*An Introduction to Many-Valued and Fuzzy Logic*

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**Added by:**Berta Grimau,**Contributed by:****Publisher’s Note:**Professor Merrie Bergmann presents an accessible introduction to the subject of many-valued and fuzzy logic designed for use on undergraduate and graduate courses in non-classical logic. Bergmann discusses the philosophical issues that give rise to fuzzy logic – problems arising from vague language – and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three-valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical systems – Lukasiewicz, Gödel, and product logics – are then presented as generalisations of three-valued systems that successfully address the problems of vagueness. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, that ask students to continue proofs begun in the text, and that engage students in the comparison of logical systems.**Comment:**In the words of the author: 'This textbook can be used as a complete basis for an introductory course on formal many-valued and fuzzy logics, at either the upper-level undergraduate or the graduate level, and it can also be used as a supplementary text in a variety of courses. There is considerable flexibility in either case. The truth-valued semantic chapters are independent of the algebraic and axiomatic ones, so that either of the latter may be skipped. Except for Section 13.3 of Chapter 13, the axiomatic chapters are also independent of the algebraic ones, and an instructor who chooses to skip the algebraic material can simply ignore the latter part of 13.3. Finally, Lukasiewicz fuzzy logic is presented independently of Gödel and product fuzzy logics, thus allowing an instructor to focus solely on the former. There are exercises throughout the text. Some pose straightforward problems for the student to solve, but many exercises also ask students to continue proofs begun in the text, to prove results analogous to those in the text, and to compare the various logical systems that are presented.' The book does include a review of classical propositional and first-order logic, but the students should've taken at least one basic logic course before getting into this material.

*An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems*

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**Added by:**Berta Grimau,**Contributed by:****Publisher’s note:**This volume is an accessible introduction to the subject of many-valued and fuzzy logic suitable for use in relevant advanced undergraduate and graduate courses. The text opens with a discussion of the philosophical issues that give rise to fuzzy logic – problems arising from vague language – and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical systems – Lukasiewicz, Godel, and product logics – are then presented as generalizations of three-valued systems that successfully address the problems of vagueness. Semantic and axiomatic systems for three-valued and fuzzy logics are examined along with an introduction to the algebras characteristic of those systems. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, ask students to continue proofs begun in the text, and engage them in the comparison of logical systems.**Comment:**This book is ideal for an intermediate-level course on many-valued and/or fuzzy logic. Although it includes a presentation of propositional and first-order logic, it is intended for students who are familiar with classical logic. However, no previous knowledge of many-valued or fuzzy logic is required. It can also be used as a secondary reading for a general course on non-classical logics.

*The Logic Book*

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**Added by:**Berta Grimau,**Contributed by:****Summary:**This book is an introductory textbook on mathematical logic. It covers Propositional Logic and Predicate Logic. For each of these formalisms it presents its syntax and formal semantics as well as a tableaux-style method of consistency-checking and a natural deduction-style deductive calculus. Moreover, it discusses the metatheory of both logics.**Comment:**This book would be ideal for an introductory course on symbolic logic. It presupposes no previous training in logic, and because it covers sentential logic through the metatheory of first-order predicate logic, it is suitable for both introductory and intermediate courses in symbolic logic. The instructor who does not want to emphasize metatheory can simply omit Chapters 6 and 11. The chapters on truth-trees and the chapters on derivations are independent, so it is possible to cover truth-trees but not derivations and vice versa. However, the chapters on truth-trees do depend on the chapters presenting semantics; that is, Chapter 4 depends on Chapter 3 and Chapter 9 depends on Chapter 8. In contrast, the derivation chapters can be covered without first covering semantics. The Logic Book includes large exercise sets for all chapters. Answers to unstarred exercises appear in the Student Solutions Manual, available at www.mhhe.com/bergmann6e, while answers to starred exercises appear in the Instructor's Manual, which can be obtained by following the instructions on the same web page.

*Logical knowledge and ordinary reasoning*

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**Added by:**Berta Grimau,**Contributed by:****Abstract:**This paper argues that the prominent accounts of logical knowledge have the consequence that they conflict with ordinary reasoning. On these accounts knowing a logical principle, for instance, is having a disposition to infer according to it. These accounts in particular conflict with so-called ‘reasoned change in view’, where someone does not infer according to a logical principle but revise their views instead. The paper also outlines a propositional account of logical knowledge which does not conflict with ordinary reasoning.**Comment:**This paper proposes a certain characterisation of what it is to have knowledge of logical principles which makes it compatible with the way in which we reason ordinarily. It can be seen as an alternative to Harman's view in 'Change in View' according to which ordinary people do not at all 'employ' a deductive logic in reasoning. Thus this paper could be used in a course on the role of logic in reasoning, following the reading of Harman's work. More generally, this reading is suitable for any advanced undergraduate course or postgraduate course on the topic of rationality.

*Proof Theory: Sequent Calculi and Related Formalisms*

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**Added by:**Berta Grimau,**Contributed by:****Publisher’s Note:**Although sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, linear logic, and modal logic. In the first chapters, the author emphasizes classical logic and a variety of different sequent calculi for classical and intuitionistic logics. She then presents other non-classical logics and meta-logical results, including decidability results obtained specifically using sequent calculus formalizations of logics.**Comment:**This book can be used in a variety of advanced undergraduate and postgraduate courses. Chapters 1, 2, 3 and 8 may be useful in an advanced undergraduate or beginning graduate course, where an emphasis is placed on classical logic and on a range of different proof calculi (mainly for classical logic). Chapters 4, 5 and 6 deal almost exclusively with non-classical logics. Chapters 7 and 9 are rich in meta-logical results, including results that have been obtained specifically using sequent calculus formalizations of various logics. These last five chapters might be used in a graduate course that embraces classical and nonclassical logics together with their meta-theory. To facilitate the use of the book as a text in a course, the text is peppered with exercises. In general, the starring indicates an increase in difficulty, however, sometimes an exercise is starred simply because it goes beyond the scope of the book or it is very lengthy. Solutions to selected exercises may be found on the web at the URL www.ualberta.ca/˜bimbo/ProofTheoryBook.

*Frege and Hilbert on Consistency*

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**Added by:**Clotilde Torregrossa,**Contributed by:**Alex Yates**Abstract:**Gottlob Frege’s work in logic and the foundations of mathemat- ics centers on claims of logical entailment; most important among these is the claim that arithmetical truths are entailed by purely logical principles. Occupying a less central but nonetheless important role in Frege’s work are claims about failures of entailment. Here, the clearest examples are his theses that the truths of geometry are not entailed by the truths of logic or of arithmetic, and that some of them are not entailed by each other. As he, and we, would put it: the truths of Eluclidean geometry are independent of the truths of logic, and some of them are independent of one another.’ Frege’s talk of independence and related notions sounds familiar to a modern ear: a proposition is independent of a collection of propositions just in case it is not a consequence of that collection, and a proposition or collection of propositions is consistent just in case no contradiction is a consequence of it. But some of Frege’s views and procedures are decidedly tinmodern. Despite developing an extremely sophisticated apparattus for demonstrating that one claim is a consequience of others, Frege offers not a single demon- stration that one claim is not a conseqtuence of others. Thus, in par- tictular, he gives no proofs of independence or of consistency. This is no accident. Despite his firm commitment to the independence and consistency claims just mentioned, Frege holds that independence and consistency cannot systematically be demonstrated.2 Frege’s view here is particularly striking in light of the fact that his contemporaries had a fruitful and systematic method for proving consistency and independence, a method which was well known to him. One of the clearest applications of this method in Frege’s day came in David Hilbert’s 1899 Foundations of Geometry,3 in which he es- tablishes via essentially our own modern method the consistency and independence of various axioms and axiom systems for Euclidean geometry. Frege’s reaction to Hilbert’s work was that it was simply a failure: that its central methods were incapable of demonstrating consistency and independence, and that its usefulness in the founda- tions of mathematics was highly questionable.4 Regarding the general usefulness of the method, it is clear that Frege was wrong; the last one hundred years of work in logic and mathemat- ics gives ample evidence of the fruitfulness of those techniques which grow directly from the Hilbert-style approach. The standard view today is that Frege was also wrong in his claim that Hilbert’s methods fail to demonstrate consistency and independence. The view would seem to be that Frege largely missed Hilbert’s point, and that a better under- standing of Hilbert’s techniques would have revealed to Frege their success. Despite Frege’s historic role as the founder of the methods we now use to demonstrate positive consequence-results, he simply failed, on this account, to understand the ways in which Hilbert’s methods could be used to demonstrate negative consequence-results. The purpose of this paper is to question this account of the Frege- Hilbert disagreement. By 1899, Frege had a well-developed view of log- ical consequence, consistency, and independence, a view which was central to his foundational work in arithmetic and to the epistemologi- cal significance of that work. Given this understanding of the logical relations, I shall argue, Hilbert’s demonstrations do fail. Successful as they were in demonstrating significant metatheoretic results, Hilbert’s proofs do not establish the consistency and independence, in Frege’s sense, of geometrical axioms. This point is important, I think, both for an understanding of the basis of Frege’s epistemological claims about mathematics, and for an understanding of just how different Frege’s conception of logic is from the modern model-theoretic conception that has grown out of the Hilbert-style approach to consistency.**Comment:**Good for a historically-based course on philosophy of logic or mathematics.

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*Frege’s Conception of Logic*

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**Added by:**Clotilde Torregrossa,**Contributed by:**Alex Yates**Publisher’s Note:**In Frege’s Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege’s understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation.The first part of the book locates the role of conceptual analysis in Frege’s logicist project. Blanchette argues that despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals.

In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.

**Comment:**This book would be a suitable resource for independent study, or for a historically oriented course on philosophy of logic, of math, or on early analytic philosophy, especially one which looks at philosophical approaches to axiomatic systems.

*Logical Consequence*

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**Added by:**Berta Grimau,**Contributed by:**Patricia Blanchette**Description:**This article is a short overview of philosophical and formal issues in the treatment and analysis of logical consequence. The purpose of the paper is to provide a brief introduction to the central issues surrounding two questions: (1) that of the nature of logical consequence and (2) that of the extension of logical consequence. It puts special emphasis in the role played by formal systems in the investigation of logical consequence.**Comment:**This article can be used as background or overview reading in a course on the notion of logical consequence. It could also be used in a general course on philosophy of logic having a section on this topic. It makes very little use of technical notation, even though familiarity with first-order logic is required. It closes with a useful list of suggested further readings.

*Models and Modality*

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**Added by:**Berta Grimau,**Contributed by:**Patricia Blanchette**Abstract:**This paper examines the connection between model-theoretic truth and necessary truth. It is argued that though the model-theoretic truths of some standard languages are demonstrably “necessary” (in a precise sense), the widespread view of model-theoretic truth as providing a general guarantee of necessity is mistaken. Several arguments to the contrary are criticized.**Comment:**This text would be best used as secondary reading in an intermediate or an advanced philosophy of logic course. For example, it can be used as a secondary reading in a section on the connection between model-theoretic truth and necessary truth.

*Ancient Logic*

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**Added by:**Berta Grimau,**Contributed by:**Giada Fratantonio**Summary:**A comprehensive introduction to ancient (western) logic from the 5th century BCE to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Topics include pre-Aristotelian logic, Aristotelian logic, Peripatetic logic, Stoic Logic and a note on Epicureans and their views on logic.**Comment:**This paper would be ideal as an introductory overview for a course on ancient logic. Alternatively, it could serve as an overview for a module on ancient logic within a more general course on the history of logic. No prior knowledge of logic is required; formalisms are for the most part avoided in the paper. Note that this is a SEP entry, so it's completely accessible to students.

*Stoic Syllogistic*

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**Added by:**Berta Grimau,**Contributed by:**Giada Fratantonio**Abstract:**For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out.**Comment:**This paper can be used as specialised/further reading for an advanced undergrad or postgraduate course on ancient logic or as a primary reading in an advanced undergrad or postgraduate course on Stoic logic. Alternatively, given that the text argues that there are important parallels between Stoic logic and Relevance logic, it could be used in a course on Relevance logic as well. It requires prior knowledge of logic (in particular, proof theory).

*Critical Thinking: A Concise Guide*

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**Added by:**Berta Grimau,**Contributed by:****Publisher’s note:**We are frequently confronted with arguments. Arguments are attempts to persuade us – to influence our beliefs and actions – by giving us reasons to believe this or that. Critical Thinking: A Concise Guide will equip students with the concepts and techniques used in the identification, analysis and assessment of arguments. Through precise and accessible discussion, this book provides the tools to become a successful critical thinker, one who can act and believe in accordance with good reasons, and who can articulate and make explicit those reasons.

Key topics discussed include:- Core concepts in argumentation.
- How language can serve to obscure or conceal the real content of arguments; how to distinguish argumentation from rhetoric.
- How to avoid common confusions surrounding words such as ‘truth’, ‘knowledge’ and ‘opinion’.
- How to identify and evaluate the most common types of argument.
- How to distinguish good reasoning from bad in terms of deductive validly and induction.

**Comment:**Appropriate for complete beginners to logic and philosophy. Adequate for an introduction to critical thinking. It doesn't presuppose any previous knowledge of logic. Moreover, there is an interactive website for the book which provides resources for both instructors and students including new examples and case studies, flashcards, sample questions, practice questions and answers, student activities and a test bank of questions for use in the classroom.

*First Order Logic: An Introduction*

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**Added by:**Berta Grimau,**Contributed by:**Matt Clemens**Publisher’s Note:**This teaching book is designed to help its readers to reason systematically, reliably, and to some extent self-consciously, in the course of their ordinary pursuits-primarily in inquiry and in decision making. The principles and techniques recommended are explained and justified – not just stated; the aim is to teach orderly thinking, not the manipulation of symbols. The structure of material follows that of Quine’s Methods of Logic, and may be used as an introduction to that work, with sections on truth-functional logic, predicate logic, relational logic, and identity and description. Exercises are based on problems designed by authors including Quine, John Cooley, Richard Jeffrey, and Lewis Carroll.**Comment:**This book is adequate for a first course on formal logic. Moreover, its table of contents follows that of Quine's "Methods of Logic", thus it can serve as an introduction or as a reference text for the study of the latter.

*Avicenna on Possibility and Necessity*

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**Added by:**Sara Peppe,**Contributed by:****Abstract:**In this paper, I raise the following problem: How does Avicenna define modalities? What oppositional relations are there between modal propositions, whether quantified or not? After giving Avicenna’s definitions of possibility, necessity and impossibility, I analyze the modal oppositions as they are stated by him. This leads to the following results:1. The relations between the singular modal propositions may be represented by means of a hexagon. Those between the quantified propositions may be represented by means of two hexagons that one could relate to each other.

2. This is so because the exact negation of the bilateral possible, i.e. ‘necessary or impossible’ is given and applied to the quantified possible propositions.

3. Avicenna distinguishes between the scopes of modality which can be either external (de dicto) or internal (de re). His formulations are external unlike al-F̄ar̄ab̄;’s ones.

However his treatment of modal oppositions remains incomplete because not all the relations between the modal propositions are stated explicitly. A complete analysis is provided in this paper that fills the gaps of the theory and represents the relations by means of a complex figure containing 12 vertices and several squares and hexagons.

**Comment:**This article is useful for eastern philosophy courses and logic courses. Even if in the first part it provides an introductory section on Avicenna's perspective, it would be better to have some pre-esxisting background on this latter one.

*Logical Self Reference, Set Theoretical Paradoxes and the Measurement Problem in Quantum Mechanics*

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**Added by:**Sara Peppe,**Contributed by:****Introduction:**From a logical point of view the measurement problem of quantum mechanics, can be described as a characteristic question of ‘semantical closure’ of a theory: to what extent can a consistent theory (in this case 2R) be closed with respect to the objects and the concepfs which are described and expressed in its metatheory?**Comment:**This paper considers the measurement problem in Quantum Mechanics from a logical perspective. Previous and deep knowledge of logics and Quantum Mechanics' theories is vital.