We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PS I-complete) POVM. We show that a measurement with 2D−1 outcomes cannot be PS I-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PS I-complete POVM has 2D outcomes. We also consider PS I-complete POVMs that have only rank-one POVM (...) elements and construct an example with 3D−2 outcomes, which is a generalization of the tetrahedral measurement for a qubit. The question of the minimal number of elements in a rank-one PS I-complete POVM is left open. (shrink)

The article treats as significant the formal coincidence between Kant’s presentation of the science of metaphysics in the “Architectonic of Pure Reason” chapter of the first Critique and Alexander Baumgarten’s presentation of the same in the Metaphysica. From his comments on Baumgarten in the metaphysics lectures, the article shows that for Kant metaphysics in its traditional form lacked completeness and systematic order. Kant fits completeness into his architectonic plan of a scientific metaphysics by converting Baumgartian ontology into (...) an “analytic of the understanding”; Kant achieves the systematicity by modeling a rational “idea of the form of the whole” after Baumgarten’s tree-like ordering of the special sciences of metaphysics. Thus, Kant realizes the completeness and systematicity in a theoretical presentation of the science of metaphysics that he finds lacking in Baumgarten precisely by borrowing from the latter his scheme for metaphysics. (shrink)

A logic of grounding where what is grounded can be a collection of truths is a “many-many” logic of ground. The idea that grounding might be irreducibly many-many has recently been suggested by Dasgupta. In this paper I present a range of novel philosophical and logical reasons for being interested in many-many logics of ground. I then show how Fine’s State-Space semantics for the Pure Logic of Ground can be extended to the many-many case, giving rise to the (...) class='Hi'>Pure Logic of Many-Many Ground. In the second, more technical, part of the paper, I do two things. First, I present an alternative formalization of plg; this allows us to simplify Fine’s completeness proof for plg. Second, I formalize plmmg using an infinitary sequent calculus and prove that this formalization is sound and complete. (shrink)

I lay down a system of structural rules for various notions of ground and establish soundness and completeness.

Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with second-order identity is equivalent to a member of a circumscribed (...) class of formulas. As a corollary, pure second-order logic with second-order identity is compact, its notion of logical truth is decidable, and it satisfies a pure second-order analogue of model completeness. We end by mentioning an extension to n th-order pure logics. (shrink)

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total (...) Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

In this paper we present a sequent calculus for the modal propositional logic GL (the logic of provability) obtained by means of the tree-hypersequent method, a method in which the metalinguistic strength of hypersequents is improved, so that we can simulate trees shapes. We prove that this sequent calculus is sound and complete with respect to the Hilbert-style system GL, that it is contraction free and cut free and that its logical and modal rules are invertible. No explicit semantic element (...) is used in the sequent calculus and all the results are proved in a purely syntactic way. (shrink)

While the presence or absence of consciousness plays the central role in the moral/ethical decisions when dealing with patients with disorders of consciousness (DOC), recently it is criticized as not adequate due to number of reasons, among which are the lack of the uniform definition of consciousness and consequently uncertainty of diagnostic criteria for it, as well as irrelevance of some forms of consciousness for determining a patient’s interests and wishes. In her article, Dr. Specker Sullivan reexamined the meaning of (...) consciousness in the DOC taxonomy and proposed to go away from the routinely used clinical definition of consciousness as “wakeful awareness”, and adopt the meaning that is common in the Eastern tradition which is a form of “pure experience” (Specker Sullivan 2018). She further argued that understanding consciousness as a “pure experience” is ethically relevant for DOC patients. This suggestion is original, novel and important since it preserves the importance of the notion of consciousness for the clinical practice while simultaneously offering an additional ethical tool for the moral decisions in medicine. At the same time, without placing such Eastern notion in the Western tradition it is difficult to see how pure experience could be usefully operationalized to make sense in the clinical practice with DOC patients. It is so because pure experience is a subjective phenomenon which is completely inaccessible in noncommunicative DOC patients and also it does not express behaviorally (Monti et al 2010), therefore some objective-like operationalization is needed. This is why the fusion of Eastern and Western traditions is required to gain the full potential of Dr. Specker Sullivan’s suggestion. We propose that such fusion could be achieved on the basis of the Operational Architectonics (OA) theory of brain-mind functioning (Fingelkurts et al. 2010; 2013a) which suits ideally the purpose due its compatibility with both Western and Eastern traditions of consciousness. (shrink)

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a (...) wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (shrink)

Recent works about ecumenical systems, where connectives from classical and intuitionistic logics can co-exist in peace, warmed the discussion on proof systems for combining logics. This discussion has been extended to alethic modalities using Simpson’s meta-logical characterization: necessity is independent of the viewer, while possibility can be either intuitionistic or classical. In this work, we propose a pure, label free calculus for ecumenical modalities, nEK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {nEK}$$\end{document}, where exactly one logical operator (...) figures in introduction rules and every basic object of the calculus can be read as a formula in the language of the ecumenical modal logic EK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EK}$$\end{document}. We prove that nEK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {nEK}$$\end{document} is sound and complete w.r.t. the ecumenical birelational semantics and discuss fragments and extensions. (shrink)

Philosophers have spilled a lot of ink over the past few years exploring the nature and significance of grounding. Kit Fine has made several seminal contributions to this discussion, including an exact treatment of the formal features of grounding [Fine, 2012a]. He has specified a language in which grounding claims may be expressed, proposed a system of axioms which capture the relevant formal features, and offered a semantics which interprets the language. Unfortunately, the semantics Fine offers faces a number of (...) problems. In this paper, I review the problems and offer an alternative that avoids them. I offer a semantics for the pure logic of ground that is motivated by ideas already present in the grounding literature, and for which a natural axiomatization capturing central formal features of grounding is sound and complete. I also show how the semantics I offer avoids the problems faced by Fine’s semantics. (shrink)

We consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability.

: In the first edition of his book on the completeness of Kant’s table of judgments, Klaus Reich shortly indicates that the B-version of the metaphysical exposition of space in the Critique of pure reason is structured following the inverse order of the table of categories. In this paper, I develop Reich’s claim and provide further evidence for it. My argumentation is as follows: Through analysis of our actually given representation of space as some kind of object, the (...) metaphysical exposition will show that this representation is secondary to space considered as an original, undetermined and as such unrepresentable intuitive manifold. Now, following Kant, the representation of any kind of object involves diversity, synthesis and unity. In the case of our representation of space as formal intuition, this involves, firstly, a manifold a priori, i.e. space as pure form, delivered by the transcendental Aesthetic, secondly, a figurative, productive synthesis of that manifold, and, thirdly, the unity provided by the categories. Analysing our given representation of space – the task of the metaphysical exposition – amounts to dismantling its unity and determine its characteristics with respect to the categories. (shrink)

Defenders of The Pure View – “Purists”, as I shall call them – maintain that perception is pure presentation. That is, a perceptual experience has no commitments that exceed what is given or presented in the experience. I argue The Pure View seems unable to offer a convincing account of amodal completion. I distinguish three Purist strategies for addressing amodal completion, and suggest that none is very promising.

Like Werner Pluhar's distinguished translation of _Critique of Judgment_ (Hackett Publishing Co., 1987), this new rendering of _Critique of Pure Reason_ reflects the elegant achievement of a master translator. This richly annotated volume offers translations of the complete texts of both the First (A) and Second (B) editions, as well as Kant's own notes. Extensive editorial notes by Werner Pluhar and James Ellington supply explanatory and terminological comments, translations of Latin and other foreign expressions, variant readings, cross-references to other (...) passages in the text and in other writings of Kant, and references to secondary works. An extensive bibliography, glossary, and detailed index are included. Patricia Kitcher's illuminating Introduction provides a roadmap to Kant's abstract and complex argumentation by firmly locating his view in the context of eighteenth-century--and current--attempts to understand the nature of the thinking mind and its ability to comprehend the physical universe. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.1 (2002) 41-62 [Access article in PDF] Pure of Heart: From Ancient Rites to Renaissance Plato Marjorie O'Rourke Boyle The philosopher who published Plato for Western thought praised him strangely. Marsilio Ficino commended his translation of the Phaedrus to his soul mate Iohannes Bessarion because in that dialogue Plato sought from God spiritual beauty. "When this gold was given to Plato by (...) God, it shone in him most brilliantly, because he was so pure in heart." The epithet "pure in heart" is not literally Ficino's diction: he wrote "in his Platonic bosom perfectly clean" (Platonico in sinu utpote mundissimo). 1 Although this philological variant insinuates his full meaning, the modern translation conveys his essential import. Ficino and Bessarion were both scholars and priests, convinced of the compatibility of Platonism with Christianity. The recipient of the letter, Bessarion, had defended the Phaedrusvigorously against protestations that, for promoting the criminal activity of pederasty, it was an obscene work of unchaste philosophy. "Love," he declared, "leads many to beatitude." 2 Ficino emphasized that "no one can attain beatitude, which consists in divine contemplation, unless he loves God ardently." 3 To that end he translated the complete works of Plato into Latin with commentaries. He also composed a speculative Theologia platonica. The pair would have approved of the application to Plato of the gospel promise "Blessed are the pure of heart [hoi katharoi t¯e kardia], for they shall see God" (Matt. 5:8). Was this promise not already fulfilled in Plato, even before Jesus' pronouncement? Did not Socrates in the Phaedrus soar to the vision of God in his myth of the charioteer, where among privileged souls he peered over the rim of intelligible being to behold the divine processional and true being? The [End Page 41] myth figured the recollection of the soul's primordial state of innocence, "when we were pure ourselves," and, beholding "the beatific vision," were initiated into its "most blessed" mystery. 4 Bessarion cited the passage defensively, declaring Socrates' speech "most holy." 5The beatitudes, from which "pure of heart" derived, emphatically comprised the opening discourse of Jesus' Sermon on the Mount. Jesus blessed the poor in spirit, the mourners, the meek, those hungry and thirsty for righteousness, the merciful, the pure of heart, the peacemakers, and those persecuted for righteousness. He pronounced a corresponding good for each type (Matt. 5:1-10). The first four beatitudes promised reversals for the unfortunate; the second four, eschatological rewards for the virtuous. 6 Ficino personalized the beatitudes. He applied them in a letter to an archbishop to his own contrasting status as a priest "of humble fortune." Indicating his slight financial means, he affirmed obedience to his lot, even contentment with his meager possessions. He then wielded the divine reversal of human judgment, so startling in the beatitudes, against his enemies. "I am certainly content with few possessions, even though I am often held in contempt by many; for these many ought to be deemed of little worth. Blessed are the poor in spirit. Blessed are the peacemakers." 7Ficino also philosophized the beatitudes. He argued in Theologia platonica that, although the celestial realm was usually considered fiery, it could be earthy, as poets imagined the Elysian Fields. Jesus' beatitude promising the possession of the earth to the meek was proof. Ficino expounded the nine grades of celestial beatitude as a belief shared by Platonists and Christians. He assigned these grades to varying degrees and combinations of three ends: the fear of divine power, the search for wisdom, and the love of good, which ultimately transformed the soul. This elaborate division Ficino compared to Jesus' beatitudes. "I think that this mystery is signified in the gospel, where the divine oracle nine times calls his disciples blessed, because through nine grades of merit, with God leading, the nine heavens are sought blessedly through nine degrees of contemplation." Despite his philosophical imposition on the biblical text, Ficino qualified his syncretism. The gospel admonished the dismissal of philosophical meanderings to pursue... (shrink)

Kant: A Complete Introduction covers the key areas that students are expected to be confident in, outlining the basics in clear English, and then providing added-value features like summaries of key books, and even lists of questions you might be asked in your seminar or exam. In doing so, Professor Wicks introduces you to all of Kant's key works, including The Critique of Pure Reason"--Page 4 of cover.

By a pure modal logic of names we mean a quantifier-free formulation of such a logic which includes not only traditional categorical, but also modal categorical sentences with modalities de re and which is an extension of Propositional Logic. For categorical sentences we use two interpretations: a “natural” one; and Johnson and Thomason’s interpretation, which is suitable for some reconstructions of Aristotelian modal syllogistic :271–284, 1989; Thomason in J Philos Logic 22:111–128, 1993 and J Philos Logic 26:129–141, 1997. In (...) both cases we use Johnson-like models. We also analyze different kinds of versions of PMLN, for both general and singular names. We present complete tableau systems for the different versions of PMLN. These systems enable us to present some decidability methods. It yields “strong decidability” in the following sense: for every inference starting with a finite set of premises we can specify a finite number of steps to check whether it is logically valid. This method gives the upper bound of the cardinality of models needed for the examination of the validity of a given inference. (shrink)

Let Λ denote the semiring of isols. We characterize existential completeness for Nerode subsemirings of Λ, by means of a purely isol-theoretic “Σ1 separation property”. Our characterization is purely isol-theoretic in that it is formulated entirely in terms of the extensions to Λ of the Σ1 subsets of the natural numbers. Advantage is taken of a special kind of isol first conjectured to exist by Ellentuck and first proven to exist by Barback . In addition, we strengthen the negative (...) part of [13] by showing that existential completeness is not secured, for a given Nerode semiring, by either a certain “functional closure” property for the extensions of partial recursive functions or the property of “pulling in” some portion of each partial recursive fiber; these latter results are perhaps a little surprising. (shrink)

The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special class of entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as ‘disguised’ quantifiers, and I think (...) that wholehearted acceptance of modal language should be considered to carry ontological commitment to something like possible worldsConsidered as two languages for describing the same subject matter, modal and purely quantificational languages show interesting differences. The operator variables of the purely quantificational languages give them more power than the modal languages, but at least some of the functions performed by the apparatus of operator variables are also performed, in a more primitive and less versatile way, by actuality operators in modal languages.A final note. Quine has written much on the inter-relations of quantifiers, identity, and the concept of existence. These, he holds, form a tightly knit conceptual system which has been evolved to a high point of perfection, but which might conceivably change yet further.29 He has also dropped hints about the possibility of a simpler, primitive or defective version of the system, in which the quantifiers are not backed up in their accustomed way by the concept of identity. He has dubbed the resulting concept a ‘pre-individuative’ concept of existence, or a concept of ‘entity without identity.’ What would a pre-individuative concept of existence be like? Quine has sometimes suggested that one might be embodied in the use of mass nouns, but the identity concept is used in connection with stuff as well as with things: “is that the same coffee that was in the cup last night?” I would submit that modality provides a better case. In view of the comparative weakness of modal languages, compared to the explicitly quantificational ones Quine takes as canonical, there is surely a sense in which the concept of existence embodied in that disguised existential quantifier, the possibility operator, is a defective one. And as we have seen, one of the differences between modal operators and explicit quantifiers is that modal operators cannot be joined with the identity predicate in the way quantifiers with operator variables can. Surely, then, there is a sense in which ordinary speech, as opposed to the metaphysical theorizing of a Leibniz or a David Lewis, conceives of possible worlds as entities without identity. (shrink)

English translation by Kneller and Losonsky of Klaus Reich, Die Vollständigkeit der Kantischen Urteilstafel -/- "This classic of Kant scholarship, whose first edition appeared in 1932, deals with one of the most controversial and difficult topics in the Critique of Pure Reason: Kant's table of judgments and their connection to the table of categories. Kant's attempt to derive the latter from the former is called the "Metaphysical Deduction," and it paves the way for the Transcendental Deduction that is universally (...) recognized as the heart of the Critique." "Many commentators have passed over the Metaphysical Deduction in silence, as if embarrassed by Kant's fatuity, and his critics are almost unanimous in finding its premise ungrounded, its argument incorrect, and its conclusion false. At the heart of the failure of the Metaphysical Deduction, it is alleged, is Kant's inability to justify the table of the forms of judgment. Critics argue that Kant simply took these forms of judgment from the logic textbooks of his day and doctored them so as to yield the required categories. Such objections were current even in Kant's time; they were repeated by Hegel, and are commonplaces of Kant criticism today." "This book is the fullest and most systematic evaluation ever made of the Metaphysical Deduction. Reich argues that its principal conclusion is correct even though Kant failed to establish it in the Metaphysical Deduction proper, and he defends the conclusion by using Kant material far removed from the text of the Metaphysical Deduction, notes by Kant that only became available in the 1920's and that remain even now untranslated into English. With the publication of this translation for English-speaking Kant scholars, a neglected and impugned part of the Critique of Pure Reason can finally receive the attention it deserves."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved. (shrink)

The present work is focussed on the completeness of physics, or what is here called the Completeness Thesis: the claim that the domain of the physical is causally closed. Two major questions are tackled: How best is the Completeness Thesis to be formulated? What can be said in defence of the Completeness Thesis? My principal conclusions are that the Completeness Thesis can be coherently formulated, and that the evidence in favour if it significantly outweighs that (...) against it. In opposition to those who argue that formulation is impossible because no account of what is to count as physical can be provided, I argue that as long as the purpose of the argument in which the account is to be used are borne in mind there are no significant difficulties. The account of the physical which I develop holds as physical whatever is needed to fix the likelihood of pre-theoretically given physical effects, and hypothesises in addition that no chemical, biological or psychological factors will be needed in this way. The thus formulated Completeness Thesis is coherent, and has significant empirical content. In opposition to those who defend the doctrine of emergentism by means of philosophical arguments I contend that those arguments are flawed, setting up misleading dichotomies between needlessly attenuated alternatives and assuming the truth of what is to be proved. Against those who defend emergentism by appeal to the evidence, I argue that the history of science since the nineteenth century shows clearly that the empirical credentials of the view that the world is causally closed at the level of a small number of purely physical forces and types of energy is stronger than ever, and the credentials of emergentism correspondingly weaker. In opposition to those who argue that difficulties with reductionism point to the implausibility of the Completeness Thesis I argue that completeness in no way entails the kinds of reductionism which give rise to the difficulties in question. I argue further that the truth of the Completeness Thesis is in fact compatible with a great deal of taxonomic disorder and the impossibility of any general reduction of non-fundamental descriptions to fundamental ones. In opposition to those who argue that the epistemological credentials of fundamental physical laws are poor, and that those laws should in fact be seen as false, I contend that truth preserving accounts of fundamental laws can be developed. Developing such an account, I test it by considering cases of the composition of forces and causes, where what takes place is different to what is predicted by reference to any single law, and argue that viewing laws as tendencies allows their truth to be preserved, and sense to be made of both the experimental discovery of laws, and the fact that composition enables accurate prediction in at least some cases. (shrink)

According to Kant, pure intuition is an indispensable ingredient of mathematical proofs. Kant‘s thesis has been considered as obsolete since the advent of modern relational logic at the end of 19th century. Against this logicist orthodoxy Cassirer’s “critical idealism” insisted that formal logic alone could not make sense of the conceptual co-evolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, played a fundamental role in the formation of the mathematical and empirical concepts. The aim (...) of this paper is to outline the basics of Cassirer’s idealizational account, and to point at some interesting similarities it has with Kant’s and Peirce’s philosophies of mathematics based on the key notions of pure intuition and theorematic reasoning, respectively. (shrink)

We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely (...) combinatorial characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper. (shrink)

Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that (...) two of the systems have the variable sharing property). I emphasize the intuitive content of the semantics and discuss how natural informational considerations underly each of the examined systems. (shrink)

We prove that S4 is complete with respect to Boolean combinations of countable unions of convex subsets of the real line, thus strengthening a 1944 result of McKinsey and Tarski 45 141). We also prove that the same result holds for the bimodal system S4+S5+C, which is a strengthening of a 1999 result of Shehtman 369).

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)

As is made plain in the critical apparatus and editorial matter appended to the original German publication of Hussed's Ideas II, I this is a text with a history. It underwent revision after revision, spanning almost 20 years in one of the most fertile periods of the philosopher's life. The book owes its form to the work of many hands, and its unity is one that has been imposed on it. Yet there is nothing here that cannot be traced back (...) to Hussed himself. Indeed, the final" clean copy" for publication, prepared by an assistant, was completely reviewed by the master three times and emended by him in detail on each occasion. Nevertheless, in the end the work was in fact not submitted for publication, and after Hussed's pen last touched the manuscript in 1928 it was set aside until posthumously edited and published by the Hussed-Archives in 1952. The story of the composition of Ideas II begins with the "pencil manuscript" of 1912. This is the ultimate textual source for both Ideas II and Ideas III. 2 It has been preserved as a folio of 84 sheets in very dense shorthand of the Gabelsberger system, written mostly with a pencil. It was composed by Hussed "in one stroke" immediately after the completion of I Edmund Husser!: Ideen zu einer reinen Phiinomenologie und phiinomenologischen Philosophie. Zweites Buch: Phiinomenologische Untersuchungen zur Konstitution. Edited by Marly Biemel. The Hague: Martinus NijhofT, 1952 (Husserliana IV). (shrink)

This new translation of the first Critique forms part of a fifteen-volume English-language edition of the works of Immanuel Kant under the general editorship of this volume’s editor-translators, Paul Guyer and Allen Wood. The edition, which is almost complete by now, comprises all of Kant’s published works along with extensive selections from his literary remains, his correspondence, and student transcripts of his lecture courses in metaphysics, ethics, logic, and anthropology. The Cambridge edition aims at a consistent English rendition of Kant’s (...) works, both within a given volume and across volumes. In terms of scope and detail, the Cambridge edition is unrivaled in any language, except for the authoritative Academy edition begun under the directorship of Wilhelm Dilthey in 1900, which, however, is still not completed and several volumes of which are in serious need of re-editing. In one case, that of the Opus postumum, the best edition currently available seems to be the one in the Cambridge edition. (shrink)

Here we investigate the classes RCA $^\uparrow_\alpha$ of representable directed cylindric algebras of dimension α introduced by Nemeti[12]. RCA $^\uparrow_\alpha$ can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, "purely cylindric algebraic" proof for the following theorems of Nemeti: (i) RCA $^\uparrow_\alpha$ is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain (...) a strong representation theorem for RCA $^\uparrow_\alpha$ if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories. (shrink)

By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as defined by (...) Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given. (shrink)

In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders |$\mathcal {L}( @, {\downarrow })$|⁠. We define the class of restricted Sahlqvist formulas for |$\mathcal {L}( @, {\downarrow })$| following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula (...) |$\varphi $| and its hybrid pure correspondence |$\pi $|⁠, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| proves |$\pi $|⁠; therefore, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| is complete with respect to the class of frames defined by |$\pi $|⁠, using a modified version |$\textsf {ALBA}^{{\downarrow }}_{\textsf {Modified}}$| of the algorithm |$\textsf {ALBA}^{{\downarrow }}$| defined in Zhao (2021, Logic J. IGPL). (shrink)

How can we prove that some fragment of a given logic has the power to define precisely all structural properties that satisfy some characteristic semantic preservation condition? This issue is a fundamental one for classical model theory and applications in non-classical settings alike. While methods differ greatly, and while the classical methods can usually not be matched for instance in the setting of finite model theory, this note surveys some interesting commonality revolving around the use and availability of tractable representatives (...) in the relevant model classes. The construction of models in which simple invariants like partial types based on some weak fragment control all the relevant structural properties, may be seen at the heart of such questions. We highlight some constructions involving degrees of acyclicity and saturation that can be achieved in finite model constructions, and discuss their uses towards expressive completeness w.r.t. bisimulation based equivalences in hypergraphs and relational structures. The emphasis is on the combinatorial challenges in such more constructive approaches that work in non-classical settings and especially in finite model theory. One new result concerns expressive completeness w.r.t. guarded negation bisimulation, a back-and-forth equivalence involving local homomorphisms. (shrink)

This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ, where x occurs only positively in γ, we obtain the flat modal fixpoint language by adding to the language of polymodal logic a connective γ for each γΓ. The term γ is meant to be interpreted as the least fixed point of the functional interpretation of the term γ. We consider the (...) following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language on Kripke structures. We prove two results that solve this problem.First, let be the logic obtained from the basic polymodal by adding a Kozen–Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective γ. Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete.Second, addressing the general case, we prove the soundness and completeness of an extension of . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for γ, of size bounded by the length of γ. Thus the axiom system is finite whenever Γ is finite. (shrink)

We present a systematic study of join-extensions and join-completions of partially ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from properties of the Dedekind–MacNeille completion to the proof of the finite embeddability property for a number of varieties of lattice-ordered algebras.

Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. One logic considered is the -free fragment of intuitionistic linear logic without the exponential !. For this fragment Petri nets form a sound and complete model. The strongest logic considered is intuitionistic linear logic, with ,&, and the exponential ! , and forms of quantification. This logic is shown sound and complete with respect to atomic nets , though only (...) once we add extra axioms specific to the Petri-net model. The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. Unfortunately, with respect to this logic, whether an assertion is true of a finite net becomes undecidable. (shrink)