Reduction and Elimination in Philosophy and the Sciences

More documents

Recommendations

Info

experience. Of course, a detailed account of the a priori on the basis of a broad conception of the enabling role of experience would be beyond the scope of this paper. Literature Arkoudas, Konstantineand Selmer Bringsjord 2007. ‘Computers, Justification, and Mathematical Knowledge.’ Minds and Machines, 17, 185-202. Borge, Steffen 2003. ‘The Word of Others.’ Journal of Applied Logic, 1, 107-118. Brown, James R. 1999. Philosophy of Mathematics:An Introduction to the World of Proofs and Pictures, Routledge.. The Four-Color Theorem, Testimony and the A Priori — Kai-Yee Wong Burge, Tyler 1998. ‘Computer Proof, A Priori Knowledge, and Other Minds.’ Philosophical Perspectives, 12, 1-37. Burge, Tyler 1993. ‘Content Preservation.’ Philosophical Review, Vol. 102, No. 4, 457-488. Chisholm, Roderick M 1980. Theory of Knowledge, 3 rd ed., Prentice Hall. Coady, C. A. J. 1992. Testimony, Clarendon Press, Oxford Detlefsen, Michael and Mark Luker (1980). ‘The Four-Color Theorem and Mathematical Proof.’ Journal of Philosophy 77, 803-820. Teller, Paul 1980. ‘Computer Proof.’ Journal of Philosophy 77, 797- 802. Tymoczko, Thomas 1979. ‘The Four-Color Problem and its Philosophical Significance.’ Journal of Philosophy, 76, 57-83. 401
The Comprehension Principle and Arithmetic in Fuzzy Logic Shunsuke Yatabe, Toyonaka, Osaka, Japan 1 Introduction Shapiro and Weir showed that Hume’s principle has a finite model (so it can not deduce Peano arithmetic PA) in Aristotelian logic which does not admit the existence of the empty property [Shapiro, Weir 2000]. This shows that Hume’s principle itself is not enough to develop arithmetic, and we should have a certain theory of properties to realize Frege’s program [Hale 2008]. In this paper, we investigate a theory of property which satisfies what Myhill called Frege’s principle [Myhill 1984], that “every formula with one free variable determines a property (not a set) which holds of all those and only those things which satisfy the formula”, and we examine how much arithmetic we can develop by it. It is well-known that it is impossible in classical logic: the comprehension principle, ∀ x[ x ∈ { y : P( y)} ≡ P( x)] for any P, implies a contradiction by the Russell paradox. However, in non-classical logic, the situation is slightly different. Many first order logics, including fuzzy logic, have been known to imply no contradiction when the comprehension principle is assumed. So we can regard any singular term as an object in such logic (and this plays an important role in Fregean Platonism). Therefore, as Myhill, we can say: 402 So if we want to take Frege’s principle seriously, we must begin to look at some kind of non-classical logic. Let us consider the case of the set theory H with the comprehension principle in Lukasiewicz infinite-valued predicate logic ∀ L . It is a version of fuzzy logic, and is a nonclassical logic weaker than classical logic which only has a weak fragment of the contraction rule. White showed that H is consistent [White 1979], and it is known as the strongest theory among set theories with the comprehension principle. We highlight two special features of sets (or properties) in H which might provide a clue of analysis: non-extensionality and full circularity. First, the basic law V does not hold in H. Let us remember the case of classical logic: Frege’s basic law V, ( ∀ P)( ∀Q)[ ext( P) = ext( Q) ≡ ( ∀x)[ P( x) ≡ Q( x)]] and the definition of membership relation imply the comprehension principle and it implies a contradiction. Neo- Fregeans modified the basic law V and adopted Restricted-V (RV) when they developed a set theory in classical logic [Shapiro 2003]. In H, the basic law V is equivalent to the axiom of extensionality (which insists the extensional equality is equivalent to the Leibniz equality), and it implies a contradiction (it is called Grisin’s paradox [Grisin 1982]). Furthermore, any version of RV has not been known to be consistent to H yet. Second, H forgives circular definitions. It is because H proves a general form of recursive definition, which is a circular definition of a very strong shape, is permitted: ( ∀ x)( ∃z)[ x ∈ z ≡ φ( x, z)] for any formula φ [Cantini 2003]. Here, we define a set z by using z itself. Since the recursive definition is an essence of computation, a certain amount of arithmetic can be developed: we can define a graph of any recursive function in H. However, the arithmetic developed in H is not a conservative extension of PA: the mathematical induction scheme implies a contradiction in H [Yatabe 2007]. In fact, we can show, in any model of H, the sentence which can be interpreted as “ω contains a nonstandard natural number” is truth-value 1. On a final note we remark about an adaptation of non-classical logic. As for an antecedent, Dummett lapsed classical logic for his own philosophical purposes, antirealism. In this sense, adherence to classical logic is not what is should be. Furthermore, this suggests that, we may argue that some rule (the law of excluded middle in his case) of classical logic corresponds a certain philosophical viewpoint (as realism). We might ask what kind of a philosophical assumption corresponds to the contraction rule as a future task 1 . 2 The set theory H, extensionality and the basic law V The comprehension principle will derive a contradiction in classical logic. However, the contraction rule is essential to derive it. Grisin proved that the comprehension principle derives no contradiction in the system Grisin logic which is classical logic minus the contraction rule [Grisin 1982]. So, the next question is, where the limit is: what is the strongest logic, between Girisin logic and classical logic, which does not derive a contradiction? Currently, the strongest logic is known to be Lukasiewicz infinite-valued predicate logic ∀ L [White 1979]. This system is known to be impossible to recursively axiomatize, so we introduce the definition of its model (because of the luck of space, here we introduce the quite informal one: we note that, this is a simplification of ( [ 0, 1], *, ⇒, 0, 1) -structure where ( [ 0, 1], *, ⇒ , 0, 1) forms a standard MV algebra [Hajek 2001]). (1) The truth value set is [0, 1] of real numbers (it is a kind of fuzzy logic). (2) ⊥ = 0 , φ → ϕ = min( 11, − φ + ϕ ) (3) ( ∀x) ϕ ( x) = inf{ ϕ( a) : a ∈ M } The rest connectives are defined by using →, ⊥ (for example ¬ A is A →⊥ and A ∧ B is ¬ ( A → ¬ ( A → B)) ). We note that, we have the multiplicative conjunction ⊗ in ∀ L : A ⊗ B = max{ 0, A + B −1} . It is easy to see that A → A ⊗ A is truth value 1 if and only if A is truth value 0 or 1. We write A ≡ B instead of ( A → B) ⊗ ( B → A) . Let H be a set theory whose only axiom scheme is the comprehension principle (i.e. ∀ x[ x ∈ { y : P( y)} ≡ P( x)] is truth value 1 for any P in any its model). We introduce two kinds of equality in H. Leibniz equality: x = y if and only if ( ∀ z)[ x ∈ z ≡ y ∈ z] , Extensional equality: x = ext y if and only if ( ∀ z)[ z ∈ x ≡ z ∈ y] 1 To answer this, we note that the contraction rule plays an essential role to imply a contradiction not only in the Russell paradox but also in the liar paradox and in the sorites paradox. Therefore we seem to need a unified framework to analyze them.
Page 1 and 2:
31 Alexander Hieke Hannes Leitgeb H
Page 3 and 4:
Reduktion und Elimination in Philos
Page 5 and 6:
Distributors Die Österreichische L
Page 7 and 8:
6 Inhalt / Contents Wittgenstein me
Page 9 and 10:
8 Inhalt / Contents The Evolution o
Page 12 and 13:
Formal Mechanisms for Reduction in
Page 14 and 15:
4. Property language and theoretica
Page 16 and 17:
imperatives, or commands in the for
Page 18 and 19:
Referential Practice and the Lure o
Page 20 and 21:
The Augustinian account confuses mo
Page 22 and 23:
hurried to fix it on the page. The
Page 24 and 25:
The Essence (?) of Color, According
Page 26 and 27:
Literature Austin, James 1980 “Wi
Page 28 and 29:
Wittgenstein’s Externalism - Gett
Page 30 and 31:
Informal Reduction E.P. Brandon, Ca
Page 32 and 33:
An Anti-Reductionist Argument Based
Page 34 and 35:
An Anti-Reductionist Argument Based
Page 36 and 37:
Did I Do It? -Yeah, You Did! Wittge
Page 38 and 39:
Literature Did I Do It? -Yeah, You
Page 40 and 41:
tween the two approaches. The task
Page 42 and 43:
On Two Recent Defenses of The Simpl
Page 44 and 45:
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
Page 76 and 77:
genstein 1993, p. 143). In order to
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
Page 106 and 107:
Betrachtung ontologischer Fragen, M
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
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
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
Page 132 and 133:
Physicalism Without the A Priori Pa
Page 134 and 135:
Literature Cartwright, Nancy 1999:
Page 136 and 137:
Wittgensteins Projektionsmethode al
Page 138 and 139:
Schlussbemerkungen Wittgensteins Pr
Page 140 and 141:
intentional states requires functio
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
Page 154 and 155:
The Modal Supervenience of the Conc
Page 156 and 157:
Literature van der Auwera, Johan an
Page 158 and 159:
3. The Determination of Form by Syn
Page 160 and 161:
Zwischen Humes Gesetz und „Sollen
Page 162 and 163:
Page 164 and 165:
Assessing Humean Supervenience Amir
Page 166 and 167:
determination thesis; it fails as a
Page 168 and 169:
Aus der obigen Definition folgt, da
Page 170 and 171:
Ding-Ontology of Aristotle vs. Sach
Page 172 and 173:
Ding-Ontology of Aristotle vs. Sach
Page 174 and 175:
How do Moral Principles Figure in M
Page 176 and 177:
“Downward Causation”: Emergent,
Page 178 and 179:
“Downward Causation”: Emergent,
Page 180 and 181:
A with distinguished subset Z of te
Page 182 and 183:
A Metaphysically Moderate Version o
Page 184 and 185:
Literature Balashov, Yuri 2002 ''Wh
Page 186 and 187:
“In der Frage liegt ein Fehler”
Page 188 and 189:
Problems with Psychophysical Identi
Page 190 and 191:
identity theory, or as a predecesso
Page 192 and 193:
of the world. It only matters that
Page 194 and 195:
Four Anti-reductionist Dogmas in th
Page 196 and 197:
Four Anti-reductionist Dogmas in th
Page 198 and 199:
notion of time and requires that so
Page 200 and 201:
Some Remarks on Wittgenstein and Ty
Page 202 and 203:
Some Remarks on Wittgenstein and Ty
Page 204 and 205:
concrete particulars can be subject
Page 206 and 207:
phenomenal concept of experiencing
Page 208 and 209:
Metaphorische Bedeutung als virtus
Page 210 and 211:
speaker is then attaching a special
Page 212 and 213:
„Vom Weißdorn und vom Propheten
Page 214 and 215:
„Die Einheit hören“ - Einige
Page 216 and 217:
„Die Einheit hören“ - Einige
Page 218 and 219:
S4) the necessity of arithmetic is
Page 220 and 221:
Getting out from Inside: Why the Cl
Page 222 and 223:
Dispensing with Particulars: Unders
Page 224 and 225:
Literature Dispensing with Particul
Page 226 and 227:
Reichenbach’s Concept of Logical
Page 228 and 229:
Defining Ontological Naturalism Mar
Page 230 and 231:
Literature Jackson, Frank 1986. “
Page 232 and 233:
properties are located. This proces
Page 234 and 235:
‘our conceptual scheme’ “is c
Page 236 and 237:
Properties and Reduction between Me
Page 238 and 239:
Does this mean that metaphysics sho
Page 240 and 241:
do not pick out pain in a system by
Page 242 and 243:
The Writing of Nietzsche and Wittge
Page 244 and 245:
his teachings. He coincides with hi
Page 246 and 247:
sense of a proposition without firs
Page 248 and 249:
Naturalistic Ethics: A Logical Posi
Page 250 and 251:
Literature Kant, Immanuel 2006 Crit
Page 252 and 253:
What we now require is a bridge the
Page 254 and 255:
Species, Variability, and Integrati
Page 256 and 257:
could mention the many sorts of iso
Page 258 and 259:
to work this way. Classic examples
Page 260 and 261:
The Key Problems of KC Matteo Pleba
Page 262 and 263:
have embraced the myth of prose: it
Page 264 and 265:
5. Prior’s Problem As Patrick Bla
Page 266 and 267:
Reductionism in Axiology: the Case
Page 268 and 269:
developing of cancer tissue in huma
Page 270 and 271:
(2) One-way Nagelian bridge princip
Page 272 and 273:
Rethinking the Modal Argument again
Page 274 and 275:
Rethinking the Modal Argument again
Page 276 and 277:
that we are to find the features we
Page 278 and 279:
Atypical Rational Agency Paul Raymo
Page 280 and 281:
Literature Anscombe, G. E. M. 1957
Page 282 and 283:
situation bieten sich keine Kriteri
Page 284 and 285:
Two Reductions of ‘rule’ Dana R
Page 286 and 287:
one that could take place, but does
Page 288 and 289:
physics is not able to make, since
Page 290 and 291:
Wittgenstein’s Attitudes Fabien S
Page 292 and 293:
view of logic, in accordance with h
Page 294 and 295:
Warum man auf transzendentalphiloso
Page 296 and 297:
Making the Mind Higher-Level Elizab
Page 298 and 299:
Literature Churchland, P. M. (1995)
Page 300 and 301:
Page 302 and 303:
Mental Causation: A Lesson from Act
Page 304 and 305:
Again, this is mistaken. Non-reduct
Page 306 and 307:
auf alle Arten von kontingenten Pro
Page 308 and 309:
Reduction, Sets, and Properties Ben
Page 310 and 311:
Literature Egan, Andy (2004): ‘Se
Page 312 and 313:
there is a tension between evaluati
Page 314 and 315:
The Elimination of Meaning in Compu
Page 316 and 317:
sufficient to explain all of the em
Page 318 and 319:
that 'A' and 'A' could never be tok
Page 320 and 321:
the necktie sense) can differ in wh
Page 322 and 323:
Supervenience and ‘Should’ Arto
Page 324 and 325:
word 'norm' can be thought to expre
Page 326 and 327:
Rule-following as Coordination: A G
Page 328 and 329:
vast majority are bent: gruesome, g
Page 330 and 331:
human life, the violent condition o
Page 332 and 333:
A Division in Mind. The Misconceive
Page 334 and 335:
A Division in Mind. The Misconceive
Page 336 and 337:
could then deduce (1c) and conclude
Page 338 and 339:
Neutral Monism. A Miraculous, Incoh
Page 340 and 341:
Neutral Monism. A Miraculous, Incoh
Page 342 and 343:
A somewhat Eliminativist Proposal a
Page 344 and 345:
Impliziert der intentionale Redukti
Page 346 and 347:
Impliziert der intentionale Redukti
Page 348 and 349:
Structure of the Paradoxes, Structu
Page 350 and 351:
Structure of the Paradoxes, Structu
Page 352 and 353: that he was working on a translatio
Page 354 and 355: Objects of Perception, Objects of S
Page 356 and 357: Objects of Perception, Objects of S
Page 358 and 359: Let us take the example of the hypo
Page 360 and 361: Reducing Sets to Modalities Rafał
Page 362 and 363: variables ai that (under an interpr
Page 364 and 365: Are Lamarckian Explanations Fully R
Page 366 and 367: A Note on Tractatus 5.521 Nuno Vent
Page 368 and 369: And he goes on saying, referring to
Page 370 and 371: The Place of Theory Reduction in th
Page 372 and 373: Ethik als irreduzibles Supervenienz
Page 374 and 375: unabhängig von denen ihrer Mitglie
Page 376 and 377: Das ‘schwierige Problem’ des Be
Page 378 and 379: The Supervenience Argument, Levels,
Page 380 and 381: The Supervenience Argument, Levels,
Page 382 and 383: their relations both to the physica
Page 384 and 385: On the Characterization of Objects
Page 386 and 387: On the Characterization of Objects
Page 388 and 389: The Functional Unity of Special Sci
Page 390 and 391: size, location, temporal characteri
Page 392 and 393: Transcendental Philosophy and Mind-
Page 394 and 395: From Topology to Logic. The Neural
Page 396 and 397: From Topology to Logic. The Neural
Page 398 and 399: The Calculus of Inductive Construct
Page 400 and 401: The Four-Color Theorem, Testimony a
Page 404 and 405: Clearly x = y → x = ext y holds,
Page 406 and 407: Intentional Fundamentalism Petri Yl
Page 408 and 409: The second dimension to be consider
Page 410 and 411: The very existence of the special s
Page 412 and 413: Heil, J. (1999), “Multiple Realiz
Page 414 and 415: independent of what is the fact, fa
Page 416 and 417: 415
Page 418: Publications of the Austrian Ludwig
show all

Reduction and Elimination in Philosophy and the Sciences

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?