Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere . The completeness theorems are used to conclude results asserting that certain (...) logics are conservatively interpretable in others. (shrink)

Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states.In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes intersection spaces.We give (...) here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set. (shrink)

In 1958 Arnould Bayart, 1911-1998, produced a semantics for first and second-order S5 modal logic, and in 1959 a completeness proof for first-order S5, and what he calls a 'quasi-completeness' proof for second-order S5. The 1959 paper is the first completeness proof for modal predicate logic based on the Henkin construction of maximal consistent sets, and indeed may be the easier application of the Henkin method even to propositional modal logic. The semantics is in terms of possible worlds, which, Bayart (...) notes, can be anything at all. The present paper provides an English translation of both these papers together with an historical introduction and a logical commentary. (shrink)

We consider a deterministic rewrite system for combinatory logic over combinators , and . Terms will be represented by graphs so that reduction of a duplicator will cause the duplicated expression to be "shared" rather than copied. To each normalizing term we assign a weighting which is the number of reduction steps necessary to reduce the expression to normal form. A lambda-expression may be represented by several distinct expressions in combinatory logic, and two combinatory logic expressions are considered equivalent if (...) they represent the same lambda-expression (up to --equivalence). The problem of minimizing the number of reduction steps over equivalent combinator expressions (i.e., the problem of finding the "fastest running" combinator representation for a specific lambda-expression) is proved to be NP-complete by reduction from the "Hitting Set" problem. (shrink)

A classical result by Słupecki states that a logic L is functionally complete for the 3-element set of truth-values THREE if, in addition to functionally including Łukasiewicz’s 3-valued logic Ł3, what he names the ‘$T$-function’ is definable in L. By leaning upon this classical result, we prove a general theorem for defining binary expansions of Kleene’s strong logic that are functionally complete for THREE.

In this paper we discuss two hybrid languages, ℒ and ℒ, and provide them with complete axiomatizations. Both languages combine features of modal and classical logic. Like modal languages, they contain modal operators and have a Kripke semantics. Unlike modal languages, in these systems it is possible to 'label' states by using A and ↓ to bind special state variables.This paper explores the consequences of hybridization for completeness. As we shall show, the challenge is to blend the modal idea of (...) canonical models with the classical idea of witnessed maximal consistent sets. The languages ℒ and ℒ provide us with two extreme examples of the issues involved. In the case of ℒ, we can combine these ideas relatively straightforwardly with the aid of analogs of the Barcan axioms coupled with a modal theory of labeling. In the case of ℒ, on the other hand, although we can still formulate a theory of labeling, the Barcan analogs are not valid. We show how to overcome this difficulty by using COV*, an infinite collection of additional rules of proof which has been used in a number of investigations of extended modal logic. (shrink)

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set (...) of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism. (shrink)

Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.

The spacing effect is among the most widely replicated empirical phenomena in the learning sciences, and its relevance to education and training is readily apparent. Yet successful applications of spacing effect research to education and training is rare. Computational modeling can provide the crucial link between a century of accumulated experimental data on the spacing effect and the emerging interest in using that research to enable adaptive instruction. In this paper, we review relevant literature and identify 10 criteria for rigorously (...) evaluating computational models of the spacing effect. Five relate to evaluating the theoretic adequacy of a model, and five relate to evaluating its application potential. We use these criteria to evaluate a novel computational model of the spacing effect called the Predictive Performance Equation. Predictive Performance Equation combines elements of earlier models of learning and memory including the General Performance Equation, Adaptive Control of Thought—Rational, and the New Theory of Disuse, giving rise to a novel computational account of the spacing effect that performs favorably across the complete sets of theoretic and applied criteria. We implemented two other previously published computational models of the spacing effect and compare them to PPE using the theoretic and applied criteria as guides. (shrink)

A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and (...) axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T→, and any logic of which this is a sublogic, is D-complete. (shrink)

The usual completeness theorem for first-order logic is extended in order to allow for a natural incorporation of real analysis. Essentially, this is achieved by building in the set of real numbers into the structures for the language, and by adjusting other semantical notions accordingly. We use many-sorted languages so that the resulting formal systems are general enough for axiomatic treatments of empirical theories without recourse to elements of set theory which are difficult to interprete empirically. Thus we provide a (...) way of applying model theory to empirical theories without tricky detours. Our frame is applied to axiomatizations of three empirical theories: classical mechanics, phenomenological thermodynamics, and exchange economics. (shrink)

We examine the relationship between two different notions of a structure being Scott set saturated and identify sufficient conditions which guarantee that a structure is uniquely Scott set saturated. We also consider theories representing Scott sets; in particular, we identify a sufficient condition on a theory T so that for any given countable Scott set there exists a completion of T that is saturated with respect to the given Scott set. These results extend Scott's characterization of countable Scott sets via (...) models and completions of Peano arithmetic. (shrink)

The logic BN4 can be considered as the 4-valued logic of the relevant conditional and the logic E4, as the 4-valued logic of entailment. The aim of this paper is to endow E4 with a 2-set-up Routley-Meyer semantics. It is proved that E4 is strongly sound and complete w.r.t. this semantics.

Brigitte Lion and Diana Stein have, in The Tablets from the Temple Precinct at Nuzi, joined forces to edit the texts of the sixty-nine Nuzi texts found in temples and to publish and edit their 282 seal impressions. This masterful collaboration excels in many areas, including in-depth prosopographical analysis and thorough integration of the local seal practice into the orbit of textual analysis. This volume sets a standard of text edition and analysis that dwarfs the achievements of the students of (...) Nuzi from the first two-thirds of the twentieth century and even since. Nevertheless, the raison d’être of this heterogeneous group of texts in the context of their storage in temples remains enigmatic. (shrink)

The prospects for a complete stochastic theory of microscopic phenomena are considered. The two traditional schools of stochastic physics, the diffusion process school and the zero-point electromagnetic field school, are reviewed. A completely relativistic theory, stochastic field theory, is proposed as an extension of the ideas of these two schools. Within the context of stochastic field theory we present the following new results: an elementary stochastization scheme which produces the zero-point electromagnetic field; a physical interpretation of the mathematical methods developed (...) by Lukosz for calculating zero-point energies; a calculation of the first-order Lamb shift which generalizes that of Welton; a new setting for a finite-temperature theory; and comments on the bag model for quark confinement. (shrink)

A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space (...) is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result. (shrink)

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity (...) types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?.The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set.The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. (shrink)

Clinicians' work depends on sincere and complete disclosures from their patients; they honour this candidness by confidentially safeguarding the information received. Breaching confidentiality causes harms that are not commensurable with the possible benefits gained. Limitations or exceptions put on confidentiality would destroy it, for the confider would become suspicious and un-co-operative, the confidant would become untrustworthy and the whole climate of the clinical encounter would suffer irreversible erosion. Excusing breaches of confidence on grounds of superior moral values introduces arbitrariness and (...) ethical unreliability into the medical context. Physicians who breach the agreement of confidentiality are being unfair, thus opening the way for, and becoming vulnerable to, the morally obtuse conduct of others. Confidentiality should not be seen as the cosy but dispensable atmosphere of clinical settings; rather, it constitutes a guarantee of fairness in medical actions. Possible perils that might accrue to society are no greater than those accepted when granting inviolable custody of information to priests, lawyers and bankers. To jeopardize the integrity of confidential medical relationships is too high a price to pay for the hypothetical benefits this might bring to the prevailing social order. (shrink)

The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of implicative expansions of Kleene’s 3-valued logic functionally including Łukasiewicz’s logic Ł3 is defined. Several properties of this class and/or some of its subclasses are investigated. Properties contemplated include functional completeness for the 3-element set of truth-values, presence of natural conditionals, variable-sharing property and vsp-related properties.

City archives, mined by Aristotle for his Didaskaliai, preserved a reasonably complete record of dramatic productions in the fifth century. But how far back did these archives go? The so-called Fasti, an inscription set up c. 346 and listing dithyrambic, comic and tragic victors year by year, must have been based on the same archives, but went back, it is thought, only as far as 502/1. Its heading πρ]τον κμοι ἦσαν τ[ι διονσ]ωι τραγωιδο δ[, however supplemented, implies an intention of (...) going back to the beginning of things, in other words to the beginning of the archival record. This raises serious doubt as to whether that record went back to the alleged date of Thespis' première, or indeed to those given for Choerilus' and Phrynichus'. (shrink)

This is the first edition to provide the complete text of both the fifth-century Calculus of Victorius of Aquitaine and Abbo of Fleury's Commentary on it. These two works shed light on the early history of mathematics, before the introduction of Arabic numerals. The wide range of Abbo's thought is reflected in the Commentary, covering the nature of wisdom, the philosophy of number, the relationship of unity and plurality, and the arithmetic of the Calculus - all of it set in (...) the broader context of his theology of Creation. This edition is crucial for our current reappraisal of Abbo of Fleury as a major figure of the tenth century, who had a profound impact on later medieval English thought. (shrink)

Objectives: Efficient procedures for obtaining informed (proxy) consent may contribute to high influenza vaccination rates in nursing homes. Yet are such procedures justified? This study’s objective was to gain insight in informed consent policies in Dutch nursing homes; to assess how these may affect influenza vaccination rates and to answer the question whether deviating from standard informed consent procedures could be morally justified. Design: A survey among nursing home physicians. Setting & Participants: We sent a questionnaire to all (356) nursing (...) homes in the Netherlands, to be completed by one of the physicians. Results: We received 245 completed questionnaires. As 21 institutions appeared to be closed or merged into other institutions, the response was 73.1% (245/335). Of all respondents 81.9% reported a vaccination rate above 80%. Almost 50% reported a vaccination rate above 90%. Most respondents considered herd immunity to be an important consideration for institutional policy. Freedom of choice for residents was considered important by almost all. Nevertheless, 106 out of 245 respondents follow a tacit consent procedure, according to which vaccination will be administered unless the resident or her proxy refuses. These institutions show significantly higher vaccination rates (p < 0.03). Conclusions: In our discussion we focus on the question whether tacit consent procedures can be morally justifiable. Such procedures assume that vaccination is good for residents either as individuals or as a group. Even though this assumption may be true for most residents, there are good reasons for preferring express consent procedures. (shrink)

Ontology engineering is mainly done by domain experts who are specialists in their domain but have, if at all, limited knowledge in logics, computer science, or analytic philosophy. The literature on formal ontologies and biomedical ontologies is neither suited nor intended to serve as an educational resource that would help domain experts to become good ontologists. Existing educational resources focus rather on ontology tools and languages than on good practice. The purpose of the GoodOD guideline is to pave the road (...) for domain experts who want (or need) to become ontology engineers. It pursues the openly pragmatic approach that ontology engineers only need to know ontology theory to the extent which supports the (measurable) outcome of their work. The GoodOD Guideline builds upon given philosophical and methodological principles as well as preexisting tools and resources. In particular (pragmatic) scientific realism, first order logics and its subset OWL-DL (description logics), set theory, an upper-level ontology with high-level categories and a canonic set of relations, constraints which limit the freedom of choice when building coordinate expressions, the ontology editor Protégé 4, a complete DL reasoner, such as HermIT, ontology design patterns, and naming conventions. -/- Given these precepts the guideline contains all the knowledge to be acquired by a user to be able to solve ontology engineering problems in a foreseeable and non-arbitrary way. The didactic concept is to convey the skills not only by theory but also by practical examples. (shrink)

This edition of John M. Lothian’s transcription of an almost com­plete set of a student’s notes on Smith’s lectures given at the University of Glasgow in 1762–63_ _brings back into print not only an important discovery but a valuable contribution to eighteenth-century rhetorical theory.

Urn models were developed by Veikko Rantala to provide a non-standard semantics for first-order logic in which the domains, over which the quantifiers range, are allowed to vary. Rantala uses game-theoretical semantics in his presentation, and the present paper is a study of urn models from a more classical, truth-conditional point of view. An axiomatic system for urn logic is set out and completeness is proved by the method of maximal consistent sets.

This is a text for a one or two semester course on axiomatic set theory; the goal is to introduce and develop one system of set theory in a complete and thorough way, presupposing only the elusive "mathematical maturity" of the reader. There are nine chapters which begin with a development of propositional and predicate logic oriented toward set theory and develop the Zermelo-Fraenkel system in exceptional detail. The book starts slowly, the first 120 pages being devoted to logical preliminaries (...) and the introduction of axioms; it gathers speed, however, and the remaining chapters include treatment of the algebra of classes, relations and functions, order of various sorts and Zorn's lemma, real numbers and equivalence classes, the equipollence of sets, similarity as a relation of sets, the ordinal numbers and their arithmetic, the cardinal numbers and their arithmetic. These later chapters are extremely detailed and comprehensive, presenting a wealth of material. There are numerous exercises, many quite difficult, at the end of each chapter, along with a summary of that chapter's content. Because of the single-mindedness of the book, there is little reference to other systems of set theory, but this is not a drawback at all. The presentation is orthodox, but not dull; careful, but not generally pedantic or repetitive. This makes it a good text for self-study.—P. J. M. (shrink)

It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from [3]. If (...) A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)

This paper is perhaps an impressionistic response to accounts of the extraordinary set-theoretical activity being undertaken by W. Hugh Woodin and colleagues in the present moment, in the context of the mathematical ontology proposed and elaborated by Alain Badiou. The argument presented is that the prevailing and sustained incoherence of the mathematical ontology underscores a contemporary deficit of humanity’s symbolic organization which, in turn, yields confusion and conflict in terms of subjective orientation. But a new axiom promises to realize a (...) coherent set theory, i.e. stable, consistent and complete. This remarkable development offers hope for the pursuit of a modern symbolic, and a consequent resolution of the general subjective disorientation. (shrink)

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however (...) complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By aframeis understood a pair 〈W, R〉 in whichWis a class (of “possible worlds”) andR⊆W2. To avoid confusion in what follows, a frame will henceforth be referred to as aKripkeframe. By contrast, ageneral frameis a pair 〈,Π〉 in whichis a Kripke frame andΠis a collection of subsets ofWclosed under the Boolean operations and satisfying the condition that ifAis inΠthen so isR−1“A. A model on a frame (of either kind) is obtained by adding a functionVwhich assigns sets of worlds to propositional variables. In the case of a general frame we require thatV(p) ∈Π. (shrink)

In his preface, Hunter explains that this volume is intended to provide for non-mathematicians an introduction to the most important results of modern mathematical logic. The reader will find here the work of Post, Skolem, Gödel, Church, Henkin, and others, presented in a terse and closely-knit style. Though acknowledging the trend toward natural deduction systems, Hunter sticks to more classical axiomatic systems on the grounds that the proofs of metatheorems are simplified by that choice. He begins with a formal system (...) of propositional logic, for which he develops proofs of completeness, consistency, and decidability. Systems of first order predicate logic are then dealt with, and Church's and Gödel's results on undecidability and incompleteness are presented. There is a brief, useful discussion indicating what parts of first order predicate logic are decidable. It is assumed that the reader has some familiarity with elementary formal logic: Hunter develops the set-theoretical results he requires. The compactness of the presentation is facilitated by the use of earlier results for propositional logic to obtain certain results for predicate logic. Especially helpful is Hunter's technique of prefacing each of the longer and more difficult proofs with a section explaining the key ideas and the general method of proof; summaries are also often provided in conclusion. Because of the orientation of the book, a more extensive discussion of the significance of the results obtained and the philosophical issues which have been raised concerning some of them, would have been welcome. Nonetheless, this is a good, concise introduction for those who are not yet prepared to tackle the more technical literature. A bibliography is provided for those interested in pursuing the subject in greater detail.--E. M. F. (shrink)

Objective: To determine whether a moral reasoning exercise can improve response quality to surveys of healthcare priorities Methods: A randomised internet survey focussing on patient age in healthcare allocation was repeated twice. From 2574 internet panel members from the USA and Canada, 2020 (79%) completed the baseline survey and 1247 (62%) completed the follow-up. We elicited respondent preferences for age via five allocation scenarios. In each scenario, a hypothetical health planner made a decision to fund one of two programmes identical (...) except for average patient age (35 vs 65 years). Half of the respondents (intervention group) were randomly assigned to receive an additional moral reasoning exercise. Responses were elicited again 7 weeks later. Numerical scores ranging from –5 (strongest preference for younger patients) to +5 (strongest preference for older patients); 0 indicates no age preference. Response quality was assessed by propensity to choose extreme or neutral values, internal consistency, temporal stability and appeal to prejudicial factors. Results: With the exception of a scenario offering palliative care, respondents preferred offering scarce resources to younger patients in all clinical contexts. This preference for younger patients was weaker in the intervention group. Indicators of response quality favoured the intervention group. Conclusions: Although people generally prefer allocating scarce resources to young patients over older ones, these preferences are significantly reduced when participants are encouraged to reflect carefully on a wide range of moral principles. A moral reasoning exercise is a promising strategy to improve response quality to surveys of healthcare priorities. (shrink)

Context: The use of first-person micro-phenomenological interviews and their productive interaction with third-person physiological data is a challenging and pressing issue in order to offer an effective and fruitful application of Varela’s neurophenomenological hypothesis. Problem: We aim at offering a generative method of analysis of first-person micro-phenomenological interviews using third-person physiological data. Our challenge is to describe this generative first-person analysis with the third-person physiological framework rather than put Varela’s hypothesis into practice in a generative way (as we did in (...) another paper. Method: The present contribution is a first pioneering study as far as the exposition of such an interactive generative methodology is concerned. It is also a new issue insofar as it deals with a case study, surprise in depression, that has not been thoroughly dealt with so far, either in philosophy or in psychopathology. Results: We show that the analysis of first-person data is an intrinsic generative one, insofar as new refined categories and multifarious circular micro- and macro-processes were discovered in the very process of analyzing. They provide the initial structural generic third-person description of surprise inherited both from philosophical phenomenological a priori categories and from the experimental startle setting with a refined micro-segmentation of the dynamic of the experience. Implications: Our article could be of interest to neurophenomenologists looking for an effective application and to researchers in quest of a method of analysis of first-person data. The present limitations are due to the still preliminary data-results we need to complete. Constructivist content: The article is directly linked to Varela’s neurophenomenological program and aims at extending and reforming it with a cardio-phenomenological approach. Keywords: First-person micro-phenomenological interviews, surprise, generative analysis of first-person data, depression, cardio-phenomenology, generative categories. (shrink)

The “representation problem” in abstract algebraic logic is that of finding necessary and sufficient conditions for a structure, on a well defined abstract framework, to have the following property: that for every structural closure operator on it, every structural embedding of the expanded lattice of its closed sets into that of the closed sets of another structural closure operator on another similar structure is induced by a structural transformer between the base structures. This question arose from Blok and Jónsson abstract (...) analysis of one of Blok and Pigozzis’s characterizations of algebraizable logics. The problem, which was later on reformulated independently by Gil-Férez and by Galatos and Tsinakis, was solved by Galatos and Tsinakis in the more abstract framework of the category of modules over a complete residuated lattice, and by Galatos and Gil-Férez in the even more abstract setting of modules over a quantaloid. We solve the representation problem in Blok and Jónsson’s original context of M-sets, where M is a monoid, and characterise the corresponding M-sets both in categorical terms and in terms of their inner structure, using the notions of a graded M-set and a generalized variable introduced by Gil-Férez. (shrink)

In this paper we investigate a number of analytical solutions to the polynomial class of nonlinear Klein-Gordon equations in multidimensional spacetime. This is done in the context of classical φ4 and φ6 field theory, the former with and without the inclusion of an external force field conjugate to φ. Both massive (m≠0) and massless (m=0) cases are considered, as well as tachyonic solutions allowed (v>c). We first present a complete set of translationally invariant solutions for the φ4 model and demonstrate (...) the role of external force fields in altering the form of these solutions. Next, spherically symmetric solutions are discussed in both φ4 and φ6 cases since they provide the most realistic models of elementary particles. (shrink)

The modal interpretation of quantum mechanics allows one to keep the standard classical definition of realism intact. That is, variables have a definite status for all time and a measurement only tells us which value it had. However, at present modal dynamics are only applicable to situations that are described in the orthodox theory by projective measures. In this paper we extend modal dynamics to include positive operator measures. That is, for example, rather than using a complete set of orthogonal (...) projectors, we can use an overcomplete set of nonorthogonal projectors. We derive the conditions under which Bell's stochastic modal dynamics for projective measures reduce to deterministic dynamics, showing that Brown and Hiley's generalization of Bohmian mechanics [quant-ph/0005026, ] cannot be thus derived. We then show how deterministic dynamics for positive operators can also be derived. As a simple case, we consider a Harmonic oscillator, and the overcomplete set of coherent state projectors. We show that the modal dynamics for this POM in the classical limit correspond to the classical dynamics, even for the nonclassical number state |n>. This is in contrast to the Bohmian dynamics, which for energy eigenstates, the dynamics are always non-classical. (shrink)

_The Collected Works of John Stuart Mill_ took thirty years to complete and is acknowledged as the definitive edition of J.S. Mill and as one of the finest works editions ever completed. Mill's contributions to philosophy, economics, and history, and in the roles of scholar, politician and journalist can hardly be overstated and this edition remains the only reliable version of the full range of Mill's writings. Each volume contains extensive notes, a new introduction and an index. Many of the (...) volumes have been unavailable for some time, but the _Works_ are now again available, both as a complete set and as individual volumes. (shrink)

Semifactual thinking about what might have been the same, e.g., ''even if Philip had not chosen the chocolate ice-cream sundae, he would have developed an allergic reaction'' has been neglected compared to counterfactual thinking about what might have been different, e.g., ''if only Philip had not chosen the chocolate ice-cream sundae, he would not have developed an allergic reaction''. We report the first systematic comparison of the two sorts of thinking in two experiments. The first experiment showed that counterfactual ''if (...) only'' thoughts about an antecedent event lead people to judge the event to be more causally related to the outcome, whereas semifactual ''even if'' thoughts lead people to judge the antecedent event to be less causally related to the outcome. In addition, the experiment showed that generating counterfactual ''if only'' thoughts increases emotional reactions such as regret, whereas generating semifactual ''even if'' thoughts decreases such reactions. The second experiment, along with a replication experiment, showed that when people complete ''if only'' and ''even if'' sentence stems, they focus on different alternative antecedents to the outcome: ''if only'' thoughts focus on alternatives that would undo the outcome whereas ''even if'' thoughts focus on alternatives that would not undo it, from among a set of available alternative antecedents in which either all, some, or none would undo the outcome. The implications of the results for theories of thinking about what might have been are discussed. (shrink)

Given the tremendous burst of activity in the philosophy of science during the last quarter century, the number of books by trained philosophers dealing with the logic of biology is surprisingly small. Simon’s book resembles Morton Beckner’s The Biological Way of Thought in its comprehensive ambitions: "trying to discover what, if anything, is distinctive about biological science, its concepts, and its mode of explaining." The most obvious difference of the two books is Simon’s long central chapter on "Theories, Models, and (...) the Concept of the Gene," which deals with the history of genetics from Mendel’s predecessors to the successors of Watson and Crick. Beckner paid little explicit attention to the history of biology. Another difference lies in Beckner’s dogged assessment of all problems in the logic of biology from the perspective of what he called the Humean Pattern of Explanation—a detailed and difficult variant of the approach of Hempel, Oppenheim, and Nagel. Simon’s philosophical position is more flexible than Beckner’s. He provides an acceptable review of the literature of the last decade and a half; some of his conclusions are most interesting—even when incompletely supported by his arguments; and the utility of his book is insured by the fact that it is necessary to go back over twelve years to find another with which it can be compared. The first chapter contains a very effective critique of J. J. C. Smart’s thesis that biology is no more an autonomous science than is electrical engineering, but is merely applied physics and chemistry. Smart had argued from the grounds that all biological generalizations implicitly refer to terrestrial evolution and are not truly universal, and Simon demonstrates the general lack of success of all efforts to demonstrate a difference in kind between so-called laws of nature and empirical generalizations of the sort found in biology. The second chapter continues in this same vein, drawing an interesting contrast between phylogenetic explanation in biology and the explanation of individual events in human history, and claiming that ‘explanatory’ status may be granted to accounts which do not meet the requirements of the deductive-nomological model, as in the case of Darwin’s theories of the geographical distribution of plants and animals and the persistence of rudimentary organs. The author’s negative reaction to much of the recent philosophical literature concerning the role of functional analysis in biology seems particularly well placed. Following Hempel, most philosophers writing in the past ten years have regarded the occurrence of the item whose function is in question as the event to be explained. Simon correctly points out that in the biological literature, statements indicating the presence of this item are part of the explanans, usually one component in an explanation of the normal functioning of the organism. He also insists that functional or teleological statements cannot effectively be replaced by nonteleological ones. A functional account is one that relates an entity to the role it plays in an organized system. In the first of several serene non-sequiturs which mar the latter half of this book, Simon insists that the above account shows that the notion of function is essentially an "extrinsic" one and that teleological principles generally cannot be used to account for experience. No better argument than an appeal to Kant’s authority backs up this conclusion. The third chapter deals with the recent history of genetics; its summary makes the claim that "at no time during this history was it correct to say that one model was replaced by another one; rather, the development appears to have been one involving successive correction and refining of earlier conceptions. There has been no revolution: geneticists have always succeeded in either repelling or accommodating critical attacks on the model." The evidence presented earlier in the chapter confers some plausibility on this conclusion, but no more: Simon takes no notice of the controversy surrounding Kuhn and the notion of scientific revolutions, and provides no criteria by which to distinguish "revolutions" from "accommodations." The precise nature of the methodological continuity to be found in the history of genetics is crucial for the view that "biology, or at least genetics, must be regarded as incapable of yielding a complete explanation of the phenomena that come within its purview; if the ultimate principles appealed to are not those of chemistry and physics and hence essentially nonbiological, the explanation cannot be regarded as complete." The text does not clearly indicate the extent to which the author agrees with the statement just quoted; perhaps he only attributes it to "many biologists" impressed by the successes of molecular genetics. It is clear the Simon regards something he calls the "Cartesian attitude" as pervasive in biology. "Biologists are committed to attempting to explain the phenomena of life in terms of the nonliving... The property of being alive or being associated with a living system is simply not treated as a logical primitive within biological science." But this seems to take us back full circle—to the position of Smart, that biology is no more than applied physics and chemistry. Simon disagrees with Smart. How does he stand vis-a-vis this last statement of the Cartesian spirit he finds so pervasive in biology? I think he disagrees with it, but it is very hard to tell. Perhaps the answer is that the resolution of the issue of the autonomy of biology is a pragmatic matter, relative to a complex set of needs and commitments: the sorts of things that determine life styles, political attitudes, and religious points of view, as well as the growth of scientific understanding. The question of the precise configuration of Simon’s pragmatism remains very vague indeed—a consequence of his over use of the style of impersonal attribution rather than direct assertion. Simon says generally sane and familiar things about Biology and the Idea of Progress and about Biology and Ethics. The really startling assertion of the penultimate chapter, that "consciousness" is neither a biological nor a behavioral predicate, and that its foundations are essentially moral and legal, is unfortunately supported by no argument whatsoever. Instead, the reader is urged to examine two of the author’s previous articles.—E. M. (shrink)

In this paper we describe a method for the specification of computationalmodels of argument using dialogue games. The method, which consists ofsupplying a set of semantic definitions for the performatives making upthe game, together with a state transition diagram, is described in full.Its use is illustrated by some examples of varying complexity, includingtwo complete specifications of particular dialogue games, Mackenzie's DC,and the authors' own TDG. The latter is also illustrated by a fully workedexample illustrating all the features of the game.

_The Collected Works of John Stuart Mill_ took thirty years to complete and is acknowledged as the definitive edition of J.S. Mill and as one of the finest works editions ever completed. Mill's contributions to philosophy, economics, and history, and in the roles of scholar, politician and journalist can hardly be overstated and this edition remains the only reliable version of the full range of Mill's writings. Each volume contains extensive notes, a new introduction and an index. Many of the (...) volumes have been unavailable for some time, but the _Works_ are now again available, both as a complete set and as individual volumes. (shrink)

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...) present some conservative translations involving classical logic, Lukasiewicz three-valued system L3, the intuitionistic system I1 and several paraconsistent logics, as for instance Sette's system P1, the D'Ottaviano and da Costa system J3 and da Costa's systems Cn, 1≤ n≤ω. (shrink)

In England, Wales and Scotland, the vast majority of abortions take place in the first trimester of pregnancy. In 2009, for example, 91% of abortions were carried out at under 13 weeks gestation for women resident in England and Wales. 1 Early abortion opens up the opportunity for a woman to have a medical abortion rather than a surgical abortion. Medical abortion is considered to be less invasive and less expensive than surgical abortion, and is increasingly becoming the preferred method. (...) 1 The proportion of medical abortions to surgical abortions has doubled in the last five years with 52% of abortions being medical abortions under 9 weeks gestation in 2009. A medical abortion typically involves taking mifepristone to block the hormones that help a pregnancy to continue, and then later on misoprostol, which makes the womb expel the embryo/fetus, usually within 4–6 h. Current legislation requires that both sets of drugs are administered in an NHS hospital or approved place, but women can go home after taking the second dose of drugs to complete the abortion. There has long been disagreement between the Department of Health and the British Pregnancy Advisory Services about what constitutes ‘any treatment for the termination of pregnancy’ under the Abortion Act 1967, specifically whether a woman taking the second set of drugs is considered to be undergoing the ‘treatment’ and is therefore required to be present in premises permitted under the Act. At the end of 2010, BPAS went to the High Court to challenge the Department of Health's view that the legislation requires women to take both sets of drugs in an approved place. BPAS argued that ‘treatment’ stops at the point of prescription and women should be able to go home, …. (shrink)

We show that without assuming completeness or continuity, a strongly independent preorder on a possibly infinite dimensional convex set can always be given a vector-valued representation that naturally generalizes the standard expected utility representation. More precisely, it can be represented by a mixture-preserving function to a product of lexicographic function spaces.