Reduction and Elimination in Philosophy and the Sciences

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Literatur Die Nichtreduzierbarkeit der klassischen Physik auf quantentheoretische Grundbegriffe — Helmut Fink Busch, Paul, Lahti, Pekka und Mittelstaedt, Peter, ²1996 The Quantum Theory of Measurement, Berlin: Springer. Joos, Erich et al. ²2003 Decoherence and the Appearance of a Classical World in Quantum Theory, Berlin: Springer. Ludwig, Günther ²1990 Die Grundstrukturen einer physikalischen Theorie, Berlin: Springer. Ludwig, Günther und Thurler, Gerald 2005 A New Foundation of Physical Theories, Berlin: Springer. Mittelstaedt, Peter 1986 Sprache und Realität in der modernen Physik, Mannheim: BI. Mittelstaedt, Peter 1998 The Interpretation of Quantum Mechanics and the Measurement Process, Cambridge: Cambridge University Press Mittelstaedt, Peter und Weingartner, Paul 2005 Laws of Nature, Berlin: Springer. Scheibe, Erhard 1997 Die Reduktion physikalischer Theorien. Teil I: Grundlagen und elementare Theorie, Berlin: Springer. Scheibe, Erhard 1999 Die Reduktion physikalischer Theorien. Teil II: Inkommensurabilität und Grenzfallreduktion, Berlin: Springer. 95
Interpretability Relations of Weak Theories of Truth Martin Fischer, Leuven, Belgium 1. Introduction Axiomatic theories of truth are understood as extensions of a syntactic base theory which is often taken to be Peano Arithmetic, PA . One way to measure the strength of a theory of truth is to take into account which formulas of arithmetic it proofs. Weak theories in this respect are theories that do not prove more than PA itself, theories that are conservative extensions of PA . The concept of conservativity has gained some interest in formalizing philosophical criteria. This is also the case in the debate on truth, in which conservativity is expected to explain the `no substance' claim of deflationism. For theories of truth conservativity over PA alone seems to be a very crude measure since it does not differentiate between different conservative theories of truth which have quite different properties and prove different formulas containing the truth predicate. A comparison of the truth-theoretic strength of theories of truth is desirable. A direct comparison of the truth-theoretic strength is the subset relation but it is only a partial order so that not all theories can be compared. Another measure of the strength of a theory of truth would be their interpretability relations to other theories especially their interpretability or noninterpretability in PA . The most famous of these interpretability relations is relative interpretability, introduced in (Tarski et al. 1953), and it is a good measure for PA as base theory. On the one hand the less restricted version of local interpretability collapses in this case into relative interpretability. On the other hand Tarskis theorem of undefinability of truth shows that there is no definitional extension of PA by a one place predicate τ , PA (τ ), so that PA (τ ) proves τ( ϕ) ↔ ϕ for all sentences ϕ of the language of arithmetic. 2. Axiomatic theories LA is the language of arithmetic and Lτ : = A {τ ∪ . The arithmetical theories are as usual. For the interpretability considerations take the arithmetic theories to be formulated with predicate- instead of functionsymbols. 96 L } ( Ind P ) P( 0) ∧ ∀x( P( x) → P( x + 1)) → ∀x( P( x)) Q is Robinson Arithmetic and PA is Peano Arithmetic, that is P A L Ind Q ∪ ) ⋅ ( , where P A L Ind ⋅ ) ( is the set of sentences that result from replacing P in ( Ind P ) by a formula of L A with at least one free variable. Accordingly, IΣ k = Q ∪ ( IndP ) ⋅ Σk . Assume that L A contains the relevant syntactical vocabulary: ‘ Ct ’ for closed term of L A , ‘ Sent ’ for sentence of L A , ‘ Form 1’ for formula of LA with one free variable, and so on, such that PA proves the relevant syntactical theorems. Especially if m is the gödelnumber of a formula ϕ (x) with one free variable x and n of a term t , then m (n) is the gödelnumber of the substitution τ of the free variable x in ϕ (x) by the numeral of t . PA is PA formulated in the language L τ . A theory of truth T is τ a Lτ -theory with PA ⊆ T . tot ( x) : ⇔ Form1( x) ∧ ∀y( τ ( x( y)) ∨τ ( ¬ & x( y ))) Disquotational theories of truth are formulated with a scheme of T-biconditionals: ( TB P ) τ ( P) ↔ P ( UTB P ) ∀x( τ ( P( x& ) ) ↔ P( x)). Compositional axioms are the universally quantified versions of the following formulas: (C 1) Ct ( x) ∧ Ct( y ) → ( τ ( x = & y) ↔ val( x) = val( y)). (C 2) Ct( x) ∧ Ct( y) → ( τ ( x ≠& y) ↔ val( x) ≠ val( y)). (C 3) Sent( x) ∧ Sent( y) → ( τ( x ∧& y ) ↔ τ( x) ∧τ ( y)). (C 4) Sent ( x) ∧ Sent( y) → ( τ ( ¬ & x ∧& y) ↔ τ( ¬ & x) ∨ τ ( ¬ & y)). (C 5) Sent( ∀& yx) → ( τ( ∀& yx) ↔ ∀z( τ ( x( z)))). (C 6) Sent ( ¬ & ∀& yx) → ( τ ( ¬ & ∀& yx) ↔ ∃z( τ ( ¬ & x( z)))). (C 7) Sent( x) → ( τ ( ¬ & ¬ & x) ↔ τ ( x)). (C 8) Sent( x) → ( τ ( ¬ & x) ↔ ¬ τ ( x)). The axiom of internal induction for total formulas is: ( It I) ∀ x( tot( x) ∧τ ( x( 0)) ∧ ∀y( τ( x( y )) → τ( x( y + 1))) → ∀y( τ( x( y )))). The relevant theories are: TB : = Q ∪ ( TBP ) ⋅ LA ∪ ( IndP ) ⋅ Lτ UTB : = Q ∪ ( UTBP ) ⋅ LA ∪ ( IndP ) ⋅ Lτ PT : PA ( C1) ( C7) r = ∪ − − PT : = PA ∪ ( C1) − ( C7) ∪ ( ItI ) PT : = PA ∪ ( C1) − ( C7) ∪ ( IndP ) ⋅ Lτ TC : PA ( C1) ( C8) r = ∪ − − TC : = PA ∪ ( C1) − ( C8) ∪ ( ItI ) r TC is also known as PA (S) and TC as T (PA) . 3. Interpretability Some basic results: (i) TB ⊂ UTB ⊂ PT = TC − (ii) IΣ1 ∪ ( C1), ( C3), ( C5), ( C8) ∪ ( It I) = TC − (iii) IΣ1 ∪ ( C1) − ( C7) ∪ ( It I) = PT − − (iv) PT , TC are finitely axiomatizable. r r (v) TB, UTB, PT , TC , PT, TC are not finitely axiomatizable. r r − (vi) TB, UTB, PT , TC , PT are conservative extensions of PA . − (vii) TC , PT, TC are nonconservative extensions of PA . Definition Let S, T be theories formulated in L S, LT . Then T is a pure extension of S iff T is an extension of S and L T = LS . T is reflexive iff T proves Con Δ for all finite Δ ⊆ T. T is essentially reflexive iff all pure extensions of T are reflexive. T has full induction iff for all formulas ϕ of L T : T proves ϕ( 0) ∧ ∀x( ϕ( x) → ϕ( x + 1)) → ∀xϕ( x) . Full induction and reflexivity are connected in the following way, as shown for example in (Hájek/Pudlák 1993, p.189): Lemma 1 If PA ⊆ T and T has full induction, then T is reflexive.
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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
Page 46 and 47: Queen Victoria’s Dying Thoughts T
Page 48 and 49: Diagonalization. The Liar Paradox,
Page 50 and 51: Proof: Given a function ƒ from con
Page 52 and 53: that I am sure that it is true that
Page 54 and 55: experiencing something as a red pen
Page 56 and 57: There can be Causal without Ontolog
Page 58 and 59: There can be Causal without Ontolog
Page 60 and 61: objects. But they also leave unansw
Page 62 and 63: The Knower Paradox and the Quantifi
Page 64 and 65: The Knower Paradox and the Quantifi
Page 66 and 67: lent. It also implies that an inven
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
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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
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Page 114 and 115: Literature Benacerraf, Paul 1965
Page 116 and 117: "PA" by using "PA* ", at least at t
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
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e invoked in response to the questi
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Do Brains Think? Christopher Humphr
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4. Conclusion So do brains think or
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How Metaphors Alter the World-Pictu
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The Modal Supervenience of the Conc
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Literature van der Auwera, Johan an
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3. The Determination of Form by Syn
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Zwischen Humes Gesetz und „Sollen
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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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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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