The author quotes some opinions according to which the valuations have no cognitive value: the valuations have no truth-value; they cannot be decided basing on experience; the have no significance in the semiotic sense of the term; no problematic of the semiotic characteristic of valuation should be taken up, as they are not problems of cognitive value; valuative expressions do not belong to the science. The author analyses mainly the 5th opinion. It can be understood in three ways: as a (...) summary of an actual development stage of the science; as a methodological thesis being a component of some methodological model of a science, and as a directive addressed to scientists and recommending them to eliminate valuations from their science. In every of the above meaning the 5th opinion is to be rejected. In the interpretation it is a trivial falsehood, in the interpretation it is ineffective as a guide line of an action, and in the interpretation it assumes an inadequate reconstruction of a science, for it is possible to carry out such a reconstruction which would make possible an occurring of valuations in the science which do not lose anything of their adequacy in other respects. Designing such a reconstruction the author takes as a point of departure a distinction between objective valuations and purely expressive valuations. The former may be used as scientific assumptions and the latter – being deprived of logical value – only as criteria in the choice of problems. It is evident, however, that the sole existence of valuations in a role of scientific assumption does not disturb a realization of the paramount aim of scientific investigations, i.e. resolutions of problems formulated by the scientists. This can only take place when some additional conditions are fulfilled which the author is trying to formulate in a general way. (shrink)

Ordinary equational logic is a connective-free fragment of first-order logic which is concerned with total functions under the relation of ordinary equality. In [AN] (see also [AN1]) and in [Cr] it has been extended in two equivalent ways into a near-equational system of logic for partial functions. The extension given in [Cr] deals with partial functions under two relationships: a relationship of existence-dependent existence and one of existence-dependent Kleene equality. For the language that involves both relationships a set of rules (...) was given that is complete. Those rules in the set that involve only existence-dependent existence turned out to be complete for the sublanguage that involves this relationship only. In the present paper we give a set of rules that is complete for the other sublanguage, namely the language of partial functions under existence-dependent Kleene equality.This language lacks a certain, often needed, power of expressing existence and fails, in particular, to be an extension of the language that underlies ordinary equational logic. That it possesses a fairly simple complete set of rules is therefore perhaps more of theoretical than of practical interest. The present paper is thus intended to serve as a supplement to [Cr] and, less directly, to [AN]. The subject is further rounded out, and some contrast is provided, by [Rob]. The systems of logic treated there are based on the weaker language in which partial functions are considered under the more basic relation of Kleene equality. (shrink)

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)

The aim of the article is to analyze the situation of contemporary logic with reference to the issue concerning connections between the pluralism of logical systems and relativism in logic. Accordingly, I seek answers to the following questions: Can the plurality of logic, more specifically, a large number and variety of systems constructed by logicians, be justified in a rational way? Does pluralism in logic imply the thesis of relativism? Is logical relativism in the contemporary philosophy of logic just a (...) superstition or is it a duly substantiated view? (shrink)

In the paper we provide an overview of issues related to the models used in the research on the logic of norms and actions. We present two models of the variability of the world: temporal (acyclic) and atemporal (cyclic). In the first one the past is always clearly defined, and the future is potentially “branched”. The second type of model allows for a return to the situation that took place. Next we describe different approaches towards agency modeling. We present the (...) most important paradigms, including the logic of actions, where actions are interpreted as transitions between the situations and STIT logic, which states only the responsibility of the agent for particular states of affairs. We discuss questions of determinism and indeterminism of actions, identification of basic actions and operations and the issue of identity. The final part of the paper is devoted to deontic categories. In particular, the applicability of the concepts of strong and weak permission and prohibition in different situational contexts, and their relationship to the two concepts of obligation: abstract and derived. (shrink)

Hegel often says that his "logic" is meant to replace metaphysics. Since Hegel's Science of Logic is so different from a standard logic, most commentators have not treated the portion of that work devoted to logical forms as relevant to this claim. This paper argues that Hegel's discussion of logical forms of judgment and syllogism is meant to be the foundation of his reformation of metaphysics. Implicit in Hegel's discussion of the logical forms is the view that the metaphysical concepts (...) discussed in Books I and II of the Science of Logic supervene on the role of subject and predicate terms in the logical forms discussed in Book III. Hegel thus has an explanation for the nature and significance of metaphysical concepts that resembles Kant's "metaphysical deduction," according to which the categories can be derived from the table of judgments. Though Hegel's metaphysics is often supposed to be influenced by Kant, prevailing interpretations do not show how Hegel's fine-grained treatment of logical forms is relevant to his critical view of metaphysics. The present interpretation provides a model for Hegel's explanation of metaphysical concepts, as well as a new picture of the structure of his Science of Logic that emphasizes the priority of its Doctrine of the Concept. (shrink)

In this essay I focus on the idea of the univocity of being, championed by Duns Scotus and given prominence more recently by Deleuze. Although I am interested in how this idea can be established, my primary concern is with something more basic: how the idea can even be properly thought. In the course of exploring this issue, which I do partly by borrowing some ideas about logical syntax from Wittgenstein's Tractatus, I try to show how there can be dialogue (...) between analytic philosophers and those of other traditions. (shrink)

Roman Ingarden, opisując rzeczywistość moralną, posługiwał się językiem wartości. Niniejszy artykuł przedstawia opis wartości moralnych i analizę terminologii stosowanej do ich charakterystyki i wyjaśniania sposobu ich istnienia. Istotne znaczenie dla teorii wartości moralnych ma doświadczenie powinności i problematyka normatywności. Zbadanie obu zagadnień prowadzi do pytań o sposób powstawania nowych klas wartości moralnych i rozpoznawania powinności do ich realizacji. Przeprowadzone badania pokazują także trudności związane ze stosowaniem języka wartości w komunikacji zagadnień moralnych.

1. STRUCTURE AND REFERENCES 1.1. The main part of the dictionary consists of alphabetically arranged articles concerned with basic logical theories and some other selected topics. Within each article a set of concepts is defined in their mutual relations. This way of defining concepts in the context of a theory provides better understand ing of ideas than that provided by isolated short defmitions. A disadvantage of this method is that it takes more time to look something up inside an extensive (...) article. To reduce this disadvantage the following measures have been adopted. Each article is divided into numbered sections, the numbers, in boldface type, being addresses to which we refer. Those sections of larger articles which are divided at the first level, i.e. numbered with single numerals, have titles. Main sections are further subdivided, the subsections being numbered by numerals added to the main section number, e.g. I, 1.1, 1.2,..., 1.1.1, 1.1.2, and so on. A comprehensive subject index is supplied together with a glossary. The aim of the latter is to provide, if possible, short defmitions which sometimes may prove sufficient. As to the use of the glossary, see the comment preceding it. (shrink)

Is logic normative for reasoning? In the wake of work by Gilbert Harman and John MacFarlane, this question has been reduced to: are there any adequate bridge principles which link logical facts to normative constraints on reasoning? Hitherto, defenders of the normativity of logic have exclusively focussed on identifying adequate validity bridge principles: principles linking validity facts—facts of the form 'gamma entails phi'—to normative constraints on reasoning. This paper argues for two claims. First, for the time being at least, Harman’s (...) challenge cannot be surmounted by articulating validity bridge principles. Second, Harman’s challenge can be met by articulating invalidity bridge principles: principles linking invalidity facts of the form 'gamma does not entail phi' to normative constraints on reasoning. In doing so, I provide a novel defence of the normativity of logic. (shrink)

Sentences about logic are often used to show that certain embedding expressions are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that determines the (...) interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk. (shrink)

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic (...) logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers ∀ new and ∃ new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels. (shrink)

Przedmiotem pracy jest logika kombinatoryczna, gałąź logiki ufundowana przez tak ich autorów, jak Schönfinkel i Curry, której najbardziej znaczącą cechą jest tworzenie języków formalnych pozbawionych zmiennych. U pierwszym rzędzie przedstawia się pojęcia bazowe tego przedsięwzięcia, tj. teorię kombinatorów i teorię funkcjonalności. Podejmując i kontynuując rezultaty, jakie osiągnął Curry, autor poddaje analizie sposób, w jaki można otrzymać zupełny zbiór aksjomatów dla rachunku zdaniowego intuicjonistycznego, wychodząc od utworzenia zupełnego zbioru kombinatorów pierwotnych.

An ontology's theory of ontic predication has implications for the concomitant predicate logic. Remarkable in its analytic power for both ontology and logic is the here developed Particularized Predicate Logic (PPL), the logic inherent in the realist version of the doctrine of unit or individuated predicates. PPL, as axiomatized and proven consistent below, is a three-sorted impredicative intensional logic with identity, having variables ranging over individuals x, intensions R, and instances of intensions $R_{i}$ . The power of PPL is illustrated (...) by its clarification of the self-referential nature of impredicative definitions and its distinguishing between legitimate and illegitimate forms. With a wellmotivated refinement on the axiom of comprehension, PPL is, in effect, a higher-order logic without a forced stratification of predicates into types or the use of other ad hoc restrictions. The Russell-Priest characterization of the classic self-referential paradoxes is used to show how PPL diagnosis and solves these antimonies. A direct application of PPL is made to Grelling's Paradox. Also shown is how PPL can distinguish between identity and indiscernibility. (shrink)

In this paper, I will explore and attempt to define one very important type of egregious discrimination of persons, racism. I will argue that racism involves a kind of logical mistake; specifically. I hope to show that racists commit the naturalistic fallacy. Finally, I will defend my account of racism against two challenges, the most important of which argues that if racism is merely a logical error then racists are not morally culpable.

Psychologism in logic holds that logic is a branch of psychology. This view has been vigorously defended by John Stuart Mill and by a number of German philosophers of logic, notably Erdmann. Its chief critics have been Husserl and Frege and, to a lesser extent, Russell. Husserl set forth a profound and detailed critique of psychologism in Logical Investigations. This paper examines this critique. First, I explain why the psychologistic theory is attractive. Then I show that Husserl's critique is not (...) convincing, partly because he does not take the theory in its most plausible form and partly because he ignores certain important distinctions (for example, between what a statement is about and what it is true in virtue of). Then I raise two new objections to the psychologistic theory. The purpose of this paper is to suggest that the psychologistic theory remains an important and serious position from which we can learn much about the status of logic. (shrink)

In 1962 a mimeographed sheet of paper fell into my possession. It had been prepared by Ernest Adams of the Philosophy Department at Berkeley as a handout for a colloquim. Headed 'SOME FALLACIES OF FORMAL LOGIC' it simply listed eleven little pieces of reasoning, all in ordinary English, and all absurd. I still have the sheet, and quote a couple of the arguments here to give the idea. • If you throw switch S and switch T, the motor will start. (...) There fore, either if you throw switch S the motor will start, or, if you throw switch T the motor will start. • It is not the case that if John passes history he will graduate. Therefore, John will pass history. The disconcerting thing about these inferences is, of course, that under the customary truth-functional interpretation of and, or, not, and if-then, they are supposed to be valid. What, if anything, is wrong? At first I was not disturbed by the examples. Having at that time consider able personal commitment to rationality in general and formal logic in par ticular, I felt it my duty and found myself easily able (or so I thought) to explain away most of them. But on reflection I had to admit that my expla nations had an ad hoc character, varying suspiciously from example to example. (shrink)

In the opening to his late essay, Der Gedanke, Frege asserts without qualification that the word "true" points the way for logic. But in a short piece from his Nachlass entitled "My Basic Logical Insights", Frege writes that the word true makes an unsuccessful attempt to point to the essence of logic, asserting instead that "what really pertains to logic lies not in the word "true" but in the assertoric force with which the sentence is uttered". Properly understanding what Frege (...) takes to be at issue here is crucial for understanding his conception of logic and, in particular, what he takes to be its normative status vis-à-vis judgement, assertion, and inference. In this paper, I focus my attention on clarifying the latter claim and Frege's motivations for making it, exposing what I take to be a fundamental tension in Frege's conception of logic. Finally, I discuss whether Frege's deployment of the horizontal in his mature Begriffsschrift helps to resolve this tension. (shrink)

Ramsey questions the distinction between subject and predicate in sentences. Since the distinction mirrors the ontological distinction object/property (particular/universal) he questions the latter distinction as well. There is a symmetry between the category of objects and the category of properties: an object can possess a number of properties and a property can be exemplified by a number of objects. Consequently, it may be claimed that the two categories are formally indiscernible. The paper develops Ramsey's thesis on the ground of logic. (...) Different senses of indiscernibility are discussed and various systems of logic are examined with respect to these senses. Most systems do not distinguish objects from properties in at least one sense. (shrink)

In a posthumous paper, Perelman discusses his decision to bring his theory of argumentation together with rhetoric rather than calling it an informal logic. This is due in part because of the centrality he gives to audience, and in part because of the negative attitude that informal logicians have to rhetoric. In this paper, I explore both of these concerns by way of considering what benefits Perelman’s work can have for informal logic, and what insights the work of informal logicians (...) might bring to the project of Perelman and Olbrechts-Tyteca. (shrink)

"A Special Case of Logical Contradiction – Content Implication Approach" The difference between so-called performative/pragmatic and logical contradiction seems to be one of the most firmly established philosophical distinctions. There are, however, some utterances, which fall outside this classification in its current form. Moreover, they reveal that the expressive power of the language of classical logic is too weak to express some form of the logical contradiction. The main aim of this paper is to characterize this special case of the (...) contradiction. By extending classical logic with a new connective, i.e. content implication, it also provides tools needed for adequate formal representation of this phenomenon. (shrink)

Logique et existenceDeleuze à propos des « conditions du réel »Pour Deleuze, l’un des problèmes fondamentaux d’une théorie de la pensée est de savoir comment la pensée peut quitter la sphère du possible pour penser le réel, c’est-àdire pour penser l’existence elle-même. La position du réel semble être hors du concept. Des pré-kantiens comme Leibniz approchaient ce problème par le biais de la distinction entre vérités d’essence et vérités d’existence, alors que des post-kantiens comme Maimon l’approchaient par la distinction entre (...) les conditions de l’expérience possible et celles de l’expérience réelle. La logique classique définit la sphère du possible par trois principes logiques – l’identité, la non-contradiction et le tiers-exclu – et la présente étude examine les trois grandes trajectoires qui, dans cette histoire de la philosophie, ont tenté d’utiliser l’un de ces trois principes classiques pour pénétrer l’existence ellemême : 1) Leibniz (et les pré-kantiens) cherchait à étendre le principe de d’identité àl’existence entière ; 2) Hegel (et les post-kantiens) cherchait à étendre le principe de non-contradiction à la totalité de l’expérience ; et 3) le groupe des penseurs appelés de manière assez large « existentialistes » cherchait à étendre le principe du tiers-exclu à la totalité de l’existence. La conclusion examine les raisons pour lesquelles Deleuze a été fasciné par chacune de ces tentatives philosophiques pour « penser l’existence », tout en pensant néanmoins qu’elles ont toutes échoué ; et pourquoi aussi il a fini par développer sa propre réponse au problème en faisant appel à un principe de différence.Logica e EsistenzaLe ‘Condizioni del reale’ in DeleuzePer Deleuze, uno dei problemi fondamentali per una teoria del pensiero è: come può il pensiero abbandonare la sfera del possibile per pensare il reale, ossia, pensare l’esistenza stessa? La posizione del reale sembra essere fuori dal concetto. Prekantiani come Leibniz affrontano questo problema in termini di distinzione fra verità dell’essenza e verità dell’esistenza, mentre post-kantiani come Maimon affrontano il problema in termini di distinzione fra condizioni dell’esperienza possibile e condizioni dell’esperienza reale. La logica classica ha definito la sfera del possibile secondo tre principi logici – identità, non-contraddizione, terzo escluso – e questo saggio analizza tre grandi ‘parabole’ della storia della filosofia che hanno tentato di usare uno di questi tre principi della logica per penetrare l’esistenza stessa: (1) Leibniz (e i pre-kantiani) hanno tentato di estendere il principio di identità a tutta l’esistenza; (2) Hegel (ei post-kantiani) hanno tentato di estendere il principio di non-contraddizione a tutta l’esistenza; (3) il gruppo di pensatori (solo approssimativamente legati fra loro) chiamati “esistenzialisti” ha tentato di estendere il principio del terzo escluso all’esistenza. La conclusione analizza sia le ragioni per le quali Deleuze era affascinato da ciascuno di questi tentativi filosofici di “pensare l’esistenza” nonostante fosse convinto che essi avessero fallito, sia i motivi per cui egli in conclusione traccia la propria risposta al problema facendo appello al principio della differenza. (shrink)

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz (...) hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field. (shrink)

“O, rocks!” Molly exclaims in impatience with Bloom’s first definition of metempsychosis, “tell us in plain words” . Looking forward, then, we remember that Bloom asks Murphy if he has seen the Rock of Gibraltar and asks further what year that would have been and if Murphy remembers the boats that plied the strait. “I’m tired of all them rocks in the sea,” replies Murphy . Bloom’s interest derives from Molly’s connection with Gibraltar, and Molly herself in her monologue remembers (...) the boats well and thinks of missing the boat at Algeciras , just before the book ends with her thoughts of the awful deepdown torrent,” the tide that moves like a river through the strait. Imaginatively she moves with that torrent, figuratively the torrent of time that plunges from the future to the past, which she accepts, with her yes, going deeply with the flow of life, and with her goes Murphy/Joyce, touching on Gibraltar at last. Molly remembers Ulysses S. Grant getting off a boat in Gibraltar, an occurrence that Adams sees as unduly stretching probability merely in order to bring Molly in incidental touch with a man named Ulysses.19 But remembering that Murphy is a “wily old customer,” we may remember also that the Ulysses of Joyce’s favorite Dante cannot rest with Penelope but, in search of knowledge and excellence, moves on through the two rocks of the straits of Gibraltar, the pillars of Hercules, to further adventure and also to destruction; and we may then think that with this reference, Joyce took pains to tell us that the Ulysses of this book here completes in hidden climax the design and purpose of his work, and sails on to oblivion, or rather to dispersion and reconstitution as everyone in the new adventure of Finnegans Wake. 19. See Adams, Surface and Symbol, p. 233. Ralph W. Rader, professor of English at the University of California, Berkeley, is the author of Tennyson’s “Maud”: The Biographical Genesis. He is currently working on a theoretical study of form in the novel and other genres. His previous contributions to Critical Inquiry include “The Literary Theoretical Contribution of Sheldon Sacks” and “The Dramatic Monologue and Related Lyric Forms”. (shrink)

The logic of quantum mechanical propositions—called quantum logic—is constructed on the basis of the operational foundation of logic. Some obvious modifications of the operational method, which come from the incommensurability of the quantum mechanical propositions, lead to the effective quantum logic. It is shown in this paper that in the framework of a calculization of this effective quantum logic the negation of a proposition is uniquely defined (Theorem I), and that a weak form of the quasimodular law can be derived (...) (Theorem II). Taking account of the definiteness of truth values for quantum mechanical propositions, the calculus of full quantum logic can be derived (Theorem III). This calculus represents an orthocomplemented quasimodular lattice which has as a model the lattice of subspaces of Hilbert space. (shrink)

Moral luck is a puzzling aspect of our psychology: Why do we punish outcomes that were not intended (i.e. accidents)? Prevailing psychological accounts of moral luck characterize it as an accident or error, stemming either from a re‐evaluation of the agent's mental state or from negative affect aroused by the bad outcome itself. While these models have strong evidence in their favor, neither can account for the unique influence of accidental outcomes on punishment judgments, compared with other categories of moral (...) judgment. Why might punishment be particularly sensitive to moral luck? We suggest that such sensitivity is easily understood from the broader perspective of punishment's ultimate adaptive goal: Changing others’ behavior by exploiting their capacity to learn. This pedagogical perspective accounts for the exceptional influence of outcomes on punitive sentiments and makes predictions for additional moderators of punishment. We review evidence supporting the pedagogical hypothesis of punishment and discuss fruitful directions for future research. (shrink)

We construct a Hilbert style system RPL for the notion of plausibility measure introduced by Halpern J, and we prove the soundness and completeness with respect to a neighborhood style semantics. Using the language of RPL, we demonstrate that it can define well-studied notions of necessity, conditionals and propositional identity.

Celem niniejszego artykułu jest omówienie niektórych problemów, które występują podczas określania koreferencji w tekście. Analizy takie były prowadzone na potrzeby projektu CORE – Komputerowe metody identyfikacji nawiązań w tekstach polskich. Głównym celem projektu było stworzenie nowatorskich metod i narzędzi informatycznych służących do automatycznego wykrywania anafor i koreferencji w tekstach pisanych w języku polskim. Główny problem z wyznaczaniem faz koreferencji w języku polskim wyłonił się w efekcie kilku czynników. Na poziomie pragmatycznym i semantycznym nie było proste zadecydować czy zachodziła (...) identyczność, czy tylko podobieństwem między dwoma obiektami. Dodatkowym utrudnieniem był brak specjalistycznej wiedzy, który sprawił, że wyznaczenie faz koreferencji było szczególnie trudne między frazami w wyjątkowo specjalistycznych tekstach. Na poziomie gramatycznym, niektóre cechy języka polskiego utrudniły anotację. Ze względu na brak rodzajników określonych i nieokreślonych bardzo trudno było określić, czy nadawca zawsze miał na myśli ten sam obiekt, czy różne obiekty należące do tej samej klasy. Wreszcie, długie zdania bez podmiotu spowodowały pewne problemy przy wyznaczaniu łańcuchów koreferencyjnych między analizowanymi frazami. (shrink)

In this paper, I consider the basis for Kant's praise of Wolff's general logic as "the best we have." I argue that Wolff's logic was highly esteemed by Kant on account of its novel analysis of the three operations of the mind (tres operationes mentis), in the course of which Wolff formulates an argument for the priority of the understanding's activity of judging.

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

What Russell regarded to be the ‘chief outcome’ of his 1914 Lowell Lectures at Harvard can only be fully appreciated, I argue, if one embeds the outcome back into the ‘classificatory problem’ that many at the time were heavily engaged in. The problem focused on the place and relationships between the newly formed or recently professionalized disciplines such as psychology, Erkenntnistheorie, physics, logic and philosophy. The prime metaphor used in discussions about the classificatory problem by British philosophers was a spatial (...) one, with such motifs as ‘standpoints’, ‘place’ and ‘perspectives’ in the space of knowledge. In fact, Russell’s construction of a perspectival space of six-dimensions was meant precisely to be a timely solution to the widely discussed classificatory problem. (shrink)