In \(\textbf{ZF}\), the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of _X_ of size \(|{\mathcal {P}} (X)|\) is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product \({\textbf{2}}^{{\mathcal {P}}(X)}\) has a dense subset of size |_X_|”. However, the latter statement turns out to be strictly weaker than \(\textbf{AC}\) while the full Hewitt–Marczewski–Pondiczery theorem is equivalent to \(\textbf{AC}\). We study the relative strengths in \(\textbf{ZF}\) between the statement “_X_ has (...) no independent family of size \(|{\mathcal {P}}(X)|\) ” and some of the definitions of “_X_ is finite” studied in Levy’s classic paper, observing that the former statement implies one such definition, is implied by another, and incomparable with some others. (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.

We show that (i) it is consistent with that there are infinite sets X on which every metric is discrete; (ii) the notion of real infinite is strictly stronger than that of metrically infinite; (iii) a set X is metrically infinite if and only if it is weakly Dedekind‐infinite if and only if the cardinality of the set of all metrically finite subsets of X is strictly less than the size of ; and (iv) an infinite set X is (...) weakly Dedekind‐infinite if and only if has infinite towers if and only if X has countable partitions. (shrink)

Many known tools for proving expressibility bounds for first-order logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is (...) one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree. (shrink)

Locality notions in logic say that the truth value of a formula can be determined locally, by looking at the isomorphism type of a small neighbourhood of its free variables. Such notions have proved to be useful in many applications. They all, however, refer to isomorphisms of neighbourhoods, which most local logics cannot test. A stronger notion of locality says that the truth value of a formula is determined by what the logic itself can say about that small (...) neighbourhood. Since the expressiveness of many logics can be characterized by games, one can also say that the truth value of a formula is determined by the type, with respect to a game, of that small neighbourhood. Such game-based notions of locality can often be applied when traditional isomorphism-based notions of locality cannot.Our goal is to study game-based notions of locality. We work with an abstract view of games that subsumes games for many logics. We look at three, progressively more complicated locality notions. The easiest requires only very mild conditions on the game and works for most logics of interest. The other notions, based on Hanf’s and Gaifman’s theorems, require more restrictions. We state those restrictions and give examples of logics that satisfy and fail the respective game-based notions of locality. (shrink)

The aim of this paper is twofold. First, I argue that Mary Shepherd has a dispositional understanding of God, matter, and finite minds. That is, she understands all of them as dispositions or powers (two terms I will use interchangeably). Second, I aim to shed light on the emanationist picture suggested by Shepherd’s remarks in her second book Essays on the Perception of an External Universe (EPEU). For instance, Shepherd calls ‘animate’ and ‘inanimate nature’ a divine ‘emanation’ (EPEU 190) (...) and describes finite beings as ‘outgoings’ that come ‘forth’ from God (EPEU 189, 190, 219). Because of this emanationist language, in conjunction with Shepherd’s dispositional understanding of God, matter, and mind, it is helpful to use Jennifer McKitrick’s (2003) distinction between bare and based disposition. For Shepherd’s remarks suggest that the powers that are matter and finite minds are akin to based dispositions because they are caused by and subsist in virtue of God. But since God – ‘THE GREAT FIRST CAUSE’ (ERCE 119) ‘who began not to be’ (EPEU 219) is also a power, the Deity functions similarly to a bare or ungrounded disposition, while matter and finite minds are similar to a ‘based dispositions’ (McKitrick 2003). (shrink)

Extending the ideas from (Hofer-Szabó and Rédei [2006]), we introduce the notion of causal up-to-n-closedness of probability spaces. A probability space is said to be causally up-to-n-closed with respect to a relation of independence R_ind iff for any pair of correlated events belonging to R_ind the space provides a common cause or a common cause system of size at most n. We prove that a finite classical probability space is causally up-to-3-closed w.r.t. the relation of logical independence iff its (...) probability measure is constant on the set of atoms of non-0 probability. (The latter condition is a weakening of the notion of measure uniformity.) Other independence relations are also considered. (shrink)

Modifying the methods of Z. Adamowicz's paper Herbrand consistency and bounded arithmetic [3] we show that there exists a number n such that ⋃m Sm (the union of the bounded arithmetic theories Sm) does not prove the Herbrand consistency of the finitely axiomatizable theory $S_{3}^{n}$.

Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...) to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension satisfying certain special identities is representable. We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density . As a consequence, every finite dimensional rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality. (shrink)

O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These include (...) all ordered structures definable in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of < is a "twisted lexicographic" order. As a consequence, for all n and linear orders ≺ definable on a subset X ⊆ Mn in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic". (shrink)

This article introduces the notion of duplex elements of the finite rings and corresponding neutrosophic rings. The authors establish duplex ring Dup(R) and neutrosophic duplex ring Dup(R)I)) by way of various illustrations. The tables of different duplicities are constructed to reveal the comparison between rings Dup(Zn), Dup(Dup(Zn)) and Dup(Dup(Dup(Zn ))) for the cyclic ring Zn . The proposed duplicity structures have several algebraic systems with dissimilar consequences. Author’s characterize finite rings with R + R is different from the (...) duplex ring Dup(R). However, this characterization supports that R + R = Dup(R) for some well known rings, namely zero rings and finite fields. (shrink)

We consider notions of boundedness of subsets of the natural numbers ℕ that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of ℕ to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and (...) sequential continuity. Most of the results do not require countable choice. (shrink)

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For (...) example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω. (shrink)

In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howard's majorizability notion; see [T, p. 456].

We are concerned with identifying by how much a finite cover of an 0-categorical structure differs from a sequence of free covers. The main results show that this is measured by automorphism groups which are nilpotent-by-abelian. In the language of covers, these results say that every finite cover can be decomposed naturally into linked, superlinked and free covers. The superlinked covers arise from covers over a different base, and to describe this properly we introduce the notion of a (...) quasi-cover.These results generalise results of the second author obtained in the case where the base of the cover is a grassmannian of a disintegrated set. They also give a complete proof of a statement of the second author extending this case to the case of a grassmannian of a modular set. To do this, we need to analyse the possible superlinked covers of such a set.We also give a combinatorial condition on the base of a cover which guarantees various chain conditions on finite covers over this base, and introduce a pregeometry which is useful in the analysis of finite covers with simple fibre groups. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality (...) set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory.

We show how the notion of full Frobenius group of finite Morley rank generalizes that of bad group, and how it seems to be more appropriate when we consider the possible existence (still unknown) of nonalgebraic simple groups of finite Morley rank of a certain type, notably with no involution. We also show how these groups appear as a major obstacle in the analysis of FT-groups, if one tries to extend the Feit-Thompson theorem to groups of finite (...) Morley rank. (shrink)

We define a formula φ in a first-order language L , to be an equation in a category of L -structures K if for any H in K , and set p = {φ;i ϵI, a i ϵ H} there is a finite set I 0 ⊂ I such that for any f : H → F in K , ▪. We say that an elementary first-order theory T which has the amalgamation property over substructures is equational if every (...) quantifier-free formula is equivalent in T to a boolean combination of equations in Mod, the category of models of T with embeddings for morphisms. Thus, we develop a theory of independence with respect to equations in general categories of structures, which is similar to the one introduced in stability but which, in our context, has an algebraic character. (shrink)

It is obvious that both Spinoza and Leibniz attach importance to the notion of expression in their philosophical writings and that both do so in a similar fashion: They agree, for example, that the mind expresses the body (although this claim has rather different meanings for each of them). Another – albeit related – use of ‘expression’ that appears in both thinkers provides a deeper insight into some metaphysical similarity as well as difference: The idea that expression is closely connected (...) with the perfection and action of individual things. While this relation is explicit in Leibniz, I will show that it is also in a less straightforward way found in Spinoza and, furthermore, that the relation of expression in regards to perfection is similar in Spinoza and Leibniz as both of them regard individuals as perfect insofar as they express the world and God. But one crucial difference in their accounts lies in the claim that, for Spinoza, what is being expressed and gives rise to perfection can be privative in nature, while such a thing cannot be the object of an expression for Leibniz. As I will argue, this not based merely on their different metaphysical views, but also on a difference in what can serve as content of an expression. (shrink)

We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom (...) of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)

On the rise over the past 20 years has been ‘moderate supernaturalism’, the view that while a meaningful life is possible in a world without God or a soul, a much greater meaning would be possible only in a world with them. William Lane Craig can be read as providing an important argument for a version of this view, according to which only with God and a soul could our lives have an eternal, as opposed to temporally limited, significance, by (...) virtue of our moral choices then making an ultimate difference. I present two major objections to this position. On the one hand, I contend that if God existed and we had souls that lived forever, then, in fact, all our lives would turn out the same. On the other hand, I maintain that, if this objection is wrong, so that our moral choices would indeed make an ultimate difference and thereby confer an eternal significance on our lives (only) in a supernatural realm, then Craig could not capture the view, aptly held by moderate supernaturalists, that a meaningful life is possible in a purely natural world. An eternal significance would be ‘too big’, reducing any meaning possible during an earthly life to nothing by comparison. I conclude that moderate supernaturalists would be wise to avoid appealing to eternal (or infinite) meaning when spelling out the way God and a soul could alone impart a great meaning to our lives; some other notion of greatness should be considered, perhaps one involving a temporary afterlife. (shrink)

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well. In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of (...) entailment, hence the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, GödelΔ, and Gödel∼ logics. (shrink)

The paper is an attempt to make sense of Hegel's notion of aufheben. The double meaning of aufheben and its alleged ?rise above the mere ?either?or?; of understanding? have been taken, by some, to constitute a criticism of the logic of either?or. It is argued, on the contrary, that Hegel's notion of aufheben, explicated in its primary and philosophical context, turns out to be a substantiation of that logic. The intelligibility of the formula of either?or depends, for example, on the (...) categories of Being and Not?Being. But if these categories are regarded as particular finite determinations themselves subject to the formula of either?or, then the formula, far from being intelligible, ?falls apart?. Hegel is arguing, in other words, that if we are to substantiate the logic of either?or, we must, at the same time, ?rise above? that logic. The role of aufheben is then considered in the special sciences. Here it is argued that we must distinguish between empirical transitions, governed by the finite determinations of things, and logical or dialectical transitions, governed by considerations of the intelligibility of the notions involved. Applying the notion of aufheben to the former transitions suggests wrongly that empirical transitions have an objective or philosophic necessity. Finally, the place of ?immanent transformation? in the context of aufheben is examined. It is concluded that if there is to be a transformation, then a distinction must be drawn between thought and its content, but then the transformation cannot be regarded as immanent. (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)

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.

Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can (...) be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

The aim of the paper is to clarify the theoretical core of Solger's thought, the foundation for his aesthetics. I first analyze Solger's dialectic of double negation. Secondly I focus on Solger's gnoseology, which is orientated toward grasping the equilibrium between the Infinite (God) and the finite (world) consisting in this double negation. Lastly I investigate the notion of sacrifice, connecting it with Solger's ironic dialectic and showing its relevance to a complete understanding of his thought.

In this paper I will examine what Blaise Pascal means by “infinite distance”, both in his works on projective geometry and in the apologetics of the Pensées’s. I suggest that there is a difference of meaning in these two uses of “infinite distance”, and that the Pensées’s use of it also bears relations to the mathematical concept of heterogeneity. I also consider the relation between the finite and the infinite and the acceptance of paradoxical relations by Pascal.

The aim of the paper is to clarify the theoretical core of Solger's thought, the foundation for his aesthetics. I first analyze Solger's dialectic of double negation. Secondly I focus on Solger's gnoseology, which is orientated toward grasping the equilibrium between the Infinite (God) and the finite (world) consisting in this double negation. Lastly I investigate the notion of sacrifice, connecting it with Solger's ironic dialectic and showing its relevance to a complete understanding of his thought.

In recent years part of the literature on probabilistic causality concerned notions stemming from Reichenbach’s idea of explaining correlations between not directly causally related events by referring to their common causes. A few related notions have been introduced, e.g. that of a “common cause system” (Hofer-Szabó and Rédei in Int J Theor Phys 43(7/8):1819–1826, 2004) and “causal (N-)closedness” of probability spaces (Gyenis and Rédei in Found Phys 34(9):1284–1303, 2004; Hofer-Szabó and Rédei in Found Phys 36(5):745–756, 2006). In this (...) paper we introduce a new and natural notion similar to causal closedness and prove a number of theorems which can be seen as extensions of earlier results from the literature. Most notably we prove that a finite probability space is causally closed in our sense iff its measure is uniform. We also present a generalisation of this result to a class of non-classical probability spaces. (shrink)

The Essay on Freedom represents a milestone in the philosophy of F. W. J. Schelling. Although Schelling already deviated from a strictly idealistic framework with his philosophy of nature, notably from that of Kant and Fichte, because he was seeking a moment of self-positing activity of the absolute subject in the object itself, it was only with The Essay on Freedom that he stepped out of that framework. The central point considered is an introduction of the second principle that was (...) complementary, but irreducible to the standard idealist principle of self consciousness. It was about the ground or nature in God. Only through that duality of principles, it was possible to grasp the positive notion of freedom as the possibility for good and evil. In such manner comprehended, the fact of human freedom in its entirety became relation with the whole of beings. The purpose of such explanation was an attempt to solve the problem that was spanning through Schellingʼs entire philosophy – the problem of the relation between finite and infinite. (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 recover the essentials of þ-forking, rosiness and super-rosiness for certain amalgamation classes K, and thence of finite-variable theories of finite structures. This provides a foundation for a model-theoretic analysis of a natural extension of the “LkLk-Canonization Problem” – the possibility of efficiently recovering finite models of T given a finite presentation of an LkLk-theory T. Some of this work is accomplished through different sorts of “transfer” theorem to the first-order theory TlimTlim of the direct limit. (...) Our results include, to start with, a recovery of the basic technology of þ-independence [15]) using a rather straightforward transfer. We also recover an analog of the “þ-Independence theorems” of Ealy and Onshuus [7] for amalgamation classes and their limits by showing how to transfer/lift an abstract independence relation on the amalgamation class to the limit theory TlimTlim. We also work out an appropriate notion of Local Character for independence relations over classes finite structures, and we use this to verify that rosiness and super-rosiness-with-finite-UþUþ-ranks coincide in these amalgamation classes and their limit theories. (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)

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics (...) from these classes have the finite model property with respect to the class of ♦-formulae, i.e. each ♦-formula has a ℒ-model iff it has a finite ℒ-model. Roughly speaking, a ♦-formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

This paper utilizes Scott domains (continuous lattices) to provide a mathematical model for the use of idealized and approximately true data in the testing of scientific theories. Key episodes from the history of science can be understood in terms of this model as attempts to demonstrate that theories are monotonic, that is, yield better predictions when fed better or more realistic data. However, as we show, monotonicity and truth of theories are independent notions. A formal description is given of (...) the pragmatic virtues of theories which are monotonic. We also introduce the stronger concept of continuity and show how it relates to the finite nature of scientific computations. Finally, we show that the space of theories also has the structure of a Scott domain. This result provides an analysis of how one theory can be said to approximate another. (shrink)

This paper aims to portray the human being as spirit, in dialogue with Levinas’ first philosophy. The relation between time and eternity is addressed in the work of both Kierkegaard and Levinas. However, in Kierkegaard’s notion of spirit there lies a discernible further development of the relation between the subject and that which transcends it. In Kierkegaard’s authorship, the absolute exteriority of the eternal does not break or suspend the finite structure of the subject. Contrary to Levinas’ critique of (...) the Danish philosopher, the possibility of a life is opened in which neither the external world nor the relation with the others is disdained. (shrink)

It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in (...) it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbach's definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle. (shrink)

We study properties of certain subclasses of the Dedekind finite sets in set theory without the axiom of choice with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows: 1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of (...) its elements are comparable. 2. The principle “Small Violations of Choice” —introduced by A. Blass—implies that the class of all Dedekind finite sets is bounded above. 3. “The class of all Dedekind finite sets is bounded above” is true in every permutation model of ZFA in which the class of atoms is a set, and in every symmetric model of ZF. 4. There exists a model of ZFA set theory in which the class of all atoms is a proper class and in which the class of all infinite Dedekind finite sets is not bounded above. 5. There exists a model of ZF in which the class of all infinite Dedekind finite sets is not bounded above. (shrink)

We investigate finitarity of unification types in locally finite varieties of Heyting algebras, giving both positive and negative results. We make essential use of finite dualities within a conceptualization for E-unification theory 733–752) relying on the algebraic notion of a projective object.

It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in (...) it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbachs definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle. (shrink)

This chapter suggests a new interpretation of Spinoza’s concept of mind claiming that the goal of the equation of the human mind with the idea of the body is not to solve the mind-body problem, but rather to show how we can, within the framework of Spinoza’s rationalism, conceive of finite minds as irreducibly distinguishable individuals. To support this view, the chapter discusses the passage from E2p11 to E2p13 against the background of three preliminaries, i.e. the notion of a (...) union between mind and body as it appears in Thomas Aquinas’ refutation of Averroism, Spinoza’s views on knowledge of actually existing things in E2p8c, and the phenomenological character of E2a2-4. It argues that while this view on the human mind does not undermine radical rationalism, it does require its amendment by some irreducibly empirical concessions. (shrink)