The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociski, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations. (shrink)

Quomodo versus lyrici legendi sint, plerumque inter metricos constat, licet de origine singulorum versuum vel colorum dubia moveantur; at ne illud quidem confirmari potest in genere illo peculiari, quo multae strophae Pindari et Bacchylidis et nonnullae poétarum scenicorum compositae sunt. Quod in talibus versibus maxime conspicuum est, hoc est, quod metra τоû σоυ et διπλασíоυ γéνоυς; in eis coniunguntur vel coniungi videntur, ita, ut ambitus utriusque partis aut par aut non multum maior minorve sit. lam antiqui metrici parum compertum habebant, (...) qua ratione eiusmodi versus explicarent. Cum maior pars ceterorum versuum lyricorum etiam post Alexandri aetatem in usu essent et doctrina grammaticorum hanc quoque materiam complecteretur, pauca versuum illorum vestigia remanebant, postquam chori lyrici et dramatici conticuerunt. Habemus quidem Meliambos Cercidae eodem genere carmina pangere conati, quo clarissimi illi veteres, sed eos perlegentes statim sentimus artem intermortuam non ab isto Megalopolitano nova vita donatam esse; nihil enim frigidius cogitari potest, quam si quis oda pindarica recitata ad Cercidae versus se convertat eosque alta voce pronuntiare conetur. Metrici graeci et romani praeter paucos versus eosque singillatim genus nostrum non tractant nee quisquam ex eorum numero de genere toto disseruit. Horatium et suamet ipsum arte prae ceteris excellentem et doctrina metrica haud mediocriter imbutum non aliter ac nos huius aevi philologos nescire, quomodo carmina pindarica ab ipso poeta componerentur, ex Oda notissima patet, in qua Pindarum quisquis studet aemulari frustra in re desperata desudare perhibetur. Ad universam lyricorum graecorum artem pertinent, quae a Cicerone in Oratore affirmantur, qui tamen clausulis orationis numerosae operam dans rem metricam non neglexit: sed in versibus res est apertior, quamquam etiam a modis quibusdam cantu remoto soluta esse videatur oratio, maximeque id in optima quoque eorum poëtarum, qui λυρικоí a graecis nominantur, quos, cum cantu spoliaveris, nuda paene remanet oratio. Causam hanc vel maximam esse puto in nullo alio carmina pangendi genere musicam, saltationem, metrum tam arte cohaesisse; cum poësis non iam pars habebatur υθμιξου, quod φθγγοδ, σχμα, συλλαβ definiebant, ut Aristoxeni terminis utar , via stabilita ambitum singulorum versuum ac colorum statuere conarentur et indoloem pedum metricorum conicendo indicarent. (shrink)

During the 1970s, prenatal screening technologies were in their infancy, but were being swiftly harnessed to uncover and prevent spina bifida. The historical rise of this screening process and prevention programme is analysed in this paper, and the role of ethical debates in key studies, editorials and letters reported in the Lancet, and other related texts and governmental documents between 1972 and 1983, is considered. The silence that surrounded rigorous ethical debate served to highlight where discussion lay—namely, within the justifications (...) offered for the prevention of spina bifida, and the efficacy and benefits of screening. In other words, the ethical justification for screening and prevention of spina bifida, when the authors are not explicitly interested in ethics, is considered. These justifications held certain notions of disability as costly to society, with an imperative to screen and prevent spina bifida for the good of the society. (shrink)

Appearing in English for the first time, Schelling’s 1842 lectures develop the idea that many philosophical concepts are born of religious-mythological notions.

A compact, coherent introduction to the study of rational belief, this text provides points of entry to such areas of philosophy as theory of knowledge, methodology of science, and philosophy of language. The book is accessible to all undergraduates and presupposes no philosophical training.

Through clear explanation of its characteristic concepts and doctrines, and paying close attention to the published works of its philosophers, the volume ...

Inleidend studieboek op universitair niveau over de Nederlandse rechtswetenschap.

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include (...) almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension U of is finitely axiomatized (provided has only finitely many inference rules); (II) U has only finitely many extensions. (shrink)

As Raymond Williams famously declared, nature is one of the most complex words in the English language – and, we may confidently predict, its Germanic relatives including Dutch. The workshop that took place in June 2007 in the Netherlands, from which this volume is derived, was based on an earlier program exploring connections between our concepts of nature and related concepts of science and religion. Though one may not immediately expect these three realms to be interrelated, countless examples suggest otherwise.

Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds (...) of PTSs. Most importantly PTSs are about statements of the form M:A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. (shrink)

This paper is based on Lectures 1, 2 and 4 in the series of ten lectures titled “Algebraic Structures for Logic” that Professor Blok and I presented at the Twenty Third Holiday Mathematics Symposium held at New Mexico State University in Las Cruces, New Mexico, January 8-12, 1999. These three lectures presented a new approach to the algebraization of deductive systems, and after the symposium we made plans to publish a joint paper, to be written by Blok, further developing these (...) ideas. That project was still incomplete when Blok died. In fact, there is no indication that he had prepared a draft of the paper, and we do not know what new material he intended to include. I am therefore not in a position to complete the project as he had envisioned it. So, I have settled for the more limited objective of presenting the material from the three lectures, leaving to others the task of adapting the techniques used there to more general situations. (shrink)

Language, Duty, and Value Jonathan Dancy, J. M. E. Moravcsik James Opie Urmson, Edited by Jonathan Dancy, J. M. E. Moravcsik, and C. C. W. Taylor. reasons in general. This is freedom in the sense of acting on reasons, yet not those ...