Reduction and Elimination in Philosophy and the Sciences

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The Calculus of Inductive Constructions as a Foundation for Semantics Piotr Wilkin, Warsaw, Poland Since Frege started his work on formalizing natural language semantics, the framework for those semantics has been set-based. First, most theories have been based on type theory, later replaced by Zermelo-Fraenkel’s set theory. Today, this paradigm is rarely questioned and even if some attempts are made to circumvent ZFC, for example to eliminate some paradoxes in quantified modal logic (Oksanen 1999), most of the development is done within that framework. This raises two major problems: set theory is structuralistic and extensional in nature. The latter needs no explanation, the former can be explained this way: in set theory, one expresses structures, not functionality. This is well seen in the definition of the ordered pair – in ZFC, one defines as {{a},{a,b}} and only later proves that this construct satisfies the properties we require of the ordered pair – namely, to be able to construct an ordered pair from two elements and to retrieve, respectively, the first or second element. What I propose here is to switch to a certain intuitionistic setting to express semantics, namely, the Calculus of Inductive Constructions. I am aware that the term “intuitionistic” will make some people, and especially philosophers, wary of the idea, since this instantly suggests a connection with Brouwer’s philosophical concept of intuitionism as a philosophy of mathematics. However, I believe that while intuitionism might be questionable as a framework for mathematics, it is wellsuited as a framework for natural-language semantics. This is because the subjects of both disciplines are fundamentally different. With natural-language semantics, we want to describe not an idealized area such as mathematics, but a very real one – how people use the language. While in mathematics it is often advisable to abstract away from the specific means of obtaining some result (namely, the calculation), it is hardly plausible to suggest that we can do the same with language expressions, as there are clearly some cases (as in the case of intensional contexts or, more specifically, propositional attitudes) where we cannot abstract from the way a specific human being reconstructs a language expression. From Frege’s times, the accepted and widespread form of a logical assertion has been C � ϕ, where C stands for the context and ϕ represents the expression to be asserted (in the given context). This is true for provability, as well as for logical consequence in a model (obviously, the two relations differ in nature, but a common characteristic is shared – they are both binary relations between a context and an evaluated expression, whether the context be a model or a theory). However, a counterproposal to this form can be submitted: C � ϕ : t, which reads: in context C, the expression ϕ is of type t. This is called a typing judgement. The concept of types dates back to the theory of types from Principia Mathematica, but this specific idea owes its roots to Church’s theory of simple types (Church 1940), which was based on the lambda calculus and proposed as an alternative. Once we enter the area of semantics, one can see the usefulness of the typing assertion over the traditional logical one. In semantics, we are often interested not only in information about true sentences, we also want to know whether a given expression is a properly constructed sentence or whether some expression is actually a definite description. Indeed, various solutions like the Montague grammar (Montague 1974) exploit this concept with the use of semantic categories, which is a concept parallel to the one by Church but designed specifically for natural language analysis in the early XX century by Polish logicians Stanislaw Lesniewski and Kazimierz Ajdukiewicz (Ajdukiewicz 1935). However, the interesting modification here lies not only in the adding of the type, but also in modifying the way we read the expression. This was first noticed by the logician Haskell Curry (Curry 1934), who recognized that in case of simply typed combinators (an analogon to Church’s simple type system), if we only assign types to basic lambda terms, the proper types correspond exactly to tautologies of a minimal (consisting only of the implication, without the other connectives) intuitionistic propositional calculus. Thus, for example, the K combinator (λxλy.x) becomes a proof of the known tautology α → (β → α). How so? According to the functional interpretation (known as the Brouwer-Heyting-Kolmogorov interpretation (Sorensen and Urzyczyn 2006)), a proof of implication is a function transforming a proof of the LHS to a proof of the RHS. So in this case, we want the function in question to take an argument being a proof of α, then return us an argument which itself is a function from β to α. The K combinator is exactly that – it takes an x (of type α), then returns us λy.x, which is itself a function – that takes an y (of type β) and returns the previously taken x (of type α). This approach, while generally interesting, is hardly revolutionary. However, this has to be coupled with a different idea – Church designed his simple type system not with this intuitionistic reading in mind, but to avoid a form of Russell’s Paradox for his untyped lambda calculus, which is represented by the Omega term: (λx.xx)(λx.xx) – a self-applying function applied to itself will never reduce to a normal form. It can be easily proven that any term that can be typed in the simple type system has the SN (strong normalization) property – any path of reductions of that term will terminate. The question raised by Curry’s discovery was – can one construct a more advanced logic than the one represented by the simple type system, but one that is still based only on strongly normalizable terms? We now know that the answer is positive, and highly so: one can represent higher-order intuitionistic logic using this sort type system, and add inductively defined predicates to the logic to make it easier to formalize certain concepts. One such system is the Calculus of Inductive Constructions (CIC), which is the theoretical basis for an interactive proof assistant program named Coq (Herbelin et al. 2006). For a sample of the expressive power of the system, one can point to the formalized version of the fourcolor theorem proof – since the proof itself is a tangible mathematical object, this removes the need to “believe” a computer-based check of each of the hundreds of cases – one only has to believe the theory behind the prover, namely, the CIC itself. For a more logically interesting example, a (constructive) proof of Gödel’s incompleteness 397
398 The Calculus of Inductive Constructions as a Foundation for Semantics — Piotr Wilkin theorem is also available (O’Connor 2005). Indeed, the expressive power of CIC is not far from that of ZFC. There is not enough space here to describe in full extent the theory behind CIC. Suffice it to say, one very important distinction between CIC and the simple-type theory (STT) is that there is no longer any clear distinction between a term and a type. Therefore, we can have both � ϕ : 2+2 = 4 and � 2+2 = 4 : Prop - the first one states that ϕ is a proof of 2 + 2 = 4, the other – that 2 + 2 = 4 is a well-formed proposition. The type hierarchy in CIC is infinite and stratified, with two base types: Set being the type of object domains and Prop being the type of propositions. One example of a proposition we have already seen, an example of a Set typing assertion is and � Nat : Set, saying that the natural numbers form an object domain (I did not want to use the terms “set” or “universe” since they seem to have various connotations, but since the term “set” is used as the type designation internally in Coq, I will adopt this and use the name “set” as short for “object domain” – this is however not to be understood as a set in the set-theoretic sense). Another important difference is that apart from non-dependent products (which can be understood as functions and thus, via the Curry-Howard isomorphism, implications), we now have dependent products (which can be thought of as generalized products and via the Curry-Howard isomorphism – represented as universal quantification). For example, the assertion � ϕ : (∀n:nat(n ≥ 0)) can be read as “ϕ is a member of a generalized product over natural numbers of domains “n ≥ 0”, which for each n contain proofs that n ≥ 0”. Presenting this entire concept would have been pointless without showing why this can be useful for semantics. Indeed, I can believe that using this framework can significantly increase the clarity of describing nonextensional contexts in semantic uses. For one quick interesting demonstration, note that this approach allows us to formally explain the difference between Frege’s notion of “function” and its “value-range”. In CIC one cannot generally deduce � f = g from � ∀x.fx = gx; extensionality is not assumed. However, this does not mean that we use some vague notion of identity – functions are constructive objects and they are the same if the constructions themselves are the same. Now, here the inductive types come in. Inductive types are generally an extension of the intuitive notion of inductive constructions to the type system. That is: you provide ways of constructing base elements of the type, as well as inductively constructing new elements from previously constructed ones. What the type system guarantees is that the construction is unique – objects of an inductive type cannot be created in any other way than by the respective constructors and the various constructors are assumed to be provably different from each other. For example, the type Nat of natural numbers is defined as Nat := O : Nat | S : Nat → Nat. This corresponds to the traditional view from Peano Arithmetic: natural numbers are either 0 or an iterated successor of 0. They are also only that: for example, you can prove in the system (with just the above definition) that 0 is different from any successor. We can use all of the mentioned instruments to illustrate how this framework can help us in formalizing semantics. For example, we can construct the operator Believes in the following way: Believes := BArith : ∀n:nat, (n = n) → Believes(n = n). This inductive definition means that what is “believed” is arithmetical equalities, but nothing else. Note that in this definition, both the truth value and the form of the expression is used – we accept only proven expressions, and only of the form n = n. We could widen the operator, for example, to ignore the truth value: take a definition of Believes := BArith : (∀n m:nat),Believes(n = m). This is a correct definition, but it no longer takes into account the logical validity of the equality. Now, we “believe” any arithmetical equality, true or not. Note that we are not taking a model-theoretic approach here – there is no need to construct a specific intensional model to formalize such a semantics. We gain the benefit of a framework especially suited for analyzing intensional constructions, but one that already possesses all the “standard” logical concepts built in. One can even add the axiom of excluded middle if needed – it is consistent with CIC. There is, however, no need for the tedious task of building a specific model for formalizing an intensional fragment of the natural language – one needs only to define the proper inductive types and predicates. All this is coupled with a system for annotating the syntax much richer than standard categorial grammars – for example, we can now assign a type to the operator very as follows: (∀s:Set),(s → Prop) → (s → Prop), which allows us to uniformly type very for predicates of any object category (as in very young with young : LivingBeing → Prop and very fast with fast : Car → Prop, so that we can both subdivide the object category and not worry about having problems with uniformly assigning categories to higher order operators). This proof-theoretic (or constructive, as you prefer) approach to natural-language semantics seems to me more suitable. After all, what we are after when formalizing a natural language is being able to express all the constructs from the natural language easily and not having to worry whether adding this or that construct will complicate the resulting model beyond manageability. Also, it places more emphasis on the functional way of our language use. I believe that the presented framework might be a viable and reasonable alternative to current settheoretic based approaches. Literature Ajdukiewicz, Kazimierz 1935 Die syntaktische Konnexität, in Studia Philosophica, I, 1935, pp. 1-27 Church, Alonzo 1940 A Formulation of the Simple Theory of Types, in The Journal of Symbolic Logic, Vol. 5, No. 2 (Jun., 1940), pp. 56-68 Curry, Haskell 1934 Functionality in Combinatory Logic, in Proceedings of the National Academy of Sciences, vol. 20, pp. 584- 590 Herbelin, Hugo et al. 2006 The Coq Reference Manual. Chapter 4: The Calculus of Inductive Constructions, http://coq.inria.fr/doc/ Reference-Manual006.html Montague, Richard 1974 The Proper Treatment of Quantification in Ordinary English, in R. Thomason (Ed.), Formal Philosophy: Selected Papers of Richard Montague. New Haven: Yale University Press. O’Connor, Russell 2005 Essential Incompleteness of Arithmetic Verified by Coq, in Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005) Oksanen, Mika 1999 The Russell-Kaplan Paradox and Other Modal Paradoxes: A New Solution, in Nordic Journal of Philosophical Logic, Vol. 4, No. 1, pp. 73-93 Sorensen, Martin Heine and Urzyczyn, Paweł 2006 Lectures on the Curry-Howard Isomorphism, Elsevier
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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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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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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
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Page 394 and 395: From Topology to Logic. The Neural
Page 396 and 397: From Topology to Logic. The Neural
Page 400 and 401: The Four-Color Theorem, Testimony a
Page 402 and 403: experience. Of course, a detailed a
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Page 406 and 407: Intentional Fundamentalism Petri Yl
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