An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebra with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there corresponds a spectral measure.

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.

A theorem of Sierpinski of 1919 characterized the cardinality of the continuum by means of lines in two orthogonal directions in the plane: CH if and only if there is a subset S of the plane such that every horizontal cross-section of S is countable and every vertical cross-section of S is co-countable. A theorem of Sikorski of 1951 characterizes the cardinality of an arbitrary set by means of hyperplanes in orthogonal directions in finite powers of that (...) set. A theorem of Davies of 1962 characterizes the cardinality of the continuum by means of lines in nonorthogonal directions in the plane, which, by another theorem of Davies of 1962, may be generalized to finite-dimensional Euclidean space. The main results of this paper unify these analogous theorems of Sikorski and Davies by characterizing the cardinality of an arbitrary set by means of hyperplanes in nonorthogonal directions in that set. (shrink)

A theorem on the extendability of certain subsets of a Boolean algebra to ultrafilters which preserve countably many infinite meets (generalizing Rasiowa-Sikorski) is used to pinpoint the mechanism of the Barwise proof in a way which bypasses the set theoretical elaborations.

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation (...) class='Hi'>theorem for complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic. (shrink)

We give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By (...) means of the extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas. (shrink)

In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. To prove the (...) completeness theorem of the L′T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′T logic and the intuitionistic and classical logics. (shrink)

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some (...) familiar results in the theory of (complete) Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold: 1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets. 2. The Boolean prime ideal theorem implies a weakened form of Sikorski's (...) theorem. (shrink)

The Value-free ideal of science (VFI) is a view that claims that scientists should not use non-epistemic values when they are justifying their hypotheses, and is widely considered to be obsolete in the philosophy of science. I will defend the ideal by demonstrating that acceptance of non-epistemic values, prohibited by VFI, necessitates legitimizing certain problematic scientific practices. Such practices, including biased methodological decisions or Questionable Research Practices (QRP), significantly contribute to the Replication Crisis. I will argue that the realizability of (...) VFI is not a necessary condition for its validity. Then, I will show how some of the prominent proposals of value-laden science legitimize problematic scientific practices, provide real-world examples, and generalize the argument. Finally, I will show how value-laden methodological decisions contribute to the Replicability Crisis and discuss two strategies for realizing VFI. (shrink)

The chapter is devoted to the probability and acceptability of indicative conditionals. Focusing on three influential theses, the Equation, Adams’ thesis, and the qualitative version of Adams’ thesis, Sikorski argues that none of them is well supported by the available empirical evidence. In the most controversial case of the Equation, the results of many studies which support it are, at least to some degree, undermined by some recent experimental findings. Sikorski discusses the Ramsey Test, and Lewis’s triviality proof, (...) with special attention dedicated to the popular ways of blocking it. Sikorski concludes that the role of the three theses in future studies of conditionals should be re-thought, and he presents alternative proposals. (shrink)

Within the epistemology of the sciences, conventionalism has been the subject of regular criticism for over six decades. Critics such as W. V. Quine and Morton White, and more recently Nathan Salmon (1992), and Paul Boghossian (1996), have attacked even the most basic tenet of conventionalism, namely its claim that the truth of certain statements is fixed not by stipulation-independent facts, but by the conventions governing the meaning of those statements and their constituents.

The overreliance on verbal models and theories in psychology has been criticized for hindering the development of reliable research programs (Harris, 1976; Yarkoni, 2020). We demonstrate how the conceptual space framework can be used to formalize verbal theories and improve their precision and testability. In the framework, scientific concepts are represented by means of geometric objects. As a case study, we present a formalization of an existing three-dimensional theory of emotion which was developed with a spatial metaphor in mind. Wundt (...) posits that the range of human emotion can be represented along three axes of basic emotions (pleasure/displeasure, excitement/inhibition, tension/relaxation), just as color vision can be represented using three basic colors. We use dimensions to represent basic emotions, points to represent emotional states, and regions to represent broader emotional concepts. We then compare our formalization to an existing structuralist formalization of Wundt’s theory. Further, we discuss the empirical predictions that our formalization generates, such as comparisons of similarity and intensity. We conclude by demonstrating how the tools developed in the conceptual space framework can be used to formulate a theory of emotion based on empirical observation. (shrink)

Alexander George has put forward a novel interpretation of the Quine-Carnap debate over analyticity. George argues that Carnap's claim that there exists an analytic-synthetic distinction was held by Carnap to be empty of empirical consequences. As a result, Carnap understood his position to be empirically indistinguishable from Quine's. Although George defends his interpretation only briefly, I show that it withstands further examination and ought to be accepted. The consequences of accepting it undermine a common understanding of Quine's criticism of Carnap, (...) and I argue that it is difficult to find a perspective from which Quine can criticize Carnap in a non-question-begging way. (shrink)

Replicability is widely regarded as one of the defining features of science and its pursuit is one of the main postulates of meta-research, a discipline emerging in response to the replicability crisis. At the same time, replicability is typically treated with caution by philosophers of science. In this paper, we reassess the value of replicability from an epistemic perspective. We defend the orthodox view, according to which replications are always epistemically useful, against the more prudent view that claims that it (...) is useful in very limited circumstances. Additionally, we argue that we can learn more about the original experiment and the limits of the discovered effect from replications at different levels. We hold that replicability is a crucial feature of experimental results and scientists should continue to strive to secure it. (shrink)

The Minimal Theory of Causation, presented in Graßhoff and May, 2001, aspires to be a version of a regularity analysis of causation able to correctly predict our causal intuitions. In my article, I will argue that it is unsuccessful in this respect. The second aim of the paper will be to defend Hitchcock’s proposal concerning divisions of causal relations against criticism made, in Jakob, 2006 on the basis of the Minimal Theory of Causation.

Indicative conditionals and tendency causal claims are closely related (e.g., Frosch and Byrne, 2012), but despite these connections, they are usually studied separately. A unifying framework could consist in their dependence on probabilistic factors such as high conditional probability and statistical relevance (e.g., Adams, 1975; Eells, 1991; Douven, 2008, 2015). This paper presents a comparative empirical study on differences between judgments on tendency causal claims and indicative conditionals, how these judgments are driven by probabilistic factors, and how these factors differ (...) in their predictive power for both causal and conditional claims. (shrink)

The nature of necessary truth was a central concern of Ludwig Wittgenstein. It was present in his early reflections on logic, a core motif of the Tractatus, and a topic he returned to over and again in his reflections on language, logic, and mathematics. This chapter explores the aspects of Wittgenstein's account of necessity and apriority, beginning with the Tractatus, where many of his core insights received their first expression. It discusses two more contemporary accounts of these topics: conceptual role (...) semantic theories of the a priori, and the arguments for divorcing apriority from necessity. Necessity in the Tractatus appears at two locations. One is at the level of individual elementary propositions. The other is at the level of the inferential connections between propositions. The chapter looks at the Tractatus theory of the proposition. Wittgenstein saw an important difference between the formulations of the rules of grammar and descriptions of states of affairs. (shrink)

In this book about the philosophy of education, Loomis and Rodriguez carefully examine the first principles of theoretic and practical reason necessary for human development and flourishing. Collaborating with the genius of C.S. Lewis, and particularly his brilliant work The Abolition of Man , the authors offer a multi-facetted, interdisciplinary investigation of perennial questions that impact human development and freedom. What is the human being? What are essential criteria for human flourishing? What is the best institutional framework for education? (...) What are the non-naturalistic, normative constraints to the ordering and functioning of a social institution like education? Are there particular institutional environments that threaten moral agency and human freedom? Are there information patterns and practices that substitute one institutional vision of reality for another? Is there a model of education that reflects truer structures of reality? Is there a particular vision embodied in the logic of institutional growth? (shrink)

Children’s movies and animated cartoons today increasingly include homosexual characters, which can be welcomed from an equal-rights perspective. Yet, an intensive public debate has been initiated regarding the (age) appropriateness of such depictions. So far, it is unclear how heterosexual adults react to the presence of gay characters in children’s animated cartoons. Drawing from social identity theory, we conducted an experiment in Germany. Using the Powtoon animation software, we created two versions of a trailer of a fictitious animated cartoon based (...) on two almost identical storylines – one involving a heterosexual couple and the other a lesbian couple. Participants were exposed to only one version. Results of the experiment revealed that adults gave higher age ratings to the version that featured the lesbian couple. Yet, this effect was moderated, with only center- to right-leaning persons being affected. No effects were detected for left-leaning individuals. An exploratory analysis further revealed a moderation effect for individuals who believed in protection myths (i. e., protecting children from exposure to homosexuality), resulting in them giving higher age ratings compared to people who did not believe in protection myths. Implications are discussed, and age-rating measures are proposed for examining reactions to gay characters in communication research. (shrink)

The suggested model is discussed with reference to two clinical populations with memory disorders – patients with misidentification syndromes and those with source memory impairment, both of whom may present with (broadly conceived) déjà vu phenomenon, without insight into false feeling of familiarity. The role of the anterior thalamic nucleus and retrosplenial cortex for autobiographical memory and familiarity is highlighted.

Antisemitism is on the rise. Recently, discussions have considered so-called “no-go zones for Jews” (city areas Jews should avoid to reduce the likelihood of being attacked). In this context and drawing from attribution theory, we examined if news consumers perceive a Jewish hate crime victim as partly responsible for being attacked when news coverage explicitly emphasizes that the victim displayed religious symbols (kippah) in a certain inner-city location. We conducted a quota-based survey experiment (N = 392) in Germany (4 groups, (...) between-subjects design) and randomly exposed participants to news coverage about an anti-Semitic attack. The article either emphasized that the Jewish victim displayed religious symbols (kippah) or not, and highlighted the specific location of the attack (Berlin synagogue vs. deprived inner-city district of Berlin). Moderated mediation analysis suggested that participants perceived the victim’s behavior to be more provocative when the news article highlighted that the victim displayed religious symbols and when the attack occurred in a deprived Berlin district. Yet, effects were only detected for individuals with low (vs. high) levels of education. Perceived provocativeness in turn increased victim blaming indicating that some individuals indirectly regarded the Jewish victim to be partly responsible for being attacked. (shrink)

This article critically examines Saul Kripke's (1972) argument for the separability of necessary truths from truths known a priori, focusing on his criticism of the standard meter case presented by Wittgenstein (1968). It attempts to show that Kripke's argument is unworkable on any of several readings. Wittgenstein's own broadly conventionalist account of necessary truth is then considered in the light of the standard meter example.

The results of forensic science are believed to be reliable, and are widely used in support of verdicts around the world. However, due to the lack of suitable empirical studies, we actually know very little about the reliability of such results. In this paper, I argue that phenomena analogous to the main culprits for the replication crisis in psychology are also present in forensic science. Therefore forensic results are significantly less reliable than is commonly believed. I conclude that in order (...) to obtain reliable estimates for the reliability of forensic results, we need to conduct studies analogous to the large-scale replication projects in psychology. Additionally, I point to some ways for improving the reliability of forensic science, inspired by the reforms proposed in response to the replicability crisis. (shrink)

Wittgenstein's Tractatus carefully distinguished the concept all from\nthe notion of a truth-function, and thereby from the quantifiers.\nI argue that Wittgenstein's rationale for this distinction is lost\nunless propositional functions are understood within the context\nof his picture theory of the proposition. Using a model Tractatus\nlanguage, I show how there are two distinct forms of generality implicit\nin quantified Tractatus propositions. Although the explanation given\nin the Tractatus for this distinction is ultimately flawed, the distinction\nitself is a genuine one, and the forms of generality that (...) Wittgenstein\nindicated can be seen in the quantified sentences of contemporary\nlogic. (shrink)