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.

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.

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)

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.

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)

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)

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)

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.

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)

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 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)

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 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)

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)

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)

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.

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)

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)

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 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)

Gödel once offered an argument for the general reflection principle in set theory that took the form of an analogy with Leibniz' Monadology. I discuss the mathematical and philosophical background to Gödel's argument, reconstruct the proposed analogy in detail, and argue that it has no justificatory force.

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 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)

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)

Daniel C. Maguire (1978, 1982) has presented a moral theory based on affective experiences. He sees this sort of theory as protection against ethical relativism as well as leading toward a morality necessarily based on religion. Along the way, ethical reductionism is discarded. This paper argues that precisely the opposite has happened. Maguire is open to the charge of ethical relativism and so loses religion as a base. A sense can be given to ethical reductionism and to what Maguire terms (...) "the intellectual fallacy" such that the reductionism is no enemy and the fallacy, no fallacy. (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.

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.

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)

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.

Leibniz's mature philosophy is monadological. The only true substances are monads which Leibniz compares to Cartesian res cogitans. In the monadological view, the only variant properties of monads are perceptions of varying degrees of distinctness; corporeal substances are phenomena that supervene on the ordered perceptions of monads. Monads continually change their perceptions, and in this minimal sense, monads are constantly active. The question that any account of Leibniz's monadological view must answer is: What causes the change in the perceptions of (...) monads? To answer this question is to gain clarity not only about Leibniz's monadological account of change, but also about Leibniz's views on causality and explanation for phenomena, since it is the changes in perceptions of monads that seem to account for changes in phenomena. (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.

A matter of significance for the history of philosophy is the question of what are the issues that underlie Leibniz's response to Locke's Essay Concerning Human Understanding, in his own New Essays on Human Understanding. Exploration of that question can contribute to interpretations of both Locke and Leibniz. Equally important, it can provide insight into problems of philosophy that have their genesis in the seventeenth century. ;The dissertation uses the Essay and the New Essays to explore what it regards as (...) the central conflict between Locke and Leibniz: the conflict about the primacy of metaphysics. It takes for its focus their differing positions on atomism, contrasting the particle theory of matter that it finds central to the Essay with Leibniz's metaphysical unit, the monad. It develops the implications of those differing positions for an idea of nature specifically--nature here conceived as the arena in which science and metaphysics meet. ;The thesis of the dissertation is that the positions of Locke and Leibniz on atomism express opposing responses of philosophy to science; and, further, that this opposition had profound significance for the subsequent direction of philosophy. The first element in the thesis is the specific concern of the dissertation. Its second element, which suggests an approach to the history of philosophy since the seventeenth century, is an issue for further exploration. ;Locke's Essay is shown as a deliberate construction of an epistemology grounded in the new science, combining that purpose with an explicit rejection of the value of metaphysical speculation as the basis for knowledge. Locke's scientific background and orientation determine his analysis of ideas, of the powers and qualities of matter, and of real essences of material substances. ;In contrast, the New Essays is seen as asserting the primacy of metaphysics for the solution of epistemological and scientific questions. Leibniz's arguments against atomism are examined, to show why the basic reason for his rejection of all particle theories of matter is their violation of specific metaphysical principles. His positive position is presented in a discussion of the monad, emphasizing its characteristics in the realm of nature. (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)

Some things have positive value and some things have negative value. The things with positive value are good and the things with negative value are bad. There are also things in-between that are neither good nor bad, which are neutral. All in all, then, there are three monadic value predicates: “good,” “bad,” and “neutral.”.

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].

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.

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

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.

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.

Words indicate concepts, which have various adicities. But words do not, in general, inherit the adicities of the indicated concepts. Lots of evidence suggests that when a concept is lexicalized, it is linked to an analytically related monadic concept that can be conjoined with others. For example, the dyadic concept CHASE(_,_) might be linked to CHASE(_), a concept that applies to certain events. Drawing on a wide range of extant work, and familiar facts, I argue that the (open class) (...) lexical items of a natural spoken language include neither names nor polyadic predicates. The paper ends with some speculations about the value of a language faculty that would impose uniform monadic analyses on all concepts, including the singular and relational concepts that we share with other animals. (shrink)

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, (...) if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt on properties which has the same relationship with the Circuit Value Problem as reach has with the Directed Reachability Problem. We use alt to show that Πn⊈ FO, Σn⊈ FO, and Δn+1⊈ FOB, solving some open problems raised in [Mat98]. (shrink)