Avec le temps, une oeuvre d'art s'éloignera fatalement du sens que, par provision, son auteur lui donne. Celui-ci, néanmoins, escompte secrètement cette méprise future comme une solution possible à son énigme. S'il est vrai que "le fondement même du discours interhumain est le malentendu" (Lacan), on devrait considérer l'art, ou la relation artistique, comme un malentendu spécialement productif, paradoxal et initiatique. Ce ne sont ni les peintres ni les regardeurs qui font les tableaux, mais la conjugaison de l'inconscience des uns (...) et des bévues des autres : ils se déchargent l'un sur l'autre de la responsabilité d'un sens qui n'en finit pas de leur échapper. Le présent ouvrage évoque quelques-unes de ces méprises en symétrie inverse, indéfiniment reconduites, et qu'on peut considérer finalement comme des "ratages réussis". Ce n'est pas le moindre intérêt de l'histoire de l'art que ces coups de théâtre qui rendent le passé lui-même imprévisible."--Page 4 of cover. (shrink)

When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the (...) recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu. (shrink)

Spinoza’s bold claim that there exists only a single infinite substance entails that finite things pose a deep challenge: How can Spinoza account for their finitude and their plurality? Taking finite bodies as a test case for finite modes in general I articulate the necessary conditions for the existence of finite things. The key to my argument is the recognition that Spinoza’s account of finite bodies reflects both Cartesian and Hobbesian influences. This recognition leads to the surprising realization there must (...) be more to finite bodies than their finitude, a claim that goes well beyond the basic substance-monism claim, namely, that anything that is, is in God. This leads to the conclusion, which may seem paradoxical, that finite bodies have both an infinite as well as a finite aspect to them. Finite bodies, I argue, both actively partially determine all the other finite bodies, thereby partially causing their existence insofar as they are finite, as well as are determined by the totality of other bodies. I articulate precisely what this infinite aspect is and how it is distinct from the general substance-monism dictum. (shrink)

Science is continually faced with describing that which is beyond. This book, through contributions from nine prominent scholars, tackles that challenge.

The SRL of King is a sound, complete and decidable logic designed specifically to support formalisms for the HPSG of Pollard and Sag. The SRL notion of modellability in a signature is particularly important for HPSG, and the present paper modifies an elegant method due to Blackburn and Spaan in order to prove that – modellability in each computable signature is 1 0 – modellability in some finite signature is 1 0 -hard, and – modellability in some finite signature is (...) decidable. (shrink)

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

Finite model theory is currently not one of the hot topics in the philosophy and history of mathematics, not even in the philosophy and history of mathematical logic. The philosophy of mathematics and mathematical logic has concentrated on infinite structures, closely related to foundational issues. In that context, finite models deserved only marginal attention because it was taken for granted that the study of finite structures is trivial compared to the study of infinite structures. In retrospect, research on finite structures (...) turned out to be neither trivial nor irrelevant but had an important impact on theoretical computer science. A closer look at the early history of the spectrum problem shows that the birth of finite model theory has roots in formal logic, at a time when computer science did not yet exist. (shrink)

This essay argues that Spinoza believed that each finite mode is absolutely necessitated by God's nature and is causally necessitated by the laws of nature in conjunction with other finite modes. A geometrical analogy from Part 2 of the Ethics is employed in order to give a more suggestive account of the ways in which all things are necessary, according to Spinoza.

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

I discuss a difficulty concerning the justification of the Axiom of Choice in terms of such informal notions such as that of iterative set. A recent attempt to solve the difficulty is by S. Lavine, who claims in his Understanding the Infinite that the axioms of set theory receive intuitive justification from their being self-evidently true in Fin(ZFC), a finite counterpart of set theory. I argue that Lavine's explanatory attempt fails when it comes to AC: in this respect Fin(ZFC) is (...) no better off than the iterative notion. (shrink)

Arrhenius’s impossibility theorems purport to demonstrate that no population axiology can satisfy each of a small number of intuitively compelling adequacy conditions. However, it has recently been pointed out that each theorem depends on a dubious assumption: Finite Fine-Grainedness. This assumption states that there exists a finite sequence of slight welfare differences between any two welfare levels. Denying Finite Fine-Grainedness makes room for a lexical population axiology which satisfies all of the compelling adequacy conditions in each theorem. Therefore, Arrhenius’s theorems (...) fail to prove that there is no satisfactory population axiology. In this paper, I argue that Arrhenius’s theorems can be repurposed. Since all of our population-affecting actions have a non-zero probability of bringing about more than one distinct population, it is population prospect axiologies that are of practical relevance, and amended versions of Arrhenius’s theorems demonstrate that there is no satisfactory population prospect axiology. These impossibility theorems do not depend on Finite Fine-Grainedness, so lexical views do not escape them. (shrink)

The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.

We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...) model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it. (shrink)

Written by a highly respected scholar of Thomas Aquinas's writings, this volume offers a comprehensive presentation of Aquinas's metaphysical thought. It is based on a thorough examination of his texts organized according to the philosophical order as he himself describes it rather than according to the theological order. -/- In the introduction and opening chapter, John F. Wippel examines Aquinas's view on the nature of metaphysics as a philosophical science and the relationship of its subject to divine being. Part One (...) is devoted to his metaphysical analysis of finite being. It considers his views on the problem of the One and the Many in the order of being, and includes his debt to Parmenides in formulating this problem and his application of analogy to finite being. Subsequent chapters are devoted to participation in being, the composition of essence and esse in finite beings, and his appeal to a kind of relative nonbeing in resolving the problem of the One and the Many. Part Two concentrates on Aquinas's views on the essential structure of finite being, and treats substance-accident composition and related issues, including, among others, the relationship between the soul and its powers and unicity of substantial form. It then considers his understanding of matter-form composition of corporeal beings and their individuation. Part Three explores Aquinas's philosophical discussion of divine being, his denial that God's existence is self-evident, and his presentation of arguments for the existence of God, first in earlier writings and then in the "Five Ways" of his Summa theologiae. A separate chapter is devoted to his views on quidditative and analogical knowledge of God. The concluding chapter revisits certain issues concerning finite being under the assumption that God's existence has now been established. -/- John F. Wippel, professor of philosophy at The Catholic University of America, was recently awarded the prestigious Aquinas Medal by the American Catholic Philosophical Association. In addition to numerous articles and papers, Wippel has coauthored or edited several other works, including Metaphysical Themes in Thomas Aquinas and The Metaphysical Thought of Godfrey of Fontaines, both published by CUA Press. (shrink)

Schutz’s references to literature and arts in his theoretical works are manifold. But literature and theory are both a certain kind of a finite province of meaning, that means they are not easily accessible from the paramount reality of everyday life. Now there is another kind of referring to literature: metaphorizing it. Using it, as may be said with Lakoff and Johnson, to understand and to experience one kind of thing in terms of another. Literally metapherein means “to carry over”. (...) Metaphorizing in this view is then a specific kind of border-crossing between different provinces of meaning. That poses two questions: 1. What means finiteness of those provinces of meaning, what kind of border crossings are possible? What is the ground for metaphorizing meaning? 2. Could this concept used for founding a theory of the constitution of the societal and of society, that overcomes the dichotomy of structure/agency? These questions will be answered with one example in view: Schutz’ report to Kaufmann of his first visit of Husserl describing his experience as feeling like Wilhelm Meister at the Society of the Tower. In a first step this metaphor is presented together with some crumbs of metaphor theory. In a second step these crumbs will be connected to Husserl’s concept of experience. After developing a short overview over Schutz’ “finite provinces of meaning,” the relation of experience, metaphors to the intersubjectivity of these provinces in their dependence from writing and printing is discussed. (shrink)

Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic (MALL) and for affine logic (LLW), i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL (which we call IMALL), and intuitionistic LLW (which we call ILLW). In addition, we shall show the finite model property for contractive linear logic (LLC), i.e., linear logic with contraction, and for its intuitionistic version (ILLC). (...) The finite model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL - -systems except FL c and GL - c of Ono [11], that will settle the open problems stated in Ono [12]. (shrink)

We write and for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality. In this paper, we investigate relationships between and for an infinite cardinal in the absence of the Axiom of Choice. We give conditions that make and comparable as well as give related consistency results.

First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice (...) of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them. (shrink)

The article explores the idea that according to Spinoza finite thought and substantial thought represent reality in different ways. It challenges “acosmic” readings of Spinoza's metaphysics, put forth by readers like Hegel, according to which only an infinite, undifferentiated substance genuinely exists, and all representations of finite things are illusory. Such representations essentially involve negation with respect to a more general kind. The article shows that several common responses to the charge of acosmism fail. It then argues that we must (...) distinguish the well-founded ideality of representations of finite things from mere illusoriness, insofar as for Spinoza we can have true knowledge of what is known only abstractly. Finite things can be seen as well-founded beings of reason. The article also proposes that within Spinoza's framework it is possible to represent a finite thing without drawing on representations of mind-dependent entities. (shrink)

Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of (...) learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work. (shrink)

What are the assumptions and tasks hidden in contemporary calls to "overcome" the metaphysical tradition? Reflecting upon the internal contradictions of the notions of "tradition" and "finiteness," Dennis J. Schmidt offers novel insights into how philosophy must relate to its traditions if it is to retain a vital sense of the plurality of "edges" that constitute its finiteness. He does this through a close examination of issues found in the work of Hegel and Heidegger, two philosophers who made the ideas (...) of both tradition and finiteness the center of their concern.Schmidt begins by asking how Heidegger can claim to have destroyed metaphysics despite Hegel's claim to have perfected its possibilities. Systematically following the development of Heidegger's critique of Hegel, Schmidt generates a dialogue between them. The topic of that dialogue is the nature of finiteness as it is articulated in time, nothing, the dialectical and hermeneutical circles, and in the notions of experience, work, technology, history, and preSocratic thought.Beginning with Heidegger's critique of Hegel in Being and Time, Schmidt's strategy is to disclose the complexities of philosophical discourse about the finite by drawing out the proximities between Hegel and Heidegger. The dialogue that results presents novel portraits of both philosophers. It also reveals that Heidegger's early, unacknowledged failure to separate himself from the Hegelian dialectic is the motive behind many of the turns and decisions of his later career.In concluding, Schmidt offers an interpretation of the wider significance of the results of that dialogue, and connects his study to other contemporary discussions of postmodernism. He expands upon the idea of the plurality of edges opened by finiteness, arguing that philosophy only understands its own past and future once it recognizes the meaning of its own finiteness.Dennis J. Schmidt is Associate Professor of Philosophy at the State University of New York at Binghamton. The Ubiquity of the Finite is included in the series Studies in Contemporary German Social Thought, edited by Thomas McCarthy. (shrink)

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional connectives of Łukasiewicz’s (...) logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete. (shrink)

A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz (...) and the Maltsev hierarchies. (shrink)

It is standard to think that in Spinoza’s system, all things are necessary and in no sense contingent. However, in his classic book, Spinoza’s Metaphysics, published in 1969, Edwin Curley argues based on the proposition 28 of the first part of the Ethics that Spinoza endorses necessitarianism of only a modest kind, according to which when it comes to finite modes, there is a sense in which they are contingent. In this paper, I revisit Curley’s argument. Commentators have responded to (...) Curley’s argument, showing that Spinoza’s remarks on infinite modes entail that finite modes can in no sense be contingent. But this alone falls short of dispelling Curley’s misgivings about the standard interpretation, for it remains unexplained why Curley is wrong in thinking that the proposition 28 supports his moderate necessitarian interpretation. In defense of the standard interpretation, I bolster the usual response to Curley in greater detail than has been done in the literature and explain why, pace Curley, the proposition 28 plays no evidential role in support of Curley’s interpretation. (shrink)

In this paper we obtain a finite Hilbert-style axiomatization of the implicationless fragment of the intuitionistic propositional calculus. As a consequence we obtain finite axiomatizations of all structural closure operators on the algebra of {–}-formulas containing this fragment.

The tangle modality is a propositional connective that extends basic modal logic to a language that is expressively equivalent over certain classes of finite frames to the bisimulation-invariant fragments of both first-order and monadic second-order logic. This paper axiomatises several logics with tangle, including some that have the universal modality, and shows that they have the finite model property for Kripke frame semantics. The logics are specified by a variety of conditions on their validating frames, including local and global connectedness (...) properties. Some of the results have been used to obtain completeness theorems for interpretations of tangled modal logics in topological spaces. (shrink)

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

We exhibit a finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees. Our method promises to lead to a full characterization of the finite lattices embeddable into the enumerable Turing degrees.

Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic and for affine logic, i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL, and intuitionistic LLW. In addition, we shall show the finite model property for contractive linear logic, i.e., linear logic with contraction, and for its intuitionistic version. The finite model property for related substructural logics also follow by our (...) method. In particular, we shall show that the property holds for all of FL and GL$^-$-systems except FL$_c$ and GL$^-_c$ of Ono [11], that will settle the open problems stated in Ono [12]. (shrink)

We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices (...) into finite trees. We prove that every insertion rule greedily generates a tree with these same structural properties; and every decreasing insertion rule generates (or admits) a tree with these same structural properties. It is also necessary and sufficient to use the same large cardinals (in the precise sense of Corollary D.25). The results suggest new areas of research in discrete mathematics called "Ramsey tree theory" and "greedy Ramsey theory" which demonstrably require more than the usual axioms for mathematics. (shrink)

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is wild, then the theory of all R-modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

The article explores the following question: which among the most often examined in the literature method of constructing Hilbert-type axiom systems for finite-valued propositional logics of Łukasi...

This work investigates the finite-time control problem for a nonlinear four-tank cross-coupled liquid level system by the port-controlled Hamiltonian model. A fixed-free methodology is exhibited which can be used to simplify the controller design procedure. To get an adjustable convergent gain of the finite-time control, a feasible technique named damping normalization is proposed. A novel parameter autotuning algorithm is given to clarify the principle of choosing parameters of the PCH method. Furthermore, a finite-time controller is designed by a state-error desired (...) Hamiltonian function, and the relationship between the settling time and a parameter is given, which can be applied in practical engineering easily to adjust the settling time according to the industrial need. Finally, simulation and experimental results verify the effectiveness of the proposed algorithm. (shrink)

In this study, we apply MOSAIC (model of syntax acquisition in children) to the simulation of the developmental patterning of children's optional infinitive (OI) errors in 4 languages: English, Dutch, German, and Spanish. MOSAIC, which has already simulated this phenomenon in Dutch and English, now implements a learning mechanism that better reflects the theoretical assumptions underlying it, as well as a chunking mechanism that results in frequent phrases being treated as 1 unit. Using 1, identical model that learns from child‐directed (...) speech, we obtain a close quantitative fit to the data from all 4 languages despite there being considerable cross‐linguistic and developmental variation in the OI phenomenon. MOSAIC successfully simulates the difference between Spanish (a pro‐drop language in which OI errors are virtually absent) and obligatory subject languages that do display the OI phenomenon. It also highlights differences in the OI phenomenon across German and Dutch, 2 closely related languages whose grammar is virtually identical with respect to the relation between finiteness and verb placement. Taken together, these results suggest that (a) cross‐linguistic differences in the rates at which children produce OIs are graded, quantitative differences that closely reflect the statistical properties of the input they are exposed to and (b) theories of syntax acquisition need to consider more closely the role of input characteristics as determinants of quantitative differences in the cross‐linguistic patterning of phenomena in language acquisition. (shrink)

The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.

This book is a rich collection of philosophical essays radically interrogating key notions and preoccupations of the phenomenological tradition. While using Heidegger’s Being and Time as its permanent point of reference and dispute, this collection also confronts other important philosophers, such as Kant, Nietzsche, and Derrida. The projects of these pivotal thinkers of finitude are relentlessly pushed to their extreme, with respect both to their unexpected horizons and to their as yet unexplored analytical potential. A Finite Thinking shows that, paradoxically, (...) where the thought of finitude comes into its own it frees itself, not only to reaffirm a certain transformed and transformative presence, but also for a non-religious reconsideration and reaffirmation of certain theologemes, as well as of the body, heart, and love. This book shows the literary dimension of philosophical discourse, providing important enabling ideas for scholars of literature, cultural theory, and philosophy. (shrink)

Tense logics in the bimodal propositional language are investigated with respect to the Finite Model Property. In order to prove positive results techniques from investigations of modal logics above K4 are extended to tense logic. General negative results show the limits of the transfer.

I show that frequentism, as an explanation of probability in classical statistical mechanics, can be extended in a natural way to a decoherent quantum history space, the analogue of a classical phase space. The result is a form of finite frequentism, in which Gibbs’ concept of an infinite ensemble of gases is replaced by the quantum state expressed as a superposition of a finite number of decohering microstates. It is a form of finite and actual frequentism (as opposed to hypothetical (...) frequentism) insofar as all the microstates exist, in keeping with the decoherence-based Everett interpretation, and some versions of pilot-wave theory. (shrink)

We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.

The partial cooperation displayed by subjects in the Centipede Game deviates radically from the predictions of traditional game theory. Even standard, infinite population, evolutionary settings have failed to provide an explanation for this behavior. However, recent work in finite population evolutionary models has shown that such settings can produce radically different results from the standard models. This paper examines the evolution of partial cooperation in finite populations. The results reveal a new possible explanation that is not open to the standard (...) models and gives us reason to be cautious when employing these otherwise helpful idealizations. *Received January 2007; revised November 2007. †To contact the author, please write to: Department of Logic and Philosophy of Science, University of California, 3151 Social Science Plaza A, Irvine, CA 92697-5100; e-mail: [email protected]. (shrink)