
Expand entry

Added by: Berta GrimauPublisher's note: This volume is an accessible introduction to the subject of manyvalued 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 threevalued systems that successfully address the problems of vagueness. Semantic and axiomatic systems for threevalued 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.

Expand entry

Added by: Berta GrimauPublisher'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 nonclassical 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 nonclassical logics and metalogical 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 nonclassical logics. Chapters 7 and 9 are rich in metalogical 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 metatheory. 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.

Expand entry

Added by: Berta Grimau, Contributed by: Giada FratantonioAbstract: 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 nonindemonstrable 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 nonindemonstrable 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).

Expand entry

Added by: Franci MangravitiAbstract:
This entry presents the framework of « dialogical logic » in the initial Lorenzen and Lorenz tradition. The rules for the game and for building strategies are provided with step by step examples, helping the reader understand how the dialogue tables reflect a dynamic process of interaction between the players. Various logics are presented within this pluralistic framework: intuitionist logic, classical logic, and modal logics, with references to various other logics. In a second part of the entry, objections against the framework are considered, together with answers provided by the « Immanent Reasoning » variant, which stays within the Lorenzen and Lorenz tradition, and by the « BuiltIn Opponent » variant first developed by Catarina Dutilh Novaes, which develops a different dialogical tradition.
Comment: Obvious overview choice for any course involving dialogical logic. Familiarity with firstorder languages is a prerequisite.

Expand entry

Added by: Sara PeppeIntroduction: 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.

Expand entry

Added by: Franci Mangraviti and Viviane FairbankAbstract: The strongest and, until recently, leastexplored approach to feminist logic holds that some formal logics have structural features that perpetuate sexism and oppression, whereas other logics are helpful for resisting and opposing these social phenomena. Our choice of logics may not be purely formal on this view: for example, some logics are preferrable to others on the grounds of feminist commitments. This strong account of feminist logic was first articulated by Val Plumwood. We will critically engage salient features of her view, especially her critique of classical logic and the centering and dominating functions she believes classical negation has. We will see that her understanding of classical negation captures neither the development of Intersectional Feminism, nor the position the concept of centering holds in transformative justice. However, Plumwood's critique of classical negation does lead us to a deeper insight regarding which logics to apply in social justice contexts. Robin Dembroff's analysis of genderqueer as a critical gender kind helps us delineate a nonclassical context in which a fourvalued logic, such as FDE, can structurally account for the critical feature of this gender kind in a way classical logic cannot. We will also observe how fourvalued logics precisely capture the destabilization of, and resistance to, the exclusive and exhaustive gender binary categories Dembroff describes.
Comment: available in this Blueprint

Expand entry

Added by: Franci MangravitiAbstract:
Val Plumwood’s 1993 paper, “The politics of reason: towards a feminist logic” (hence forth POR) attempted to set the stage for what she hoped would begin serious feminist exploration into formal logic – not merely its historical abuses, but, more importantly, its potential uses. This work offers us: (1) a case for there being feminist logic; and (2) a sketch of what it should resemble. The former goal of Plumwood’s paper encourages feminist theorists to reject antilogic feminist views. The paper’s latter aim is even more challenging. Plumwood’s critique of classical negation (and classical logic) as a logic of domination asks us to recognize that particular logical systems are weapons of oppression. Against antilogic feminist theorists, Plumwood argues that there are other logics besides classical logic, such as relevant logics, which are suited for feminist theorizing. Some logics may oppress while others may liberate. We provide details about the sources and context for her rejection of classical logic and motivation for promoting relevant logics as feminist.
Comment (from this Blueprint): This is an ideal companion piece to Plumwood's paper: it provides an accessible summary, and discusses both objections to the paper and possible responses.

Expand entry

Added by: Franci MangravitiAbstract:
From the newsletter's introduction: "Lauren Eichler [...] examines the resonances between feminist and Native American analyses of classical logic. After considering the range of responses, from overly monolithic rejection to more nuanced appreciation, Eichler argues for a careful, pluralist understanding of logic as she articulates her suggestion that feminists and Native American philosophers could build fruitful alliances around this topic."
Comment: available in this Blueprint

Expand entry

Added by: Jie GaoPublisher's Note: The first systematic exposition of all the central topics in the philosophy of logic, Susan Haack's book has established an international reputation (translated into five languages) for its accessibility, clarity, conciseness, orderliness, and range as well as for its thorough scholarship and careful analyses. Haack discusses the scope and purpose of logic, validity, truthfunctions, quantification and ontology, names, descriptions, truth, truthbearers, the settheoretical and semantic paradoxes, and modality. She also explores the motivations for a whole range of nonclassical systems of logic, including manyvalued logics, fuzzy logic, modal and tense logics, and relevance logics.
Comment: This textbook is intended particularly for philosophy students who have completed a first course in elementary logic. But, though the book is clearly written, such students still may find the content difficult, as it addresses difficult topics in the foundations of logic the primary literature for which is very technical. That said, it has been a widely used textbook for courses on philosophy of logic. Chapters of it can be used individually in accordance with the arrangements of the course.

Expand entry

Added by: Franci Mangraviti and Viviane FairbankAbstract: Val Plumwood charged classical logic not only with the invalidity of some of its laws, but also with the support of systemic oppression through naturalization of the logical structure of dualisms. In this paper I show that the latter charge  unlike the former  can be carried over to classical mathematics, and I propose a new conception of inconsistent mathematics  queer incomaths  as a liberatory activity meant to undermine said naturalization.
Comment: available in this Blueprint

Expand entry

Added by: Franci MangravitiAbstract:
The author argues that there is a strong connection between the dualisms that have strengthened and naturalized systematic oppression across history (man/woman, reason/emotion, etc.), and "classical" logic. It is suggested that feminism's response should not be to abandon logic altogether, but rather to focus on the development of alternative, less oppressive forms of rationality, of which relevant logics provide an example.
Comment (from this Blueprint): This is a seminal text of feminist logic, and thus a natural pick for any course wanting to discuss the topic. It could however also be assigned in a course on relevant logics interested in discussing particular applications, especially if such a course has previously spent time on the arguments in Plumwood's "False laws of logic" (or more generally, in Sylvan&co's "Relevant logics and their rivals"). Eckert and Donahue's "Towards a Feminist Logic" is a useful reading companion.

Expand entry

Added by: Franci MangravitiAbstract:
Being a pragmatic and not a referential approach to semantics, the dialogical formulation of paraconsistency allows the following semantic idea to be expressed within a semiformal system: In an argumentation it sometimes makes sense to distinguish between the contradiction of one of the argumentation partners with himself (internal contradiction) and the contradiction between the partners (external contradiction). The idea is that external contradiction may involve different semantic contexts in which, say A and not A have been asserted. The dialogical approach suggests a way of studying the dynamic process of contradictions through which the two contexts evolve for the sake of argumentation into one system containing both contexts. More technically, we show a new, dialogical, way to build paraconsistent systems for propositional and ﬁrstorder logic with classical and intuitionistic features (i.e. paraconsistency both with and without tertium nondatur) and present their corresponding tableaux.
Comment: This paper would fit well in a course on dialogical formulations of logic (as either main or further reading, depending on the time dedicated to Lorenzstyle approaches), or in a course on paraconsistent logic (as an alternative way of thinking about paraconsistency); both topics are introduced in an accessible enough way. If students have no familiarity with tableaux systems, sections 4 and 5.2 can be skipped.

Expand entry

Added by: Franci MangravitiAbstract:
The problems of the meaning and function of negation are disentangled from ontological issues with which they have been long entangled. The question of the function of negation is the crucial issue separating relevant and paraconsistent logics from classical theories. The function is illuminated by considering the inferential role of contradictions, contradiction being parasitic on negation. Three basic modelings emerge: a cancellation model, which leads towards connexivism, an explosion model, appropriate to classical and intuitionistic theories, and a constraint model, which includes relevant theories. These three modelings have been seriously confused in the modern literature: untangling them helps motivate the main themes advanced concerning traditional negation and natural negation. Firstly, the dominant traditional view, except around scholastic times when the explosion view was in ascendency, has been the cancellation view, so that the mainstream negation of much of traditional logic is distinctively nonclassical. Secondly, the primary negation determinable of natural negation is relevant negation. In order to picture relevant negation the traditional idea of negation as otherthanness is progressive) refined, to nonexclusive restricted otherthanness. Several pictures result, a reversal picture, a debate model, a record cabinet (or files of the universe) model which help explain relevant negation. Two appendices are attached, one on negation in Hegel and the Marxist tradition, the other on Wittgenstein's treatment of negation and contradiction.
Comment: Can be used in a course on relevant logic or on negation. The emphasis on comparing different models makes it ideal for discussion. No familiarity with relevant logic is required.

Expand entry

Abstract:
From the introduction: "we argue that the semantics of the first degree paradoxfree implication system FD supports the claim it is superior to strict implication as an analysis of entailment at the first degree level. The semantics also reveals that Disjunctive Syllogism, [...] far from being a paradigmatic entailment, is invalid, and allows the illegitimate suppression of tautologies"
Comment: The paper introduces some of the central ideas in the relevance logic literature, e..g the connection between suppression and sufficiency, and the modeling of negation via the Routley star. It is a natural pick for a specialized course on relevance logic, but it can also be used as an introduction to (or further reading about) relevance logic in a general course on nonclassical logics. Some familiarity with classical and modal logic (in particular, the notion of strict implication) is required.

Expand entry

Added by: Franci MangravitiAbstract:
Some writers object to logical pluralism on the grounds that logic is normative. The rough idea is that the relation of logical consequence has consequences for what we ought to think and how we ought to reason, so that pluralism about the consequence relation would result in an incoherent or unattractive pluralism about those things. In this paper I argue that logic isn’t normative. I distinguish three diﬀerent ways in which a theory – such as a logical theory – can be entangled with the normative and argue that logic is only entangled in the weakest of these ways, one which requires it to have no normativity of its own. I use this view to show what is wrong with three diﬀerent arguments for the conclusion that logic is normative.
Comment: Appropriate for any course touching on the normativity of logic question. Familiarity with the question and with logical pluralism is helpful, but not required. Could be paired with a defense of normativity for discussion.
 1
 2
Comment: This book is ideal for an intermediatelevel course on manyvalued and/or fuzzy logic. Although it includes a presentation of propositional and firstorder logic, it is intended for students who are familiar with classical logic. However, no previous knowledge of manyvalued or fuzzy logic is required. It can also be used as a secondary reading for a general course on nonclassical logics. In the words of the author: 'The truthvalued 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.'