TRACTARIAN FIRST-ORDER LOGIC: IDENTITY AND THE N-OPERATOR

Recently, the design and development of nanozyme-based logic gates have received much attention. In this work, by engineering the stability of the nanozyme-catalyzed product, we demonstrated that the chromogenic system of 3, 3′, 5, 5′-tetramethylbenzidine (TMB) can act as a visual output signal for constructing various Boolean logic operations. Specifically, cerium oxide or ferroferric oxide-based nanozymes can catalyze the oxidation of colorless TMB to a blue color product (oxTMB). The blue-colored solution of oxTMB could become colorless by some reductants, including the reduced transition state of glucose oxidase and xanthine oxidase. As a result, by combining biocatalytic reactions, the color change of oxTMB could be controlled logically. In our logic systems, glucose oxidase, β-galactosidase, and xanthine oxidase acted as inputs, and the state of oxTMB solution was used as an output. The logic operation produced a colored solution as the readout signal, which was easily distinguished with the naked eye. More importantly, the study of such a decolorization process allows the transformation of previously designed AND and OR logic gates into NAND and NOR gates. We propose that this work may push forward the design of novel nanozyme-based biological gates and help us further understand complex physiological pathways in living systems.

In this paper, we share our experiences concerning the introduction of the Android-based self-study app AXolotl within the first-semester logic course offered at our university. This course is mandatory for students majoring in Computer Science and Artificial Intelligence. AXolotl was used as part of an optional lab assignment bridging clausal reasoning and SAT solving with classical reasoning, proof construction, and first-order logic. The app provides an intuitive interface for proof construction in various logical calculi and aids the students through rule application. The goal of the lab assignment was to help students make a smoother transition from clausal and decompositional reasoning used earlier in the course to inferential and contextual reasoning required for proof construction and first-order logic. We observed that the lab had a positive influence on students' understanding and end the paper with a discussion of these results.

We formalize in Isabelle/HOL soundness and completeness of a one-sided sequent calculus for first-order logic. The completeness is shown via a translation from a semantic tableau calculus, whose completeness proof we base on the theory entry ‘First-Order Logic According to Fitting’ by Berghofer in the Archive of Formal Proofs. The calculi and proof techniques are taken from Ben-Ari’s textbook Mathematical Logic for Computer Science (Springer, 2012). We thereby demonstrate that Berghofer’s approach works not only for natural deduction but also constitutes a framework for mechanically checked completeness proofs for a range of proof systems.

Analytical approach of developing the expression of output of all-optical frequency encoded different logical units and a way-out to implement the logic gates

This review covers the content of reports and discussions at the 12th Kant Readings Conference held in April 2019 and organised by the research unit of the Academia Kantiana of the Immanuel Kant Baltic Federal University, Kaliningrad. Traditionally, Kant Readings have been thematically universal, embracing all the areas of Kant’s legacy. This time the conference focused on practical philosophy, i.e. the historical grounds and modern significance of Kant’s ethical thought as compared to other philosophical projects of the Enlightenment era. Due attention was paid to the reception of the ethics of Kant and the Enlightenment by philosophers in Russia and the West. Breakout groups discussed aspects of interconnection between the Enlightenment ethics and esthetics as well as interdisciplinary problems at the interface of philosophy of politics and philosophy of education, including ways to counter various forms of intellectual enslavement. The possibilities of applying Kant’s ethical principles were discussed in close connection with the interpretation of the latest trends in the development of science and technology. It was noted that the intellectual and social communication environment of today has much in common with that of the Enlightenment era, which makes the philosophical strategy proposed during that era amenable to adaptation and development.

The paper overviews various methods that are used for the analysis of output signals generated by enzyme-based logic systems. The considered methods include optical techniques (optical absorbance, fluorescence spectroscopy, surface plasmon resonance), electrochemical techniques (cyclic voltammetry, potentiometry, impedance spectroscopy, conductivity measurements, use of field effect transistor devices, pH measurements), and various mechanoelectronic methods (using atomic force microscope, quartz crystal microbalance). Although each of the methods is well known for various bioanalytical applications, their use in combination with the biomolecular logic systems is rather new and sometimes not trivial. Many of the discussed methods have been combined with the use of signal-responsive materials to transduce and amplify biomolecular signals generated by the logic operations. Interfacing of biocomputing logic systems with electronics and "smart" signal-responsive materials allows logic operations be extended to actuation functions; for example, stimulating molecular release and switchable features of bioelectronic devices, such as biofuel cells. The purpose of this review article is to emphasize the broad variability of the bioanalytical systems applied for signal transduction in biocomputing processes. All bioanalytical systems discussed in the article are exemplified with specific logic gates and multi-gate networks realized with enzyme-based biocatalytic cascades.

Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences.

In this paper, we investigate an extended first-order Belnap-Dunn logic with classical negation. We introduce a Gentzen-type sequent calculus FBD+ for this logic and prove theorems for syntactically and semantically embedding FBD+ into a Gentzen-type sequent calculus for first-order classical logic. Moreover, we show the cut-elimination theorem for FBD+ and prove the completeness theorems with respect to both valuation and many-valued semantics for FBD+.

Sören Halldén. On the logic of ‘better.’ Library of Theoria, no. 2. C. W. K. Gleerup, Lund, and Ejnar Munksgaard, Copenhagen, 1957, 112 pp. - Lennart Åqvist. Deontic logic based on a logic of ‘better.’Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23–26 August, 1962, Acta philosophica Fennica, no. 16, Helsinki1963, pp. 285–290. - Mark Fisher. A three-valued calculus for deontic logic. Theoria (Lund), vol. 27 (1961), pp. 107–118. - Lennart Åqvist. Postulate sets and decision procedures for some systems of deontic logic. Theoria (Lund), vol. 29 (1963), pp. 154–175. - Volume 31 Issue 2

In this paper we show how to extend the standard cut-elimination procedure from first-order intuitionistic stable logic to a class of intuitionistic stable theories. Building on previous works by Negri and von Plato, we aptly modify the underlying calculus for first-order intuitionistic logic so as to preserve the admissibility of all the structural rules, including cut, in the presence of a restricted version of the rule of classical reductio ad absurdum and of a special case of universal rules.

For a reasonable sound and complete proof calculus for firstorder logic consider the problem to decide, given a sentence ϕ of firstorder logic and a natural number n, whether ϕ has no proof of length ≤ n. We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT of tautologies of propositional logic. This equivalence is an instance of a general result linking the complexity of so-called slicewise monotone parameterized problems with the existence of an optimal proof system for TAUT.

Concerning the three fundamental first-order fuzzy logics, the set of logically valid formulae is \(\varPi _2\)-complete for Łukasiewicz logic, \(\varPi _2\)-hard for Product logic, and \(\varSigma _1\)-complete for Godel logic, as with classical first-order logic. Among these fuzzy logics, only Godel logic is recursively axiomatisable. Hence, it was necessary to provide a hyperresolution-based proof method suitable for automated deduction, as one has done for classical logic. As another step, we can incorporate a countable set of intermediate truth constants of the form \(\bar{c}\), \(c\in (0,1)\), together with the equality, Open image in new window, strict order, Open image in new window, projection, Open image in new window, operators in Godel logic; and modify the hyperresolution calculus, inferring over so-called order clauses. We shall investigate the deduction problem of a formula from a countable theory in this expansion and the so-called canonical standard completeness, where the semantics of Godel logic is given by the standard \(\varvec{G}\)-algebra as well as truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. We get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in this expansion of Godel logic.

Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work, it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as first-order logic, the λ-calculus or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitely supported permutation action); previous work proved soundness and completeness. The HSP theorem characterizes the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to so-called freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this article, we give the constructions which show that, after all, a ‘nominal’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products and an atoms-abstraction construction specific to nominal-style semantics.

In recent decades, plural logic has established itself as a well-respected member of the extensions of first-order classical logic. In the present paper, I draw attention to the fact that among the examples that are commonly given in order to motivate the need for this new logical system, there are some in which the elements of the plurality in question are internally singularized (e.g. ‘Whitehead and Russell wrote Principia Mathematica’), while in others they are not (e.g. ‘Some philosophers wrote Principia Mathematica’). Then, building on previous work, I point to a subsystem of plural logic in which inferences concerning examples of the first type can be adequately dealt with. I notice that such a subsystem (here called ‘discrete plural logic’) is in reality a mere variant of first-order logic as standardly formulated, and highlight the fact that it is axiomatizable while full plural logic is not. Finally, I urge that greater attention be paid to discrete plural logic and that discrete plurals are not used in order to motivate the introduction of full-fledged plural logic—or, at least, not without remarking that they can also be adequately dealt with in a considerably simpler system.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.