Reduction and Elimination in Philosophy and the Sciences

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Literature Egan, Andy (2004): ‘Second-Order Quantification and the Metaphysics of Properties’, Australasian Journal of Philosophy 82, 48– 66. Fine, Kit (1994): ‘Essence and Modality’, in: Philosophical Perspectives 8, 1–16. Fine, Kit (1995): ‘Ontological Dependence’, Proceedings of the Aristotelian Society 95, 269–90. Reduction, Sets, and Properties — Benjamin Schnieder Jackson, Frank (1998), From Metaphysics to Ethics, Oxford: Clarendon Press. Künne, Wolfgang (2003): Conceptions of Truth, Oxford: Clarendon Press. Lewis, David (1986): On the Plurality of Worlds, Oxford: Blackwell. Schnieder, Benjamin (2004): Substanzen und (ihre) Eigenschaften, Berlin: de Gruyter. 309
Context-Based Approaches to the Strengthened Liar Problem Christine Schurz, Salzburg, Austria Kripke´s paper (Kripke 1975) was the starting point for numerous formal languages that are able to express a theory of their own concept of truth without producing a liar paradox. However these theories are not regarded as modeling the concept of truth of natural language. The main reason for this is the strengthened liar paradox, or rather a formalized version of this problem. Whether it is possible to solve this problem without being committed to something (more or less) like Tarski´s hierarchical proposal (Tarski 1935), is still a highly disputed matter. The majority of those who seek for alternatives to Tarski´s proposal attempt to analyze the strengthened liar problem by uncovering certain context-dependent elements in a strengthened liar sentence. This paper consists of two parts. First, I shall outline a formal version of the strengthened liar. Thereafter I take a look at the context-based approach to this problem. From now on, I shall assume a formal language L of first order logic that is interpreted by a structure M. M is assumed to contain names that are assigned to the sentences of L, and functions that represent a certain amount of the syntax of L. The symbol Tr is a one-place predicate of L that is intended to represent “truth in L”. By the fixed-point-lemma a strengthened liar sentence, i.e., a sentence φ of the form ¬Tr(l) such that M╞ l = ‘¬Tr(l)’ can be formed (the term ‘¬Tr(l)’ is the object-language-name of the formula ¬Tr(l)). I shall assume a theory T of “truth in L” such that neither T├ φ nor T├ ¬φ. Such a theory results e.g. from the axioms of (Friedman and Sheard 1987) which define Tr together with some axioms of a theory of M (if, for instance, M is a model of arithmetic, then we can take the axioms of Robinson´s arithmetic). The theory T matches with the intuition that the strengthened liar sentence is neither true nor false in as much as it gives us no information concerning the truthvalue of φ. But in context of natural language we are also capable of expressing this intuition by a meaningful sentence and furthermore infer other statements from such a sentence (which together lead to the natural language strengthened liar argument). Thus, if we seek T to be as close as possible to the concept “is true” of natural language, we have to investigate whether it is possible for T to contain any formula of L which represents a semantical diagnosis as “φ is neither true nor false” of φ. Is it possible to add such a formula to the axioms of T without causing T to be inconsistent? This is indeed an intricate problem. The first problem: Usually the formula ¬Tr(‘φ’) is taken to be a most appropriate candidate to represent a diagnostic statement about φ. Since φ fails to be derivable from T, it also fails to be true according to T (otherwise it would be derivable from T), which can in L be represented by the formula ¬Tr(‘φ’). So ¬Tr(‘φ’) should be derivable from T. But, on the other hand, we have a “falsity-intuition” concerning ¬Tr(‘φ’), taking ¬Tr(‘φ’) to represent the sentence “φ is false”. Since T gives us no information whether φ is true or false, also no formula representing “φ is false” will be derivable from T. Therefore ¬Tr(‘φ’) should not be derivable from T. So in the end we have a “failureintuition” and a “falsity-intuition” concerning the sentence ¬Tr(‘φ’). Both intuitions display features of how we 310 informally reason about the strengthened liar in natural language. So they do not have to be considered as rival intuitions of which we have to select only one that represents our “actual” reasoning or the way we should rationally reason. Maybe, if L is supposed to account for natural language, one has to find a way to model both of these intuitions and explain how T├ ¬Tr(‘φ’) as well as not T├ ¬Tr(‘φ’) can be the case. But obviously this can only be consistently realized if we assume some context-sensitive element in the sentence ¬Tr(‘φ’). Before I turn to the second problem, let me note that one could object against the analysis I just gave that it shows that ¬Tr(‘φ’) is after all no appropriate choice for a formula representing a semantical diagnosis of φ. But then the only alternative is to introduce a new “untrueness”predicate U to express a formula belonging to T that represents a statement about φ. But of course this gives rise to a new strengthened liar sentence φ' of the form U(l') such that M╞ l' = ‘U(l')’ that leads to just the same problems. So in the end we would have just the same conflict of failure- against falsity-intuition. The second problem: This problem results from our attempt to model the ‘failure-intuition’ for ¬Tr(‘φ’). According to this attempt, T contains the sentence ¬Tr(‘φ’), but, on the other hand, it should not contain φ. These conditions contradict to the rule of substitutivity of identity (I shall from now on write “(SoI)” to refer to this rule): (SoI) Substitutivity of identity: Let t1 and t2 be any two terms, let P be any oneplace predicate. Then the following rule is valid: t1= t2, P(t1) P(t2) We have to make a decision whether to give up (SoI) or again put forward an explanation in terms of a contextshift. We might even again employ the context-shift we have already posited in “¬Tr(‘φ’)” occurring in the argument based on the failure-intuition and the same formula occurring in the argument based on the falsity-intuition. Then we not only suppose two different contextual interpretations of “¬Tr(‘φ’)”, but also two different contextual interpretations of φ. Indeed some proponents of contextual approaches (e.g. (Burge 1979) and (Simmons 1993)) argue for another tension between two semantical views of the strengthened liar sentence φ. But this point is controversial. After all, φ is in the first place assumed to be an entirely pathological sentence to which no classical truth value can be assigned. ¬Tr(‘φ’) is intended to reflect this assumption. So if we finally refute that φ is entirely pathological but consider φ to be also true according to another reading, then we refute the basic observation of our whole reasoning. To conclude, in context of our formal language L, the strengthened liar paradox is constituted by (at least) two tensions in our metalinguistic reasoning about φ. Firstly, the tension between evaluating ¬Tr(‘φ’) as true according the failure-intuition, and evaluating it as neither true nor false according to the falsity-intuition. Secondly,
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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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lent. It also implies that an inven
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The Scapegoat Theory of Causality M
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We may have evolved a specialized c
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easoning have their source in the l
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Wittgenstein on Frazer and Explanat
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genstein 1993, p. 143). In order to
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an adequate syntactic analysis of o
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Wittgenstein meets ÖGS: Wovon man
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Wittgenstein meets ÖGS: Wovon man
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3. Der Elementarsatz wird als Relat
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Literatur Glock, Hans Johann 2006
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Explaining the Brain: Ruthless Redu
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Occam’s Razor in the Theory of Th
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Literature Campbell, Donald T. 1987
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Die Nichtreduzierbarkeit der klassi
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Literatur Die Nichtreduzierbarkeit
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Let T I be the minimal theory of tr
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Does Bradley’s Regress Support No
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(3) Mutatis mutandis, realism has t
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Zeitliche Ontologie und zeitliche R
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Betrachtung ontologischer Fragen, M
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The decisive consequences of this a
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Benacerraf and Bad Company (An Atta
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offer a solution to the bad company
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Literature Benacerraf, Paul 1965
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"PA" by using "PA* ", at least at t
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Hard Naturalism and its Puzzles Ren
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The Mind-Body-Problem and Score-Kee
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mental properties, for instance, or
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kommt es zu einem gravierenden Miss
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Reduction Revisited: The Ontologica
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4. Conclusion Reduction Revisited:
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All in all, we have seen that reduc
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Physicalism Without the A Priori Pa
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Literature Cartwright, Nancy 1999:
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Wittgensteins Projektionsmethode al
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Schlussbemerkungen Wittgensteins Pr
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intentional states requires functio
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Relating Theories. Models and Struc
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Literature Relating Theories. Model
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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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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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Page 272 and 273: Rethinking the Modal Argument again
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Page 276 and 277: that we are to find the features we
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Page 280 and 281: Literature Anscombe, G. E. M. 1957
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Page 284 and 285: Two Reductions of ‘rule’ Dana R
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Page 290 and 291: Wittgenstein’s Attitudes Fabien S
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Page 296 and 297: Making the Mind Higher-Level Elizab
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Page 302 and 303: Mental Causation: A Lesson from Act
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Page 342 and 343: A somewhat Eliminativist Proposal a
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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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