Reduction and Elimination in Philosophy and the Sciences

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Literature Benacerraf, Paul 1965 “What Numbers Could Not Be”, The Philosophical Review, 74(1). Frege, Gottlob 1953 The Foundations of Arithmetic, tr. by J. L. Austin, Blackwell, Oxford. Heck. Richard 1992 “On the Consistency of Second-order Contextual Definitions”, Nous, 26. Benacerraf and Bad Company (An Attack on Neo-Fregeanism) — Michael Gabbay Weir, Alan 2003 “Neo-Fregeanism: An Embarrassment of Riches”, Notre Dame Journal of Formal Logic, 44(1). Wright, Crispin 1983 Frege’s Conception of Numbers as Objects. Aberdeen University Press. Wright, Crispin 2000 “Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint”, Notre Dame Journal of Formal Logic, 41. 113
Deflationism and Conservativity: Who did Change the Subject? Henri Galinon, Paris, France 1. The Problem Deflationists about truth hold that truth is not a substantial property. But what counts as a substantial property? We shall be interested in the thesis that the following is a necessary and sufficient condition for the non-substantiality of truth: 114 (Conservativity) The theory of truth of any given theory A is a conservative extension of A. Suppose that (Conservativity) holds, then the deflationist would have some right to claim that truth is an explanatorily thin property: for it would show that whenever non semantical facts can be explained by a theory being true, they can also be explained by the theory. Suppose (Conservativity) does not hold; then there is a theory A, and sentences in the A-vocabulary that witness nonconservativity; the provability in our theory of truth of those A-unprovable true LA -sentences would constitute some evidence against the deflationary thesis that truth is explanatorily dispensable. As a matter of logical fact, some putative theories of truth have the conservativeness property over some given base theories, whereas others don't. But, the conservativity argument against deflationism continues, conservative theories of truth are not acceptable, because they fail to meet an essential requirement. To be sure, the concept of truth features in all these theories, that is to say a predicate satisfying Tarski's convention-T. But having the concept of truth is not enough for a theory to be the theory of the truth of a given theory. For truth ascriptions come with epistemic commitments, and theories of truth must account for them. Ketland (1999), in particular, has argued that holding a theory to be true is not only to hold all its theorems to be true (distributively, so to speak) but also to hold that all of its theorems are true (resp. collectively). Consequently the truth theory of A has to prove reflection principles for A: For all x, if Pr(x) then T(x). Shapiro (1998) has an argument for a related conclusion. He offers a perfectly natural explanation (I'll say of what in a minute) involving the concept of truth, and argues that in any good theory of truth for A we should be able to carry out the reasonning. The example, unsurprinsingly, involves gödelian phenomena: the Gödel sentence conPA, it is well known, is a true and undecidable statement of PA; but why is it that conPA is true? A natural explanation goes like this, according to Shapiro: […] all the axioms of PA are true, and inference rules preserve truth. Thus every theorems of PA are true. It follows that 0=1 is not a theorem and so PA is consistent. (Shapiro(1998), p.505. Shapiro uses « A » where I write « PA »). As we know, there are arithmetical sentences expressing (under codings) the consistency of PA, and these sentences are not theorems of PA. Shapiro's argument is then that they should be provable in the theory of the truth of PA. To sum up: theories of truth come with some epistemic commitments, and those commitments yield nonconservativity results of truth theories over their base theory; hence truth is not explanatorily thin: knowledge of the truth of an arithmetical theory T yields new arithmetical knowledge beyond T. We agree that from truth ascriptions consistency ascriptions should follow. We shall argue, however, that the conservativity argument against deflationism is flawed 1 . 2. The Fable Let us go into the fantasy of imagining a concurring civilization where people call themselves earthlungs. Earthlungs resemble us in every respect, except that in mathematics they not only study arithmetic, but also have come to recognize the interest and significance of arithmutic. In fact they have come to believe that natural numburs are the real elementary blocks constituting the universe, and they are for this reason much interested in studying them. A partial axiomatization of arithmutic is obtained by PA + ¬conPA, where ¬conPA denotes the negation of a given sentence of the language of PA that is true in N (the standard model of PA) if and only if PA is consistent. Further axioms have been proposed but they are much debated at the moment and so we leave them aside. As it happens, earthlungs mostly use only the PA-part of arithmutic. Moreover, they use the same conventions for formalism as we do when doing logic and, believe it or not, they call PA the partial axiomatisation of arithmutic which is identical to our PA (a nice starting point for a philosophical vaudeville). We will sometimes write PA * to denote their axiomatization and distinguish it from our PA. That is PA * and PA are formally identical but intentionaly different, the first being intended as speaking about numburs, while the second is to be understood as speaking about numbers. Those people also have two Gudule Theorems that, I have to admit, are just as good as our Gödel Theorems. They usually state them as follows: First theorem: If T is a consistent, recursively enumerable and sufficiently rich theory, then T is incomplete. Second theorem If PA is consistent then: PA does not prove ∀x (¬PrPA (x), ⎡0=1⎤) All this is standard on Urth. The second theorem has especially been welcome since it had long been an open question whether ¬conPA* was independent of PA*. Now when they hear us say that G-d-l's theorems show that there are true statements undecidable in PA, they agree, but they do not agree that conPA is one of them 2 ! Now Peter, a guy from here who doesn't know much about earthlungs, once decided to engage Puter, one of theirs, in a conversation about the explanatory power of truth (it was raining hard that Sunday). Here's the conversation. (Caveat: I have tried to disambiguate occurences of 1 Field (1999) has an answer to the conservativity argument and we basically agree with the general lines developped there. We can think of our argument as a variation on his own. We think our version is worth developing, though, since it crucially avoids to take a stand on contentious claims about which axioms are "essential to truth" and which are not (especially in connection with the problem raised by induction axioms involving the truth predicate). 2 This is just because N is not the salient interpretation of PA in earthlung conversational contexts.
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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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Page 68 and 69: The Scapegoat Theory of Causality M
Page 70 and 71: We may have evolved a specialized c
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 78 and 79: an adequate syntactic analysis of o
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
Page 90 and 91: Occam’s Razor in the Theory of Th
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
Page 100 and 101: Does Bradley’s Regress Support No
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
Page 112 and 113: offer a solution to the bad company
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Page 118 and 119: Hard Naturalism and its Puzzles Ren
Page 120 and 121: The Mind-Body-Problem and Score-Kee
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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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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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