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.

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)

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)

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)

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

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.

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)

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)

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.

In lieu of an abstract, here is a brief excerpt of the content:R. D. Ackerman BELIEVING IN A FICTION: WALLACE STEVENS AT THE LIMITS OF PHENOMENOLOGY The "ring of men" of "Sunday Morning" will chant their "devotion to the sun, / Not as a god, but as a god might be, / Naked among them, like a savage source" (CP, pp. 69-70).' Solar nakedness is deferred even as it is named. The problem for belief is the question of appearance and (...) representation. Things appear as and like.... The phenomenological return to things is also a turn to fiction, as Husserl understood: "If anyone loves a paradox, he can really say... that the element which makes up the life of phenomenology... is fiction. ' " The sun is divine, and it is not; it is both naked and clothed, savage and civilized, source (or end) and means. But such paradox is ultimately unsatisfying, and as we will see with Stevens, the need for belief—for an image or idea of totality—forces the issue of fiction and reality. The Stevens we will have in view is not the Stevens—as if there were such a creature. Our Stevens is the one most persuaded by promises such as those of phenomenology, the one most concerned with decreation (reduction) and fresh beginnings, with discovery and recognition, with imagination, the body, and the earth. This is the Stevens for whom the barrier of language presages at last not absence and difference but fictive recovery of presence and identity, the one for whom poetry represents a way of belief beyond mysticism even as it bodes mythic proximity. Finally, this is the Stevens who predominates throughout the prose (as well as in much of the poetry), who tries mightily to secure a belief while disencumbering himself of metaphysics—but alas, I think, with small success. Although Stevens's association of the problem of belief with the idea of fiction occurs in early poems such as "A High-Toned Old Christian 79 80Philosophy and Literature Woman" and "To the One of Fictive Music," his most intensive involvement with fiction as belief dates from the early 1940s and spans the period of his major essays and his last four volumes of poems. He recounts, for instance, a 1942 conversation with a student: "I said that I thought that we had reached a point at which we could no longer really believe in anything unless we recognized that it was a fiction. The student said that that was an impossibility, that there was no such thing as believing in something that one knew was not true. It is obvious, however, that we are doing that all the time" (L, p. 430). What does it mean to believe in a fiction? That is my main subject here, as we seek out the limits of the phenomenological Stevens. His letters of this period are preoccupied with the problem of fictive belief: "If one no longer believes in God (as truth), it is not possible merely to disbelieve; it becomes necessary to believe in something else.... A good deal of my poetry recently has concerned an identity for that thing.... In one of the short poems that I have just sent to the harvard advocate, I say that one's final belief must be in a fiction. I think that the history of belief will show that it has always been in a fiction" (L, p. 370). The poem referred to, "Asides on the Oboe," begins: The prologues are over. It is a question now, Of final belief. So, say that final belief Must be in a fiction. It is time to choose. (CP, p. 250) Acknowledging that "in the long run, poetry would be the supreme fiction" (L, pp. 430-31), Stevens nonetheless singles out the nominative function of poetry as the essence of its claim to supremacy: "Poetry means not the language of poetry but the thing itself.... The subject matter is what comes to mind when one says of the month of August... 'Thou art not August, unless I make thee so'" (L, p. 377). The quoted line (also from "Asides") underscores the paradox that for Stevens the essence of fiction (the nontrue) is its capacity to nominate the true or the real. This... (shrink)

Initial responses to questionnaires used to assess participants' understanding of informed consent for malaria vaccine trials conducted in the United States and Mali were tallied. Total scores were analyzed by age, sex, literacy (if known), and location. Ninety-two percent (92%) of answers by United States participants and 85% of answers by Malian participants were correct. Questions more likely to be answered incorrectly in Mali related to risk, and to the type of vaccine. For adult participants, independent predictors of higher scores (...) were younger age and female sex in the United States, and male sex in Mali. Scores in the United States were higher than in Mali (P = 0.005). Despite this difference participants at both sites were well informed overall. Although interpretation must be qualified because questionnaires were not intended as research tools and were not standardized among sites, these results do not support concerns about systematic low understanding among research participants in developing versus developed countries. (shrink)

Hicks' edition of the De Anima contains valuable commentary from Hicks as well as useful summaries of the views of earlier commentators.

Requests by patients or their families for treatment which the patient's physician considers to be "inappropriate" are becoming more frequent than refusals of treatment which the physician considers appropriate. Such requests are often based on the patient's religious beliefs about the attributes of God (sovereignty, omnipotence), the attributes of persons (sanctity of life), or the individual's personal relationship with God (communication, commands, etc). We present four such cases and discuss some of the basic religious tenets of the three Abrahamic faith (...) traditions as they relate to such requests. We suggest that religious reasons for requesting "inappropriate" treatment are "special" and deserve serious consideration. We offer guidance to assist clinicians and clinical ethicists as they attempt to resolve these conflicts, emphasising the importance of understanding the religious beliefs of the patient/surrogate and suggesting the assistance of a religious interpreter. We suggest open discussion with patients and families of both the clinical situation and the theological basis for these requests. We also suggest that clinicians use additional religious doctrines or principles from patients' own traditions to balance the reasons behind the requests. We conclude that most persistent requests for "inappropriate" treatment should be honoured. (shrink)

Originally published in 1907, this book contains the ancient Greek text of Aristotle's De Anima, his treatise on the differing souls of living things. An English translation is provided on each facing page, and Hicks supplies a very detailed commentary on each line at the end of the book, as well as a summary of each section. This book will be of value to anyone with an interest in Greek philosophy and the history of classical scholarship.

This work is available on its own or as part of the 7 volume set Selected Works of R. D. Laing.

Open peer commentary on the article “Exploring the Depth of Dream Experience: The Enactive Framework and Methods for Neurophenomenological Research” by Elizaveta Solomonova & Xin Wei Sha. Upshot: A neurophenomenological approach to the enactive account of consciousness in general is supported by an account of how the brain functions in creating imagery of non-present objects and situations. Three types of non-sensory imagery are needed to ground our consciousness of sensory imagery: proprioceptive imagery, motor imagery, and what Eugene Gendlin calls the (...) “felt sense” of a situation. Dreams show clearly how we image situations without sensory input, a process that is clearly enactive rather than reactive. This enactive account of imagery then supports Gendlin’s method of interpreting dreams by comparing their “felt sense” to the felt sense of waking situations. (shrink)

We show there is only one consistent way to update a probability assignment, that given by Bayes's rule. The price of inconsistent updating is a loss of efficiency. The implications of this for the problem of induction are discussed.