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18 results in Cambridge Elements

Nonmonotonic Logic

Nonmonotonic Logic
  • Christian Straßer
  • Published online:
    18 October 2025
    Print publication:
    13 November 2025
  • Nonmonotonic logics serve as formal models of defeasible reasoning, a type of reasoning where conclusions are drawn absent absolute certainty. Defeasible reasoning takes place when scientists interpret experiments, in medical diagnosis, and in practical everyday situations. Given its wide range of applications, nonmonotonic logic is of interest to philosophy, psychology, and artificial intelligence. This Element provides a systematic introduction to the multifaceted world of nonmonotonic logics. Part I familiarizes the reader with basic concepts and three central methodologies: formal argumentation, consistent accumulation, and semantic methods. Parts II–IV provide a deeper understanding of each of these methods by introducing prominent logics within each paradigm. Despite the apparent lack of unification in the domain of nonmonotonic logics, this Element reveals connections between the three paradigms by demonstrating translations among them. Whether you're a novice or an experienced traveler, this Element provides a reliable map for navigating the landscape of nonmonotonic logic.
Logical Pluralism

Logical Pluralism
  • Colin R. Caret
  • Published online:
    30 September 2025
    Print publication:
    23 October 2025
  • Logical pluralism is the view that there is more than one correct logic. This view emerged in a dialectical context in which certain laws of logic were hotly debated by philosophers. For example, philosophers have spilled a great deal of ink over the logical principle of explosion ('from a contradiction, everything follows'). One side in the debate accepts this principle, the other side rejects it. It is exceedingly natural to assume that these rival points of view are incompatible, hence one side of the debate is correct while the other is incorrect. This is logical monism: the view that there is exactly one correct logic. Pluralists argue that the monistic assumption is subtly and surprisingly wrong. According to the pluralist, some logics that appear to be irreconcilable rivals are, in fact, both correct in their own ways. This Element will explain the debate over logical pluralism in an accessible manner.
Logic and Science

Logic and Science
  • An Exploration of Logical Anti-Exceptionalism
  • Filippo Ferrari, Massimiliano Carrara
  • Published online:
    10 April 2025
    Print publication:
    10 April 2025
    • Element
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  • This Element delves into the relationship between logic and the sciences, a topic brought to prominence by Quine, who regarded logic as methodologically and epistemologically akin to the sciences. For this reason, Quine is seen as the forefather of anti-exceptionalism about logic (AEL), a stance that has become prevalent in the philosophy of logic today. Despite its popularity and the volume of research it inspires, some core issues still lack clarity. For one thing, most works in the debate remain vague on what should count as logic and what should count as a science. Furthermore, the terms of the comparison are rarely specified and discussed in a systematic way. This Element purports to advance the debate on these crucial issues with the hope of fostering our understanding of the fundamentals of AEL. This title is also available as Open Access on Cambridge Core.
Probability and Inductive Logic

Probability and Inductive Logic
  • Antony Eagle
  • Published online:
    30 January 2025
    Print publication:
    30 January 2025
  • Reasoning from inconclusive evidence, or 'induction', is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This element explores the prospects of using probability theory to provide an inductive logic: a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that the overall standing of a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated by the hypothesis having a higher probability conditional on some evidence than it does unconditionally. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge of inductive scepticism, and the objection that the probabilistic approach makes evidential support arbitrary.
Free Logic

Free Logic
  • A Generalization
  • Greg Frost-Arnold
  • Published online:
    09 January 2025
    Print publication:
    30 January 2025
  • Classical logic assumes that names are univocal: every name refers to exactly one existing individual. This Principle of Univocality has two parts: an existence assumption and a uniqueness assumption. The existence assumption holds that every name refers to at least oneindividual, and the uniqueness assumption states that every name refers to at most one individual. The various systems of free logic which have been developed and studied since the 1960s relax the existence assumption, but retain the uniqueness assumption. The present work investigates violations of both halves of the Principle of Univocality. That is, whereas the free logics developed from the 1960s are called 'free' because they are free of existential assumptions, the current Element generalizes this idea, to study logics that are free of uniqueness assumptions. We explore several versions of free logic, comparing their advantages and disadvantages. Applications of free logic to other areas of philosophy are explored.
Meinongianism

Meinongianism
  • Maria Elisabeth Reicher
  • Published online:
    09 January 2025
    Print publication:
    16 January 2025
  • Meinongianism (named after Alexius Meinong) is, roughly, the view that there are not only existent but also nonexistent objects. In this book, Meinong's so-called object theory as well as “neo-Meinongian” reconstructions are presented and discussed, especially with respect to logical issues, both from a historical and a systematic perspective. Among others, the following topics are addressed: basic principles and motivations for Meinongianism; the distinction between “there is” (“x”) and “exists” (“E!”); interpretations and kinds of quantification; Meinongianism, the principle of excluded middle and the principle of non-contradiction; the nuclear-extranuclear distinction and modes of predication; varieties of neo-Meinongianism and Meinongian logics.
The Logic of Grounding

The Logic of Grounding
  • Fabrice Correia
  • Published online:
    16 December 2024
    Print publication:
    16 January 2025
  • The concept of grounding – of a fact obtaining in virtue of other facts – has been a topic of intensive philosophical and logical investigation over roughly the past two decades. Many philosophers take grounding to deserve a central place in metaphysical theorizing, in great part because it is thought to do a better job than other concepts – e.g., reduction and supervenience – at capturing certain phenomena. Studies on the logic of grounding have largely been conducted with this philosophical background in mind. In this Element, I try to give a faithful picture of the contemporary development of the logic of grounding in a way that is both reasonably comprehensive and reasonably systematic.
Logic and Information

Logic and Information
  • Edwin Mares
  • Published online:
    20 May 2024
    Print publication:
    13 June 2024
  • This Element looks at two projects that relate logic and information: the project of using logic to integrate, manipulate and interpret information and the proect of using the notion of information to provide interpretations of logical systems. The Element defines 'information' in a manner that includes misinformation and disinformation and uses this general concept of information to provide an interpretation of various paraconsistent and relevant logics. It also integrates these logics into contemporary theories of informational updating, probability theory and (rather informally) some ideas from the theory of the complexity of proofs. The Element assumes some prior knowledge of modal logic and its possible world semantics, but all the other necessary background is provided.
Propositional Quantifiers

Propositional Quantifiers
  • Peter Fritz
  • Published online:
    23 April 2024
    Print publication:
    16 May 2024
  • Propositional quantifiers are quantifiers binding proposition letters, understood as variables. This Element introduces propositional quantifiers and explains why they are especially interesting in the context of propositional modal logics. It surveys the main results on propositionally quantified modal logics which have been obtained in the literature, presents a number of open questions, and provides examples of applications of such logics to philosophical problems.
Relevance Logic

Relevance Logic
  • Shay Allen Logan
  • Published online:
    28 March 2024
    Print publication:
    18 April 2024
  • Relevance logics are a misunderstood lot. Despite being the subject of intense study for nearly a century, they remain maligned as too complicated, too abstruse, or too silly to be worth learning much about. This Element aims to dispel these misunderstandings. By focusing on the weak relevant logic B, the discussion provides an entry point into a rich and diverse family of logics. Also, it contains the first-ever textbook treatment of quantification in relevance logics, as well as an overview of the cutting edge on variable sharing results and a guide to further topics in the field.
The Many Faces of Impossibility

The Many Faces of Impossibility
  • Koji Tanaka, Alexander Sandgren
  • Published online:
    28 February 2024
    Print publication:
    28 March 2024
  • Possible worlds have revolutionised philosophy and some related fields. But, in recent years, tools based on possible worlds have been found to be limited in many respects. Impossible worlds have been introduced to overcome these limitations. This Element aims to raise and answer the neglected question of what is characteristically impossible about impossible worlds. The Element sheds new light on the nature of impossible worlds. It also aims to analyse the main features and utility of impossible worlds and examine how impossible worlds can capture distinctions which are unavailable if we limit ourselves to possible world-based tools.
Temporal Logics

Temporal Logics
  • Valentin Goranko
  • Published online:
    08 September 2023
    Print publication:
    05 October 2023
  • Temporal Logics are a rich variety of logical systems designed for formalising reasoning about time, and about events and changes in the world over time. These systems differ by the ontological assumptions made about the nature of time in the associated models, by the logical languages involving various operators for composing temporalized expressions, and by the formal logical semantics adopted for capturing the precise intended meaning of these temporal operators. Temporal logics have found a wide range of applications as formal frameworks for temporal knowledge representation and reasoning in artificial intelligence, and as tools for formal specification, analysis, and verification of properties of computer programs and systems. This Element aims at providing both a panoramic view on the landscape of the variety of temporal logics and closer looks at some of their most interesting and important landmarks.
Logical Consequence

Logical Consequence
  • Gila Sher
  • Published online:
    20 August 2022
    Print publication:
    08 September 2022
  • To understand logic is, first and foremost, to understand logical consequence. This Element provides an in-depth, accessible, up-to-date account of and philosophical insight into the semantic, model-theoretic conception of logical consequence, its Tarskian roots, and its ideas, grounding, and challenges. The topics discussed include: (i) the passage from Tarski's definition of truth (simpliciter) to his definition of logical consequence, (ii) the need for a non-proof-theoretic definition, (iii) the idea of a semantic definition, (iv) the adequacy conditions of preservation of truth, formality, and necessity, (v) the nature, structure, and totality of models, (vi) the logicality problem that threatens the definition of logical consequence (the problem of logical constants), (vii) a general solution to the logicality, formality, and necessity problems/challenges, based on the isomorphism-invariance criterion of logicality, (viii) philosophical background and justification of the isomorphism-invariance criterion, and (ix) major criticisms of the semantic definition and the isomorphism-invariance criterion.
Classical First-Order Logic

Classical First-Order Logic
  • Stewart Shapiro, Teresa Kouri Kissel
  • Published online:
    26 April 2022
    Print publication:
    19 May 2022
  • One is often said to be reasoning well when they are reasoning logically. Many attempts to say what logical reasoning is have been proposed, but one commonly proposed system is first-order classical logic. This Element will examine the basics of first-order classical logic and discuss some surrounding philosophical issues. The first half of the Element develops a language for the system, as well as a proof theory and model theory. The authors provide theorems about the system they developed, such as unique readability and the Lindenbaum lemma. They also discuss the meta-theory for the system, and provide several results there, including proving soundness and completeness theorems. The second half of the Element compares first-order classical logic to other systems: classical higher order logic, intuitionistic logic, and several paraconsistent logics which reject the law of ex falso quodlibet.
Gödel's Incompleteness Theorems

Gödel's Incompleteness Theorems
  • Juliette Kennedy
  • Published online:
    05 April 2022
    Print publication:
    14 April 2022
  • This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
Proofs and Models in Philosophical Logic

Proofs and Models in Philosophical Logic
  • Greg Restall
  • Published online:
    25 March 2022
    Print publication:
    21 April 2022
  • This Element is an introduction to recent work proofs and models in philosophical logic, with a focus on the semantic paradoxes the sorites paradox. It introduces and motivates different proof systems and different kinds of models for a range of logics, including classical logic, intuitionistic logic, a range of three-valued and four-valued logics, and substructural logics. It also compares and contrasts the different approaches to substructural treatments of the paradox, showing how the structural rules of contraction, cut and identity feature in paradoxical derivations. It then introduces model theoretic treatments of the paradoxes, including a simple fixed-point model construction which generates three-valued models for theories of truth, which can provide models for a range of different non-classical logics. The Element closes with a discussion of the relationship between proofs and models, arguing that both have their place in the philosophers' and logicians' toolkits.
Higher-Order Logic and Type Theory

Higher-Order Logic and Type Theory
  • John L. Bell
  • Published online:
    10 March 2022
    Print publication:
    31 March 2022
  • This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory (also known as higher-order intuitionistic logic), an important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined.
Set Theory

Set Theory
  • John P. Burgess
  • Published online:
    21 January 2022
    Print publication:
    10 March 2022
  • Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.