Naturalness, Vagueness, and Sortals

In the past few years, deflationary positions in the debate on the nature of composite material objects have become prominent. According to Ted Sider these include the thesis of quantifier variance, against which he has defended ontological realism. Recently, Sider has considered the possibility of rejecting his arguments against the vagueness of the unrestricted quantifiers in terms of translation functions. Against this strategy, he has presented an intuitive complaint and has argued that it can only be resisted if quantifier variance is accepted. But this is false. In this paper I argue, against Sider, that there is a coherent way to combine the rejection of quantifier variance with the vagueness of the unrestricted quantifiers. I sketch a model to show this, and then I consider, on the basis of it, several versions of the indeterminacy argument against the vagueness of the unrestricted quantifiers that Sider has formulated over the years.     

Demiquantifiers on $$\ell $$-groups

In this paper we introduce two kinds of unary operations on abelian \(\ell \)-groups with a positive distinguished element u. One of them, called demiquantifier of type I, behaves like an existential quantifier (in the sense of Cimadamore and Varela) in the negative cone, and like a universal quantifier in the positive cone. The other kind of unary operation we introduce, called demiquantifier of type II, satisfies analogous properties to demiquantifiers of type I via a translation of the negative cone, by means of the element u. These unary operations are interdefinable with the usual existential quantifiers, provided the distinguished element u is a strong unit. Moreover, if G is an abelian \(\ell \)-group, then the restriction of a demiquantifier of type II to the MV-algebra \(\Gamma (G,u)\) yields a different type of quantifier, where \(\Gamma \) is Mundici’s functor. These quantifiers are interdefinable with the usual existential quantifiers on MV-algebras given by Rutledge, provided that the involution of the corresponding MV-algebras have a fixed point.     

Hierarchies of Partially Ordered Connectives and Quantifiers

Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form what is a strengthening of the Walkoe result. MSC: 03C80.     

Quantifier elimination for the theory of algebraically closed valued fields with analytic structure

The theory of algebraically closed non‐Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz, and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper. This theorem also has other consequences in the geometry of definable sets. The method of proving quantifier elimination in this paper for an analytic language does not require the algebraic quantifier elimination theorem of Weispfenning, unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

素数阶群理论的量词消去算法及其上界

在以前的一些工作中,作者已经证明语言(?)={+,0,e)上素数阶群的理论T有量词消去性质并研究了它的判定问题的复杂性.本文在此基础上将利用T的判定问题的复杂性结果给出理论T的量词消去的一个算法,同时给出该算法的复杂性上界.     

Modal Meinongianism for Fictional Objects

Drawing on different suggestions from the literature, we outline a unified metaphysical framework, labeled as Modal Meinongian Metaphysics (MMM), combining Meinongian themes with a non-standard modal ontology. The MMM approach is based on (1) a comprehension principle (CP) for objects in unrestricted, but qualified form, and (2) the employment of an ontology of impossible worlds, besides possible ones. In §§1–2, we introduce the classical Meinongian metaphysics and consider two famous Russellian criticisms, namely (a) the charge of inconsistency and (b) the claim that naïve Meinongianism allows one to prove that anything exists. In §3, we have impossible worlds enter the stage and provide independent justification for their use. In §4, we introduce our revised comprehension principle: our CP has no restriction on the (sets of) properties that can characterize objects, but parameterizes them to worlds, therefore having modality explicitly built into it. In §5, we propose an application of the MMM apparatus to fictional objects and defend the naturalness of our treatment against alternative approaches. Finally, in §6, we consider David Lewis’ notorious objection to impossibilia, and provide a reply to it by resorting to an ersatz account of worlds.     

Vectorization hierarchies of some graph quantifiers

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier in , the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two -equivalent models with certain additional structure, yields a pair of -equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity. Received: 15 October 1996     

Humean Supervenience in the Light of Contemporary Science

It is shown that Lewis’ ontological doctrine of Humean supervenience incorporates at its foundation the so-called separability principle of classical physics. In view of the systematic violation of the latter within quantum mechanics, the claim that contemporary physical science may posit non-supervenient relations beyond the spatiotemporal ones is reinforced on a foundational basis concerning constraints on the state representation of physical systems. Depending on the mode of assignment of states to quantum systems — unit state vectors versus statistical density operators — we distinguish between strongly and weakly non-Humean, non-supervenient relations. It is demonstrated that in either case, the relations of quantum entanglement constitute prototypical examples of irreducible physical relations that do not supervene upon a spatiotemporal arrangement of Humean qualities, weakening, thereby, the thesis of Humean supervenience. In this respect, the status of Lewis’ recombination principle is examined, whereas his conception of lawhood is critically investigated. It is concluded that the assumption of ontological reductionism, as expressed in Lewis’ Humean doctrine, cannot be regarded as a reliable code of the nature of the physical world and its contents. It is proposed instead that due to the undeniable existence of non-supervenient relations, a metaphysic of relations of a moderate kind ought to be acknowledged as an indispensable part of our understanding of the natural world at a fundamental level.     

Inhabitation of polymorphic and existential types

This paper shows that the inhabitation problem in the lambda calculus with negation, product, polymorphic, and existential types is decidable, where the inhabitation problem asks whether there exists some term that belongs to a given type. In order to do that, this paper proves the decidability of the provability in the logical system defined from the second-order natural deduction by removing implication and disjunction. This is proved by showing the quantifier elimination theorem and reducing the problem to the provability in propositional logic. The magic formulas are used for quantifier elimination such that they replace quantifiers. As a byproduct, this paper also shows the second-order witness theorem which states that a quantifier followed by negation can be replaced by a witness obtained only from the formula. As a corollary of the main results, this paper also shows Glivenko’s theorem, Double Negation Shift, and conservativity for antecedent-empty sequents between the logical system and its classical version.     

On first-order definitions of subgraph isomorphism properties

Let φ(F) be the property of containing (as a subgraph) an isomorphic copy of a graph F. It is easy to show that this property cannot be defined in a first-order language by a sentence with a quantifier depth (or variable width) strictly less than the number of vertices in F. Nevertheless, such a definition exists in some classes of graphs. Three classes of graphs are considered: connected graphs with a large number of vertices, graphs with large treewidth, and graphs with high connectivity.     

Three Flawed Distinctions in the Philosophy of Time

The distinctions between A-series and B-series, between synchronic and diachronic identity and between perdurance and endurance are basic in the philosophy of time; yet they are flawed. McTaggart’s claim that the B-series is static and that a series has to be changing to be really temporal arises from a misunderstanding of temporal relations and of the task of ontological analysis. The dynamic appearance of the A-series results from the incompleteness of the analysis. “Synchronic identity” is synonymous with “strict identity”, which has nothing to do with simultaneity. “Diachronic Identity” is another designation for persistence of an ordinary thing through time and change. Now, strict self-identity holds independently of whether a thing has a short or a long duration. Hence, diachronic identity is synchronic identity. Lewis’ distinguishes two kinds of ontological analyses of persistence, the perdurance and the endurance analysis. This dichotomy is in several respects not exhaustive. Above all, his definition of “persist” is inadequate being based on the notion of multiple temporal localisation which is apt with interrupted but misplaced with persistent, i.e., temporally continuous objects.     

A Deflationary Metaphysics of Morality

The metaphysical dispute between moral realists and antirealists is cast in terms of properties: the realist holds that moral properties exist, the antirealist denies this claim. There is a longstanding philosophical dispute over the nature of properties, and the obscurity of properties may make the realist/antirealist dispute even more obscure. In the spirit of deflationary theories of truth, we can turn to a deflationary theory of properties in order to clarify this issue. One might reasonably worry that such an account of properties would not be capable of properly characterizing disputes regarding the existence or nonexistence of genuine moral properties. In this paper, I will show that, within this framework, the traditional disputes over the existence of moral properties can be characterized in a far clearer fashion than is usually the case. A deflationary account of properties, along with an explanatory hierarchy of properties, makes the dispute in ontology clear.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

Interface of global and local semantics in a self-navigating system based on the concept lattice

If the syntax–semantics interaction is driven by the interface, it also interrupts the interaction on its own right. Because the syntax is verified to be isomorphic to the semantics, the interaction is open to the diagonal argument leading to a contradiction. That is why it is necessary to introduce a particular interface to drive the interface to make the interaction possible despite the contradiction. In this context we propose the system implemented with the syntax–semantics loop by using a concept lattice and a particular weak quantifier. This system is expressed as the self-navigating system which wanders in a two-dimensional space, encounters some landmarks, constructs the relationship among landmarks to which decision making with respect to the move is referred. The syntax of this system is defined as two-dimensional move and the semantics is defined as a concept lattice [B. Ganter, R. Wille, Formal Concept Analysis, Springer, Berlin, 1999] constructed by the binary relation between landmarks and some properties of landmarks, and by Galois connection. To implement the interface driving and interrupting the interaction between syntax and semantics, we divided semantics into local and global concept lattices, and introduce a weak quantifier to connect a local with a global lattice. Because the contradiction results from diagonal argument or using a normal quantifier ∀, the use of a quantifier is restricted dependent on the situation to avoid a contradiction. It is shown that due to the role of a weak quantifier our self-navigating system is both robust and open to the emergent property through simulating studies.     

Particularist semantic normativity

We sketch the view we call contextual semantics. It asserts that truth is semantically correct affirmability under contextually variable semantic standards, that truth is frequently an indirect form of correspondence between thought/language and the world, and that many Quinean commitments are not genuine ontological commitments. We argue that contextualist semantics fits very naturally with the view that the pertinent semantic standards are particularist rather than being systematizable as exceptionless general principles.     

Kant, Spinoza, and the Metaphysics of the Ontological Proof

This paper provides an interpretation and evaluation of Spinoza’s highly original version of the ontological proof in terms of the concept of substance instead of the concept of perfection in the first book of his Ethics. Taking the lead from Kant’s critique of ontological arguments in the Critique of Pure Reason, the paper explores the underlying ontological and epistemological presuppositions of Spinoza’s proof. The main topics of consideration are the nature of Spinoza’s definitions, the way he conceives of the relation between a substance and its essence, and his conception of existence. Once clarity is shed upon these fundamental issues, it becomes possible to address the proof in its own terms. It is then easy to see that Kant’s objections miss their target and that the same is true of those advanced by another of the ontological argument’s most famous critics, Bertrand Russell. Finally, several interpretations of Spinoza’s proof are proposed and critically evaluated; on all of them, the argument turns out to be either invalid or question-begging.     

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.     

Mining information from time series in the form of sentences of natural language

The goal of this paper is to provide a more detailed explanation of the principles how special formulas that characterize properties of trend of time series can be formed and how they are interpreted. Then we show how these formulas can be used in a tectogrammatical tree that construes special sentences of natural language, using which information on behavior of time series is provided. We also outline the principles of mining this information. The last part is devoted to application of the theory of intermediate quantifiers to mining summarized information on time series also in sentences of natural language.     

Stationary logic of ordinals

The L(aa)-theory of ordinals—Th_aa(On)—is studied. It is shown that Th_aa(On) is primitive recursive. In a suitable language it is possible to eliminate quantifiers. L(aa)-equivalence invariants are given. Both the complete L(aa)-theories of ordinals and the complete extensions of Th_aa(On) are characterized. An ordering is L(aa)-inductive if every L(aa)-definable subset (with suitable parameters) has a least element. The models of Th_aa(On) are the L(aa)-inductive orderings. A variant of the back and forth method is introduced in order toprove primitive recursive decidability and elimination of quantifier results.     

More examples of invariance under twisting

The so-called “invariance under twisting” for twisted tensor products of algebras is a result stating that, if we start with a twisted tensor product, under certain circumstances we can “deform” the twisting map and we obtain a new twisted tensor product, isomorphic to the given one. It was proved before that a number of independent and previously unrelated results from Hopf algebra theory are particular cases of this theorem. In this article we show that some more results from literature are particular cases of invariance under twisting, for instance a result of Beattie-Chen-Zhang that implies the Blattner-Montgomery duality theorem.