Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of (...) which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987. (shrink)

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndon's discovery of nonrepresentable relation algebras. This paper introduces the calculus of relations and the (...) theory of relation algebras through a review of these historical developments. (shrink)

There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations.

Conjecture (1) of [Ma83] is confirmed here by the following result: if $3 \leq \alpha < \omega$, then there is a finite relation algebra of dimension α, which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form $S \subseteq T$ (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not (...) provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations). (shrink)

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

Consequence is at the heart of logic, and an account of consequence offers a vital tool in the evaluation of arguments. This text presents what the authors term as 'logical pluralism' arguing that the notion of logical consequence doesn't pin down one deductive consequence relation; it allows for many of them.

This is a supplement to the paper “Finitary Algebraic Logic” [1]. It includes corrections for several errors and some additional results. MSC: 03G15, 03G25.

What happens when someone acts? A familiar answer goes like this. There is something that the agent wants, and there is an action that he believes conducive to its attainment. His desire for the end, and his belief in the action as a means, justify taking the action, and they jointly cause an intention to take it, which in turn causes the corresponding movements of the agent's body. I think that the standard story is flawed in several respects. The flaw (...) that will concern me in this paper is that the story fails to include an agent-or, more precisely, fails to cast the agent in his proper role. (shrink)

For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of (...) algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φ n has a proof with only n variables. To show that φ n has no proof with only n - 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic. (shrink)

Providing an original and systematic treatment of foundational issues in philosophy of religion, J. L. Schellenberg's new book addresses the structure of..

If K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism Lwx is an undecidable theory. A stronger algebraic result shows that the set of logically valid sentences in Lwx forms a hereditarily undecidable theory in Lwx. These results generalize (...) similar theorems, due to Tarski, concerning relation algebras and the formalism Lx. (shrink)

Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order.

The article describes a capstone engineering course for preservice and inservice secondary science and mathematics teachers and shows teacher attitudes toward engineering before and after the course and results of a comparison with a convenience sample. It also gives the results of the attitudes of high school student experimental and control groups toward engineering in a pretest-posttest design. Four findings were made: (a) The science and mathematics preservice and inservice teacher attitudes toward engineering became more favorable after the capstone course; (...) (b) the attitudes of the convenience sample of secondary graduate students were less favorable toward engineering than they were among the treatment group in the capstone course; (c) the attitudes of the experimental group of high school students became significantly more favorable toward engineering after a 3-week unit on engineering; and (d) the attitudes of the experimental group became significantly more favorable toward engineering than did those of the control group after the 3-week unit. (shrink)

Abstract I argue that meaning in life is importantly influenced by bioloical ties. More specifically, I maintain that knowing one's relatives and especially one's parents provides a kind of self-knowledge that is of irreplaceable value in the life-task of identity formation. These claims lead me to the conclusion that it is immoral to create children with the intention that they be alienated from their bioloical relatives?for example, by donor conception.

Retributive theories of punishment argue that punishing a criminal for a crime she committed is sufficient reason for a justified and morally permissible punishment. But what about when the state gets lucky in its decision to punish? I argue that retributive theories of punishment are subject to “Gettier” style cases from epistemology. Such cases demonstrate that the state needs more than to just get lucky, and as these retributive theories of punishment stand, there is no anti-luck condition. I’ll argue that (...) Gettier-style cases demonstrate an impermissible instance of punishment, even though they meet the conditions of retributive theories of punishment. Retributive theories are, therefore, too weak. The safety condition from epistemology provides the anti-luck condition needed for permissible punishment. I argue that two forms of retributivism, rights-forfeiture and what I call standard retributivism, are both subject to Gettier-style cases. Unlike the literature on standards of proof, this paper argues that safety is a condition on punishment itself, i.e., in all nearby possible worlds, the accuser must correctly accuse the convicted for the crime they actually committed. (shrink)

J.L. Schellenberg offers a path to a new kind of religious outlook. Reflection on our early stage in the evolutionary process leads to skepticism about religion, but also offers a new answer to the problem of faith and reason, and the possibility of a new, evolutionary form of religion.

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement (...) calculus, descriptions, and equality. Succeeding chapters explore abstract set theory—with examinations of class membership as well as relations and functions—cardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use. (shrink)

Ultimism and the aims of human immaturity -- Faith without details, or how to practice skeptical religion -- Simple faith and the complexities of tradition -- The structure of faith justification -- How skeptical faith is true to reason -- Anselm's idea -- Leibniz's ambition -- Paley's wonder -- Pascal's wager -- Kant's postulate -- James's will -- Faith is positively justified : the many modes of religious vision.

It's an achievement of the last couple of decades that people who work in linguistic semantics and people who work in the philosophy of language have arrived at a friendly, de facto agreement as to their respective job descriptions. The terms of this agreement are that the semanticists do the work and the philosophers do the worrying. The semanticists try to construct actual theories of meaning (or truth theories, or model theories, or whatever) for one or another kind of expression (...) in one or another natural language; for example, they try to figure out how the temperature could be rising compatibly with the substitutivity of identicals. The philosophers, by contrast, keep an eye on the large, foundational issues, such as: what's the relation between sense and denotation; what's the relation between thought and language; whether translation is determinate; and whether life is like a fountain. Every now and then the philosophers and the semanticists are supposed to get together and compare notes on their respective progress. Or lack thereof. (shrink)

This paper offers an alternate explanation of cases from the doxastic wronging literature. These cases violate what I call the degree of inquiry right—a novel account of zetetic obligations to inquire when interests are at stake. The degree of inquiry right is a moral right against other epistemic agents to inquire to a certain threshold when a belief undermines one’s interests. Thus, the agents are sometimes obligated to leave inquiry open. I argue that we have relevant interests in reputation, relationships, (...) and the well-being of our social groups. These interests generate obligations against others to “do their homework” before closing inquiry. This alternate account makes better sense of puzzles that accounts of doxastic wronging fall prey to. (shrink)

The aim of this paper is to elaborate on the phenomenological approach to expertise as proposed by Dreyfus and Dreyfus and to give an account of the extent to which their approach may contribute to a better understanding of how athletes may use their cognitive capacities during high-level skill execution. Dreyfus and Dreyfus's non-representational view of experience-based expertise implies that, given enough relevant experience, the skill learner, when expert, will respond intuitively to immediate situations with no recourse to deliberate actions (...) or mental representations. The paper will subsequently outline some implications and consequences of such an approach and will also examine to what extent Dreyfus and Dreyfus's skill model is capable to resist different attacks that have been made against their view, and in particular regarding the practical application of their approach to the skill domain of competitive sport. (shrink)

Prologue: Toward a new Enlightenment -- From bounded to relational being -- Bounded being -- In the beginning is the relationship -- The relational self -- The body as relationship : emotion, pleasure and pain -- Relational being in everyday life -- Multi-being and the adventure of everyday life -- Bonds, barricades, and beyond -- Relational being in practice -- Knowledge as co-creation -- Education in a relational key -- Therapy as relational recovery -- Organizing : the precarious balance -- (...) From the moral to the sacred -- Beyond moral pluralism -- All our relations, approaching the sacred -- Epilogue: The coming of relational consciousness. (shrink)

It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RA n and wRRA contains the other.

A common objection to Sosa’s epistemology is that it countenances, in an objectionable way, unsafe knowledge. This objection, under closer inspection, turns out to be in far worse shape than Sosa’s critics have realised. Sosa and his defenders have offered two central response types to the idea that allowing unsafe knowledge is problematic: one response type adverts to the animal/reflective knowledge distinction that is characteristic of bi-level virtue epistemology. The other less-discussed response type appeals to the threat of dream scepticism, (...) and in particular, to the idea that many of our everyday perceptual beliefs are unsafe through the nearness of the dream possibility. The latter dreaming response to the safety objection to Sosa’s virtue epistemology has largely flown under the radar in contemporary discussions of safety and knowledge. We think that, suitably articulated in view of research in the philosophy and science of dreaming, it has much more going for it than has been appreciated. This paper further develops, beyond what Sosa does himself, the dreaming argument in response to those who think safety (as traditionally understood) is a condition on knowledge and who object to Sosa’s account on the grounds that it fails this condition. The payoffs of further developing this argument will be not only a better understanding of the importance of insights about dreaming against safety as a condition on knowledge, but also some reason to think a weaker safety condition, one that is relativised to SSS (i.e., skill/shape/situation) conditions for competence exercise, gets better results all things considered as an anti-luck codicil on knowledge. (shrink)

The phenomenon of ambivalence is an important one for any philosophy of action. Despite this importance, there is a lack of a fully satisfactory analysis of the phenomenon. Although many contemporary philosophers recognize the phenomenon, and address topics related to it, only Harry Frankfurt has given the phenomenon full treatment in the context of action theory – providing an analysis of how it relates to the structure and freedom of the will. In this paper, I develop objections to Frankfurt's account, (...) all revolving around the charge that his account contains a serious ambiguity between willing and identifying. With such objections in place, I then develop an analysis that avoids the difficulties and ambiguities that Frankfurt's analysis is prey to. I briefly distinguish ambivalence from other types of internal conflict. This paper aims to offer conceptual clarification on the phenomenon of ambivalence, which will then allow for discussions about the normative merits and demerits of ambivalence, the effects of ambivalence on autonomous action, and methods of resolution of ambivalence. (shrink)