Through reading the early work of Walter Benjamin—up to and including the Trauerspiel, author Monad Rrenban elicits a cohesive conception of the wild, inforgettable form, philosophy, as inherent in everything. This book, distinct in its analysis and depth of analysis, elaborates the wild, unforgettable form—philosophy in relation to language, the discipline and the practice of philosophy, criticism, and the politics of death.

Through reading the early work of Walter Benjamin—up to and including the Trauerspiel, author Monad Rrenban elicits a cohesive conception of the wild, inforgettable form, philosophy, as inherent in everything. This book, distinct in its analysis and depth of analysis, elaborates the wild, unforgettable form—philosophy in relation to language, the discipline and the practice of philosophy, criticism, and the politics of death.

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of (...) "relatively functional" algebra, and show that every member of MBA is isomorphic to a relatively functional one. (shrink)

Aiming for applications in monadic second-order model theory, we study first-order theories without definable pairing functions. Our main results concern forking-properties of sequences of indiscernibles. These turn out to be very well-behaved for the theories under consideration.

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed.

In this paper we extend Mundici’s functor Γ to the category of monadic MV-algebras. More precisely, we define monadic ℓ -groups and we establish a natural equivalence between the category of monadic MV-algebras and the category of monadic ℓ -groups with strong unit. Some applications are given thereof.

This article discusses the final development of Gottfried Wilhelm Leibniz’s metaphysics: the theory of monads. It examines Leibniz’s arguments for monads as mindlike “simple substances,” his description of the properties of monads, and the distinction he draws among different types of monads. The remainder of the article focuses on two problems that attend Leibniz’s claim that reality ultimately consists solely of monads and their internal states (perceptions and appetitions). The first problem is whether a relation among monads can account for (...) the supposed unity of a living body or corporeal substance; the second is whether the metaphysics of monads supports a plausible explanation of the reality of matter. With regard to the second problem, the article explores Leibniz’s thesis that monads are, in two senses, “requisites” of matter. It concludes with reflections on the limits of his attempt to explain the physical world in terms of monads alone. (shrink)

We define and study monadic MV-algebras as pairs of MV-algebras one of which is a special case of relatively complete subalgebra named m-relatively complete. An m-relatively complete subalgebra determines a unique monadic operator. A necessary and sufficient condition is given for a subalgebra to be m-relatively complete. A description of the free cyclic monadic MV-algebra is also given.

In the following paper I would like to try to expound on a concept quite important in the philosophy of Leibniz – that of the “Monas Dominans”. In particular, I would like to approach this subject in the first place by means of considerations of a “historical-genetic” nature, while in the second part of my work I propose to put forward some possible interpretations of it. In both cases I will try to compare my ideas with those of recent studies (...) on this theme. The difficulties are numerous; first of all, the texts published to date in which Leibniz introduces and presents his doctrine of the “Monas Dominans” are relatively few, especially if compared to the space given to other elements of his philosophical system; secondly, if on the one hand the general intention put forward by Leibniz through the introduction of this doctrine is clear enough (the Monas Dominans renders “One” in the Machine the “corporeal substance”, that is the “Animal”, as found in the letter to De Volder of June 1703 to which scholars often make reference), on the other hand, when one tries to clarify its details and consider its implications one immediately runs into the same difficulties already pointed out in other studies (regarding, for example, the definition of “corporeal substance”, the relationship between the “Monas dominans” and the “innumerable subordinate monads” that form the “Organic Machine”, etc.). A primary idea I will try to present in this work regards the highlighting in the work of Leibniz of a close correlation between the appearance of the expression “organism” and that of the expression “Monas Dominans”. (shrink)

The Gödel translation provides an embedding of the intuitionistic logic$\mathsf {IPC}$into the modal logic$\mathsf {Grz}$, which then embeds into the modal logic$\mathsf {GL}$via the splitting translation. Combined with Solovay’s theorem that$\mathsf {GL}$is the modal logic of the provability predicate of Peano Arithmetic$\mathsf {PA}$, both$\mathsf {IPC}$and$\mathsf {Grz}$admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions$\mathsf {MIPC}$,$\mathsf {MGrz}$, and$\mathsf {MGL}$of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version (...) of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds$\mathsf {M^{+}IPC}$into$\mathsf {M^{+}Grz}$and the splitting translation embeds$\mathsf {M^{+}Grz}$into$\mathsf {MGL}$. As proven by Japaridze, Solovay’s result extends to the monadic system$\mathsf {MGL}$, which leads us to a provability interpretation of both$\mathsf {M^{+}IPC}$and$\mathsf {M^{+}Grz}$. (shrink)

In our paper, monadic modal pseudocomplemented De Morgan algebras are considered following Halmos’ studies on monadic Boolean algebras. Hence, their topological representation theory is used successfully. Lattice congruences of an mmpM is characterized and the variety of mmpMs is proven semisimple via topological representation. Furthermore and among other things, the poset of principal congruences is investigated and proven to be a Boolean algebra; therefore, every principal congruence is a Boolean congruence. All these conclusions contrast sharply with known results for monadic (...) De Morgan algebras. Finally, we show that the above results for mmpM are verified for monadic tetravalent modal algebras. (shrink)

In this paper, we investigate universal and existential quantifiers on NM-algebras. The resulting class of algebras will be called monadic NM-algebras. First, we show that the variety of monadic NM-algebras is algebraic semantics of the monadic NM-predicate logic. Moreover, we discuss the relationship among monadic NM-algebras, modal NM-algebras and rough approximation spaces. Second, we introduce and investigate monadic filters in monadic NM-algebras. Using them, we prove the subdirect representation theorem of monadic NM-algebras, and characterize simple and subdirectly irreducible monadic NM-algebras. (...) Finally, we present the monadic NM-logic and prove its completeness with respect to monadic NM-algebras. These results constitute a crucial first step for providing an algebraic foundation for the monadic NM-predicate logic. (shrink)

One of the main obstacles for panpsychism, the view that consciousness is fundamental and ubiquitous, is the difficulty of explaining how simple subjects could combine to form complex subjects. Known as the subject combination problem, it poses a possibly insurmountable challenge to the view. In this paper, I will assume that this challenge cannot be overcome and instead present a version of panpsychism that completely avoids talk of combination. Inspired by Gottfried Wilhelm Leibniz’s metaphysics of monads, I will focus on (...) a relational explanation of how simple subjects could constitute complex experiences, without them having to combine in virtue of their subjectivity. I call this view _monadic panpsychism_. Additionally, my proposal will not rely on emergence and so it will circumvent problems commonly faced by emergentist accounts. As I will argue, monadic panpsychism is preferable to combinatory and emergentist panpsychism because it faces a significantly less worrisome set of objections. Apart from being unaffected by the seemingly insuperable issue of subject combination, I will demonstrate that monadic panpsychism also has tools to address other kinds of the combination problem. That alone justifies the need for a new formulation of panpsychism, one which faces unique difficulties but also offers unique solutions. (shrink)

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of (...) MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $2^{3.2^{r.} } ^{2^{{\rm{2}}^{{\rm{r - 1}}} } } $ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r. (shrink)

Cappelen and Hawthorne present a powerful critique of fashionable relativist accounts of truth, and the foundational ideas in semantics on which the new relativism draws. They argue compellingly that the contents of thought and talk are propositions that instantiate the fundamental monadic properties of truth and falsity.

In this paper, we are concerned with the development of a general set theory using the single axiom version of Leśniewski’s mereology. The specification of mereology, and further of Tarski’s geometry of solids will rely on the Calculus of Inductive Constructions (CIC). In the first part, we provide a specification of Leśniewski’s mereology as a model for an atomless Boolean algebra using Clay’s ideas. In the second part, we interpret Leśniewski’s mereology in monadic second-order logic using names and develop a (...) full version of mereology referred to as CIC-based Monadic Mereology (λ-MM) allowing an expressive theory while involving only two axioms. In the third part, we propose a modeling of Tarski’s solid geometry relying on λ-MM. It is intended to serve as a basis for spatial reasoning. All parts have been proved using a translation in type theory. (shrink)

The beginning of the twenty-first century saw something of a comeback for relativism within analytical philosophy. Relativism and Monadic Truth has three main goals. First, we wished to clarify what we take to be the key moving parts in the intellectual machinations of self-described relativists. Secondly, we aimed to expose fundamental flaws in those argumentative strategies that drive the pro-relativist movement and precursors from which they draw inspiration. Thirdly, we hoped that our polemic would serve as an indirect defence of (...) a traditional and natural picture concerning truth. According to this picture, what we call ‘Simplicity’, the fundamental structure of semantic reality is best revealed by construing truth as a simple monadic property of propositions that serve as the objects of belief, assertion, meaning and agreement. Our project was not a straightforward one. So-called relativists are not uniform in their key ideology, are often sloppy, casual, obscure or confused in their self-characterization, and differ in their argumentative emphasis among themselves and over time, thereby presenting a target that is both amorphous and shifty. This is an area where parties will frequently claim not to understand each other and where certain parties will sometimes accuse others of failing to make any sense at all. In such a situation, any effort to impose order will inevitably strike some parties as tendentious and unfair. That said, we felt that we had enough of …. (shrink)

This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].

The genesis for this volume was in the bombing of Japan during World War II, where the author, as a young boy, watched the bombers overhead, speculating about the lives of the pilots and their relationship with those huddled on the ground._ From this disturbing diorama, Professor Hiroshi Kojima, the translator of Martin Buber into Japanese, unfolds a new approach to Buber's “I-Thou” relation, drawing upon insights from Husserl, Heidegger, and others in the tradition of continental philosophy to extend and (...) deepen Buber's thought. In chapters that reflect upon a wide range of phenomena—from religion, science, and technology, to imagination, embodiment, and power—Professor Kojima articulates a conception of what it means to be a human being that stands as an alternative to atomism and alienation of the modern world. Analyses of haiku and other aspects of Japanese culture demonstrate how Kojima's theory can illuminate the spiritual traditions of both East and West. Original in its thought and revealing in its insight into Japanese thought and culture, _Monad and Thou__ represents the life's work of one of Japan's great thinkers. (shrink)

In spite of their materialist aspirations, both classical and neoclassical economic theories rely on non-material notions of value to explain market activity. André Orléan calls this commitment of orthodox economics "the substance hypothesis." In this essay, I show how the substance hypothesis mirrors Gottfried Wilhelm Leibniz's account of monads, which he called the "true atoms of nature." I argue that value is the atom of economic nature in orthodox economic theories. Like monads, it is a fantasy. The atom of economic (...) nature that governs our actual, material lives, I argue, is money. (shrink)

I shall proceed in the following way. In parts II and III of this paper, I shall discuss the strengths and weaknesses of the interpretation put forward by Robert Merrihew Adams in his recent book, and I shall expand upon this account, discussing a crucial but hitherto unexamined aspect of the relation between dominant and subordinate monads, reconstructed from Leibniz's letters to Des Bosses and his essays of 1714, _Principles of Nature and Grace and Monadology. In part IV of this (...) paper, I shall examine various difficulties that arise both for my interpretation and, more generally, for Leibniz's account of the relation of domination and subordination. (edited). (shrink)

Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: (i) For every $S \subseteq \omega, \mathrm{ZFC} +$ "S and the monadic theory of ω 2 are recursive each in the other" is consistent; and (ii) ZFC + "The full second-order theory of ω 2 is interpretable in the monadic theory of ω 2 " is consistent.

Cappelen and Hawthorne present a powerful critique of fashionable relativist accounts of truth, and the foundational ideas in semantics on which the new relativism draws. They argue compellingly that the contents of thought and talk are propositions that instantiate the fundamental monadic properties of truth and falsity.

We investigate monadic fragments of Intuitionistic Control Logic, which is obtained from Intuitionistic Propositional Logic by extending language of IPL by a constant distinct from intuitionistic constants. In particular we present the complete description of purely negational fragment and show that most of monadic fragments are finite.

Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.

In 1928 Edmund Husserl wrote that “The ideal of the future is essentially that of phenomenologically based (“philosophical”) sciences, in unitary relation to an absolute theory of monads” (“Phenomenology”, Encyclopedia Britannica draft) There are references to phenomenological monadology in various writings of Husserl. Kurt Gödel began to study Husserl’s work in 1959. On the basis of his later discussions with Gödel, Hao Wang tells us that “Gödel’s own main aim in philosophy was to develop metaphysics—specifically, something like the monadology of (...) Leibniz transformed into exact theory—with the help of phenomenology.” (A Logical Journey: From Gödel to Philosophy, p. 166) In the Cartesian Meditations and other works Husserl identifies ‘monads’ (in his sense) with ‘transcendental egos in their full concreteness’. In this paper I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel and to aspects of Leibniz’s original monadology. (shrink)

Most interpreters think that for Leibniz, teleology is goodness-directedness. Explaining a monadic action teleologically, according to them, simply means explaining it in terms of the goodness of the state at which the agent aims. On some interpretations, the goodness at issue is always apparent goodness: an action is end-directed iff it aims at what appears good to the agent. On other interpretations, the goodness at issue is only sometimes apparent goodness and at other times merely objective goodness: some actions do (...) not aim at what appears good to the agent, but merely at what is objectively good—that is, at what God knows to be good—and that is sufficient for teleology. My paper, on the other hand, argues that both of these interpretations are mistaken. Monadic teleology, I contend, does not have to consist in striving for the good; neither goodness nor God is required to make monadic actions teleological. (shrink)

Epistemic monadic Boolean algebras are obtained by enriching monadic Boolean algebras with a knowledge operator. Epistemic monadic logic as the monadic fragment of first-order epistemic logic is introduced for talking about knowing things. A Halmos-style representation of epistemic monadic Boolean algebras is established. Relativizations of epistemic monadic algebras are given for modelling updates. These logics are semantically complete.

In this new work, Richard T. W. Arthur offers a fresh interpretation of Leibniz's theory of substance. He goes against a long trend of idealistic interpretations of Leibniz's thought by instead taking seriously Leibniz's claim of introducing monads to solve the problem of the composition of matter and motion.

We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

Many concepts, which can be constituents of thoughts, are somehow indicated with words that can be constituents of sentences. But this assumption is compatible with many hypotheses about the concepts lexicalized, linguistic meanings, and the relevant forms of composition. The lexical items simply label the concepts they lexicalize, and that composition of lexical meanings mirrors composition of the labeled concepts, which exhibit diverse adicities. If a phrase must be understood as an instruction to conjoin monadic concepts that correspond to the (...) constituents, lexicalization must be a process in which non-monadic concepts are used to introduce monadic analogues. But given such analogues, along with some thematic concepts, conjunctions can mimic the effect of saturating polyadic concepts. The lexical items efface conceptual adicity distinctions, making it possible to treat a recursive combination of expressions as a sign of monadic predicate conjunction. (shrink)

We investigate the monadic logic of trees with ω + 1 levels, the monadic topology of the product space ω λ and a strengthening of monadic logic for trees with ω levels.

In this paper we extend Mundici’s functor Γ to the category of monadic MV-algebras. More precisely, we define monadic ℓ-groups and we establish a natural equivalence between the category of monadic MV-algebras and the category of monadic ℓ-groups with strong unit. Some applications are given thereof.

The mature Leibniz often describes monads as having two types of modifications: perceptions and appetites. But why would monads need appetites? When reading secondary literature about Leibniz, it can easily look as if appetites are superfluous: some scholars describe the inner workings of monads without saying much, if anything, about appetites. Instead, they focus on perceptions and explain the transition to new perceptions by reference to prior perceptions together with the underlying primitive force or law of the series. These interpretations (...) appear to be doing just fine without invoking appetites. And, in fact, there are texts in which Leibniz himself neglects appetites almost completely. Some passages suggest that perceptions cause other perceptions, and other texts indicate that the monads themselves cause perceptions directly. If that is the case, it is not clear what purpose appetites serve; they do not appear to be needed for explaining monadic change. One possible response to the question why monads need appetites is the following. Appetites, according to Leibniz, are monadic tendencies to transition to new perceptions. Perhaps without such tendencies, no change would occur in a monad. On this interpretation, a full explanation of monadic change needs to invoke tendencies, and those in turn are (or include) appetites. I think that this response is partially correct, as I will argue later. Yet, it does not fully explain the function of appetites. How precisely are appetites (viewed as tendencies) supposed to figure into the explanation—do they function as causes of change, and if so, what kinds of causes? Are they efficient, final, or formal causes, or perhaps causes of yet another type? Or might they figure into the explanation in some non-causal way? This paper explores possible answers to the question what role appetites play in the lives of monads. There is not enough time to discuss these answers in much detail or exhaust the logical space, but I will briefly consider the options that strike me as the most promising contenders. These options are: (1) appetites are superfluous; there is no sense in which they explain or cause monadic change; (2) appetites are formal causes; (3) appetites are final causes; (4) appetites are efficient causes; (5) appetites are ultimately the same as perceptions (or aspects of perceptions), and perceptions in turn cause or explain monadic change; and (6) appetites are not causally involved in monadic change, but they are explanatory in a loose, derivative sense. I will argue for a version of the last option. The resulting picture of monadic change is admittedly radical and strange but there are excellent reasons to embrace it. (shrink)

Daniel Garber presents a study of Leibniz's conception of the physical world, elucidating his puzzling metaphysics of monads, mind-like simple substances.

Husserl's transcendental phenomenology is not a mere egology, but gets its concrete accomplishment only as a phenomenology of 'transcendental intersubjectivity'. However, the subjective centers of any transcendentality and thus of every constitution — even of intersubjectivity itself — have to be such unities as Leibniz' 'monads', that is, individually concrete subjects producing all their representations of one another completely out of themselves, respectively. Thus the problem arises, how the genuine transcendental status of each monadic subject in all its constitutive achievements (...) could be maintained so that they develop their own intentional lives in a universal mutual accordance, without presupposing again a supra-monadic ground for their 'harmony', i.e. a higher-levelled ordering unity. I shall argue that th peculiar Husserlian 'transcendental monadology' is inevitably bound to this intrinsic paradox which can at best be reduced to a mere postulate but never be resolved. (shrink)

This paper examines a widely accepted reading of monads as the most fundamental elements of reality. Garber [Leibniz – Body, Substance, Monad, Oxford: Oxford University Press, 2009] argues that simple monads – seen as mind-like atoms without parts and extension – replace the corporeal substance of Leibniz’s middle period. Phemister [Leibniz and the Natural World – Activity, Passivity and Corporeal Substances in Leibniz’s Philosophy, Dordrecht: Springer, 2005] argues that monads figure also at the top as complete corporeal substances. Building (...) on Fichant [‘L’invention métaphysique’ in G.W. Leibniz, Discours de métaphysique suivi de Monadologie et autres textes, edited by Michel Fichant, Paris: Gallimard, 2004], I argue that, for Leibniz, monads function not only as building blocks at the bottom level of composition but also at the top, as grounding the unity and hence the being of complete substances and organic unities. Since organic unities or living beings are seen by Leibniz as natural machines with a nested structure, and since monads are likened to living beings, this would imply that the use of the concept ‘monad’ holds not only at the bottom, and not only at the top, but all over the range between them. (shrink)

In every undirected graph or, more generally, in every undirected hypergraph of bounded rank, one can specify an orientation of the edges or hyperedges by monadic second-order formulas using quantifications on sets of edges or hyperedges. The proof uses an extension to hypergraphs of the classical notion of a depth-first spanning tree. Applications are given to the characterization of the classes of graphs and hypergraphs having decidable monadic theories.

In this paper, I show that by “Monade” the later Husserl means roughly what he meant by “das reine Bewußtsein” in the period of Ideas I. Of both consciousness and Monade, Husserl claims that objects of perception are immanent to them. I describe this claim as “quasi-representationalist” just because it bears enough similarity to some versions of contemporary representationalism. Since Husserl also claims that perceptual objects are publicly accessible, the inevitable conclusion seems to be that parts of perceptual consciousness must (...) also be publicly accessible. So strong internalist, Cartesian or idealist, interpretations of Husserl must be wrong. By the same token, if perceptual objects are necessarily physical, it then seems Husserl must agree that the physical can be, in some sense, immanent to consciousness. (shrink)

Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: For every $S \subseteq \omega, \mathrm{ZFC} +$ "S and the monadic theory of ω 2 are recursive each in the other" is consistent; and ZFC + "The full second-order theory of ω 2 is interpretable in the monadic theory of ω 2 " is consistent.

To escape from the labyrinth of the continuum, Leibniz maintains, we must think very differently about the nature of space, time, bodies, and substances; and in particular we must posit an infinity of simple substances or monads. The main aim of this historically rich and interpretively provocative book is to explain why Leibniz says such things by examining his purported solution and how he arrived at it. Each of the book’s seven chapters focuses on a different “Ariadnean thread” that supposedly (...) helped Leibniz find his way out of the labyrinth. They concern the themes of “composition, aggregation, atoms, forms, motion, substance, and continuation in existence” (6). (shrink)

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for$\forall $) and “sometime in the past” (for$\exists $). It is well known that Prior’s intuitionistic modal logic${\sf MIPC}$axiomatizes the monadic fragment of the intuitionistic predicate logic, and that${\sf MIPC}$is translated fully and faithfully into the monadic fragment${\sf MS4}$of the predicate${\sf S4}$via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension${\sf TS4}$of${\sf S4}$and provide a full and faithful translation (...) of${\sf MIPC}$into${\sf TS4}$. We compare this new translation of${\sf MIPC}$with the Gödel translation by showing that both${\sf TS4}$and${\sf MS4}$can be translated fully and faithfully into a tense extension of${\sf MS4}$, which we denote by${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for${\sf MS4.t}$using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. (shrink)

Assume ZFC is consistent then for everyB⫅ω there is a generic extension of the ground world whereB is recursive in the monadic theory ofω 2.

Let S2S [WS2S] espectively be the storn [weak] monadic second order theory of the binary tree T in the language of two successor functions. An S2S-formula whose free variables are just individual variables defines a relation on T (rather than on the power set of T). We show that S2S and WS2S define the same relations on T, and we give a simple characterization of these relations.

In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...) line H, which is a hyperfinite initial segment of the hyperintegers. The U-monad topology of H is the quotient topology of the U-topological space H modulo U. In this paper we answer a question of Keisler and Leth about the U-monad topologies by showing that when H is κ-saturated and has cardinality κ, (1) if the coinitiality of U1 is uncountable, then the U1-monad topology and the U2-monad topology are homeomorphic iff both U1 and U2 have the same coinitiality; and (2) H can produce exactly three different U-monad topologies (up to homeomorphism) for those U's with countable coinitiality. As a corollary H can produce exactly four different U-monad topologies if the cardinality of H is ω1. (shrink)

Let ( * X, * T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in * X is defined as m(x) = μ T (st(x)). Consider the relation $R_{\mathrm{ns}} = \{\langle x, y \rangle \mid x, y \in \mathrm{ns} (^\ast X) \text{and} y \in m(x)\}.$ Frank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of R ns to the whole of * X. (...) Wattenberg's extensions of R ns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of R ns to * X should be possible in any topological space. A study of such extensions of R ns is the purpose of the present paper. We call a binary relation $W \subseteq ^\ast X \times ^\ast X$ an infinitesimal on * X if it is monadic and reflexive on * X. We prove, among other things, that the existence of an infinitesimal on * X that extends R ns is equivalent to the condition that the space (X, T) be regular. (shrink)

In recent years large amounts of electronic texts have become available. While the first of these corpora had only a low level of annotation, the more recent ones are annotated with refined syntactic information. To make these rich annotations accessible for linguists, the development of query systems has become an important goal. One of the main difficulties in this task consists in the choice of the right query language, a language which at the same time should be powerful enough to (...) let users formulate the queries they want and which should be efficiently evaluable to keep query response times short. There is a widespread belief that such a query language does not exist. It is therefore the aim of this paper to show that there is indeed a powerful query language that can be efficiently evaluated. We propose the use of monadic second-order logic as a query language. We show that a query in this language can be evaluated in linear time in the size of a tree in the corpus. We also provide examples of complicated linguistic queries expressed in monadic second-order logic thereby demonstrating the high expressive power of the language. (shrink)

Büchi, J. R. The monadic second order theory of [omega symbol]₁.--Büchi, J. R. and Siefkes, D. Axiomatization of the monadic second order theory of [omega symbol]₁.