We give a negative solution to the problem, posed by A. Chagrov and M. Zakharyaschev, of whether every quasi-normal propositional modal logic can be axiomatized by an independent set of axioms, with the inference rules of Substitution and Modus Ponens.

Let $\mathcal {I}$ be an ideal on $\omega $. For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$. Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions).We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal (...) {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$, respectively. For a maximal ideal $\mathcal {I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$ ) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$.We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially (...) in formal argumentation, (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)

The paper develops an attack on quasi-realism in ethics, according to which expressivism about ethical discourse—understood as the thesis that the states that discourse expresses are non-representational—is consistent with some of the discourse's familiar surface features, thus ‘saving the ethical appearances’. A dilemma is posed for the quasi-realist. Either ethical discourse appears, thanks to those surface features, to express representational states, or else there is no such thing as its appearing to express such states. If the former then, (...) by expressivism, the appearance presented by ethical discourse is false, so the ethical appearances are not saved. If the latter, it is unintelligible why an appeal to projection should be needed to explain how the surface features come to express non-representational states if no explanation is needed—as evidently none is—to explain how they come to express representational states. The conclusion of this argument is then argued to converge with some other considerations which show that there is no gap between ethical discourse's possessing the surface features in question and its expressing representational states. (shrink)

IN the right circumstances and the right frame of mind we are prepared to make aesthetic appraisals of almost anything, from hills, cottages and cars, to symphonies, people and poems. My problem is to try and set a boundary in at least one direction to the catholicity of this kind of judgement. I want to argue that when we use a word from our aesthetic vocabulary for appraising a theory in science or a proof in mathematics we are not properly (...) to be described as making an aesthetic appraisal, though there are close parallels with our normal use of these words. (shrink)

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras.algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety ������ the lattice of congruences of A is isomorphic to the lattice of deductive filters (...) on A of the τ-assertional logic of ������. Moreover, if ������ has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ ������ of the 1-assertional logic of ������ coincide with the ������-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics. (shrink)

The paper questions the ability of current university systems to respond appropriately to the complex demands of an Information Economy. It argues that new relationships between creative subjects and technology require new thinking about the nature and purpose of universities per se. In particular, attention is drawn to the growing involvement of the private sector in higher education. It is argued that it may not be appropriate to think of the `university of the future' in terms of current public sector (...) and quasi public sector institutions, but rather in terms of an emporium, based on an international trade in educational services, and with the `University' as we now understand it occupying the functions of licensing, quality assurance and cultural custodianship. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Converging on an Open QuestErnesto Laclau (bio)I very much enjoyed the exchange in which Judith Butler and I engaged last year, through an e-mail correspondence between what Borges would have called the “unlikely geographies” of Berkeley and London. The points of convergence of our respective approaches are clear: as Butler points out, the process of gender formation that she describes and the logic of hegemony as presented in (...) my work (and in that which I wrote in collaboration with Chantal Mouffe) coincide in several of their assumptions. Neither we nor Butler see identities—political in one case, gendered in the other—as the expression of an intemporal mechanism or principle, but as products of the enactment of contingent norms; and both of us deny those norms a transcendental status, an a priori “hardness” which would reproduce itself unchanged in all historical instances. On the contrary, we see them as submitted to historical variations and as penetrated by a constitutive indetermination and ambiguity.I have used the word “transcendental” ex professo, because it is the status of the transcendental which is at the root of many of the most crucial problems in contemporary theory. Most people would agree that transcendentalism, in its classical formulations, is today unsustainable, but there is also a generalized agreement that some kind of weak transcendentalism is unavoidable. In the deconstructionist tradition, for instance, the notion of “quasi-transcendentals” has acquired considerable currency. But most theoretical approaches are haunted by the perplexing question of the precise status of that “quasi.” The problem touches on, on the one hand, the question of “metalanguage”; on the other, the status, in theory building, of categories that apparently refer to empirical events but that in practice have a quasi-transcendental status, operating as the a priori conditions of intelligibility of a whole discursive domain. What is the status in psychoanalysis, for instance, of categories such as “phallus,” of the “castration complex”? Because of the undecided status of the “quasi,” we are confronted with a plurality of alternatives, whose two polar extremes would be a total hardening of those categories, which would thus become a priori conditions of all possible human development, and a no less extreme historicism which sees in them only contingent events, products of particular cultural formations. The first extreme is confronted with the whole array of problems emerging from any transcendentalization of empirical conditions; the second, with the difficulties derived from not dealing with those conditions which make possible even a historicist discourse. The logic of the “quasi” tries to avoid both extremes, but it is extremely unclear in what that logic would consist of. These are questions which have not been dealt with enough, in either Butler’s approach or in mine; but they are issues to which both of us will have to return—perhaps in future exchanges.Let me now move to two central points contained in Butler’s latest piece, published in this issue of Diacritics as an addendum to our original exchange. They relate to the relationship between hegemony and iteration, and to the role of names in fixing meaning. I think that iteration belongs to the structure of any hegemonic operation, but that the latter stresses a double dimension of both repetition and displacement of meaning. To understand how these two dimensions interact with each other is crucial in order to grasp the [End Page 17] logic of the political. (I understand hegemony to be the central category in political analysis.) As Derrida has shown, iterability (the possibility of repetition in a plurality of instances) is something which belongs to the essence of the sign. What would something be which occurs in only one instance? It would simply not be a sign. But if the sign has to be the same, it also has to be different each time—and in that case the instance of its differential use has to be as part of its internal structure as the dimension of sameness. Now, in that case, it can only remain the same by becoming something constantly different from itself. This is the point where the usefulness of this analysis for a theory of the hegemonic becomes visible: for... (shrink)

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by such a (...) matrix, wherenow is a modal algebra, andFis a filter of. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. (shrink)

Let $R$ be a convergent term rewriting system, and let $CR$-equality on combinatory logic terms be the equality induced by $\beta \eta R$-equality on terms of the lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates $CR$-equality and whose irreducibles are exactly the translates of long $\beta R$-normal forms. The classical notion of strong normal form in combinatory logic (...) is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined first-order algebraic and higher-order setting the classical combinatory logic descriptions of the translates of long $\beta$-normal forms in the lambda calculus. As a consequence, the translates of long $\beta R$-normal forms are easily seen to serve as canonical representatives for $CR$-equivalence classes of combinatory logic terms for non-empty, as well as for empty, $R$. (shrink)

It is shown that the axioms Cubitt and Sugden (1994; Economic J. 104: 798) impose on a theory of rationally justifiable play (TRJP) do not prevent the possibility that two players necessarily disagree concerning the probability they ascribe to the choice of a third player. This appears to indicate that those axioms are not sufficient for defining a `reasonable' TRJP. In addition, for the case in which a player's beliefs are statistically independent, conditions for a TRJP are suggested under which (...) the existence of a quasi-strict equilibrium is sufficient, but the existence of a consistent n-pair is not, for defining a TRJP meeting those requirements. (shrink)

Atoms and molecules are particular kinds of restricted n-body systems, which generally behave as quasi-separable, unlike other n-body systems, e.g., Newtonian ones. The Coulomb repulsion and the Pauli exclusion principle in atoms and molecules are responsible for that separability. Additionally, chemical bonds, especially covalent bonds, enhance the separability of molecules. Independent particle models do not describe atoms and molecules since first-order energy corrections are high. However, these corrections obtained by the first-order perturbation or mean-field strongly converge, implying a one-electron (...) effective potential description. Consequently, stable states of atoms and molecules can be reasonably described by one-electron effective potentials, which strongly differ from other n-body problems. We discuss the peculiarities of the correlation motion of generic systems in the context of the four fundamental forces. In particular, we have shown that the two components (attraction and repulsion) of the electromagnetic force confer a relatively low correlation motion to atoms and molecules. We discuss the physical and chemical nature of atoms and molecules, comparing the degrees of separability between different systems. For example, the separability of Newtonian systems is generally possible in particular classes of restricted systems due to relative mass differences. However, for atoms and molecules, the separability is much broader. (shrink)

A biological function is supposed to be performed adequately, and hence may fail to do so: this is dysfunction. This raises two questions. One is how to make explicit the way in which function can be discriminated from dysfunction without confusing dysfunction with non-function. The second question is how what is “right” and “wrong” can be legitimated by natural regulatory norms. A function can be viewed as a quality to which at least one variable with a definite set of values (...) is associated. Accordingly, function and dysfunction are the same quality of the same trait, but differ in the values associated to this variable. Dysfunction occurs when the associated set of values does not match this normal state. Biological systems have the epistemic singularity that their existence is the consequence of two distinct causal regimes, the so-called “proximate” and “ultimate” causes, whose convergence defines a system’s prescriptive normality. Each cause imposes restrictive rules that limit the possible ways these systems can putatively exist. When a system is insensitive to ultimate cause, it is determined by proximate causation alone, and hence escapes its own prescriptive norms. In conclusion: The normality of a biological system is defined by the convergence of proximate and ultimate causes. A function is a variable-associated quality whose values are defined relative to the norms of the system. Dysfunction occurs when the set of values of the same variable does not match the normality of the system, which can occur when the system is insensitive to ultimate causation. (shrink)

Two recent papers (Cubitt and Sugden, 1994; Samuelson, 1992) have established impossibility results which cast doubt on the coherence of the assumption of ’common knowledge of rationality'. It is shown that the Cubitt–Sugden result is the more powerful of the two impossibilities. Second, it is proved that the existence of a quasi-strict equilibrium is sufficient to construct sets which satisfy the Cubitt–Sugden axioms. This fact is used to establish that their impossibility result cannot arise in 2-player games. Finally, it (...) is shown that if a weak symmetry postulate is added, a new impossibility result arises for this class of games. (shrink)

We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...) is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: $\Delta _{3}^{1}{\rm CA}_{0}+{\rm AQI}\nvdash \Sigma _{3}^{0}$ -Determinacy. Hence the theories: ${\mathrm{\Pi }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0},\text{\hspace{0.17em}}{\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\text{\hspace{0.17em}}{\mathrm{\Sigma }}_{3}^{0}-\mathrm{D}\mathrm{e}\mathrm{t}\phantom{\rule{0ex}{0ex}}$ -Det, ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\mathrm{A}\mathrm{Q}\mathrm{I}$ , and ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ are in strictly descending order of strength. (shrink)

Contemporary diagnostic categories and various modes of treatment of mood disorders contribute to the development of a managed form of selfhood in contemporary society, particularly as articulated with management in the workplace. This produces a new iteration of the normal in relation to psychopathology; instead of the normal as an absence of disorder or distress, normalcy becomes the private management, often stemming from an external or internalized social injunction, of symptoms through various available techniques of self-care. I support (...) this claim through an analysis of the development of the category of the mood disorder under DSM-III and the functioning of common contemporary treatments such as antidepressant medications, cognitive-behavioral therapy, and alternative techniques such as mindfulness meditation. A new social problematic emerges, in part out of this convergence, in which the notion of mood becomes generalized and an object of vigilance. Everyone has moods; we always have moods; and moods are in principle always subject to the need of being managed. (shrink)

In the context of modal logics one standardly considers two modal operators: possibility ) and necessity ) [see for example Chellas ]. If the classical negation is present these operators can be treated as inter-definable. However, negative modalities ) and ) are also considered in the literature [see for example Béziau ; Došen :3–14, 1984); Gödel, in: Feferman, Collected works, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300; Lewis and Langford ]. Both of them can be (...) treated as negations. In Béziau a logic \ has been defined on the basis of the modal logic \. \ is proposed as a solution of so-called Jaśkowski’s problem [see also Jaśkowski ]. The only negation considered in the language of \ is ‘it is not necessary’. It appears that logic \ and \ inter-definable. This initial correspondence result between \ and \ has been generalised for the case of normal logics, in particular soundness-completeness results were obtained [see Marcos :279–300, 2005); Mruczek-Nasieniewska and Nasieniewski :229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski it has been proved that there is a correspondence between \-like logics and regular extensions of the smallest deontic logic. To obtain this result both negative modalities were used. This result has been strengthened in Mruczek-Nasieniewska and Nasieniewski :261–280, 2017) since on the basis of classical positive logic it is enough to solely use \ to equivalently express both positive modalities and negation. Here we strengthen results given in Mruczek-Nasieniewska and Nasieniewski by showing correspondence for the smallest regular logic. In particular we give a syntactic formulation of a logic that corresponds to the smallest regular logic. As a result we characterise all logics that arise from regular logics. From this follows via respective translations a characterisation of a class of logics corresponding to some quasi-regular logics where \ is the smallest element. Moreover, if a given quasi-regular logic is characterised by some class of models, the same class can be used to semantically characterise the logic obtained by our translation. (shrink)

In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgement of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry – Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such (...) as when Gödel observes something quasi-spatial character of sets. It is shown that Dooyeweerd’s theory of modal aspects provides a philosophical framework that exceeds his own restrictive understanding of infinity )to the potential infinite) and at the same time makes it possible to account for key insights found in the thought of Bernays and Gödel. When Laugwitz says that discreteness rules within the sphere of the numerical, he says nothing more than what Dooyeweerd had in mind with his idea that discrete quantity, as the meaning-nucleus of the arithmetical aspect, qualifies every element within the structure of the quantitative aspect. And when Bernays says that analysis expresses the idea of the continuum in arithmetical language his mode of speech is equivalent to saying that mathematical analysis could seen as being founded upon the spatial anticipation within the modal structure of the arithmetical aspect. The view of the actual infinite (the at once infinite) in terms of an “as if” approach (Bernays), that is, as appreciated as a regulative hypothesis through which every successively infinite multiplicity of numbers could be envisaged as being giving all at once, as an infinite totality, provides a sound understanding of the at once infinite and makes it plain why every form of arithmeticism fails. Such attempts have to call upon Cantor’s proof on the non-denumerability of the real numbers – and this proof pre-supposes the use of the at once infinite which, in turn, pre-supposes the (irreducibile) spatial order of simultaneity and the patial whole-parts relation. (shrink)

Thach's target article presents a remarkable overview and integration of animal and human studies on the functions of the cerebellum and makes clear theoretical predictions for both the normal operation of the cerebellum and for the effects of cerebellar lesions in the mature human. Commentary is provided on three areas, namely, spatial navigation, implicit learning, and cerebellar agenesis to elicit further development of the themes already present in Thach's paper, [THACH].

This paper explores the idea that the structure of intentionality is fundamentally the structure of a practice, not the structure of a language, or some quasi-linguistic system of representational entities. I show how and why neo-Fregean theory of content is committed to this practical turn. Mis-representation is often thought to be problematic for the neo-Fregean, but I show not only that it accommodates the phenomena better than the representationalist position, but also that the idea of error that the representationalist (...) wants with empty singular terms is redundant, for it has no role to play in explaining the systematicity of thought. (shrink)

Objective Apeiron was founded in 1966 and has developed into one of the oldest and most distinguished journals dedicated to the study of ancient philosophy, ancient science, and, in particular, of problems that concern both fields. Apeiron is committed to publishing high-quality research papers in these areas of ancient Greco-Roman intellectual history; it also welcomes submission of articles dealing with the reception of ancient philosophical and scientific ideas in the later western tradition. The journal appears quarterly. Articles are peer-reviewed on (...) a double-blind basis. Topics ancient philosophy ancient science reception of ancient philosophical and scientific ideas in the later western tradition Article formats Original Articles. (shrink)

Philologus, founded in 1846, is one of the oldest and most respected periodicals in the field of Classics. It publishes articles on Greek and Latin literature, historiography, philosophy, history of religion, linguistics, reception, and the history of scholarship. The journal aims to contribute to our understanding of Greco-Roman culture and its lasting influence on European civilization. The journal Philologus, conceived as a forum for discussion among different methodological approaches to the study of ancient texts and their reception, publishes original scholarly (...) contributions with a strong international focus. Publications languages are: German, English, Italian, French. As of 2014 the publication of supplementary volumes will be resumed. The series Philologus. Supplemente / Philologus. Supplementary Volumes will publish monographs and edited volumes pertaining to all aspects of the study of ancient literature and its reception, with a special focus on topics concerning literary theory, philosophy and the history of scholarship from an interdisciplinary perspective. (shrink)

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that (...) lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed. (shrink)

Philologus, founded in 1846, is one of the oldest and most respected periodicals in the field of Classics. It publishes articles on Greek and Latin literature, historiography, philosophy, history of religion, linguistics, reception, and the history of scholarship. The journal aims to contribute to our understanding of Greco-Roman culture and its lasting influence on European civilization. The journal Philologus, conceived as a forum for discussion among different methodological approaches to the study of ancient texts and their reception, publishes original scholarly (...) contributions with a strong international focus. Publications languages are: German, English, Italian, French. As of 2014 the publication of supplementary volumes will be resumed. The series Philologus. Supplemente / Philologus. Supplementary Volumes will publish monographs and edited volumes pertaining to all aspects of the study of ancient literature and its reception, with a special focus on topics concerning literary theory, philosophy and the history of scholarship from an interdisciplinary perspective. (shrink)

In a text written during his stay in Paris, Leibniz, to deny ontological reality to relations, employs an argument well known to the medieval thinkers and which later would be revived by Francis H. Bradley. If one assumes that relations are real and that a relation links any property to a subject – so runs the argument – then one falls prey to an infinite regress. Leibniz seems to be well aware of the consequences that this argument has for his (...) own metaphysical views, where the relation of inherence (‘inesse’) plays such a central role. Thus, he attempts first to interpret the relation of inherence as something ‘metaphoric’, originating from our ‘spatial way’ of looking at the surrounding world; and then he tries to reduce it to the part-whole relation which clearly he considers weaker, from the ontological point of view, than that of ‘being in’. (shrink)

In a text written during his stay in Paris, Leibniz, to deny ontological reality to relations, employs an argument well known to the medieval thinkers and which later would be revived by Francis H. Bradley. If one assumes that relations are real and that a relation links any property to a subject – so runs the argument – then one falls prey to an infinite regress. Leibniz seems to be well aware of the consequences that this argument has for his (...) own metaphysical views, where the relation of inherence plays such a central role. Thus, he attempts first to interpret the relation of inherence as something ‘metaphoric’, originating from our ‘spatial way’ of looking at the surrounding world; and then he tries to reduce it to the part-whole relation which clearly he considers weaker, from the ontological point of view, than that of ‘being in’. (shrink)

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are (...) not normal. The set of extensions of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix A is (...) Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (shrink)

The addition of "actually" operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing "actually" operators has concentrated entirely upon extensions of KT5 and has employed a particular modeltheoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing "actually" operators, the weakest of which are conservative (...) extensions of K, using a novel generalisation of the standard semantics. (shrink)

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are (...) not normal. The set of extensions of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

This paper gives a characterization of those quasi-normal extensions of the modal system S4 into which intuitionistic propositional logic Int is embeddable by the Gödel translation. It is shown that, as in the normal case, the set of quasi-normal modal companions of Int contains the greatest logic, M*, for which, however, the analog of the Blok-Esakia theorem does not hold. M* is proved to be decidable and Halldén-complete; it has the disjunction property but does not (...) have the finite model property. (shrink)

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary (...) expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH . We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH . Finally, we apply these results to give bases for “small” subvarieties of BDQDSH , DQSSH , and DblSH. (shrink)

Some conditions are given to ensure that for a jump homogeneous Markov process $\{X(t),t\ge 0\}$ the law of the integral functional of the process $T^{-1/2} \int^T_0\varphi(X(t))dt$ converges to the normal law $N(0,\sigma^2)$ as $T\to \infty$, where $\varphi$ is a mapping from the state space $E$ into $\bbfR$.

Normal 0 false false false MicrosoftInternetExplorer4 st1:*{behavior:url(#ieooui) } /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} One of the most celebrated biologists of the past century, Edward O. Wilson has received virtually every scientific award and recognition for his provocative and innovative enquiry into the nature of the relationship between moral behavior and biology which the scientific community can offer. For over (...) twenty-five years, his development of a field of study at Harvard University called sociobiology and the analytical concept he created called conscilience have revolutionized our thinking about ethics and human behavior. Exploring the interrelatedness of the chemistry of the mind and the genetic bases of culture, he has challenged the philosophical, theological, and sociological world to embrace a creatively converging view of science and ethics, of biology and human behavior. This essay is a critical review of Wilson’s major works written during the past three decades which have caused both international exuberance among scientists and an exclamation of outrage among many in the philosophical and theological world. This is the first comprehensive review and critical assessment of his major works. (shrink)

Simon Blackburn has not shied away from the use of vivid imagery in developing, over a long and prolific career, a large-scale philosophical vision. Here one might think, for instance, of ‘Practical Tortoise Raising’ or ‘Ramsey's Ladder’ or ‘Frege's Abyss’. Blackburn develops a ‘quasi-realist’ account of many of our philosophical and everyday commitments, both theoretical (e.g., modality and causation) and practical (e.g., moral judgement and normative reasons). Quasi-realism aims to provide a naturalistic treatment of its targeted phenomena while (...) earning the right to deploy all of the ‘trappings’ of realism—i.e., while eschewing any idea that our normal thought and talk about such phenomena are pervasively in error. The quasi-realist project is that of explaining how (as Huw Price puts it here) ‘the folk come to “talk the realist talk” without committing ourselves—us theorists, as it were—to “walking the metaphysical walk”’ (p. 136). Quasi-realism, too, can speak of truth, facts, properties, belief, knowledge, and so on. The imagery in this collection also abounds, though, in capturing a different view of quasi-realism: No fewer than three of the contributors picture Blackburn as wanting to have his cake and eat it too (Louise Antony asking, in addition, ‘Who doesn't? It's cake’ [p. 19]). (shrink)

En temps normal, les contrefactuels sont conçus comme produisant des énoncés portant sur des états de choses, mais des états de choses se trouvant dans des mondes simplement possibles ou alternes. Analysés ainsi, il s’avère que presque tous les contrefactuels sont incohérents. Tout contrefactuel analysé de la sorte exige qu’il y ait un monde métaphysiquement (et pas épistémiquement seulement) possible w où les lois sont les mêmes qu’ici, et où la quasi-totalité des faits sont les mêmes qu’ici. (Les (...) différences factuelles sont liées à l’antécédent et au conséquent du contrefactuel.) Mais comme je vais le montrer, cette exigence implique de manière typique la postulation de mondes dont on peut démontrer la nécessaire non-existence au moyen de déductions assez élémentaires. De plus, l’analyse des contrefactuels en termes de mondes possibles est coupable de circularité cachée. Analysés ainsi, les contrefactuels ne peuvent effectivement être compris qu’en termes de lois de la nature (les lois qui s’appliquent ici sont présupposées dans le monde hypothétique — excepté dans le cas atypique où le contrefactuel est également un contre-nomique). Mais il y a un argument répandu voulant que le concept de loi ne puisse pas être défini lui-même, sinon dans les termes de la notion de contrefactuel (une loi est, entre autres, quelque chose qui vient à l’appui des contrefactuels). Je donnerai une analyse purement épistémique des contrefactuels, déclarant que ce sont des propositions pseudo-probabilistes. Je déclarerai également que le type de probabilité en question peut être défini intégralement en termes de ce qui a été le cas (non pas de ce qui aurait été le cas, ni même de ce qui doit être le cas en un sens nomique.) Mon analyse n’est donc coupable d’aucune sorte de circularité. (shrink)

Many classic philosophical debates converge on the twin questions ‘What is man?’ and ‘What is his place in nature?’, in the sense that taking up a position in those debates normally commits one to a certain range of answers to these questions. Such answers typically lie near the centre of one's web of belief, deeply entrenched in the structure of one's concepts, and thus remain remarkably resistant to the standard techniques of confirmation and refutation.

This target article discusses the verbal working memory system used in sentence comprehension. We review the concept of working memory as a short-duration system in which small amounts of information are simultaneously stored and manipulated in the service of accomplishing a task. We summarize the argument that syntactic processing in sentence comprehension requires such a storage and computational system. We then ask whether the working memory system used in syntactic processing is the same as that used in verbally mediated tasks (...) that involve conscious controlled processing. Evidence is brought to bear from various sources: the relationship between individual differences in working memory and individual differences in the efficiency of syntactic processing; the effect of concurrent verbal memory load on syntactic processing; and syntactic processing in patients with poor short-term memory, patients with poor working memory, and patients with aphasia. Experimental results from these normal subjects and patients with various brain lesions converge on the conclusion that there is a specialization in the verbal working memory system for assigning the syntactic structure of a sentence and using that structure in determining sentence meaning that is separate from the working memory system underlying the use of sentence meaning to accomplish other functions. We present a theory of the divisions of the verbal working memory system and suggestions regarding its neural basis. (shrink)

In his analysis of “the essential tension between tradition and innovation” Thomas S. Kuhn focused on the apparent paradox that, on the one hand, normal research is a highly convergent activity based upon a settled consensus, but, on the other hand, the ultimate effect of this tradition-bound work has invariably been to change the tradition. Kuhn argued that, on the one hand, without the possibility of divergent thought, fundamental innovation would be precluded. On the other hand, without a strong (...) emphasis on convergent thought, science would become a mess created by continuous theory changes and scientific progress would again be precluded. On Kuhn’s view, both convergent and divergent thought are therefore equally necessary for the progress of science. In this paper, I shall argue that a similar fundamental tension exists between the demands we see for novel insights of an interdisciplinary nature and the need for established intellectual doctrines founded in the classical disciplines. First, I shall revisit Kuhn’s analysis of the essential tension between tradition and innovation. Next, I shall argue that the tension inherent in interdisciplinary research between, on the one hand, intellectual independence and critical scrutiny and, on the other hand, epistemic dependence and trust is a complement to Kuhn’s essential tension within mono-disciplinary science between convergent and divergent thought. (shrink)