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)

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)

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.

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.

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)

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.

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.

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)

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)

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.

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.

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)

This volume is not restricted to papers presented at the 1988 Colloquium, but instead aims to provide the reader with a (relatively) coherent reading on Algebraic Logic, with an emphasis on current research. To help the non-specialist reader, the book contains an introduction to cylindric and relation algebras by Roger D. Maddux and an introduction to Boolean Algebras by Bjarni Joacute;nsson.

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research (...) on increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Fichte’s Doctrine of Science of 1811 offers a sound model for explaining the conditions of semantics in its connection with the idea of freedom. Following Wolfram Hogrebe’s suggestion that the principle of contradiction works as an archaeological semantic postulate, i.e., is the implicit condition for any sentence to be meaningful, we argue that in Fichte’s definition of the phenomenon we find such a semantic postulate at a higher genetic level than the principle of contradiction indicated by Hogrebe. Moreover, the Doctrine (...) of Science of 1811 offers an analysis of the steps through which this phenomenon displays itself – i.e. Vermögen, als, Soll – which can be understood as the conditions for establishing a meaningful sentence. Fichtes Wissenschaftslehre 1811 kann uns einen soliden Modell zur Erklärung der Entstheung von Semantik anbieten. Mein Beitrag fangt mit der Darstellung Wolfram Hogrebes Hypothese an, dass der Satz vom Widerspruch als archäologisches Bedeutungspostulat fungiert, d.h. als implizite Bedingung der Möglichkeit jeglicher bedeutenden Äußerung. Ausgehend dieser Idee Hogrebes, hier schlage ich vor, dass man im Fichtes Begriffe der Erscheinung ein genetisch höheres archäologisches Bedeutungspostulat als der Satz vom Widerspruch finden kann. Daraufhin bietet die Wissenschaftslehre 1811 einen Einsicht in alle Schritte, wodurch die Erscheinung sich entfaltet – Vermögen, Durch, Als, Soll – die wir als die entsprechende Bedingungen für die Herstellung eines bedeutungsvollen Satzes verstehen können. (shrink)

Contemporary Continental Philosophy steps back from current debates comparing Continental and analytic philosophy and carefully, yet critically outlines the tradition’s main philosophical views on epistemology and ontology. Forgoing obscure paraphrases, D’Amico provides a detailed, clear account and assessment of the tradition from its founding by Husserl and Heidegger to its challenge by Derrida and Foucault. Though intended as a survey of this tradition throughout the twentieth century, this study’s focus is on the philosophical problems which gave it birth and even (...) now continue to shape it.The book reexamines Husserl as an early critic of epistemological naturalism whose grasp of the philosophical importance of the theory of meaning was largely ignored. Heidegger’s contrasting effort to revive ontology is examined in terms of his distinction between ontic and ontological questions. In contrast with many earlier studies, the author outlines confusions engendered by the misappropriation of the distinct philosophical agendas of Husserl and Heidegger by such famous figures as Sartre and Merleau-Ponty. The book is also original in its emphasis on how social externalism in epistemology, inspired by Karl Mannheim, influenced this tradition’s structuralist and Marxist phases. The philosophical defenses of a theory of interpretation by Gadamer and Habermas are closely examined and assessed and the study concludes with a a probing yet balanced account of Foucault and Derrida as critics of philosophical autonomy. The book concludes by reassessing this century-long divide between the analytic and Continental traditions and its implication for the future of philosophy. (shrink)

Writing On the Sublime and Beautiful, Edmund Burke tells us the ideas most capable of making an impression are those related to self-preservation and society. Such ideas are bound to our passions. Passions belonging to self-preservation turn on pain or danger.1 Those belonging to society do as well, although in this work, Burke dwells exclusively on pain. Because he tells us the king of terrors is death, we might infer pain is inferior to danger, the latter more formidable. We experience (...) delight in both when they are remote and do not affect us. But this delight, or indeed pleasure of any kind, is less powerful in terms of passion. Possibly for this reason Burke declines to qualify delight or pleasure as a source... (shrink)

In his article , 97–105) R. Reneham rightly classes Sail. Cat.20.9 as a conscious imitation of Cic.Cat.1.1, but adopts the unsatisfactory explanation of parody. Such parody is, as he notes, without parallel in Sallust and ineptly distracts attention from the vigorous development of Catiline's rhetoric. Elsewhere mimesis is regularly a compliment to the author imitated, often closely functional by reinforcing a point from the parallel of a similar context . Similarly I suggest that here Sallust recalls Cicero's words to illustrate (...) that perversion of vocabulary which is the keynote of Catiline's speech: just as he misuses, for example, the terms virtus fidesque at the beginning of his speech, in stark contrast to Sallust's own definition, so he perverts the famous words of the attack which revealed his true villainy in similar savage indignatio. (shrink)