Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an L ω 1ω -interpretability theory. We also give a survey of this theory.

We give classical proofs, strengthenings, and generalizations of Lecomte’s characterizations of analytic ω-dimensional hypergraphs with countable Borel chromatic number.

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c (...) to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

Let X be a separable Banach space and Q be a coanalytic subset of . We prove that the set of sequences in X which are weakly convergent to some eX and is a coanalytic subset of . The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 60–64], this result leads to the following dichotomy theorem: if X is a Banach space, (...) is a regular method of summability and is a bounded sequence in X, then there exists a subsequence iL such that either there exists eX such that every subsequence iH of iL is weakly summable w.r.t. to e and QiH,e); or for every subsequence iH of iL and every eX with QiH,e)the sequence iH is not weakly summable to e w.r.t. . This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 232–234]. Both theorems obtain some considerable generalizations. (shrink)

In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy strict non-imposition (citizens’ sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s (1972) impossibility (...) theorem is stronger than Murakami’s impossibility theorem, both by virtue of replacing Murakami’s assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami’s dichotomy and impossibility theorems to the setting of incomplete social preference. We prove that if one drops completeness from Murakami’s assumptions, his remaining assumptions imply (i) that a collective choice rule is either Paretian, anti-Paretian, or dis-Paretian (unanimous individual preference implies noncomparability) and (ii) that adding proposed constraints on noncomparability, such as the regularity axiom of Eliaz and Ok (2006), restores Murakami’s dictator-or-inverse-dictator result. (shrink)

In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form \. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.

We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.

The aim of this paper is to offer a conceptual analysis of Weinberg's proof of the spin-statistics theorem by comparing it with Pauli's original proof and with the subsequent textbook tradition, which typically resorts to the dichotomy positive energy for half-integral spin particles/microcausality for integral-spin particles. In contrast to this tradition, Weinberg's proof does not directly invoke the positivity of the energy, but derives the theorem from the single relativistic requirement of microcausality. This seemingly innocuous difference marks an important (...) change in the conceptual basis of quantum physics. Its historical, theoretical, and conceptual roots are here reconstructed. The link between Weinberg's proof and Pauli's original is highlighted: Weinberg's proof turns out to do justice to Pauli's anti-Dirac lines of thought. The work of Furry and Oppenheimer is also surveyed as a “third way” between the textbook tradition established by Pauli and Weinberg's approach. (shrink)

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the (...) mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In (...) recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails. (shrink)

We prove a dichotomy theorem for minimal structures and use it to prove that the number of non-isomorphic countable elementary extensions of an arbitrary countable, infinite first-order structure is infinite.

In [9] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [9] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions).

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. (...) We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable [Formula: see text]-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. (shrink)

The results of this paper concern the effective cardinal structure of the subsets of [ω1]<ω1, the set of all countable subsets of ω1. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.

We show that every ℵα can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1 but whose automorphism group fails the topological Vaught conjecture on analytic sets. We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.

The results of this paper concern the effective cardinal structure of the subsets of [ω1]<ω1, the set of all countable subsets of ω1. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.

We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For (...) example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets. (shrink)

A set-theoretical proof of Gowers’ Dichotomy Theorem is presented together with its application to another dichotomy concerning asymptotic l 2 basic sequences.

We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is N₀. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In (...) particular, this holds for "the" minimum analytic graph G₀ with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ ∈ {2, 3...., N₀. c}, there is a treeing of E₀ with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm—Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (R<N, ⊇). (shrink)

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding (...) [Formula: see text] where [Formula: see text] is a saturated separably closed field. (shrink)

We introduce notions of strong and eventual strong non-isolation for types in countable, stable theories. For T superstable or small stable we prove a dichotomy theorem: a regular type over a finite domain is either eventually strongly non-isolated or is non-orthogonal to a NENI type . As an application we obtain the upper bound for Lascar’s rank of a superstable theory which is one-based or trivial, and has fewer than 20 non-isomorphic countable models.

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open (...) question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of (...) energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

Abstract.The Judeo-Christian, Enlightenment, and postmodernist paradigms have become intellectually and ethically exhausted. They are obviously failing to provide a conceptual framework conducive to eliminating some of humanity's worst scourges, including war and environmental destruction. This raises the issue of a successor, which necessitates a reexamination of first principles, starting with our concept of God. Pantheism, which is differentiated from panentheism, denies the existence of a transcendent, supernatural creator and instead asserts that God and the universe are one and the same. (...) Understood via intuition, modern cosmology, and other natural sciences, it offers an alternative worldview that posits the divine and sacred nature of the universe/creation. By asserting the fallacy of the creator/creation dichotomy and any attempts to anthropomorphize or personalize God, pantheism precludes hubris stemming from erroneous notions of divine favoritism. The links between Judeo-Christianity and the Enlightenment are traced and a case made that the latter has resulted in the equally erroneous and hubristic notion of human ascendancy to a Godlike status, with the concept of progress providing a secular version of the Christian belief in salvation. By reestablishing the natural sciences’metanarrative, even as it asserts the divinity of the material universe, pantheism simultaneously demotes postmodernism and reconciles science with religion. Pantheism provides a theological foundation for deep ecology and also stakes out a viable third position in relation to the ongoing dispute between advocates of intelligent design and the scientific establishment. (shrink)

We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the (...) Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (finitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property. (shrink)

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver [20] and Harrington, Kechris and Louveau [6] show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on , the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on . In this article we examine the effective (...) content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene’s O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is that of the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal α the existence of singletons whose α-jumps are Turing incomparable. (shrink)

We present a way of classifying the logically possible ways out of Gärdenfors' inconsistency or triviality result on belief revision with conditionals. For one of these ways—conditionals which are not descriptive but which only have an inferential role as being given by the Ramsey test—we determine which of the assumptions in three different versions of Gärdenfors' theorem turn out to be false. This is done by constructing ranked models in which such Ramsey-test conditionals are evaluated and which are subject to (...) natural postulates on belief revision and acceptability sets for conditionals. Along the way we show that in contrast with what Gärdenfors himself proposed, there is no dichotomy of the form: either the Ramsey test has to be given up or the Preservation condition. Instead, both of them follow from our postulates. (shrink)

SummaryInstead of telling us why one should or should not derive an ‘ought’ from an ‘is’, most philosophers try to see whether this can or cannot be done. Since there are no clearly stated rules of inference for the ordinary language, the is‐ought dichotomy is re‐examined within the context of the deontic logic. This paper shows that rules like ‐AB ÓOB enables us to adopt the theorems of the propositional logic normatively, because logical standards for exactness, truth, completeness, (...) etc. are all values and conventions accepted by the deontic logicians right from the beginning.RésuméLa plupart des philosophes reste attachée à une manière incorrecte de poser la question de la dichotomie entre ce qui est et ce qui devrait être. Au lieu de se demander, avec Hume, si nous avons de bonnes raisons ?on;autoriser ou ?on;interdire des raisonnements qui concluent de ľêtre au devoir être, des philosophes tels que Searle se demandent si de tels raisonnements sont possibles dans une langue naturelle. Comme, dans une telle langue, nous n'avons pas de règie univoque pour la déduction logique, j'examine la règie de la logique déontique AB OAOB selon la formalisation de Azizah al‐Hibri. Je montre que la règie ne viole pas du tout ľinterdiction de conclure ?on;énonces de faits à des énoncés normatifs. Car dans la logique déontique, les règies et les théorèmes de la logique des propositions sont utilisés dans un sens normatif.ZusammenfassungIn bezug auf die ‘Sein‐Sollen‐Dichotomie’ scheinen die meisten Philosophen bis jetzt auf einer falschen Fragestellung zu beharren. Anstatt wie Hume zu fragen, ob wir gute Gründe haben, Schlüsse von Sein und Sollen zuzulassen oder zu verbieten, versuchen Philosophen wie etwa Searle festzustellen, ob wir dies im Rahmen der natürlichen Sprache tun konnen. Da wir in der natürlichen Sprache keine eindeutige Regel für logische Schlüsse haben, wird die Regel der deontischen Logik nach der Formalisierung von Azizah al‐Hibri untersucht. Es wird gezeigt, dass die Regel keineswegs das Verbot, von 1st‐Sätzen auf Soll‐Sätze zu schliessen, verletzt. Denn in der deontischen Logik werden die Regeln und Theoreme der Satzlogik in einem norrnativen Sinn gebraucht. (shrink)

Suppose that is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where is a family of (...) continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences. (shrink)

The brain displays both the anatomical features of a vast amount of interconnected topological mappings as well as the functional features of a nonlinear, metastable system at the edge of chaos, equipped with a phase space where mental random walks tend towards lower energetic basins. Nevertheless, with the exception of some advanced neuro-anatomic descriptions and present-day connectomic research, very few studies have been addressing the topological path of a brain embedded or embodied in its external and internal environment. Herein, by (...) using new formal tools derived from algebraic topology, we provide an account of the metastable brain, based on the neuro-scientific model of Operational Architectonics of brain-mind functioning. We introduce a “topodynamic” description that shows how the relationships among the countless intertwined spatio-temporal levels of brain functioning can be assessed in terms of projections and mappings that take place on abstract structures, equipped with different dimensions, curvatures and energetic constraints. Such a topodynamical approach, apart from providing a biologically plausible model of brain function that can be operationalized, is also able to tackle the issue of a long-standing dichotomy: it throws indeed a bridge between the subjective, immediate datum of the naïve complex of sensations and mentations and the objective, quantitative, data extracted from experimental neuro-scientific procedures. Importantly, it opens the door to a series of new predictions and future directions of advancement for neuroscientific research. (shrink)

Philosophers have often bluntly said, and more often tacitly assumed that careful and reasonable men will confine themselves to two very rigid ways of talking. Vile must either show that what we say is a theorem deducible from assumed axioms and postulates, or we must show that what we say is made probable by evidence. This book is at heart an attack upon the idea that rationality requires any such straitjacket, and it repudiates the dichotomy between "analytic" philosophy and (...) philosophy “in the grand tradition.” Rationality is here conceived as a subtle and complex temper of deciding, most needed precisely where what we have to say cannot be stuffed into the two narrow pigeonholes in question. Originally published in 1957. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

In the 1951 Gibbs lecture, Gödel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naïve proof, to the effect that Gödel’s first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naïve proof, in agreement with Gödel’s conception, superseding the criticisms against the usual notion of naïve proof used by real working mathematicians. We explore the connection (...) between Gödel’s theorem and naïve proof so understood, both from a classical and a dialetheic perspective. (shrink)

We study the weak truth table degree spectra of first-order relational structures. We prove a dichotomy among the possible wtt degree spectra along the lines of Knight’s upward-closure theorem for Turing degree spectra. We prove new results contrasting the wtt degree spectra of finite- and infinite-signature structures. We show that, as a method of defining classes of reals, the wtt degree spectrum is, except for some trivial cases, strictly more expressive than the Turing degree spectrum.

This paper grew as a continuation of [Sh462] but in the present form it can serve as a motivation for it as well. We deal with the same notions, all defined in 1.1, and use just one simple lemma from there whose statement and proof we repeat as 2.1. Originally entangledness was introduced, in [BoSh210] for example, in order to get narrow boolean algebras and examples of the nonmultiplicativity of c.c-ness. These applications became marginal when other methods were found and (...) successfully applied (especially Todorčevic walks) but after the pcf constructions which made their début in [Sh-g] and were continued in [Sh462] it seems that this notion gained independence.Generally we aim at characterizing the existence of strong and weak entangled orders in cardinal arithmetic terms. In [Sh462, §6] necessary conditions were shown for strong entangledness which in a previous version was erroneously proved to be equivalent to plain entangledness. In §1 we give a forcing counterexample to this equivalence and in §2 we get those results for entangledness (certainly the most interesting case). A new construction of an entangled order ends this section. In §3 we get weaker results for positively entangledness, especially when supplemented with the existence of a separating point (Definition 2.2). An antipodal case is defined in 3.10 and completely characterized in 3.11. Lastly we outline in 3.12 a forcing example showing that these two subcases of positive entangledness comprise no dichotomy. The work was done during the fall of 1994 and the winter of 1995. The second author proved Theorems 1.2, 2.14, the result that is mentioned in Remark 2.11 and what appears in this version as Theorem 2.10(a) with the further assumptionden(I)θ< μ. The first author is responsible for waving off this assumption (actually by showing that it holds in the general case), for Theorems 2.12 and 2.13 in Section 2 and for the work which is presented in Section 3. (shrink)

This chapter critiques the notion of identity, a philosophical reaction in the Quinean sense that vindicates another significant hypothesis, which is called the “No entity without identity” theorem. Here, the identification of an entity relies upon its identity. The first segment of this chapter revisits the following question: How do we encounter the concept of a subjective “mind”? Could we “infer” or “experience” the concept of “mind” in our daily lives? Every entity has some characteristics that prompt comparison with similar (...) entities. Although I think that other minds have a subjective periphery and that other minds have an objective periphery lurking behind this (privileged access), we can construe the relationship in a subjective way (individual mind). In the second section of the chapter, I address a different point by considering why the idea of “other minds” causes philosophical dilemmas. Can “other minds” exist as a substance in one’s own mind? The category of human knowledge encompasses the reality of minds and other minds. The other minds remain an accompanying part of the individual mind since the expression, belief, and mental content of the individual mind merge in the linguistic expression in a split manner that we may ascribe as privileged access (subjective incorrigible knowledge) and corrigible knowledge. But I think that the relation between minds and other minds sounds especially idiosyncratic in its own identity. Despite becoming diverse, the individual mind can still be intimate by blending and attunement between the subjective and objective dichotomy. (shrink)

There are powerful fairness and efficiency arguments for making charitable donations to soup kitchens 100% deductible. These arguments have no purchase for donations to fund opulent church organs, yet these too are 100% deductible under the current tax code. This stark dichotomy is only the tip of the iceberg. Looking at a wider sampling of charitable gifts reveals a charitable continuum. Based on sliding scales for efficiency, multiple theories of fairness, pluralism, institutional competence and social welfare dictate that charitable (...) deductions should in most cases be fractions between zero and one. Moreover, the Central Limit Theorem strongly suggests that combining this welter of largely independent criteria with the wide variety of charitable gifts results in a classic bell-shaped normal curve of optimal deductions, with a peak at some central value and quickly decaying to zero at the extremes of 0% and 100%. Given that those are the only two options under the current tax code, the current charitable deduction regime inevitably makes large errors in most cases. Actually calculating a precise optimal percentage for each type of charitable donation is of course impractical. This Article suggests, however, that we can do much better than the systematically erroneous current charitable deduction. Granting a 100% deduction only for donations to the desperately poor, along with 50%, 25%, and 0% for gifts yielding progressively fewer efficiency, fairness, pluralism, and institutional competence benefits, promises to deliver a socially more desirable charitable deduction. (shrink)

Believing that the enterprise of constructing "artificial intelligence" transcends the bounds of any one discipline, the editors of Mechanical Bodies, Computational Minds have brought together researchers in AI and scholars in the humanities to reexamine the fundamental assumptions of both areas. The AI community, for example, could benefit from explorations of human intelligence and creativity by philosophers, psychologists, anthropologists, literary critics, and others, while analysis of AI's theoretical struggles and technical advances could yield insights into such traditional humanist concerns as (...) the nature of rationality and the mind-body dichotomy.The contributions include a continuation of the famous Hubert Dreyfus-Daniel Dennett debate over Kasparov's defeat by IBM's Deep Blue; Philip Agre's tracing of difficulties in AI research to the inherited tensions of Cartesian dualism; Evelyn Fox Keller's examination of the development of computer technology in relation to biology; Douglas Hofstadter's argument that thinking is more than the theorem-solving activities of AI; and Alison Adam's discussion of the implicitly male universal subject used in AI. (shrink)

The no-free-lunch theorems promote a skeptical conclusion that all possible machine learning algorithms equally lack justification. But how could this leave room for a learning theory, that shows that some algorithms are better than others? Drawing parallels to the philosophy of induction, we point out that the no-free-lunch results presuppose a conception of learning algorithms as purely data-driven. On this conception, every algorithm must have an inherent inductive bias, that wants justification. We argue that many standard learning algorithms should (...) rather be understood as model-dependent: in each application they also require for input a model, representing a bias. Generic algorithms themselves, they can be given a model-relative justification. (shrink)

The important recent book by Schurz ( 2019 ) appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior—they prove that there are “as many priors” (loosely speaking) for which any induction algorithm _A_ out-generalizes some induction algorithm _B_ as vice-versa. Importantly though, in addition to the NFL theorems, there are many (...) _free lunch_ theorems. In particular, the NFL theorems can only be used to compare the expected performance of an induction algorithm _A_, considered in isolation, with the expected performance of an induction algorithm _B_, considered in isolation. There is a rich set of free lunches which instead concern the statistical _correlations_ among the generalization errors of induction algorithms. As I describe, the meta-induction algorithms that Schurz advocates as a “solution to Hume’s problem” are simply examples of such a free lunch based on correlations among the generalization errors of induction algorithms. I end by pointing out that the prior that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference. (shrink)

Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves (...) to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic. (shrink)