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.

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)

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)

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.

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.

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)

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)

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)

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

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

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While (...) the conjecture is known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

In this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ω m + 1.

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)

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)

Preface to the fifth edition -- A world of difference -- A universe charged with the grandeur of God : Christian theism -- The clockwork universe : deism -- The silence of finite space : naturalism -- Zero point : nihilism -- Beyond nihilism : existentialism -- Journey to the east : eastern pantheistic monism -- A separate universe : the New Age spirituality without religion -- The vanished horizon : postmodernism -- A view from the Middle East : (...) Islamic theism -- The examined life. (shrink)

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)

In the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) _Existence_: Settings where no countably additive probability exists though finitely additive probabilities do. (2) _Complete Additivity_: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then (...) probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the _Comparative_ principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the _Comparative_ principle [ \({\mathcal{C}}\) ]. For numerical probabilities, principle \({\mathcal{C}}\) contrasts pairs, P 1 and P 2, defined over a common partition \(\prod\) = {a _i_ : _i_ ∈ I} of measurable events. \({\mathcal{C}}\) requires that no P 1 may be pointwise dominated, i.e., no (finitely additive) probability P 2 exists such that for each _i_ ∈ I, P 2 (a _i_ ) > P 1 (a _i_ ). By design, the cardinality of \(\prod\) is not limited in \({\mathcal{C}}\), which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle \({\mathcal{C}}\) in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle \({\mathcal{C}}\) is subject to each of the above concerns, (1) and (2). Also, Easwaran considers principle \({\mathcal{C}}\) with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle \({\mathcal{C}}\) with qualitative probabilities and show that \({\mathcal{C}}\) restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle \({\mathcal{C}}\). (shrink)

Since the publication of his first book in 1953, Yves Bonnefoy has become one of the most important French poets of the postwar years. At last, we have the long-awaited English translation of Yves Bonnefoy’s celebrated work, _L’Arrière-pays_, which takes us to the heart of his creative process and to the very core of his poetic spirit. In his poem, “The Convex Mirror,” Bonnefoy writes: “Look at them down there, at that crossroads, / They seem to hesitate, then go on.” (...) The idea of the crossroads haunts Bonnefoy’s work, as he is troubled by the idea that the path not taken may lead to the _arrière-pays_, a place of greater plenitude, and of more authentic being—an “elsewhere in the absolute.” Seized by this fear that what he terms “presence” exists always somewhere else, a little further on, Bonnefoy here sets out on a labyrinthine quest to find traces of this “original place,” which he locates not only in objects of knowledge and experience as diverse as the deserts of Asia, a hill fort in India, a church in Armenia, the painting of Piero della Francesca but also, crucially, in the undivided intensity of his experiences as a child. Written with a visionary grace, _The Arrière-pays_ is a spiritual testament to art, philosophy, and poetry. Enriched by a new preface by the poet, this volume also includes three recent essays in which he returns to his original account of an ethical and aesthetic haunting, one that recounts the struggle between our instinct to idealize—what he deems our eternal Platonism—and the equally strong need to combat this and to be reconciled with our nature as finite beings, made of flesh and blood, in the world of the here and now. (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)

Finite Transcendence: Existential Exile and the Myth of Home introduces and situates “existential exile” as an experience of the fundamental finitude of human existence and demonstrates how a particular way of responding in faith may enable one to find home in exile. Using the literary and philosophical oeuvre of Albert Camus as a model, this book demonstrates the manner in which mythic literature can both present and engage the condition of exile toward its possible transcendence.

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)

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC (...) property is first order. (shrink)

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

The philosophy of superdeterminism is based on a single scientific fact about the universe, namely that cause and effect in physics are not real. In 2020, accomplished Swedish theoretical physicist, Dr. Johan Hansson published a physics proof using Albert Einstein’s Theory of Special Relativity that our universe is superdeterministic meaning a predetermined static block universe without cause and effect in physics. In the absence of cause and effect in physics, there can be no actual energy in our universe but only (...) the illusion of energy. This is consistent with the zero energy universe theory, which says that the positive matter energy is exactly balanced by the negative gravitational energy, so that the total energy of the universe is zero. An infinite universe would have an infinite amount of positive matter energy exactly cancelling out an infinite amount of negative gravitational energy. However, mathematically infinity minus infinity is not equal to zero, but rather is undefined. Consequently, an infinite universe would be inconsistent with the perfect cancelling balance of positive matter energy and negative gravitational energy under the zero energy universe theory. As a result, only a finite universe is consistent with the zero energy universe theory and the philosophy of superdeterminism. (shrink)

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice (...) of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

A partition $\{C_i\}_{i\in I}$ of a Boolean algebra Ω in a probability measure space (Ω, p) is called a Reichenbachian common cause system for the correlation between a pair A,B of events in Ω if any two elements in the partition behave like a Reichenbachian common cause and its complement; the cardinality of the index set I is called the size of the common cause system. It is shown that given any non-strict correlation in (Ω, p), and given any (...) class='Hi'>finite natural number n > 2, the probability space (Ω,p) can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. (shrink)

This paper investigates the question of how we manage to single out the natural number structure as the intended interpretation of our arithmetical language. Horsten submits that the reference of our arithmetical vocabulary is determined by our knowledge of some principles of arithmetic on the one hand, and by our computational abilities on the other. We argue against such a view and we submit an alternative answer. We single out the structure of natural numbers through our intuition of the absolute (...) notion of finiteness. (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.

Several treatments of the Shooting Room Paradox have failed to recognize the crucial role played by its involving a number of players unbounded in expectation. We indicate Reflection violations and/or Dutch Book vulnerabilities in extant ``solutions''and show that the paradox does not arise when the expected number of participants is finite; the Shooting Room thus takes its place in the growing list of puzzles that have been shown to require infinite expectation. Recognizing this fact, we conclude that prospects for (...) a ``straight solution" are dim. (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)

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)

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)

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.

In this book, written in the wake of such influential European thinkers as Nietzsche, Heidegger, Foucault, Derrida, and Vattimo, Simon Oliai argues that unless the “European” affirmation of man’s finite existence becomes universal, we shall never rid ourselves of the repressive shadow of a long dead metaphysical idol.

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to any finitely accessible (...) category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

In this paper we investigate those extensions of the bimodal provability logic ${\vec CSM}_{0}$ (alias ${\vec PRL}_{1}$ or ${\vec F}^{-})$ which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing ${\vec CSM}_{0}$ are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not (...) even complete with respect to Kripke semantics. (shrink)

A nonnegative interger is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of $\alpha, \subset$ for inclusion and $\subset_+$ for proper inclusion. Let α, β 1 ,...,β k be subsets of some set ρ. Then α' stands for ρ-α and β 1 ⋯ β k for β 1 ∩ ⋯ ∩ β k . For (...) subsets α 1 ,..., α r of ρ we write: $\alpha_\sigma - \{x \in v \ (\nabla i) \lbrack i \in \sigma \Rightarrow x \in \alpha_i\rbrack\} \text{for} \sigma \subset (1, \ldots, r),\\ s_i = \sum \{N(\alpha_\sigma) \mid \sigma \subset (1,\ldots, r) \& N(\sigma) = i\}, \text{for} 0 \leqq i \leqq r$ . Note that α 0 = v, hence s 0 = N(v). If the set v is finite, the classical inclusion-exclusion principle (abbreviated IEP) states $(a) N(\alpha_1 \cup \cdots \cup \alpha_r) = \sum^r_{t=1} (-1)^{t-1} s_t = \sum_{o \subset_+\sigma \subset (1,\ldots,r)}$ (b) N(α' 1 ⋯ α' r ) = ∑ r t=0 (-1) t s t = ∑ (-1) N(σ) N (α σ ). In this paper we generalize (a) and (b) to the case where α 1 ,⋯, α r are subsets of some countable but isolated set v. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalization of (a) and (b) are proved in § 3. Since they involve recursive distinctness, this notion is discussed in § 2. In § 4l we consider a natural extension of "the sum of the elements of a finite set σ" to the case where σ is countable. § 5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection λ of all isols which permit us to further generalize IEP by substituting μ(α) for $\operatorname{Req} \alpha$. (shrink)

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.

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.

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP ≠ coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to (...) be special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences. -/- We have introduced the most fundamental conjectures in our earlier works. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before. (shrink)

The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constraining the space of computational‐level theories of cognition. To utilize this constraint, a precise and workable definition of “computational tractability” is needed. Following computer science tradition, many cognitive scientists and psychologists define computational tractability as polynomial‐time computability, leading to the P‐Cognition (...) thesis. This article explains how and why the P‐Cognition thesis may be overly restrictive, risking the exclusion of veridical computational‐level theories from scientific investigation. An argument is made to replace the P‐Cognition thesis by the FPT‐Cognition thesis as an alternative formalization of the Tractable Cognition thesis (here, FPT stands for fixed‐parameter tractable). Possible objections to the Tractable Cognition thesis, and its proposed formalization, are discussed, and existing misconceptions are clarified. (shrink)

ABSTRACT The rapid increase in human population is one of the underlying factors driving the ecological crisis. Despite efforts on the part of educators to raise awareness of environmental issues, the ecological impact of a burgeoning population – and the ethical implications of having children – remains an unbroachable topic. Nevertheless, the increase in human numbers is central to questions of sustainability: How can a species expect to survive in a finite terrestrial environment without limits to its population? Since (...) most of the world’s ecological impact can be traced to capitalist-industrial- consumer societies in over-developed nations, the middle and upper classes in rich countries must weigh the ecological consequences of their family-planning decisions. In this paper, I argue that educational programs that are concerned with environmental ethics should have students examine the assumptions and implications of having children. I consider the risks associated with this proposal and respond to a series of possible objections. This paper does not advocate coercive measures for population control, but rather enjoins a pedagogical responsibility to view having children as an act with ecological consequences, an act that must be subject to careful examination. (shrink)