Reduction and Elimination in Philosophy and the Sciences

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"PA" by using "PA* ", at least at the begining of the conversation. I wrote PA(*) when I was not sure which one one had in mind.) Peter: Do you believe that the axioms of PA are true? Deflationism and Conservativity: Who did Change the Subject? — Henri Galinon Puter: Yes, I believe that the axioms of PA* are true. Peter: And do you believe inference rules to be truth-preserving? Puter: I do. Peter: You believe then that all of PA's theorems are true? Puter: Indeed. Peter (getting excited): Hence, since PA proves 0≠1, you believe it is true, and thus you believe that PA does not prove 0=1, that is you believe PA to be consistent. Puter: Yes, I do! Peter: You agree, then, that your commitment to the truth of PA is a commitment to its consistency, and that a good theory of truth for PA should account for that? Puter: Yes, it would be nice. Peter (proud) : Look, Tarski's theory of truth for PA, T(PA), is doing precisely this 3 . Puter (sincerly): I know, that's great indeed! 4 Peter: But look: PA does not prove the consistency of PA, while T(PA) does. Truth has an explanatory power, it explains new facts that PA can't account for, facts that are expressible in the language of PA. My acceptance of PA does not logically commit me to the acceptance of conPA, but once I have acknowledged the truth of PA the acceptance of conPA is forced upon me. There's a new purely arithmetical fact which is explained by my truthattribution to PA. Puter (embarrassed): But the consistency of PA (*) is not an arithmutical fact. Peter: I beg you pardon? Puter: Well, I agree that your argument is a sound arithmetic reasoning, but it is not an arithmutic reasoning. First, it is false that the sentence conPA expresses the consistency of PA * in arithmutic. And second, you cannot carry your inductive reasoning out in any sound axiomatization of arithmutic, be it PA * or another theory. This is fortunate, since the negation of conPA is a true arithmutical fact! Peter: I'm not with you here. 3 For reference, T(PA) is the theory obtained by extending PA with the Tarkian recursive axioms for truth, letting the the truth predicate enter the induction scheme. Equivalently one could get a theory of satisfaction. Moreover, such recursive definitions can be turned into explicit definitions provided that we ascend to a richer theory (when it exists) allowing higher-order variables, or proving existence of sets of higher rank than than any set the existence of which is provable in the base theory . See for instance McGee (1991). It is not very important here which of those strong « theory of truth » for PA one has in mind 4 Of course he had understood Peter's claim as: T(PA* ) proves the consistency of PA* . Puter: Well, ok. First things first: the sentence conPA, it is well-known, expresses the consistency of $PA* $ in the sense that it is true in N if and only if PA(*) is consistent. But it is of course not the case that conPA is true in arithmutic if and only if PA is consistent! Now the second point. In your argument you apply induction in the following manner: axioms are true, rules are truth preserving, hence all theorems are true. This is a perfectly correct inference of course, but it belongs to arithmetic, not arithmutic, since the induction involves vocabulary beyond the language of PA. To carry this argument out on the theory PA*, we usually use an axiomatic metatheory containing PA-arithmetic to talk about strings of PA*, plus a recursive truth theory (Tarski style), and we let the truth predicate appear in the meta-induction scheme. Sometimes we also enrich the logical-mathematical part of our metatheory so as to be able to explicitely define the truth predicate for our base theory. In any case, there is in the metalanguage a sentence conPA* which expresses the consistency of PA* in the sense of being true in the intended model of the metatheory if and only if PA* is consistent, which is provable in the metatheory. Peter: I'm not sure that I have understood your point correctly. For arithmetic, arithmutic, or what have you, it remains true that the truth-theory of PA nonconservatively extends PA, doesn't it? Puter: As you can see from my example, T(PA* ) is not conservative over PA*. But my point is that in this case it does not mean anything interesting. Although non-conservative over PA*, T(PA* ) does not explain anything more than PA* in the sense that arithmutic is left the same before and after we ascend to its theory of truth. It is easy to see that under disambiguation of the vocabulary of PA between arithmutic in the base theory and arithmetic in the metatheory, the non-conservativity phenomenon vanishes. Peter: Ok. There may have been a misundertanding. I agree with you that the nonconservativity of T(PA* ) over PA* is meaningless and may just result from a fallacy of equivocation between PA and PA*. But the point remains that the theory of the truth of PA (and by this I now mean explicitely arithmetic PA) should prove the consistency of PA (idem), which PA does not. And in that case, since PA is thought of as a theory of arithmetic, and since you admit that consistency of a formal system is an arithmetical fact, you have to admit that the non-conservativity of T(PA) over PAarithmetic shows truth to have an explanatory power after all! Puter: I don't think so, for the situation is in fact exactly the same as before. Suppose I'm told that the theory PA is true, but that I'm not sure whether it is PA or PA* which is under discussion. I will in any case be able to prove that the theory is consistent in my truth-theoretic metatheory. For not only is T(PA) non-conservative over PA, but so is T(PA* ). It is as it should be since the consistency of PA has nothing to do with the the way we think of PA, it has to do only with its formal features. So whether I interpret PA as arithmetic, arithmutic, or whatever in the metatheory, consistency will follow from truth. Now the further claim that we have so derived a truth pertaining to the field of investigation of our base 115
116 Deflationism and Conservativity: Who did Change the Subject? — Henri Galinon theory, that is arithmetic in the case in point, that claim can only be sustained by an argument to the effect that our metatheory is sound as an arithmetical theory: for in general, it is just not true that the restriction of the truth theory of a theory A to the vocabulary of A is sound for the intended model of A! (remember it was not sound as an arithmutical theory). But how do we know that our metatheory is arithmetically sound? There's nothing in our base theory that can guarantee this. Clearly our recognition of T(PA) as arithmetically sound amounts to our acknowledgement of some systems stronger than PA as being arithmetical systems (T(PA) with unduction on unrestricted vocabulary, or second-order arithmetic in the case of an explicitely defined truth-predicate, etc.). In other words, a claim that T(PA) is non-conservative over PA and arithmetically sound amounts to a bold statement of new axioms for arithmetic above PA. It is not our commitment to truth which is doing the relevant nonconservative job, but our commitment to PA being arithmetic and to T(PA) being a stronger, arithmetically sound, theory. 3. The Moral The moral of this story is simple. If someone knows that PA is true, he can conclude that PA is consistent. But he won't be able to convert this into arithmetical knowledge, that is to derive any new arithmetical fact, unless his arithmetical knowledge outreaches PA from the start. And it seems inevitable then to say that it is this hidden knowledge, which is unfolded in the course of building the truththeory, that does the explanatory work. More generally, knowledge that an interpreted theory A is true yields knowledge that A is consistent. But it will never in itself give any new insights into A-facts, unless one knew from the beginning that A was somehow expressively defective relative to his actual knowledge concerning the intended field of T and had some ways to recognize some extensions of T as being sound relative to his knowledge of Tfacts. Were this last condition not met, how could he be sure that one did not change the subject? It seems fair to conclude that the conservativity argument does not show that truth has any explanatory power by itself. On the contrary, close inspection of the argument tells in favor of the thesis that "true" is a genuine expressive device. Literature Field, Hartry. 1999 "Deflating the conservativness argument", Journal of Philosophy 96, 533–540. Ketland, Jeffrey, 1999 "Deflationism and Tarski’s paradise". Mind 108, 69–94. McGee,Van 1991 Truth, Vagueness and Paradox. Indianalpolis: Hackett. Shapiro, Stewart 1998 "Proof and truth: Through thick and thin", Journal of Philosophy 95, 493–521.
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31 Alexander Hieke Hannes Leitgeb H
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Reduktion und Elimination in Philos
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Distributors Die Österreichische L
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6 Inhalt / Contents Wittgenstein me
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8 Inhalt / Contents The Evolution o
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Formal Mechanisms for Reduction in
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4. Property language and theoretica
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imperatives, or commands in the for
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Referential Practice and the Lure o
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The Augustinian account confuses mo
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hurried to fix it on the page. The
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The Essence (?) of Color, According
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Literature Austin, James 1980 “Wi
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Wittgenstein’s Externalism - Gett
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Informal Reduction E.P. Brandon, Ca
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An Anti-Reductionist Argument Based
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An Anti-Reductionist Argument Based
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Did I Do It? -Yeah, You Did! Wittge
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Literature Did I Do It? -Yeah, You
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tween the two approaches. The task
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On Two Recent Defenses of The Simpl
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On Two Recent Defenses of The Simpl
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Queen Victoria’s Dying Thoughts T
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Diagonalization. The Liar Paradox,
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Proof: Given a function ƒ from con
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that I am sure that it is true that
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experiencing something as a red pen
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There can be Causal without Ontolog
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There can be Causal without Ontolog
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objects. But they also leave unansw
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The Knower Paradox and the Quantifi
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The Knower Paradox and the Quantifi
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Page 68 and 69: The Scapegoat Theory of Causality M
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Page 72 and 73: easoning have their source in the l
Page 74 and 75: Wittgenstein on Frazer and Explanat
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Page 80 and 81: Wittgenstein meets ÖGS: Wovon man
Page 82 and 83: Wittgenstein meets ÖGS: Wovon man
Page 84 and 85: 3. Der Elementarsatz wird als Relat
Page 86 and 87: Literatur Glock, Hans Johann 2006
Page 88 and 89: Explaining the Brain: Ruthless Redu
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Page 92 and 93: Literature Campbell, Donald T. 1987
Page 94 and 95: Die Nichtreduzierbarkeit der klassi
Page 96 and 97: Literatur Die Nichtreduzierbarkeit
Page 98 and 99: Let T I be the minimal theory of tr
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Page 102 and 103: (3) Mutatis mutandis, realism has t
Page 104 and 105: Zeitliche Ontologie und zeitliche R
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Page 108 and 109: The decisive consequences of this a
Page 110 and 111: Benacerraf and Bad Company (An Atta
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Page 114 and 115: Literature Benacerraf, Paul 1965
Page 118 and 119: Hard Naturalism and its Puzzles Ren
Page 120 and 121: The Mind-Body-Problem and Score-Kee
Page 122 and 123: mental properties, for instance, or
Page 124 and 125: kommt es zu einem gravierenden Miss
Page 126 and 127: Reduction Revisited: The Ontologica
Page 128 and 129: 4. Conclusion Reduction Revisited:
Page 130 and 131: All in all, we have seen that reduc
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Page 134 and 135: Literature Cartwright, Nancy 1999:
Page 136 and 137: Wittgensteins Projektionsmethode al
Page 138 and 139: Schlussbemerkungen Wittgensteins Pr
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Page 142 and 143: Relating Theories. Models and Struc
Page 144 and 145: Literature Relating Theories. Model
Page 146 and 147: e invoked in response to the questi
Page 148 and 149: Do Brains Think? Christopher Humphr
Page 150 and 151: 4. Conclusion So do brains think or
Page 152 and 153: How Metaphors Alter the World-Pictu
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Page 156 and 157: Literature van der Auwera, Johan an
Page 158 and 159: 3. The Determination of Form by Syn
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Page 164 and 165: Assessing Humean Supervenience Amir
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determination thesis; it fails as a
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Aus der obigen Definition folgt, da
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Ding-Ontology of Aristotle vs. Sach
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Ding-Ontology of Aristotle vs. Sach
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How do Moral Principles Figure in M
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“Downward Causation”: Emergent,
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“Downward Causation”: Emergent,
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A with distinguished subset Z of te
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A Metaphysically Moderate Version o
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Literature Balashov, Yuri 2002 ''Wh
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“In der Frage liegt ein Fehler”
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Problems with Psychophysical Identi
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identity theory, or as a predecesso
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of the world. It only matters that
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Four Anti-reductionist Dogmas in th
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Four Anti-reductionist Dogmas in th
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notion of time and requires that so
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Some Remarks on Wittgenstein and Ty
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Some Remarks on Wittgenstein and Ty
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concrete particulars can be subject
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phenomenal concept of experiencing
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Metaphorische Bedeutung als virtus
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speaker is then attaching a special
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„Vom Weißdorn und vom Propheten
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„Die Einheit hören“ - Einige
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„Die Einheit hören“ - Einige
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S4) the necessity of arithmetic is
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Getting out from Inside: Why the Cl
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Dispensing with Particulars: Unders
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Literature Dispensing with Particul
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Reichenbach’s Concept of Logical
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Defining Ontological Naturalism Mar
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Literature Jackson, Frank 1986. “
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properties are located. This proces
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‘our conceptual scheme’ “is c
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Properties and Reduction between Me
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Does this mean that metaphysics sho
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do not pick out pain in a system by
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The Writing of Nietzsche and Wittge
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his teachings. He coincides with hi
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sense of a proposition without firs
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Naturalistic Ethics: A Logical Posi
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Literature Kant, Immanuel 2006 Crit
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What we now require is a bridge the
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Species, Variability, and Integrati
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could mention the many sorts of iso
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to work this way. Classic examples
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The Key Problems of KC Matteo Pleba
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have embraced the myth of prose: it
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5. Prior’s Problem As Patrick Bla
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Reductionism in Axiology: the Case
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developing of cancer tissue in huma
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(2) One-way Nagelian bridge princip
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Rethinking the Modal Argument again
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Rethinking the Modal Argument again
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that we are to find the features we
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Atypical Rational Agency Paul Raymo
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Literature Anscombe, G. E. M. 1957
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situation bieten sich keine Kriteri
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Two Reductions of ‘rule’ Dana R
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one that could take place, but does
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physics is not able to make, since
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Wittgenstein’s Attitudes Fabien S
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view of logic, in accordance with h
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Warum man auf transzendentalphiloso
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Making the Mind Higher-Level Elizab
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Literature Churchland, P. M. (1995)
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Zwischen Humes Gesetz und „Sollen
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Mental Causation: A Lesson from Act
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Again, this is mistaken. Non-reduct
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auf alle Arten von kontingenten Pro
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Reduction, Sets, and Properties Ben
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Literature Egan, Andy (2004): ‘Se
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there is a tension between evaluati
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The Elimination of Meaning in Compu
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sufficient to explain all of the em
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that 'A' and 'A' could never be tok
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the necktie sense) can differ in wh
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Supervenience and ‘Should’ Arto
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word 'norm' can be thought to expre
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Rule-following as Coordination: A G
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vast majority are bent: gruesome, g
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human life, the violent condition o
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A Division in Mind. The Misconceive
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A Division in Mind. The Misconceive
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could then deduce (1c) and conclude
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Neutral Monism. A Miraculous, Incoh
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Neutral Monism. A Miraculous, Incoh
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A somewhat Eliminativist Proposal a
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Impliziert der intentionale Redukti
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Impliziert der intentionale Redukti
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Structure of the Paradoxes, Structu
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Structure of the Paradoxes, Structu
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that he was working on a translatio
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Objects of Perception, Objects of S
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Objects of Perception, Objects of S
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Let us take the example of the hypo
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Reducing Sets to Modalities Rafał
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variables ai that (under an interpr
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Are Lamarckian Explanations Fully R
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A Note on Tractatus 5.521 Nuno Vent
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And he goes on saying, referring to
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The Place of Theory Reduction in th
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Ethik als irreduzibles Supervenienz
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unabhängig von denen ihrer Mitglie
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Das ‘schwierige Problem’ des Be
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The Supervenience Argument, Levels,
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The Supervenience Argument, Levels,
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their relations both to the physica
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On the Characterization of Objects
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On the Characterization of Objects
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The Functional Unity of Special Sci
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size, location, temporal characteri
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Transcendental Philosophy and Mind-
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From Topology to Logic. The Neural
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From Topology to Logic. The Neural
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The Calculus of Inductive Construct
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The Four-Color Theorem, Testimony a
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experience. Of course, a detailed a
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Clearly x = y → x = ext y holds,
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Intentional Fundamentalism Petri Yl
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The second dimension to be consider
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The very existence of the special s
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Heil, J. (1999), “Multiple Realiz
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independent of what is the fact, fa
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415
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