An introduction to logic from the perspective of algebra.

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let <A> be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to <A> and <antiA>, and one corresponding to neutral (indeterminate) <neutA> (also denoted <neutroA>) between the opposites}, which may (...) or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation). (shrink)

In this article, I examine the relationship between physical quantities and physical states in quantum theories. I argue against the claim made by Arageorgis that the approach to interpreting quantum theories known as Algebraic Imperialism allows for “too many states.” I prove a result establishing that the Algebraic Imperialist has very general resources that she can employ to change her abstract algebra of quantities in order to rule out unphysical states.

This main aim of this paper is to prove that in a direct commutative BCK-algebra an ideal I is nitely generated if and only if I is a principal ideal. This result generalizes the result obtained by E. Y. Deeba in [2]. We also give an answer to the question posed by E. Y. Deeba in [1]: for what class of BCK-algebras is every Noetherian algebra a principal ideal algebra ?

Mary Everest, Boole's wife, claimed after the death of her husband that his logic had a psychological, pedagogical, and religious origin and aim rather than the mathematico-logical ones assigned to it by critics and scientists. It is the purpose of this paper to examine the validity of such a claim. The first section consists of an exposition of the claim without discussing its truthfulness; the discussion is left for the sections 2?4, in which some arguments provided by the examination of (...) the inner consistency of Mary Everest's writings, Boole's own writings, and other sources, lead to the conclusion that there are sound reasons to accept Mary Everest's viewpoint. (shrink)

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on (...) the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)

Name der Zeitschrift: Semiotica Jahrgang: 2016 Heft: 208 Seiten: 133-154.

We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

Some varieties that are extensions of relational algebras with two constants that play the role of projections are studied. The classes have as a subvariety the abstract fork algebra equivalent variety involving projections. They are obtained by weakening some laws valid in AFA. Some applications of the varieties in the literature and in the specification of abstract data types are exhibited. For each of the classes obtained, an answer is given to the question: 'Is the relational reduct of the class (...) representable?'. For the subvarieties formed with the models that have a representable relational reduct, a representation theorem is proved. For them the finitization problem is studied. Next the varieties presented are compared by means of the inclusion order. For each class the problem of characterizing finite models is considered. Simple models in the varieties are studied. Finally the existence of equivalent classes with a binary operation like fork is studied. (shrink)

We prove completeness of the propositional modal logic S 4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and . Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, , and (...) we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in with an open representative. We prove completeness of the modal logic S 4 for the algebra . A corollary to the main result is that non-theorems of S 4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in. (shrink)

It is shown in this article that the two sides of an equation in the medieval Arabic algebra are aggregations of the algebraic “numbers” (powers) with no operations present. Unlike an expression such as our 3x + 4, the Arabic polynomial “three things and four dirhams” is merely a collection of seven objects of two different types. Ideally, the two sides of an equation were polynomials so the Arabic algebraists preferred to work out all operations of the enunciation to a (...) problem before stating an equation. Some difficult problems which involve square roots and divisions cannot be handled nicely by this basic method, so we do find square roots of polynomials and expressions of the form “A divided by B” in some equations. But rather than initiate a reconsideration of the notion of equation, these developments were used only for particularly complex problems. Also, the algebraic notation practiced in the Maghreb in the later middle ages was developed with the “aggregations” interpretation in mind, so it had no noticeable impact on the concept of polynomial. Arabic algebraists continued to solve problems by working operations before setting up an equation to the end of the medieval period. (shrink)

In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of Bellantoni–Cook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we develop a semantics of SLR using BCK -algebras consisting of certain polynomial-time algorithms. It will follow from this semantics that safe recursion with arbitrary result type built up from N and (...) ⊸ as well as recursion over trees and other data structures remains within polynomial time. In its original formulation SLR supported only natural numbers and recursion on notation with first-order functional result type. (shrink)

Schwinger’s algebra of microscopic measurement, with the associated complex field of transformation functions, is shown to provide the foundation for a discrete quantum phase space of known type, equipped with a Wigner function and a star product. Discrete position and momentum variables label points in the phase space, each taking \(N\) distinct values, where \(N\) is any chosen prime number. Because of the direct physical interpretation of the measurement symbols, the phase space structure is thereby related to definite experimental configurations.

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest (...) arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

ArgumentFrom the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply$\sqrt {18} $byx, for instance, the result was expressed as the rhetorical equivalent of$\sqrt {18{x^2}} $. The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or “number,” of a term was thought of as how many of that term are present, and not as the scalar (...) multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff was the first to admit them in special cases, and subsequently they appear more liberally in Cardano, Scheubel, Bombelli, and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took place entirely in the conceptual context of premodern, “cossic” algebra, and persisted in the sixteenth century independent of the development of the new algebra of Viète, Decartes, and Fermat. This was a formal innovation violating prevailing concepts that we propose could only be introduced because of the growing autonomy of notation from rhetorical text. (shrink)

The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum σ- and γ-matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it (...) is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed. (shrink)

We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as (...) a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. (shrink)

The algebra of sets has, basically, two different types of symbols. One type of symbol (∩, ?, +, ?) defines another set from two other sets. A second type of symbol (?, ?, =, ?) makes a proposition about two sets. When the construction of these two types of symbols is based on the same four-dot matrix as the logic symbols described in a previous paper, the three symbol types then dovetail together into a harmonious whole that greatly simplifies derivation (...) in the algebra of sets. (shrink)

De Morgan's Formal Logic, which was published on virtually the same day in 1847 as Boole's The Mathematical Analysis of Logic, contains a logic of complex terms (LCT) which has been sadly neglected. It is surprising to find that LCT contains almost a full theory of Boolean algebra. This paper will: (1) provide some background to LCT; (2) outline its main features; (3) point out some gaps in it; (4) compare it with Boole's algebra; (5) show that it is a (...) lattice-theoretical formulation of Boolean algebra; (6) discuss some issues of historical priority; and (7) conclude with the puzzle of LCT's lack of influence. (shrink)

We present the history of the conference Applications of Algebra in Logic and Computer Science, whose twenty-third edition will be held in March, 2019. At the end we outline some plans for the future.

This paper argues that Noether's axiomatic method in algebra cannot be assimilated to Weyl's late view on axiomatics, for his acquiescence to a phenomenological epistemology of correctness led Weyl to resist Noether's principle of detachment.

It is well known that, in Peano arithmetic, there exists a formula Theor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Theor (˹p˺) ≡ p implies p is a theorem ∼Theor (˹p˺) ≡ p implies p is provably equivalent to Theor (˹0 = 1˺). Therefore, the considered "equations" admit, up to provable equivalence, only one solution. In this paper we prove (Corollary 1) that, in general, if P (x) is an arbitrary formula built (...) from Theor (x), then the fixed-point of P (x) (which exists by the diagonalization lemma) is unique up to provable equivalence. This result is settled referring to the concept of diagonalizable algebra (see Introduction). (shrink)