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Deductive Systems in Traditional and Modern Logic
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Deductive Systems in Traditional and Modern Logic

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Logic".

Deadline for manuscript submissions: closed (15 August 2019) | Viewed by 52728

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Urszula Wybraniec-Skardowska

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Guest Editor
Department of Philosophy, Cardinal Stefan Wyszyński University in Warsaw, Wójcickiego 1/3 bl. 23 II, O1-938 Warsaw, Poland
Interests: logic and its applications; metalogic; history of logic; philosophy; mathematics; information sciences; theory of deductive systems and foundations of mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Alex Citkin

E-Mail
Guest Editor
Metropolitan Telecommunications, New York, NY 10041, USA
Interests: intermediate logics; varieties and quasivarieties of Heyting algebras
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

I have the intention of launching a Special Issue of Axioms devoted to (1) the presentation of some new deductive systems, modified known systems and little-known systems with their specifics, intuitive foundation, and methodological or metamathematical properties (2) comparison of systems of the same scientific discipline with each other; as well as (3) a collective review of current and updated systems of the same type, taking into account discussed below all aspects and explaining their functioning in the past, present, and future (deductive systems will be understood as deductive theories and distinguished from their syntactic characterization by appropriate ordered pairs or triples).

Contemporary understanding of science, as a science of a high degree of exactness, requires treating it as a deductive theory (a deductive system). Loosely speaking, such a theory (system) is a set of its language sentential expressions that includes the set of all its expressions which are derivable (are deducible) from some expressions of the set by means of deduction (inference) rules, i.e. which are its consequences and have a proof on the basis of the expressions of the set. The feature of deductive systems (theories) is the deducibility and provability of their theorems. These systems are built using a method of deduction employing (a) the axiomatic method or (b) the natural deduction method (Jaśkowski-Słupecki-Borkowski, Gentzen or semantic tables). Method (b) leads to natural deduction systems, and the most often used method—(a)—leads to axiomatic systems and to the presentation or characterization of logical and mathematical theories as axiomatic deductive systems. Methods (a) and (b) may be used to build some other 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 and are not always immediately built as axiomatic systems. Depending on the degree of methodological precision, three of their forms are distinguished: pre-axiomatic, non-formalized axiomatic, and formalized axiomatic. As we know, the pre-axiomatic form was typical of arithmetic and geometry, and later set theory, and probability theory and its axiomatization was carried out only in the 19th century, while such mathematical theories as the Boolean system and theories of groups, rings, fields were immediately built as formalized axiomatic ones. Logical systems (calculi) and theories based on them and constructed by means of a deduction method are most often formalized axiomatic systems, but they can also be considered as formalized natural deduction systems.

Formalized axiomatic systems have a tradition originating from 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 famous monograph on Aristotle’s syllogistic (1951)—was Aristotle’s syllogistic. J. Łukasiewicz initiated the construction of the first systems of syllogistic corresponding to the contemporary requirements, and thus the formalized axiomatic systems. He conducted formalization of syllogistic on two levels, using, for the first time, in addition to the usual axiomatic method by means of proof, a new axiomatic method by rejection proof—the so-called axiomatic rejection (or refutation) method. He, and continuators of his ideas (mainly J. Słupecki and the co-workers from his research circle), applied this method to the bi-level formalization of some classical and non-classical logical deductive systems of sentences or names to define two disjoint sets of language expressions of the given system: the set of all its theses (theorems); asserted, accepted, intuitively true expressions (called the assertion system); and the set of all the other non-accepted, or intuitively false, rejected expressions of the system (called the rejection or the refutation system). In this way, the bi-aspectul formalization of deductive systems provides some new inspiration to build different sciences.

Prof. Dr. Urszula Wybraniec-Skardowska
Dr. Alex Citkin
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Metalogic and metamathematics
  • Syntax
  • Well-formed expression
  • Deduction and natural deduction
  • Deductive sciences
  • Axiomatization
  • Formalisation
  • System of axioms
  • Inference rule
  • Proof and methods of proof
  • Consequence operation
  • Deductive system (theory)
  • Assertion system
  • Refutation (rejection) system
  • Classical and non-clasical logical and mathematical systems
  • Deductive systems of different sciences
  • Methodology of deductive systems

Published Papers (19 papers)

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Editorial

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4 pages, 150 KiB  
Editorial
Deductive Systems in Traditional and Modern Logic
by Alex Citkin and Urszula Wybraniec-Skardowska
Axioms 2020, 9(3), 108; https://doi.org/10.3390/axioms9030108 - 13 Sep 2020
Viewed by 2158
Abstract
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 [...] Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)

Research

Jump to: Editorial, Other

13 pages, 263 KiB  
Article
The Formal Framework for Collective Systems
by Rafał Palak and Krystian Wojtkiewicz
Axioms 2021, 10(2), 91; https://doi.org/10.3390/axioms10020091 - 15 May 2021
Cited by 2 | Viewed by 1713
Abstract
Automated reasoning is becoming crucial for information systems. Building one uniform decision support system has become too complicated. The natural approach is to divide the task and combine the results from different subsystems into one uniform answer. It is the basic idea behind [...] Read more.
Automated reasoning is becoming crucial for information systems. Building one uniform decision support system has become too complicated. The natural approach is to divide the task and combine the results from different subsystems into one uniform answer. It is the basic idea behind the system approach, where one solution is a composition of multiple subsystems. In this paper, the main emphasis is on establishing the theoretical framework that combines various reasoning methods into a collective system. The system’s formal abstraction uses graph theory and provides a discussion on possible aggregation function definitions. The proposed framework is a tool for building and testing specific approaches rather than the solution itself. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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16 pages, 342 KiB  
Article
Kripke-Style Models for Logics of Evidence and Truth
by Henrique Antunes, Walter Carnielli, Andreas Kapsner and Abilio Rodrigues
Axioms 2020, 9(3), 100; https://doi.org/10.3390/axioms9030100 - 19 Aug 2020
Cited by 8 | Viewed by 2461
Abstract
In this paper, we propose Kripke-style models for the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment (FDE) [...] Read more.
In this paper, we propose Kripke-style models for the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment (FDE) with a classicality operator ∘ that recovers classical logic for formulas in its scope. According to the intended interpretation here proposed, these models represent a database that receives information as time passes, and such information can be positive, negative, non-reliable, or reliable, while a formula ∘A means that the information about A, either positive or negative, is reliable. This proposal is in line with the interpretation of N4 and FDE as information-based logics, but adds to the four scenarios expressed by them two new scenarios: reliable (or conclusive) information (i) for the truth and (ii) for the falsity of a given proposition. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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29 pages, 424 KiB  
Article
Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic
by Sopo Pkhakadze and Hans Tompits
Axioms 2020, 9(3), 84; https://doi.org/10.3390/axioms9030084 - 21 Jul 2020
Cited by 2 | Viewed by 2458
Abstract
Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given [...] Read more.
Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given additional premisses. This nonmonotonic aspect is in contrast to valid inference relations, which are monotonic. Although nonmonotonic reasoning has been extensively studied in the literature, only few works exist dealing with a proper proof theory for specific logics. In this paper, we introduce sequent-type calculi for two variants of default logic, viz., on the one hand, for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default logic, due to Gelfond, Lifschitz, Przymusinska, and Truszczyński. The first variant of default logic employs Łukasiewicz’s three-valued logic as the underlying base logic and the second variant generalises defaults by allowing a selection of consequents in defaults. Both versions have been introduced to address certain representational shortcomings of standard default logic. The calculi we introduce axiomatise brave reasoning for these versions of default logic, which is the task of determining whether a given formula is contained in some extension of a given default theory. Our approach follows the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti, which employs a rejection calculus for axiomatising invalid formulas, taking care of expressing the consistency condition of defaults. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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28 pages, 338 KiB  
Article
Minimal Systems of Temporal Logic
by Dariusz Surowik
Axioms 2020, 9(2), 67; https://doi.org/10.3390/axioms9020067 - 16 Jun 2020
Cited by 1 | Viewed by 2264
Abstract
The article discusses minimal temporal logic systems built on the basis of classical logic as well as intuitionistic logic. The constructions of these systems are discussed as well as their basic properties. The K t system was discussed as the minimal temporal logic [...] Read more.
The article discusses minimal temporal logic systems built on the basis of classical logic as well as intuitionistic logic. The constructions of these systems are discussed as well as their basic properties. The K t system was discussed as the minimal temporal logic system built based on classical logic, while the IK t system and its modification were discussed as the minimal temporal logic system built based on intuitionistic logic. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
16 pages, 291 KiB  
Article
Aristotle’s Syllogistic as a Deductive System
by Piotr Kulicki
Axioms 2020, 9(2), 56; https://doi.org/10.3390/axioms9020056 - 19 May 2020
Cited by 2 | Viewed by 6255
Abstract
Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the [...] Read more.
Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
9 pages, 264 KiB  
Article
Distribution Tableaux, Distribution Models
by J.-Martín Castro-Manzano
Axioms 2020, 9(2), 41; https://doi.org/10.3390/axioms9020041 - 17 Apr 2020
Cited by 5 | Viewed by 2557
Abstract
The concept of distribution is a concept within traditional logic that has been fundamental for the syntactic development of Sommers and Englebretsen’s term functor logic, a logic that recovers the term syntax of traditional logic. The issue here, however, is that the semantic [...] Read more.
The concept of distribution is a concept within traditional logic that has been fundamental for the syntactic development of Sommers and Englebretsen’s term functor logic, a logic that recovers the term syntax of traditional logic. The issue here, however, is that the semantic counterpart of distribution for this logic is still in the making. Consequently, given this disparity between syntax and semantics, in this contribution we adapt some ideas of term functor logic tableaux to develop models of distribution, thus providing some alternative formal semantics to help close this breach. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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11 pages, 251 KiB  
Article
A Note on Fernández–Coniglio’s Hierarchy of Paraconsistent Systems
by Janusz Ciuciura
Axioms 2020, 9(2), 35; https://doi.org/10.3390/axioms9020035 - 30 Mar 2020
Cited by 3 | Viewed by 2943
Abstract
A logic is called explosive if its consequence relation validates the so-called principle of ex contradictione sequitur quodlibet. A logic is called paraconsistent so long as it is not explosive. Sette’s calculus P 1 is widely recognized as one of the most important [...] Read more.
A logic is called explosive if its consequence relation validates the so-called principle of ex contradictione sequitur quodlibet. A logic is called paraconsistent so long as it is not explosive. Sette’s calculus P 1 is widely recognized as one of the most important paraconsistent calculi. It is not surprising then that the calculus was a starting point for many research studies on paraconsistency. Fernández–Coniglio’s hierarchy of paraconsistent systems is a good example of such an approach. The hierarchy is presented in Newton da Costa’s style. Therefore, the law of non-contradiction plays the main role in its negative axioms. The principle of ex contradictione sequitur quodlibet has been marginalized: it does not play any leading role in the hierarchy. The objective of this paper is to present an alternative axiomatization for the hierarchy. The main idea behind it is to focus explicitly on the (in)validity of the principle of ex contradictione sequitur quodlibet. This makes the hierarchy less complex and more transparent, especially from the viewpoint of paraconsistency. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
10 pages, 211 KiB  
Article
Term Logic
by Peter Simons
Axioms 2020, 9(1), 18; https://doi.org/10.3390/axioms9010018 - 10 Feb 2020
Cited by 4 | Viewed by 2573
Abstract
The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, [...] Read more.
The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, a constant empty term, and term conjunction and negation. The idea of basing term logic on existence or non-existence, outlined by Brentano, is here carried through in modern guise. It is shown how categorical syllogistic reduces to just two forms of inference. Tree and diagram methods of testing validity are described. An obvious translation into monadic predicate logic shows the system is decidable, and additional expressive power brought by adding quantifiers enables numerical predicates to be defined. The system’s advantages for pedagogy are indicated. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
7 pages, 230 KiB  
Article
The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term
by Eugeniusz Wojciechowski
Axioms 2020, 9(1), 6; https://doi.org/10.3390/axioms9010006 - 31 Dec 2019
Cited by 1 | Viewed by 2112
Abstract
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the [...] Read more.
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the original natural number (N) term is replaced by the term Anzahl (A). The functor of the successor (S) is defined in it. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
25 pages, 397 KiB  
Article
Synthetic Tableaux with Unrestricted Cut for First-Order Theories
by Dorota Leszczyńska-Jasion and Szymon Chlebowski
Axioms 2019, 8(4), 133; https://doi.org/10.3390/axioms8040133 - 29 Nov 2019
Cited by 4 | Viewed by 3117
Abstract
The method of synthetic tableaux is a cut-based tableau system with synthesizing rules introducing complex formulas. In this paper, we present the method of synthetic tableaux for Classical First-Order Logic, and we propose a strategy of extending the system to first-order theories axiomatized [...] Read more.
The method of synthetic tableaux is a cut-based tableau system with synthesizing rules introducing complex formulas. In this paper, we present the method of synthetic tableaux for Classical First-Order Logic, and we propose a strategy of extending the system to first-order theories axiomatized by universal axioms. The strategy was inspired by the works of Negri and von Plato. We illustrate the strategy with two examples: synthetic tableaux systems for identity and for partial order. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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19 pages, 327 KiB  
Article
Hybrid Deduction–Refutation Systems
by Valentin Goranko
Axioms 2019, 8(4), 118; https://doi.org/10.3390/axioms8040118 - 21 Oct 2019
Cited by 5 | Viewed by 2814
Abstract
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. [...] Read more.
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. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
19 pages, 366 KiB  
Article
Deduction in Non-Fregean Propositional Logic SCI
by Joanna Golińska-Pilarek and Magdalena Welle
Axioms 2019, 8(4), 115; https://doi.org/10.3390/axioms8040115 - 17 Oct 2019
Cited by 4 | Viewed by 2722
Abstract
We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that [...] Read more.
We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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13 pages, 1246 KiB  
Article
Deontic Logics as Axiomatic Extensions of First-Order Predicate Logic: An Approach Inspired by Wolniewicz’s Formal Ontology of Situations
by Andrzej Malec
Axioms 2019, 8(4), 109; https://doi.org/10.3390/axioms8040109 - 06 Oct 2019
Cited by 3 | Viewed by 2137
Abstract
The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events [...] Read more.
The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events are understood as sequences (strings) of elementary situations in Wolniewicz′s sense. On the other hand, acts are understood as two-element legal events: the first element of a sequence is a choice situation (a situation that will be changed by an act), and the second element of this sequence is a chosen situation (a situation that arises as a result of that act). In this approach, legal rules (i.e., orders, bans, permits) are treated as sets of legal events. The article presents four deontic systems for legal events: AEP, AEPF, AEPOF, AEPOFI. In the first system, all legal events are permitted; in the second, they are permitted or forbidden; in the third, they are permitted, ordered or forbidden; and in the fourth, they are permitted, ordered, forbidden or irrelevant. Then, we present a deontic logic for acts (AAPOF), in which every act is permitted, ordered or forbidden. The theorems of this logic reflect deontic relations between acts as well as between acts and their parts. The direct inspiration to develop the approach presented in the article was the book Ontology of Situations by Boguslaw Wolniewicz, and indirectly, Wittgenstein’s Tractatus Logico-Philosophicus. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
17 pages, 381 KiB  
Article
A Kotas-Style Characterisation of Minimal Discussive Logic
by Krystyna Mruczek-Nasieniewska and Marek Nasieniewski
Axioms 2019, 8(4), 108; https://doi.org/10.3390/axioms8040108 - 01 Oct 2019
Cited by 4 | Viewed by 1822
Abstract
In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the [...] Read more.
In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
14 pages, 2298 KiB  
Article
Logic of Typical and Atypical Instances of a Concept—A Mathematical Model
by Jean-Pierre Desclés and Anca Christine Pascu
Axioms 2019, 8(3), 104; https://doi.org/10.3390/axioms8030104 - 04 Sep 2019
Cited by 2 | Viewed by 2812
Abstract
In this paper, we give a mathematical model of the logic of determination of objects (LDO) based on preordered sets, and a mathematical model of the logic of typical and atypical instances (LTA). We prove that LTA is an extension of LDO. It [...] Read more.
In this paper, we give a mathematical model of the logic of determination of objects (LDO) based on preordered sets, and a mathematical model of the logic of typical and atypical instances (LTA). We prove that LTA is an extension of LDO. It can manipulate several types of “exceptions”. Finally, we show that the structural part of LTA can be modeled by a quasi topology structure (QTS). Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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14 pages, 673 KiB  
Article
On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers
by Urszula Wybraniec-Skardowska
Axioms 2019, 8(3), 103; https://doi.org/10.3390/axioms8030103 - 04 Sep 2019
Cited by 4 | Viewed by 2903
Abstract
The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including [...] Read more.
The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
23 pages, 405 KiB  
Article
Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property
by Alex Citkin
Axioms 2019, 8(3), 100; https://doi.org/10.3390/axioms8030100 - 30 Aug 2019
Cited by 2 | Viewed by 2674
Abstract
Using the defined notion of the inference with multiply-conclusion rules, we show that in the logics enjoying the disjunction property, any derivable rule can be inferred from the single-conclusion rules and a single multiple-conclusion rule, which represents the disjunction property. Also, the conversion [...] Read more.
Using the defined notion of the inference with multiply-conclusion rules, we show that in the logics enjoying the disjunction property, any derivable rule can be inferred from the single-conclusion rules and a single multiple-conclusion rule, which represents the disjunction property. Also, the conversion algorithm of single- and multiple-conclusion deductive systems into each other is studied. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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Other

Jump to: Editorial, Research

3 pages, 168 KiB  
Book Review
Review of “The Significance of the New Logic” Willard Van Orman Quine. Edited and Translated by Walter Carnielli, Frederique Janssen-Lauret, and William Pickering. Cambridge University Press, Cambridge, UK, 2018, pp. 1–200. ISBN-10: 1107179025 ISBN-13: 978-1107179028
by Alfredo Roque Freire
Axioms 2019, 8(2), 64; https://doi.org/10.3390/axioms8020064 - 22 May 2019
Cited by 1 | Viewed by 2688
Abstract
In this review, I will discuss the historical importance of “The Significance of the New Logic” by Quine. This is a translation of the original “O Sentido da Nova Lógica” in Portuguese by Carnielli, Janssen-Lauret, and Pickering. The American philosopher wrote this book [...] Read more.
In this review, I will discuss the historical importance of “The Significance of the New Logic” by Quine. This is a translation of the original “O Sentido da Nova Lógica” in Portuguese by Carnielli, Janssen-Lauret, and Pickering. The American philosopher wrote this book in the beginning of the 1940s, before a major shift in his philosophy. Thus, I will argue that the reader must see this book as an introduction to an important period in his thinking. I will provide a brief summary of the chapters, remarking on valuable features in each of them and positions Quine abandoned in his later work. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
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