Formal Logic

This work describes an algorithm of automata construction for LTL safety properties, suitable for bounded model checking. Existing automata construction methods are tailored to BDD-based symbolic model checking. The novelty of our approach is that we construct deterministic automata, unlike the standard approach, which constructs nondeterministic automata. We show that the proposed method has significant advantages for bounded model checking over traditional methods.

The Protein Ontology (PRO) provides a formal, logically-based classification of specific protein classes including structured representations of protein isoforms, variants and modified forms. Initially focused on proteins found in human, mouse and Escherichia coli, PRO now includes representations of protein complexes. The PRO Consortium works in concert with the developers of other biomedical ontologies and protein knowledge bases to provide the ability to formally organize and integrate representations of precise protein ...

This paper presents an information retrieval methodology which uses Formal Concept Analysis in conjunction with semantics to provide contextual answers to users' queries. User formulates a query on a set of heterogeneous data sources. This set is semantically unified by the proposed notion conceptual context. A context can be global: it defines a semantic space the user can query -or instantaneous-it defines the current position of the user in the semantic space. Our methodology consists first in a pre-treatment providing the global conceptual context and then in an online contextual processing of users' requests, associated to an instantaneous context. This methodology can be applied to heterogeneous data sources such as web pages, databases, email, personal documents and images, etc. One interest of our approach is to perform a more relevant and refined information retrieval and contextual navigation, closer to the users' expectation.

Abstract Programmable Logic Controllers (PLC) have been used in industrial environment to control several different tasks. Such devices can be programmed using one of the languages specified in the IEC 61131–3 standard, such as Ladder or Functional Logic ...

We present a logical framework for modeling and reasoning about the evolution of requirements. We demonstrate how a sufficiently rich meta level logic can formally capture intuitive aspects of managing changes to requirements models, while maintaining completeness and consistency. We consider a theory as the deductive closure of a given set of axioms and conclude that software engineering is concerned, in essence, with, building and managing large theories. This theory construction commences with the development of the requirements model which we view as a theory of some nonmonotonic logic. Requirements evolution then involves the mapping of one such theory to another. Exploiting the deductive power of the theory of belief revision and nonmonotonic reasoning we develop a formal description of this mapping, as well as the requirements engineering process itself. This work thus offers a rigorous approach to reasoning about requirements evolution and a important focus for defining semantically well founded methods and tools for the effective management of changing requirements

Praca stanowi pomoc dydaktyczną, w szczególności dla studentów kierunków humanistycznych i społecznych na początku kursu logiki oraz dla uczniów stykających się z jej podstawami. Ma charakter rozszerzonego o liczne polecenia dodatkowe zbioru zadań związanych z podstawami systemu klasycznego rachunku zdań z alfabetem ośmiofunktorowym. W zbiorze uwzględniono przykładowe rozwiązania wszystkich zadań.

The Web Service Modeling Language (WSML) is a language for the specification of different aspects of Semantic Web Services. It provides a formal language for the Web Service Modeling Ontology WSMO which is based on well-known logical formalisms, specifying one coherent language framework for the description of Semantic Web Services, starting from the intersection of Datalog and the Description Logic SHIQ. This core language is extended in the directions of Description Logics and Logic Programming in a principled manner with strict layering. WSML distinguishes between conceptual and logical modeling in order to facilitate users who are not familiar with formal logic, while not restricting the expressive power of the language for the expert user. IRIs play a central role in WSML as identifiers. Furthermore, WSML defines XML and RDF serializations for inter-operation over the Semantic Web.

In “A Defense of Presentism,” Ned Markosian addresses the problem of singular propositions about non-present objects. The proposed solution uses a paraphrasing strategy that differentiates between two kinds of meaning in declarative sentences, and also distinguishes between two truth-conditions for singular propositions. The solution, however, is unsatisfactory. I demonstrate that both truth-conditions suffer from the same problems in spite of the examples used to support the claim that one is a proper treatment for singular propositions. Part of the difficulty is in the limited expressivity of logical formalisms, a of time, but one which calls for greater attention.

Praca stanowi pomoc dydaktyczną, w szczególności dla studentów WPiA UJ. Ma charakter rozszerzonego o drobne uwagi zbioru zadań dotyczących sprawdzania tautologiczności formuł na gruncie systemu klasycznego rachunku zdań z alfabetem ośmiofunktorowym. W zbiorze uwzględniono przykładowe rozwiązania wszystkich zadań.
Nie jest to wersja ostateczna zbioru - w przyszłości będą się pojawiać znaczące jego uzupełnienia.

This article is a fragment of scientific research devoted to the analysis of modern approaches to the understanding of architecture. The author opposes the productivity of the analytical approach to the architectural theory and the methodological inconsistency of the declarative approach. In this regard, the problem of demarcation of strict analytical (rational) and publicistic (declarative) formats seems to be relevant.

Статья представляет собой фрагмент научного исследования, посвященного анализу современных подходов к теоретическому осмыслению феноменологии архитектуры. Противопоставляются содержательность и продуктивность аналитического подхода к
архитектурной тематике и методологическая несостоятельность подхода декларативного (программного). В этой связи представляется актуальной проблематика демаркации
строгого аналитического (общерационального) и публицистического (декларативного) форматов.

Для цитирования: Фигурный Г.Н. Парадигматика современной архитектуры: аналитика и декларации // Architecture and Modern Information Technologies. – 2019. – №2(47). – С. 71-82 [Электронный ресурс]. – Режим доступа: https://marhi.ru/AMIT/2019/2kvart19/PDF/04_figurnyj.pdf

In this paper we present a framework for software architecture veri cation using machine-assisted formal proof. Our approach is based on the translation of an existing architecture description language (ADL) based speci cation to an alternate mathematical representation. We use higher-order logic as mechanized by the Prototype Veri cation System (PVS) as the formal basis of our framework. Our approach is not tied to any particular ADL. Rather, we de ne an ADL-independent model of architecture description which formalizes the fundamental design concepts of architecture modeling notations. A key feature of our framework is its exibility; the architect can choose the design concepts that are modeled. Moreover, since the model is generic to many ADLs, it allows veri cation of systems that are speci ed using more than one notation. We introduce our model of architecture description, and illustrate the utility of our approach by verifying properties of an example architecture, a simple compiler speci ed in a pipe-and-lter architectural style.

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

A new IEEE compliant floating-point rounding algorithm for computing the rounded product from a carry-save representation of the product is presented. The new rounding algorithm is compared with the rounding algorithms of Yu and Zyner and of Quach et al. . For each rounding algorithm, a logical description and a block diagram is given and the latency is analyzed.

This book exposes the representation, developed by the author, of the logical operators using matrix algebra and describes some applications. After an elementary introduction to logical calculus and to matrix algebra, the book describes how neurocomputational models inspired this matrix-vector formalism. Then, the book presents the basic matrix logical operators, the generation of many-valued logics, the variety of modal logics produced by this formalism, the differential and integral calculus of logic operations and, inspired in the properties of reversible logic Fredkin gate, the book ends showing some possible signatures of dynamical complexity in elementary cellular automata.

Charles L. Dodgson's reputation as a significant figure in nineteenth-century logic was firmly established when the philosopher and historian of philosophy William Warren Bartley, III published Dodgson's 'lost' book of logic, Part II of Symbolic Logic, in 1977. Bartley's commentary and annotations confirm that Dodgson was a superb technical innovator. In this paper, I closely examine Dodgson's methods and their evolution in the two parts of Symbolic Logic to clarify and justify Bartley's claims. Then, using more recent publications and unpublished letters, I argue that Dodgson approached the elimination problem in class logic differently than his contemporaries, and in doing so, anticipated several important concepts and techniques in automated deductive reasoning. These materials also provide additional insight into his reasons for writing this book.

My book, "Deflationism & Semantic Theories of Truth",  is available as a paperback via Amazon:

https://www.amazon.com/dp/0993594549/

https://www.amazon.co.uk/dp/0993594549/

To say that the sentence “snow is white” is true, is to say that snow is white, and nothing more.  Because of this apparent redundancy of the truth predicate for sentences, deflationists maintain that truth is a notion lacking in content and therefore not a worthy subject for philosophical analysis.  This book builds on the work of Jeffrey Ketland in demonstrating that semantic definitions of truth for formalised languages, in particular those of Alfred Tarski and Saul Kripke, are richer than their deflationary rivals, in the sense that they enable us to derive results that cannot be derived via deflationary theories.  This deductive power of semantic theories of truth suggests that truth is, contrary to the deflationists’ view, a substantive notion that warrants continued philosophical analysis.

O estudo da lógica é algo essencial para a filosofia, de forma que podemos considerá-la como a metodologia da par excellence. Assim sendo, a compreensão das noções básicas da lógica torna-se algo imprescindível para qualquer estudante da área. Não somente direcionado a tais discentes, o estudo da lógica também se faz presente e necessário nos mais variados campos do saber, como nas questões que fundamentam a matemática e até nas bases relativas à argumentação jurídica. É tendo essa visão da abrangência do estudo da lógica que este livro foi pensado.

Escrito de forma sucinta e não pretendendo esgotar em nenhuma hipótese a vasta gama de teorias presentes hoje na área da lógica, o presente manual introduz o aluno na leitura das principais ferramentas da linguagem formal. Além de estimular o leitor, a linguagem adotada nos textos buscou aproximar as teses lógicas da linguagem cotidiana. Vale lembrar que, tal como formulava Aristóteles, a lógica não é nenhuma ciência em si, mas apenas o instrumento da ciência, i.e., o caminho para o raciocínio correto. Desse modo, espera-se que todos os leitores façam bom uso desse instrumento e que ele sirva para a busca
filosófica do conhecimento.

Пропонований навчальний посібник призначений для практичного освоєння логічної теорії. За допомогою перспективної методики у ньому розглядаються головні теми і ключові проблеми логіки як теоретичної та навчальної дисципліни, а також наводяться логічні завдання і методики їх розв’язання, що забезпечує набуття логічного досвіду, підвищення рівня логічної культури мислення. Для студентів, аспірантів, викладачів та усіх, хто вивчає логіку самостійно, вдосконалює мислення й розвиває інтелект.

Mathematical logic is one of the scientific foundations of information technology and computer engineering. Therefore, engineering students must be proficient in logical operations. Firstyear engineering students answered four logical reasoning tasks as part of a larger survey on scientific thinking, producing more incorrect responses than previous studies would have predicted, most notably in international student groups. This paper presents and analyzes these results, and compares and discusses the differences between natural human reasoning and engineering logic. It seeks to explain failures in logical reasoning that occur despite formal training and considers several possible cultural and psychological explanations to this phenomenon. The conclusion highlights the need for stronger emphasis in developing problem-solving and metacognitive skills in engineering education. The importance of fluency in the language of study emerges as a significant factor in learning technical subjects.

The issue of the relationship between formal and informal logic depends strongly on how one understands these two designations. While there is very little disagreement about the nature of formal logic, the same is not true regarding informal logic, which is understood in various (often incompatible) ways by various thinkers. After reviewing some of the more prominent conceptions of informal logic, I will present my own, defend it and then show how informal logic, so understood, is complementary to formal logic.

LAWS OF FORM CONFERENCE 2022

Submissions for papers, panel sessions, interactive presentations, workshops, performance sessions, and creative contributions inspired by George Spencer-Brown's work and life - and particularly his key work, Laws of Form (LoF) - are now open and welcomed from participants keen to contribute to LoF22 which will be held from Thursday 4 August to Saturday 6 August, 2022 at the University of Liverpool.

Invited Keynotes

Keynotes will be given by Dr. Francis Jeffrey, Barry Smith, and Stephen Wolfram.

https://lof50.com/

Set Theory: An open introduction is a brief introduction to the philosophy of set theory. It is written for students with a little background in logic (such as one might get from forallx:Cambridge), and some high school mathematics. By the end of this book, students reading it might have a sense of:

    why set theory came about;
    how to embed large swathes of mathematics within set theory + arithmetic;
    how to embed arithmetic itself within set theory;
    what the cumulative iterative conception of set amounts to;
    how one might try to justify the axioms of ZFC.

Originally, I released this as Open Set Theory. I subsequently made some corrections, and have handed all of it over to the Open Logic Project. You can find the source code at the dedicated site. Alternatively, you can just download a ready-to-go free pdf of the book.

By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised.

The Equivalence Analogy which was first propounded in Euclid's book of Principles consists of two premises (Φ=Ψ and Ψ=Θ) and a conclusion (Φ=Θ). This form of syllogism is greatly similar to the first form of Aristotelian categorical syllogism because Φ is the primitive term; Ψ is the middle term, and Θ is the final term, and the conclusion is established between the two side terms. Thus it appears that the equivalence analogy must be a part of categorical syllogisms of the first form; however, the important problem here is that the basis of this syllogism does not follow Aristotle's definition of syllogism. In his view, a syllogism is a statement in which nothing is required for arriving at a conclusion except the premises of the syllogism. However, this syllogism requires an external premise which can justify the transfer of the equality relation from the premises to the conclusion. Following different approaches, Arab logicians have tried to establish the validity of the equivalence analogy. Here, the writers evaluate these approaches and, while demonstrating their shortcomings, refer to a new approach in this regard and prove that equality and subsumption (predication) are both of the type of transitive relations. Such relations are transferable from the premises to the conclusion, whereas non-transitive relations, such as a capacity equal to half of something or twice as much as it lacks this feature. Therefore, the correctness of the equivalence analogy and the first form of Aristotelian categorical conjunctive syllogism can be considered to be dependent on type possibility of transition in equality and subsumption relations.

The traditional conflict in epistemology has been between rationalism and empiricism. Philosophers have often tried to align themselves in one of these schools even though a close reading of some of their writings may suggest they may often have used both varyingly, one more than the other. This paper demonstrates that both rationalism and empiricism work together and are thus called the rational-empiric model. The paper extends the scope of the rational-empiric model to also evolving cognitive developments and other aspects of reasoning such as computational thought and makes an effort towards summarizing the nature of human reason by explaining the functional intellect and the structural brain since the time of Aristotle to date drawing from theology, philosophy, logic, computer science, and psychology. The paper presents the rational-empiric model in a broader in-depth context with supporting ideas and evidence. Various literature on theology, logic, philosophy, and psychology has been used to elaborate on the concepts.  The paper concludes that the purpose of the rational empiric model is to discover knowledge by finding and understanding ontologies as they exist and function or as they ought to exist and function.

This discussion reviews the thinking of some prominent philosophers of argument to extract principles common to their thinking. It shows that a growing concern with dialogical pragmatics is better appreciated as a part of applied ethics than of applied epistemology. The discussion concludes by indicating a possible consequence for philosophy of argument and invites further discussion by asking whether argumentation philosophy has an implicit, underlying moral, or even political, posture

In this overview article, we first explain what we take informal logic to be, discussing misconceptions and distinguishing our conception of it from competing ones; second, we briefly catalogue recent informal logic research, under 14 headings; third, we suggest four broad areas of problems and questions for future research; fourth, we describe current scholarly resources for informal logic; fifth, we discuss three implications of informal logic for philosophy in particular, and take note ofpractical consequences of a more general sort.

Argumentation logicians have recognized a specter of relativism to haunt their philosophy of argument. However, their attempts to dispel pernicious relativism by invoking notions of a universal audience or a community of model interlocutors have not been entirely successful. In fact, their various discussions of a universal audience invoke the context-eschewing formalism of Kant's categorical imperative. Moreover, they embrace the Kantian method for resolving the antinomies that continually vacillates between opposing extremes -here between a transcendent universal audience and a context-embedded particular audience. This tack ironically restores the very external mediation they thought to obviate in their aim to 'dethrone' the absolutism and totalitarianism of formal logic with a democratic turn to audience adherence, the acceptability of premises and inferential links, and a contextual, or participant-relative, notion of cogency.

Since its inception, logic has studied the acceptable rules of reasoning, the rules that allow us to pass from certain statements, serving as premises or assumptions, to a statement taken as a conclusion.
The first kinds of such rules were distilled by Aristotle and are known as moduses. Stoics ramified Aristotle’s system, and for centuries, the syllogistic remained the main tool for logical deduction.
With the birth of formal logic, new types of deduction emerged, and to support this new kind of inference, the deductive systems were used. Since then, the deductive systems have been at the heart of logical investigations. In one form or the other, they are used in all branches of logic.
Contemporary understanding of science, as a theory of a high degree of exactness, requires treating it as a deductive theory (a deductive system). Generally speaking, such a theory (system) is a set of its sentential expressions which are derivable (deducible) from some expressions of the set selected as axioms, by means of deduction (inference) rules. The expressions obtained as a result of derivation from a given set of expressions are consequences; that is, they have a proof. The principal feature of deductive systems (theories) is the deducibility or provability of their theorems. From a verygeneral point of view, there are two m ethods of deduction: (a) the axiomatic method (Hilbert style
method) and (b) the natural deduction method (Jaśkowski–Słupecki–Borkowski, Gentzen or semantic tableaux). Method (b) leads to natural deduction systems, while the most often used method (a) leads to
presentation (or characterization) of logical and mathematical theories as axiomatic deductive systems.
Methods (a) and (b) are used in different scientific disciplines, such as physics, chemistry, sociology, philosophical and psychological sciences, information sciences, discursive sciences, computer science,
and some technical sciences.
Deductive sciences have not always been built explicitly as axiomatic systems. Depending on the degree of methodological precision, three of their forms have been distinguished: pre-axiomatic, non-formalized axiomatic, and formalized axiomatic. As we know, a pre-axiomatic form was commonly used in arithmetic and geometry, and later in set theory and probability theory. Their axiomatization was carried out only at the end of the 19th century and the beginning of the 20th century. In contrast, such mathematical theories as the Boolean system and theories of groups, rings, and fields were built as formalized axiomatic systems since inception. The deductive method (calculi) is most often used for formalizations of theories, but these theories also admit formalization as natural deductive systems. Formalized axiomatic systems are rooted in a tradition originated by G. Frege (1891, 1903), but the first axiomatic system (non-formalized) in the history of science—as it was disclosed by Jan Łukasiewicz in his seminal monograph on Aristotle’s syllogistic (1951)—was Aristotle’s syllogistic system.
J. Łukasiewicz initiated the construction of the first systems of syllogistic satisfying the contemporary requirements, and thus, the requirements of formalized axiomatic systems. He constructed a formalization of syllogistic logic on two levels using (in addition to the commonly used axiomatic method by means of proof) a new axiomatic method by means of rejection—the so-called axiomatic rejection, or refutation, method. He and his disciples (mainly J. Słupecki and his collaborators) applied this method to the bi-level formalization of
some classical and non-classical logical deductive systems of sentences or names. This approach allows one to define two disjoint sets of language expressions of a given system: the set of all its theses (theorems), which are asserted, accepted, intuitively true expressions (called the assertion system), and the set of all the other expressions—non-accepted, or intuitively false, refuted, rejected expressions of the system (called the rejection or the refutation system). In such a way, the bi-level formalization of deductive systems provides some new inspiration to build different sciences.
This book is a collection of articles included in the special issue “Deductive Systems” of Axioms regarding mainly the logical deductive system. They are ordered in accordance with the well-known division of logic into term logic (logic of names) and propositional logic (propositional calculus), which correspond to two historical stages of the development of logic, namely, Aristotelian logic and the logic of stoics, with the latter being a contemporary counterpart of propositional logic. Deductive systems for classical propositional logic are broadly known, and one of them is most often assumed for the term logics. Systems for non-classical propositional logics, which are inspired by philosophy, are introduced in the book later than systems related to term logics. Term logic can be interpreted in predicate logic, that is, the second part of contemporary logic. Predicate logic is the basis of
mathematical deductive systems (theories).

An introductory textbook in logic, covering the syntax and semantics of propositional logic (including proof techniques and refutation trees), the syntax and semantics of predicate logic (including proof techniques and refutation trees), inductive logic, the probability calculus, the analysis of fallacies, and further developments in formal logic including the theory of descriptions, higher-order logics, modal logic, and formal arithmetic. Features over 500 problems with complete solutions.

The single most important statement that can be made with regard to the logical status of human and physical geographical reasoning is that it belongs to the class of non-monotonic reasoning. In other words,  geographical reasoning is defeasible (i.e. earlier drawn conclusions can be defeated by the addition of new facts, some of which are generated by the very application of our ideas outside academe) and non-demonstrative (i.e. our beliefs about reality cannot be demonstrated once and for ever; we must do with building provisional arguments based on the current weight of the evidence, which itself changes with the times). Indeed, unlike mathematics, which is a closed world of formalisms that relies heavily on deduction, demonstration, and proof, geography studies and attempts to improve the real, messy world out there. Because that wild, unruly world is always in a process of becoming, the conclusions drawn yesterday may have to be abandoned today, and so on, without end in sight. The core virtue of geographical reasoning is not certainty, but adaptability. As the world changes, so do geography’s entertained beliefs about how to improve it. The point we are trying to make is that an alternative window into the nature of geographical reasoning, besides the one provided by the discipline’s key concepts, offers an opening toward the more subtle problematic of the mechanisms by which the updating of geographical beliefs operate.

International Graduate Student Conference.
26th–28th Nov, 2020.
Humboldt-Universität zu Berlin. Online.
Co-organized with Paul Hasselkuß (Düsseldorf), Tiago Hirth (Lisbon), Deborah Kant (Konstanz), Deniz Sarikaya (Hamburg), Tobias Schütz (Mainz), Anna Kiel Steensen (Zurich), and Benjamin Wilck (Berlin)

The paper surveys different notions of implicit definition. In particular , we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such "structural definitions" can be semantically understood in two different ways, namely (i) as specifications of the meaning of the primitive terms of a theory and (ii) as definitions of higher-order mathematical concepts or structures. We analyze these two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning.

Dopo i preliminari filosofici sulla dialettica da usare nella trattazione della relatività, la si analizza considerandone gli aspetti concettuali, formali, linguistici, fisici e matematici, con accenni di speciazione biologica, sino all'unificazione in un unico sistema.