The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS r , the variablesX, Y, Z ... represent the subsets ofS R . The formulacxy designates an implication withx (...) as its antecedent andy as its consequent,cxy is always an element ofS R δ(X) means, thatX is closed with respect to the modus ponens rule.A R designates the class of allS R -substitutions of the axioms of the sentential calculusR. (shrink)

Here is a deft and new introduction to Gentzen proof techniques in axiom systems and to the analysis of formal axiom systems; in short, axiomatics inside and out. Treating of deduction in propositional and predicate logic, metatheoretical problems about both set theory and its paradoxes, the book is flexibly structured for selective use as a text. Yet the discussion is unified and motivated by the concept of the axiomatic system--the history of its use and analysis, and (...) its present practical and theoretical status. Well-chosen illustrations convey an historical overview of Euclid, non-Euclidean geometers, Peano, Boole, Hilbert and Gödel.--P. D. J. (shrink)

In a fluent, clear, and lively style this translation by two of Maslov's junior colleagues brings the work of the late Soviet scientist S. Yu. Maslov to a wider audience. Maslov was considered by his peers to be a man of genius who was making fundamental contributions in the fields of automatic theorem proving and computational logic. He published little, and those few papers were regarded as notoriously difficult. This book, however, was written for a broad audience of readers and (...) describes elegant examples of applications in such fields as computer science, artificial intelligence, operations research, economic modeling, and biological modeling, among others. The book also brings to light the work by the American mathematician E. L. Post, which inspired Maslov's own work in the development of a general theory and which has been long neglected by mathematicial logicians and systems theorists in the United States. The book's first chapter introduces the Rules of the Game. Part I, Mathematics of Calculi, covers E. L. Post's canonical systems, deductive systems and algorithms, and probabilistic calculi and deductive information. Part II, Horizonal Modeling, takes up a "toy" economy, the calculi of technological possibilities, and the development of rules. Part III, Vertical Modeling, deals with the topics of "to fight and to search" and the consequences of the asymmetry of cognitive mechanisms. Vladimir Lifschitz is affiliated with the Department of Computer Science at Stanford University, and Michael Gelfond with the Department of Electrical Engineering and Computer Science at the University of Texas, El Paso. Theory of Deductive Systems and Its Applicationsis included in the Foundation of Computing Series, edited by Michael Garey. (shrink)

Many philosophers who do not analyze laws of nature as the axioms and theorems of the best deductive systems nevertheless believe that membership in those systems is evidence for being a law. This raises the question, “If the best systems analysis fails, what explains the fact that being a member of the best systems is evidence for being a law?” In this essay I answer this question on behalf of Leibniz. I argue that although Leibniz’s (...) philosophy of laws is inconsistent with the best systems analysis, his philosophy of nature’s perfection enables him to explain why membership in the best systems is evidence for being a law of nature. (shrink)

The promised mathematical system—the Constructibility Theory—is presented as an axiomatized deductive theory formalized in a many‐sorted first‐order logical language. The axioms of the theory are specified and a justification for each of the axioms is given. Objections to the theory are considered.

This paper deals with a foundational aspect of Integrated Information Theory of consciousness: the nature of the relation between the axioms of phenomenology and the postulates of cause-effect power. There has been a lack of clarity in the literature regarding this crucial issue, for which IIT has received much criticism of its axiomatic method and basic tenets. The present contribution elucidates the problem by means of a categorial analysis of the theory’s foundations. Its main results are that: IIT has a (...) set of nine fundamental concepts of reason, called categories, which constitute its categorial lexicon and through which it formulates a system of principles incorporating the axioms, the postulates, and the central identity; and the connection between the axioms and postulates is grounded by their common root in this categorial lexicon, the categories of which find their justification by means of a phenomenological and transcendental deduction. Some further results are the unique origin of axioms and postulates in the categories; the distinction between conceptual and formalized postulates; a clarification of the uniqueness problem of categorial lexica in general; and an IIT account of objectivity by explicating how the physical is defined by means of categories. All of this is put to use against various criticism targeting IIT’s theoretical core. If successful, the proposed interpretation illuminates a central issue in the contemporary study of consciousness and contributes to an environment of mutual understanding between defenders and critics of the theory. (shrink)

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn + that, in the intuitive sense, are infallible operations, always leading from accepted (true) sentences of a deductive system to accepted (true) sentences of the deductive system (see Tarski in Monatshefte für Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Słupecki (...) in Stud Logic 9:163–176, 1960, Stud Logic 10:77–95, 1960). The second kind are dual consequence operations of the type Cn − that can be regarded as anti-infallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see Słupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria Matematyka 4(81):35–61, 1983, Ann Pure Appl Logic 127:243–266, 2004, in On the notion and function of rejected propositions, 179–202, 2005). The operations of the types Cn + and Cn − can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple: $$\begin{array}{ll}{\rm by\,the\,triple}:\hspace{1.4cm} (+ , -)\hspace{0,6cm} \\ {\rm or\,by\,the\,triple}:\hspace{1.0cm} (-, +)\hspace{0,6cm} .\end{array}$$ We compare axiom systems for operations of the types Cn + and Cn −, give some methodological properties of deductive systems defined by means of these operations (e.g. consistency, completeness, decidability in Łukasiewicz’s sense), as well as formulate different metatheorems concerning them. (shrink)

The idea of rejection of some sentences on the basis of others comes from Aristotle, as Jan Łukasiewicz states in his studies on Aristotle's syllogistic [1939, 1951], concerning rejection of the false syllogistic form and those on certain calculus of propositions. Short historical remarks on the origin and development of the notion of a rejected sentence, introduced into logic by Jan Łukasiewicz, are contained in the Introduction of this paper. This paper is to a considerable extent a summary of papers (...) which are not easily available, even to the Polish reader: (1) J. Słupecki, Funkcja Łukasiewicza (Łukasiewicz’s function), Zeszyty Naukowe Uniwersytetu Wrocławskiego, Seria B, nr 3 (1959), 33-40; (2) U. Wybraniec-Skardowska, Teoria zdań odrzuconych (Theory of Rejected Sentences), (doctoral dissertation under the supervision of Jerzy Słupecki, published as a monograph), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Monografie, Nr 22 (1969), 5-131; (3) G. Bryll, Kilka uzupełnień teorii zdań odrzuconych (Some supplements to the theory of rejcted sentences), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Mnografie, Nr 22 (1969), 153-154. The paper also contains a good number of results which have not been published yet. Chapter I contains results presented in the papers (1)-(3). It provides an extension of Alfred Tarski’s theory of deductive systems presented by him in the papers [1930]: Fundamentale Begriffe der Methodologie der deduktiven Wssenschaften, I and Über einige fundamentale Begriffe der Metamathematik. The enriched theory is marked with T. The most essential new concept in T is the function Cn-1, whose definition was given by Słupecki in (1) on the basis of the so-called Tarski’s general theory of deductive systems. It has the form: y e Cn-1X iff Ex e X (x e Cn {y}), where Cn is Tarski’s consequence operation. In accordance with the definition: A sentence y is rejected on the basis of the sentences of the set X iff at least one of sentences of X is a consequence of y (is deducible from y). The intuitive meaning of the rejection function Cn-1: on the basis of false sentences we can reject false sentences only (while by means of the consequence operation Cn on the basis of true sentences we can deduce true sentences only). The function Cn-1 is a generalization of the notion of rejected sentences which was introduced into logic by Łukasiewicz. The essential property of the rejection function Cn-1 is that it satisfies the axioms of general Tarski’s consequence Cn, so it is a consequence operation, called the rejection consequence. In addition, it is an additive and normal operation. In Chapter I, there are given notions analogous to those of the theory of deductive systems, but they are written down by means of the symbol ‘Cn-1’ and not ‘Cn’. There are established the properties of introduced notions and differences and analogies taking place between them and properties of respective notions of the theory T. There are also given generalizations of the notions of ‘decidable system’ and ‘consistent system’ used by Łukasiewicz. The short Chapter II contains axioms of the system T’ which is equivalent to the system T. The only difference between sets of primitive notions of these systems consists in the appearance of the function Cn-1 in the system T’ instead of the function Cn. This chapter reproduces the results given in (2), but they are partially simplified. (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking (...) just one syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and (...) intuition of geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques. (shrink)

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and (...) intuition of geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques. (shrink)

This paper is a continuation of Part I under the same title. Its Chapter III contains results given in the following publications: U. Wybraniec-Skardowska, Teoria zdań odrzuconych (Theory of Rejected Sentences), (doctoral dissertation under the supervision of Jerzy Słupecki, published as a monograph), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Monografie, Nr 22 (1969), 5-131. G. Bryll, Związki logiczne pomiędzy zdaniami nauk empirycznych (Logical relations between sentences of empirical sciences). Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i (...) Monografie, Nr 22 (1969), 155-216. This chapter contains an attempt to formalize some questions of methodology of empirical sciences. The theory T is enriched here not only by new definitions but also by a new primitive term and new axioms. The most important notion defined in this chapter is the set IndX of all inductive conclusions obtained on the basis of sentences of the set X. The function CnI defined by the equality: CnI X = X u IndX, satisfies the axioms of the general theory of deductive systems. The set CnI X can be understood as a system of empirical sciences in a narrower sense. This paper is a development of the considerations of G. Bryll, included in the above-listed article by this author. (shrink)

Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional (...) logic, for which I show soundness and completeness for both deductions and refutations. (shrink)

Sahlqvist theory is extended to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic.

This paper deals with Tarski's first axiomatic presentations of the syntax of deductive system. Andrzej Grzegorczyk's significant results which laid the foundations for the formalization of metalogic, are touched upon briefly. The results relate to Tarski's theory of concatenation, also called the theory of strings, and to Tarski's ideas on the formalization of metamathematics. There is a short mention of author's research in the field. The main part of the paper surveys research on the theory of deductive (...) class='Hi'>systems initiated by Tarski, in particular research on the axiomatization of the general notion of consequence operation, axiom systems for the theories of classic consequence and for some equivalent theories, and axiom systems for the theories of nonclassic consequence. In this paper the results of Jerzy Supecki's research are taken into account, and also the author's and other people belonging to his circle of scientific research. Particular study is made of his dual characterization of deductive systems, both as systems in regard to acceptance and systems in regard to rejection . Comparison is made, therefore, with axiomatizations of the theories of rejection and dual consequence, and the theory of the usual consequence operation. (shrink)

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. Hybrid logics are here divided into two groups: standard hybrid logics involving nominals as expressions of a separate sort, and non-standard hybrid logics, which do not involve nominals but whose expressive power matches the expressive power of binder-free standard hybrid logics.The original results of this book are split into two parts. This division reflects the (...) division of the book itself. The first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, the efforts focused on hybrid logics referred to as non-standard in this book. Non-standard hybrid logics are understood as modal logics with global counting operators ) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise: 1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators ), which can be easily extended onto other frame classes, 2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames ), MT), MD), MB), MK4) with numerical subscripts coded in binary. Establishing the exponential-size model property for this logic defined over the classes of Euclidean and equivalential frames ), MS5).Results of the second type consist of designing concrete deductive systems for standard and non-standard hybrid logics. More precisely, they include: 1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics. An interesting feature of indicated calculi is the nonbranching character of the rule, 2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics. The internalization technique applied to a tableau calculus for the modal logic with global counting operators is novel in the literature, 3. Devising the first hybrid algorithm involving an inequality solver for modal logics with global counting operators. Transferring the arithmetical part of reasoning to an inequality solver turned out to be sufficient in ensuring termination.The book is directed to philosophers and logicians working with modal and hybrid logics, as well as to computer scientists interested in deductive systems and decision procedures for logics. Extensive fragments of the first part of the book can also serve as an introduction to hybrid logics for wider audience interested in logic.The content of the book is situated in the areas of formal logic and theoretical computer science with some elements of the theory of computational complexity. (shrink)

The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of (...) class='Hi'>deductive systems. In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper. (shrink)

Basin-Matthews-Viganò approach to construction of labelled deduction systems for normal modal logics is adapted to „Fitch proof-format“, and it is applied to the language of deontic-praxeological logic. Segerberg's suggestion on how to asses the adequacy of a logic for Kanger's theory of rights is being formally explicated and it is proved that herewith proposed system of labelled deduction satisfies Segerberg's criteria of adequacy. For the purpose of building the proof a semantics is given, which connects „the simplest semantics of (...) action“ with standard semantics of deontic logic. Soundness and completeness of the proposed labelled deduction with respect to aforementioned semantics is proved. (shrink)

The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in that non-triviality (...) is provable by strictly finitary methods. This result does not contradict Gödel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods. (shrink)

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order (...) to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

Classic of Changes is a Chinese cultural classic born more than 3000 years ago. Its profound philosophical thoughts and the use of divination have brought Classic of Changes to a strong oriental mysticism. The view of the heaven and man of yin and yang and the five elements states of Classic of Changes are completely different from the Western elemental theory of ancient Greece. The latter gave birth to classical and modern scientific theories, and the yin and yang and (...) the eight trigrams symbol has become synonymous with oriental mysticism. In fact, the cosmology of the Holism of Classic of Changes is a precious scientific heritage of mankind. The axiomatization of the symbolic formal system of Classic of Changes aims to unveil the veil of oriental mysticism and provide oriental wisdom for the development of modern science. Transforming the symbolic system of Classic of Changes into a formal axiom system is the crystallization of the fusion of wisdom between the East and the West. The axiomatization of the symbolic formal system of Classic of Changes shows that the oriental and the western scientific tradition harmony but not sameness and there is no conflict. Classic of Changes can also be interpreted by the axiomatic system like Euclid’s Elements. The main contribution of this paper is that the author skillfully uses mathematical language to formulate the system of Classic of Changes, reconstructs the ideological system of Classic of Changes with the axiomatic method and realizes the scientificalization of Chinese classical natural philosophy. The formal axiom system of Classic of Changes may give us such a revelation: the gap between the oriental and western scientific traditions is mainly in the axiomatization, but the lack of oriental scientific tradition may be the bottleneck of the development of modern science. (shrink)

This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the two (...) systems of natural deduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)

In this paper, we investigate the possibility of translating a fragment of natural deduction system (NDS) for natural language semantics into modern type theory (MTT), originally suggested by Luo (2014). Our main goal will be to examine and translate the basic rules of NDS (namely, meta-rules, structural rules, identity rules, noun rules and rules for intersective and subsective adjectives) to MTT. Additionally, we will also consider some of their general features.

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished]. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of [Tarski 1959]: a predicate of betwenness and (...) a four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane - which obeys the Euclidean axioms in [Tarski and Givant, 1999] - and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagora’s theorem. We conclude with a Representation Theorem relating models of our system that satisfy second order continuity to the mathematical structure called ‘Minkowski spacetime’ in physics textbooks. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first (...) order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin and Malament. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski : a predicate of betwenness and a four (...) place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant, 175–214 1999)—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models \ of our system \ that satisfy second order continuity to the mathematical structure \, called ‘Minkowski spacetime’ in physics textbooks. (shrink)

Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.

Contemporary Humeans treat laws of nature as statements of exceptionless regularities that function as the axioms of the best deductive system. Such ‘Best System Accounts’ marry realism about laws with a denial of necessary connections among events. I argue that Hume’s predecessor, George Berkeley, offers a more sophisticated conception of laws, equally consistent with the absence of powers or necessary connections among events in the natural world. On this view, laws are not statements of regularities but the most general (...) rules God follows in producing the world. Pace most commentators, I argue that Berkeley’s view is neither instrumentalist nor reductionist. More important, the Berkeleyan Best System can solve some of the problems afflicting its Humean rivals, including the problems of theory choice and Nancy Cartwright’s ‘facticity’ dilemma. Some of these solutions are available in the contemporary context, without any appeal to God. Berkeley’s account deserves to be taken seriously in its own right. (shrink)

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in (...) the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell). (shrink)

In 1951 Lukasiewicz [[sic]] linked Aristotle's Prior Analytics with modern formal logic. This book attempts to analyze Aristotle's syllogistic theory in the light of Lukasiewcz's work and the whole tradition of classic interpretations of Aristotle's logic. The first of the book's five chapters shows that for Aristotle the syllogism is basically a relationship of terms couched in conditional form; a relationship of variables rather than concrete terms; and a relationship that sees S linked with P not by the copula but (...) by less metaphysical expressions. The classical S is P form is not Aristotelian at all but a much later renovation. The second chapter maintains that Aristotle distinguishes "relative necessity" from "absolute necessity." Aristotle himself ignores his own distinction at times and thus creates logical difficulties for himself. The oft-fought sea battle is fought again in this chapter. Chapter three deals with the controversial notions of "perfect" and "imperfect" syllogisms. The author asserts that by "perfect syllogism" Aristotle means self-evident syllogisms that serve as axioms for deducing other proofs. Chapters four and five deal with the figures of the syllogism and the syllogism as a deductive system respectively. Patzig concludes that Aristotle's syllogistic theory is purely formal in its structure and can only be properly understood if it is not tied to a metaphysics and converted into a philosophical logic. He denies that it is a universal logic, but rather sees it as a specialized theory of relations.--J. J. R. (shrink)

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom (...) of infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)

We present a new axiomatization of classical mereology in which the three components of the theory—ordering, composition, and decomposition prin-ciples—are neatly separated. The equivalence of our axiom system with other, more familiar systems is established by purely deductive methods, along with additional results on the relative strengths of the composition and decomposition axioms of each theory.

The issue of the emergence of formal axiomatic logical systems due to the emergence of logical antinomies in formal axiomatic systems, specifically the issue of developing formal logical axiomatics in the calculus of considerations was investigated in the considered research. At the same time, in order to determine the characteristics of the implementation of the logical-methodological principles and provisions of the deductive reasoning obviously, conceptual-logical foundations of the calculus of considerations was studied and the main propositions of (...) the calculus of considerations and the analysis of the initial logical operations on them were given in the study. The basic laws of logic and the expression of one logical act with another one were also explained in the research work by giving the concepst of proportional form and tautology in the calculus of considerations. At the same time, main properties and the procedure of setting of the formal deductive theory in the calculus of considerations were studied in the article. In order to provide an adequate characterization of the conceptual and methodological bases of the calculus of considerations in the research work, the basic logical laws of the calculus of considerations developed by the German mathematician and thinker P. Hilbert were also given. In addition, the main principles and methodological provisions of the deductive method were investigated, and the rules of deriving new conclusions from the axioms were interpreted. At the end of the reviewed article, the main principles and methods of the formal axiomatic mathematical systems built on a deductive basis, specifically the calculus of considerations, were studied and a logical-methodological analysis of the calculus of considerations was carried out on this basis. In this framework, the syntactic and semantic analysis of formalized language in formal axiomatic logical systems was carried out separately, the constituent parts of those systems, including the construction scheme, were given. (shrink)

. Dzik [2] gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems, here called Dzik systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the (...) conditional . A partial solution is provided. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I (...) apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

The aim of the paper is to present two natural deduction systems for Intuitionistic Sentential Calculus with Identity ( ISCI ); a syntactically motivated \(\mathsf {ND}^1_{\mathsf {ISCI}}\) and a semantically motivated \(\mathsf {ND}^2_{\mathsf {ISCI}}\). The formulation of \(\mathsf {ND}^1_{\mathsf {ISCI}}\) is based on the axiomatic formulation of ISCI. Its rules cannot be straightforwardly classified as introduction or elimination rules; ISCI -specific rules are based on axioms characterizing the identity connective. The system does not enjoy the standard subformula property, but (...) due to the normalization procedure non-subformulas can label only leaves of proofs. In \(\mathsf {ND}^2_{\mathsf {ISCI}}\), we propose only two general identity-related rules, in reference to the treatment of the identity connective in First-Order Logic. (shrink)