you also, reasonably, hedged your bet on that, as I mentioned. And my point with you is that it is not supported that Frege is a formal logician that holds the view Odden mentions. What logician or mathematician had such a view, let alone what support is there that such a view was common?
  6. I said, "The probability is 0. That is a triviality that follows from the fact that there is no object that is both square and round." To say that x is a round square is tantamount to a conjunction: x is round and x is square. Each predicate, 'round' and 'square' is meaningful, so the conjunction is meaningful, and, in this case, false, since there is no x such that x is round and x is square. I don't know what problem you find in this.
  7. On that very last point, are you quite sure it is Frege's view about true at one time and not another? Nothing you've said entails that Frege held that truth "has NOTHING TO DO WITH consciousness" [emphasis added] nor that Frege regarded truth as "a [certain kind of] method". Perhaps somewhere Frege wrote something that, in his views, entails such claims, but I'd like to know where. Nor have you supported that Frege held that, among the things propositions are about, reality is not one of them. That does NOT entail that these things have "NOTHING to do with consciousness" [emphasis added] not that truth is "a [certain kind of] method". And if a method, then a method performed by whom or what? The question was not whether some philosophers believe this or that, but rather whether formal logicians in general hold that "propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method." Indeed, your own remarks lead to observing that, for example, Frege does not regard truth as "purely symbol transformation".
  8. The probability is 0. That is a triviality that follows from the fact that there is no object that is both square and round.
  9. David Odden wrote, "For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method." I should think not; quite the contrary. I don't know how anyone would infer that Frege held that "propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method." I am not a Frege expert, but even my modest knowledge of him allows me to recognize that the above quote is quite opposed to Frege's notions, aside from the matter of finding whatever it means to say that truth is a certain kind of method. What specific passages of Frege's do you have in mind? Though, I did see that later in your post you recognized that likely this is not anything of Frege's.
  10. P.S., you wrote, "symbolic logic is totally divorced from cognition, which is why it holds little interest for us." So I asked who 'us' refers to there. Objectivists? And I asked, "You don't think there are Objectivists who are interested in symbolic logic?" You replied, "In fact I myself am interested in a reality-based formalization of reasoning and cognition.' But that's not what I asked. Rather I am interested to know whether by "it holds little interest for us" you mean Objectivists generally, mostly, or uniformly, and whether you rule out that certain Objectivists might be interested in the symbolic logic you find uninteresting.
  11. You wrote, "For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method." I am curious whether you have found this in the literature of the subject; and if so, where. In any case, if by 'formal logician' you mean one who works in the field of formal logic, then it is not a uniform position of formal logicians that "propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method" and probably not even a very prevalent view in the field, moreover that I don't know what author on the subject would describe truth as a [certain kind of] METHOD. No, I made no such claim. As to the tautologies, whatever one holds about deductions being necessarily truth preserving (or even more simply that certain deductive systems are truth preserving), it is not required to subscribe to anything about necessity simply to see that "((P->Q)&P)->Q" is a tautology upon a certain definition of 'tautology'. Yes, and the notion of 'tautology' (in a particular technical sense), in ordinary classical mathematical logic, refers both to semantical and syntactical considerations, but can be reduced to purely syntactical (That is, the set of tautologies can be specified purely syntactically. Note: That is a quite limited technical matter and does NOT in itself to make such a broad claim as "truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method"). Anyway, your comment does not refute any remark I've made. I'll leave to you to say what a "dysfunctional function" is in mathematics. In any case, your * operation is just a garden variety Boolean function. To regard it as "dysfunctional" seems to be an odd way of anthroprormorphizing it. Whatever I think logicians are in their closets, I don't see an attempt to deceive merely by using special terminology, especially as I described what many an introductory textbook says on such subjects as the material conditional. Please don't presume. I say what I mean as clearly as I can. And what I have said is quite limited. I've not made the claimed any sweeping views such as what may be useful to ALL people and as to what "actual" logic is. It was not my intent in that remark to justify a system. I am not interested in performing excercises for you. I just asked what you have in mind with a certain statement of yours. More specifically, I wonder what actual text from book or article in the field of formal logic you are referring to. It's fine if you'd rather not say; I am interested though.
  12. I don't know which particular writers on the subject of formal logic you have in mind. I've never encountered such a view in my own (admittedly limited) readings of authoritative authors on the subject. Though, of course, there are amateurs in the subject who post endorsements of the extreme view you just mentioned. That crude view is not a philosophy of formalism I have seen endorsed by better informed writers; while, quite likely formalism is not even the most prevelent view of modern logicians who deal with formal logic, and certainly not the only one. No, "(P->Q)&(P)&(Q)" is not regarded as a tautology. Maybe you have in mind "((P->Q)&P)->Q)"? That the defintion of 'tautology' is stipulative in that regard is correct. But given that definition, it is proven that "((P->Q)&P)->Q)" is a tautology. That "((P->Q)&P)->Q)" is a tautology as confirmed by a truth table doesn't require any stipulation about necessity. There are 16 binary Boolean functions. That logic books usually focus on only a few of them is arbitrary in a certain sense but reflects mainly that certain of these functions more often enter into everyday argumentation, particular mathematical arguments. Meanwhile, it is fully recognized by logicians that your * operation is indeed a full fledged member of the set of 16 binary Boolean functions. Dishonesty, I would think, is attempt to decieve or mislead. I've not seen grounds to infer such a motivation. Of course, for many words, logicians have special technical definitions that in different ways differ from ordinary, everyday, non-technical definitions of those words. But I've not seen any intent by logicians to fool anyone about that, especially as so many books on logic indeed discuss differences between the technical sense and everyday senses, especially about difference between the technical sense of material implication (that associated with the horseshoe symbol) and the sense of 'implies' in which causality or relevence is at play. Indeed, any informed logician will quite quite readily grant that the notion of material implication does not capture the notion of relevance between antecedent and consequent; that sense is investigated in the study of relevance logic. I don't know of anyone who claims that sentential connectives capture the scope of human cognition. However, it seems that the classical first order system of connectives and quantifiers does permit a formalization of a basic mathematical reasoning; or, at least, that portion of mathematical basic reasoning that we find in the proof of mathematical theorems in the literature. (Of course, that is not to deny the importance of alternatives to classical first order logic.) Such an invention is not REQUIRED for studying, understanding, or using formal logic as formal logic is conveyed in ordinary modern study. I wonder what specifically you have in mind there. Of course, one may find certain things unsuited or interesting, but "totally divorced" strikes as overstatement here. Also, who do you mean by 'us'? Objectivists? You don't think there are Objectivists who are interested in symbolic logic? As to reality, symbolic sentential logic is applied to such subjects as electronic switching circuits, et. al. You are yourself typing at a computer whose advent was enabled by such developments.
  13. Both chords and melody. Both a G minor chord and a G minor melody (one built on a G minor scale) have a "darker" sound than major. I already told you. Because a piece in C major has a cadence G7 to C major while a piece in A minor has a cadence E7 to A minor. Those are both different notes, involved, and different chord structures, and sound different. And the cadences and melodic figures leading to the final cadence are different too. Of course; it would BE the same. But a piece in C major ends on a C note (usually, nearly invariably, I mean) and a piece in A minor ends on an A note, so if both start on an E note then they WON'T be the same pattern of intervals since a path from E to C is not the same as a path from A to C. It's not a matter so much of octaves; and not a matter so much of STARTING note or chord (more important is ENDING note and chord). And two things: a minor chord sounds different from a major chord, because the intervals in a minor chord are different from those in a major chord. Again, two axes of comparison: C major triad: C E G. vs. C minor triad: C Eb G. (same tonic, different modality - called 'parallel minor') C major triad: C E G. A minor triad : A C E. (different tonic, different modality - called 'relative minor') Notice C Eb G and A C E are transpositions of one another. Also, past triads and past even sevenths, minor chords may get different alterations to the extensions. For example, (usually) the eleventh of a major chord needs to be augmented to avoid a minor ninth clash with the third of the chord, whereas the eleventh of a minor chord can be perfect since that forms an acceptable major ninth with the third of the chord. Also, typically in jazz, the ninth of the dominant in minor is different from the ninth of the dominant in major. So these cadences: G7b9 Cm - minor cadence G7 CM - major cadence Also the fifth: G+7 Cm - minor cadence G7 CM - major cadence or both: G+7b9 Cm - minor cadence G7 CM - major cadence
  14. It's not just a matter of the starting note. Do the experiment both times starting on C. That is C natural minor vs. C major. So C D Eb F G Ab Bb C vs. C D E F G A B C They have a very different sound not just for having different notes, but for being a different PATTERN of INTERVALS. And, as I mentioned before, even with the same notes, but starting and ending differently, there is a different sound because a different pattern of intervals: So A B C D E F G A vs. C D E F G A B C sound different even though the same notes, since the pattern of intervals is different. As to a piece just running a scale, of course melody is not just running scales, but the point is that melody is a PATTERN of INTERVALS not just a sequence of notes. I think it would help if you reconceptualize what the basic unit of melody is. The basic unit of melody is not notes, but rather INTERVALS. For example, a melody that goes C D G Eb D, is transposed, e.g., as F G C Ab G. Because the melody is not so much distinguished by the sequence of notes C D G Eb D, but rather by the sequence: up whole step, up perfect 4th, down major 3rd, down half step. That's why we don't need perfect pitch to recognize a melody. We only need relative pitch, the ability to recognize INTERVALS (the difference in pitch between two notes). Perfect pitch is the ability to recognize each note. Most musicians DON'T have perfect pitch. If an ordinary musician is away from his instrument, and you play a note on a piano, then the musician can't tell you whether it's a C or any of the other eleven pitches. But if you play a C and then an Eb, then the musican CAN tell you that is a minor third INTERVAL. And this is why we recognize a melody no matter what key it is played in. If I sing a melody you know but I start on a different note from the one Nat 'King' Cole starts on - but keep the same intervals - you'll recognize the melody sure enough. Your recognition of a melody is not a recognition of certain notes, but rather of the pattern of intervals (I'm not mentioning here the length of the notes and pauses between them, since, for now, I'm just isolating to the question of pitch). As to chords, usually not just two notes played at the same time, but rather, at least three notes (and often more) and not just any three or more notes, but certain combinations.
  15. Yes, it is true, that the set of notes in the A natural minor scale is the same set of notes as in the C major scale (that is, A natural minor is a mode of C major), but the PATTERN OF INTERVALS is different. Melodies are made from INTERVALS not just from notes onto themselves. If you play the A natural minor scale, starting on A and ending on A, then play the C major scale starting on C and ending on C, you will hear the difference, since the PATTERN OF INTERVALS is different, even though the same set of seven notes is used. As to a piece of music being in A minor as opposed to C major, when the piece is in A minor, its final chord is an A minor chord, and when in C major, it's final chord is a C major chord. And the A minor chord is a different set of notes from the C major chord. And it's not just the final chord is such and such but that a piece of music has a certain sound as it is a "story" about reaching its final chord. The piece "moves toward" its final chord; this is the "harmonic movement" which gives the piece so much of its character. And that is most prominent not just in the final chord, but in the final CADENCE (a cadence is a movement through a series of chords toward a certain chord). The final cadence in A minor is E7 to A minor. The final cadence in C major is G7 to C major. The sound of those cadences is distinctly different. And a piece has cadences throughout, not just final cadences. The cadences leading up to the final cadence give the piece its character; and a piece in minor will have different cadences from a piece in major. The best way to understand this is to hear it in conjunction with the theory. Go to a piano. Play the A natural minor scale, then the C major scale. You'll hear the difference. Play an A minor chord, then a C major chord. You'll hear the difference. Play a cadence E7 to A minor, then G7 to C major. You'll hear the difference.
  16. I"m not sure what is meant. Could be referring to the inauguration. Four days late though to match the title of the Rand play "Night Of January 16th".
  17. I haven't read his calculus book. I have gotten some benefit from his three volume history. However, I think his 'Mathematics: The Loss Of Certainty' is not very sharp in its view of incompleteness and related results in mathematical logic.
  18. Intuitionistic logic cannot be 3-valued, nor n-valued for any natural number n. I do not personally know all the details of the proof, but it is a famous result of Godel's. However, there are different 3-valued logics that one can look up in the literature.
  19. Right, and easily provable by induction on the number of variables.
  20. I take it that by xor, you mean exclusive or, which has this truth table: A B AxorB T T F T F T F T T F F F By the way, this is equivalent to ~(P<->Q). The operation is associative, and you're correct that AxorBxorC evaluates as true when A, B, and C are all true. I don't understand. Are you seeking to define a binary operation (an operation on A and that has among three different values (O, T, U)? A truth table for that would like this (fill in your column below 'AxB'): A B AxB T T T O T U O T O O O U U T U O U U / P.S. You've not responded for a long time to my post in the thread in which I gave a proof that there is no function from a set onto its power set. May I take it that your qualms about that matter were thus satisfied as consistent with my last post?
  21. No, we don't suppose that for some b we have d in f(. Rather, we suppose that for some b we have that d IS f(. That is simply the supposition that d is in the range of f, which is required if f is onto Px. I'll restate the proof with more detail: Theorem: There is no function from x onto Px. Proof: Suppose f is a function from x to Px. Let d be the set of b such that b in x and b not in f(. d is a subset of x. If f is onto Px, then, since d is in Px, we have d in the range of f, so for some b in x we have d is f(. Suppose b in f(, then since f( is d, we have b in d, but then b not in f(. So if b in f( then b not in f(. Suppose b not in f(, then since f( is d, we have b not in d, so, since b in x, we have b in f(. So if b not in f( then b in f(. So b in f( iff b not in f(, which is a contradiction. So f is not onto Px. The proof uses only minimal logic (which is even weaker than intuitionistic logic), identity theory, and the axiom schema of separation. Any other proof from axioms of ZFC set theory that there exists a set x and function f from x onto Px would prove that ZF is inconsistent (since if ZFC is inconsistent then ZF is inconsistent). ZF has been under intensely detailed examination by mathematicians of all kinds going on about 90 years now, which makes it at least seem unlikely that any time soon you'll prove that ZF is inconsistent.
  22. Even stronger, that there is no function at all (injective or not) from x onto Px.
  23. No, that's not the argument. However, just the first part, that we can put x 1-1 with a subset of Px is correct (and in the manner you described), and is one half of showing that Px strictly dominates x. And that does use the pairing axiom and the axiom of extensionality (to prove the existence of singletons), also (unless there's a way I'm not thinking of), we need the union axiom and the power set axiom to get a Cartesian product from which to use the axiom schema of separation to form the injection. What I showed was that there is no surjection from x onto Px. Therefore, there is no 1-1 between x and Px. Now, that result and also taking the result you just mentioned, that there is an injection from x into Px, we get that Px strictly dominates x. At what exact step in the proof do you think there are other set theoretical axioms used merely to show that there is no surjection from x onto Px? / Note: I turned off icons (smilely faces) for my proof, but smiley faces are showing up in your quote of my proof.
  24. What I just gave is the diagonal argument. I gave it semi-formally, since that seemed to be the context of the discussion at the time. And the poster wanted to see exactly what axioms are used. Also, I want to impress the fact that there is a proof that is indeed a formal mathematical object, so that 'proof' is taken in that context and not necessarily in certain other senses of the word. The semi-formal proof I gave is sufficient to see that there is a formal mathematical proof of the statement. For just a relaxed natural language rendering: Theorem: There is no function from x onto Px. Proof: Suppose f is a function from x to Px. Let d be the set of b such that b in x and b not in f(. d is a subset of x. Suppose for some b, we have d is f(. So b in f( iff b not in f(, which is a contradiction. So, for no b do we have that d is f(. So d is not in the range of f.
  25. Definition: Px = y <-> Az(zey <-> z subset of x) Theorem: ~Ef(f is a function from x onto Px) Proof: 1. Suppose f is a function from x to Px. 2. Let d = {b | bex & ~b e f(} ... from axiom schema of separation 3. d subset of x ... from 2 and by definition of 'subset of' 4. Suppose Eb(bex & d=f() 5. b e f( <-> ~b e f( ... from 2 and 4 by identity 6. ~Eb(bex & d=f() ... from 4, 5, by contradiction Notes: (1) The definition of 'P' is properly enabled courtesy the power set axiom. But, as I mentioned previously, we could still obtain the gist of the theorem without the power set axiom. (2) In the proof, the only set theoretic axiom used is an instance of the axiom schema of separation. The only logic used is intuitionisic logic (even weaker than classical logic); and, if I'm not mistaken, only minimal logic (a system even weaker than intuitionistic logic) is used.
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