Provides comprehensive coverage of logical structure as well as the techniques of formal reasoning.

Provides comprehensive coverage of logical structure as well as the techniques of formal reasoning.

Argumentation, which can be abstractly defined as the interaction of different arguments for and against some conclusion, is an important skill to learn for everyday life, law, science, politics and business. The best way to learn it is to try it out on real instances of arguments found in everyday conversational exchanges and legal argumentation. The introductory chapter of this book gives a clear general idea of what the methods of argumentation are and how they work as tools that (...) can be used to analyze arguments. Each subsequent chapter then applies these methods to a leading problem of argumentation. Today the field of computing has embraced argumentation as a paradigm for research in artificial intelligence and multi-agent systems. Another purpose of this book is to present and refine tools and techniques from computing as components of the methods that can be handily used by scholars in other fields. (shrink)

Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by (...) many researchers to try to put some order in the present logical jungle. Thus Cl91], Ep90] and Wo88] are three recent books in which an attempt is made to develop a general theoretical framework for the study of logics. On the more pragmatic side, several systems have been developed with the goal of providing a computerized logical framework in which many di erent logical systems can be implemented in a uniform way. An example is the Edinburgh LF( HHP91]). (shrink)

In the second half of Book One of the Prior Analytics, Aristotle reflects on the application of the formalized logic has developed in the first half, focusing particularly on the non-modal or assertoric syllogistic developed in the first seven chapters. These reflections lead Alexander of Aphrodisias, who was a great exponent of Aristotelianism in the late second century, to explain and sometimes argue against subsequent developments of Aristotle's logic and alternatives and objections to it, ideas associated mainly with (...) his colleague Theophrastus and with the Stoics. The other main topic of this part of the Prior Analytics is the specification of a method for discovering true premises needed to prove a given proposition. Aristotle's presentation is sometimes difficult to follow, and Alexander's discussion is extremely helpful to the uninitiated reader. In his commentary on the final chapter translated in this volume, Alexander provides an insightful account of Aristotle's criticism of Plato's method of division. (shrink)

Epistemic logic proceeds axiomatically. ”An agent knows that A” is formalized as a modal operator in a formal language which is interpreted using the standard apparatus of modal logic.

This paper develops an account of Wittgenstein’s method of language-games as a method of logic that exhibits important continuities with Russell’s and the early Wittgenstein’s conceptions of logic and logical analysis as the method of philosophy. On the proposed interpretation, the method of language-games is a method for isolating and modeling aspects of the uses of linguistic expressions embedded in human activities that enables one to make perspicuous complex uses of expressions by gradually building up the complexity of (...) clarificatory models. Wittgenstein’s introduction of the language-game method constitutes an attempt to overcome certain limitations of calculus-based logical methods, and to respond in this way to problems with Russell’s and his own early philosophy of logic. The method is nevertheless compatible with the employment of calculus-based methods in logic and philosophy, and makes no exclusive claim to being the correct method. (shrink)

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction (...) method and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

The aim of this paper is to present the method of Socratic proofs for seven modal propositional logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. This work is an extension of [10] where the method was presented for the most common modal propositional logics: K, D, T, KB, K4, S4 and S5.

Jaśkowski’s discussive logic D2 was formulated with the help of the modal logic S5 as follows (see [7, 8]): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \in {D_{2}}}$$\end{document} iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}$$\end{document}, where (–)• is a translation of discussive formulae from Ford into the modal language. We say that a modal logic L defines D2 iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$${{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}$$\end{document}. In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (†): have the same theses beginning with ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond}$$\end{document}’ as S5. Of course, all logics fulfilling the above condition, define D2. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D2 is equivalent to having the property (†). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (†). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D2. (shrink)

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill (1806–73) disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, (...) the work constitutes essential reading for anyone seeking a full understanding of Mill's thought. Volume 1 contains Mill's introduction, which elaborates upon his definition of logic as 'not the science of Belief, but the science of Proof, or Evidence'. It also features discussions of the central components of logical reasoning - propositions and syllogisms - in relation to Mill's theories of inductive reasoning and experimental method. (shrink)

Jaśkowski’s discussive logic D2 was formulated with the help of the modal logic S5 as follows (see [7, 8]): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \in {D_{2}}}$$\end{document} iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}$$\end{document}, where (–)• is a translation of discussive formulae from Ford into the modal language. We say that a modal logic L defines D2 iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$${{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}$$\end{document}. In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (†): have the same theses beginning with ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond}$$\end{document}’ as S5. Of course, all logics fulfilling the above condition, define D2. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D2 is equivalent to having the property (†). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (†). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D2. (shrink)

1. STRUCTURE AND REFERENCES 1.1. The main part of the dictionary consists of alphabetically arranged articles concerned with basic logical theories and some other selected topics. Within each article a set of concepts is defined in their mutual relations. This way of defining concepts in the context of a theory provides better understand ing of ideas than that provided by isolated short defmitions. A disadvantage of this method is that it takes more time to look something up inside an extensive (...) article. To reduce this disadvantage the following measures have been adopted. Each article is divided into numbered sections, the numbers, in boldface type, being addresses to which we refer. Those sections of larger articles which are divided at the first level, i.e. numbered with single numerals, have titles. Main sections are further subdivided, the subsections being numbered by numerals added to the main section number, e.g. I, 1.1, 1.2,..., 1.1.1, 1.1.2, and so on. A comprehensive subject index is supplied together with a glossary. The aim of the latter is to provide, if possible, short defmitions which sometimes may prove sufficient. As to the use of the glossary, see the comment preceding it. (shrink)

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, the (...) work constitutes essential reading for anyone seeking a full understanding of Mill's thought. Continuing the discussion of induction, Volume 2 concludes with Book VI, 'On the Logic of the Moral Sciences', in which Mill applies empirical reasoning to human behaviour. A crucial early formulation of his thinking regarding free will and necessity, this book establishes the centrality of 'the social science' to Mill's philosophy. (shrink)

Peirce’s triadic logic has been under discussion since its discovery in the 1960s by Fisch and Turquette. The experiments with matrices of three-valued logic are recorded in a few pages of unpublished manuscripts dated 1909, a decade before similar systems have been developed by logicians. The purposes of Peirce’s work on such logic, as well as semantical aspects of his system, are disputable. In the most extensive work about it, Turquette suggested that the matrices are related in (...) dual pairs of axiomatic Hilbert-style systems. In this paper, we present a simple tableau proof for a fragment of Peirce three-valued logic, called P3, based on similar approaches in many-valued literature. We demonstrated that this proof is sound and complete. Besides that, taking the false as the only undesignated value and adding non-classical negations to the calculus, we can explore paraconsistent and paracompleteness theories into P3. Keywords: Charles S. Peirce, many-valued logics, theory of proof, tableau method. (shrink)

I consider two cognitive phenomena: inquiring and justifying, as complementary processes running in opposite directions. I explain on an example that the former process is driven by questions and the latter is a codification of the results of the first one. Traditionally, proof theory focuses on the latter process, and thus describes the former, at best, as an example of a backward proof search. I argue that this is not the best way to analyze cognitive processes driven by questions, and (...) that proof-theoretical analysis of questions can bring mutual benefits for both proof theory and erotetic logic. In the second part of my paper (sect. 6.3) I introduce the structural rules of weakening and contraction. Results concerning their admissibility are discussed. I also sketch an erotetic interpretation of the rule of weakening as reducing information noise, and that of contraction as securing information. (shrink)

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional connectives of Łukasiewicz’s (...) logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete. (shrink)

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, the (...) work constitutes essential reading for anyone seeking a full understanding of Mill's thought. Volume 1 contains Mill's introduction, which elaborates upon his definition of logic as 'not the science of Belief, but the science of Proof, or Evidence'. It also features discussions of the central components of logical reasoning - propositions and syllogisms - in relation to Mill's theories of inductive reasoning and experimental method. (shrink)

The method of doubt has been used in philosophy and theology by both philosophers and theologians, among them al-Ghazālī. Al-Ghazālī’s method conveys the process of how he was cured of his epistemological and existential crisis. This study analyzes each phase of the process in terms of epistemology and logic; it explains the problems and how they appeared to al-Ghazālī.

The method of doubt has been used in philosophy and theology by both philosophers and theologians, among them al-Ghazālī. Al-Ghazālī’s method conveys the process of how he was cured of his epistemological and existential crisis. This study analyzes each phase of the process in terms of epistemology and logic; it explains the problems and how they appeared to al-Ghazālī.

The Tractatus revolves around the connection between two central topics – the preconditions of symbolic representation and the nature of logic-cum-philosophy. Proper philosophy is an activity, namely of revealing the hidden structures that allow language to represent reality by way of logical analysis. At the same time the main purpose of such logical analysis consists in revealing metaphysical statements to be nonsensical. In the subsequent development of analytic philosophy, these two ideas parted company. The positive aim of revealing the (...) logical form of sentences resulted in a program closely associated with Davidson, namely a theory of meaning for natural languages that yields metaphysical corollaries. The negative aim of overcoming metaphysics ushered in the activity of dissolving conceptual confusions through conceptual rather than logical analysis, propelled by the later Wittgenstein. My presentation pursues these historical lines of influence. But the ultimate aim is a substantive one, namely to establish whether the two projects can be kept apart. With the later Wittgenstein, I criticize the claim that formal calculi constitute the hidden structure of natural languages. But I also contend that the hope of engaging in anti-metaphysical dialectic without relying on logico-conceptual analysis of some kind falls prey to a “myth of mere method”. (shrink)