Introductions to Logic

This is an introduction to the Isabelle proof assistant aimed at philosophers and their students.

There are obvious benefits to be gained from the study of logic: heightened ability to express ideas clearly and concisely, increased skill in defining one's terms, enlarged capacity to formulate arguments rigorously and to analyze them critically. But the greatest benefit, in my judgment, is the recognition that reason can be applied in every aspect of human affairs.

CAT4 is proposed as a general method for representing information, enabling a powerful programming method for large-scale information systems. It enables generalised machine learning, software automation and novel AI capabilities. This is Part 3 of a five-part introduction. The focus here is on explaining the semantic model for CAT4. Points in CAT4 graphs represent facts. We introduce all the formal (data) elements used in the classic semantic model: sense or intension (1st and 2nd joins), reference (3rd join), functions (4th join), (...) time and truth (logical fields), and symbolic content (name/value fields). Concepts are introduced through examples alternating with theoretical discussion. Some concepts are assumed from Part 1 and 2, but key ideas are re-introduced. The purpose is to explain the CAT4 interpretation, and why the data structure and CAT4 axioms have been chosen: to make the semantic model consistent and complete. We start with methods to translate information from database tables into graph DBs and into CAT4. We conclude with a method for translating natural language into CAT4. We conclude with a comparison of the system with an advanced semantic logic, the hyper-intensional logic TIL, which also aims to translate NL into a logical calculus. The CAT4 Natural Language Translator is discussed in further detail in Part 4, when we introduce functions more formally. Part 5 discusses software design considerations. (shrink)

Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

This text is a short introduction to logic that was primarily used for accompanying an introductory course in Logic for Linguists held at the New University of Lisbon (UNL) in fall 2010. The main idea of this course was to give students the formal background and skills in order to later assess literature in logic, semantics, and related fields and perhaps even use logic on their own for the purpose of doing truth-conditional semantics. This course in logic does not replace (...) a proper introduction to semantics and is not intended as such, although parts of Chapter 1 and 4 could be used to supplement an introductory course in semantics. In contrast to other introductions it has a certain focus on ‘writing things down correctly.’ Proofs of metatheorems are omitted, though. -/- This is work in progress. Please send suggestions and corrigenda to [email protected]. Have fun! (shrink)

A "good" arguer is like an architect with a penchant for civil and civic engineering. Such an arguer can design and present their reasons artfully about a variety of topics, as good architects do with a plenitude of structures and in various environments. Failures in this are rarely hidden for long, as poor constructions reveal themselves, often spectacularly, so collaboration among civical engineers can be seen as a virtue. Our logical virtues should be analogous. When our arguments fail due to (...) being uncivil and demagogic, since we inhabit the arguments we build, we are all crushed beneath our flawed reasoning. This mixed metaphor takes us to a self-referential analysis of argumentation, analogy, and humor. I argue that good argumentation strives to collaboratively convince rather than belligerently persuade. A convincing means toward this end is through humorous analogical arguments, whether the matter at hand is ethical, logical, theological, phenomenological, epistemological, metaphysical, political, or about baseball. (shrink)

forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, soundness, and functional completeness. Exercises with solutions are available. (...) It is provided in PDF (for screen reading, printing, and a special version for dyslexics), HTML, and in LaTeX source code. (shrink)

Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines. It considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, (...) independent study as a course text, it covers more material than is typically covered in an introductory course. -/- Topics include: translation, proofs, and trees for sentential and predicate logic with identity, definite descriptions, and functional terms; normal forms; proofs and trees for the principal normal modal systems; an introduction to second order and quantified modal logic; formal syntax and semantics; mathematical induction; metatheory for many of these systems; semantics and trees for free logic, intuitionistic logic, and three-valued and paraconsistent logics. -/- A companion website contains a detailed student solutions manual with a running commentary on all starred exercises and a set of editable slides for instructors to customize their courses. (shrink)

Introduction: Logic & Arguments What is a Proposition / Statement? What is an Argument? Premises(s) & Conclusion Resources: Slides, Examples, Indicator Words, & YouTube Video Lecture.

Deductively Valid Argument Forms Modus Ponens & Tollens Disjunctive & Hypothetical Syllogisms Destructive & Constructive Dilemmas Deductively Invalid Argument Forms Denying the Antecedent (DA) & Affirming the Consequent (AC) Resources: Slides, Examples, Helpful Hints, & "at-a-glance" Handout.

Identifying Arguments: Deduction & Induction Evaluating Arguments Deductive: Valid, Invalid, (Un)Sound Inductive: Strong, Weak, (Un)Cogent Inference to the Best Explanation [IBE] Identifying & Evaluating COMPLEX Arguments How to Present, Explain, and Evaluate an Argument Resources: Slides, Examples.

Propositional Logic: Introduction & Symbols Logical Operators & Translation Symbols Helpful Hints Translating: Simple Statements Complex Statements Conditional Statements Conjunctive Statements Disjunctive Statements Translating with Multiple Operators Identifying the Main Operator How To Translate into Propositional Logic in 4 EASY STEPS Resources: Slides, Examples, & Helpful Hints.

Basic Concepts in Logic Identifying & Evaluating Arguments Valid Argument Forms Complex Arguments Propositional Logic: Symbols & Translation Truth Tables: Statements Classifying & Comparing Statements.

Argumente begegnen uns überall: in der Politik, in Schule und Hochschule, Beruf und Alltag. Dabei werden wir immer auch mit der Frage konfrontiert, wie die behaupteten Aussagen begründet werden. Angelehnt an Thomas Nagels Einführung in die Philosophie beginnt der Band mit Hinweisen, wie wir Argumente interpretieren können. Es folgt eine Einführung in die logische Analyse von Argumenten und ihre Strukturen: Bedingungen, Alternativen und andere logische Zusammenhänge. Immer wieder werden die vorgestellten Inhalte durch Beispiele und Gegenbeispiele erläutert, so dass die Themen (...) anschaulich und praktisch anwendbar werden – ein vorzügliches Einführungsbuch für Schule, Studium und Selbststudium. (shrink)

By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero.We investigate the (...) validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants. (shrink)

We generalize Kada’s definable strengthening of Dilworth’s characterization of the class of quasi-orders admitting an antichain of a given finite cardinality.

Elementary Symbolic Logic: Concepts, Techniques, and Context introduces symbolic logic in a way that is accessible and yet rigorous enough to provide an adequate foundation for students who intend to further pursue studies in logic, or who work in areas of study—for example, philosophy or linguistics—where a serious understanding of logic is nonnegotiable. Moreover, while it is not a history book, it aims to provide some context for the development of symbolic logic.

The Dao Companion to Chinese Philosophy of Logic is the first general overview of Chinese logical philosophy in English, presumably providing students and researchers of Chinese thought in the Engl...

forall x: Dortmund is an adaptation and German translation of forall x: Calgary. As such, it is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity, the syntax of truth-functional (propositional) logic and truth-table semantics, the syntax of first-order (predicate) logic with identity and first-order interpretations, formalizing German in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as the (...) syntax and (Kripke-)semantics of modal logic. The book is provided in PDF and in LaTeX source code. A booklet with solutions for the exercises in the book is available. (shrink)

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.

We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure G expanding a group, and left invariant NIP formula δ...

In order to decide whether a discursive product of human reason corresponds or not to the logical order, one must analyze it in terms of syntactic correctness, consistency, and validity. The first step in logical analysis is formalization, that is, the process by which logical forms of thoughts are represented in different formal languages or logical systems. Although each thought can be properly formalized in different ways, the formalization variants are not equally adequate. The adequacy of formalization seems to depend (...) on several essential features: parsimony, accuracy, transparency, fertility and reliability. Because there is a partial antinomy between these traits, it is impossible to find a perfectly adequate variant of formalization. However, it is possible and preferable to reach a reasonable compromise by choosing the variant of formalization which satisfies all of these fundamental characteristics. (shrink)

This special issue contains three papers on evolutions in logic during the so-called ‘post-medieval’ period (roughly the years 1450–1700). The papers discuss the following topics: (1) traditions of logic in the British Isles during the sixteenth and seventeenth centuries; (2) approaches to validity in medieval and post-medieval logic; and (3) scholastic and humanist interpretations of the prologue to Galen’s Art of Medicine. All papers provide an original contribution to research on post-medieval logic, which to date is still in the early (...) stages. Together they testify to the great diversity of the discourse on logic during the period under consideration, something that has not been sufficiently appreciated in the scholarship. (shrink)

How Logic Works is an introductory logic textbook that is different by design. Rather than teaching elementary symbolic logic as an abstract or rote mathematical exercise divorced from ordinary thinking, Hans Halvorson presents it as the skill of clear and rigorous reasoning, which is essential in all fields and walks of life, from the sciences to the humanities—anywhere that making good arguments, and spotting bad ones, is critical to success. Instead of teaching how to apply algorithms using “truth trees,” as (...) in the vast majority of logic textbooks, How Logic Works builds on and reinforces the innate human skills of making and evaluating arguments. It does this by introducing the methods of natural deduction, an approach that teaches students not only how to carry out a proof and solve a problem but also what the principles of valid reasoning are and how they can be applied to any subject. The book also allows students to transition smoothly to more advanced topics in logic by teaching them general techniques that apply to more complicated scenarios, such as how to formulate theories about specific subject matter. How Logic Works shows that formal logic—far from being only for mathematicians or a diversion from the really deep questions of philosophy and human life—is the best account we have of what it means to be rational. By teaching logic in a way that makes students aware of how they already use it, the book will help them to become even better thinkers. A concise, readable, and user-friendly introduction to elementary symbolic logic that primarily uses natural deduction rather than algorithmic “truth trees” Draws on more than two decades’ experience teaching introductory logic to undergraduates Provides a stepping stone to more advanced topics. (shrink)

This is a basic logic text for first-time logic students. Custom-made texts from the chapters is an option as well. And there is a website to go with text too.

This paper is an introduction to the volume Language, Logic and Mathematics in Schopenhauer. It shows the basic interpretations discussed in Schopenhauer’s research, explains the aims and tasks of Schopenhauer’s philosophy and shows the importance of language, logic and mathematics in Schopenhauer’s system.

I present in this note an alternative answer to the princesses' earring problem, found in the first chapter of Cezar Augusto Mortari's book "Introdução à Lógica”.

Derek Turner, Professor of Philosophy, has written an introductory logic textbook that students at Connecticut College, or anywhere, can access for free. The book differs from other standard logic textbooks in its reliance on fun, low-stakes examples involving dinosaurs, a dog and his friends, etc. This work is published in 2020 under a Creative Commons AttributionNonCommercial-NoDerivatives 4.0 International License. You may share this text in any format or medium. You may not use it for commercial purposes. If you share it, (...) you must give appropriate credit. If you remix, transform, add to, or modify the text in any way, you may not then redistribute the modified text. (shrink)

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: 1. If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich (...) will enroll. There are many different things one can say about this argument, but many agree that if we do not equivocate (if the terms mean the same thing in the premises and the conclusion) then the argument is valid, that is, the conclusion follows deductively from the premises. This does not mean that the conclusion is true. Perhaps the premises are not true. However, if the premises are true, then the conclusion is also true, as a matter of logic. This entry is about the relation between premises and conclusions in valid arguments. (shrink)

Philosophy of logic is a fundamental part of philosophical study, and one which is increasingly recognized as being immensely important in relation to many issues in metaphysics, metametaphysics, epistemology, philosophy of mathematics, and philosophy of language. This textbook provides a comprehensive and accessible introduction to topics including the objectivity of logical inference rules and its relevance in discussions of epistemological relativism, the revived interest in logical pluralism, the question of logic's metaphysical neutrality, and the demarcation between logic and mathematics. Chapters (...) in the book cover the state of the art in contemporary philosophy of logic, and allow students to understand the philosophical relevance of these debates without having to contend with complex technical arguments. This will be a major new resource for students working on logic, as well as for readers seeking a better understanding of philosophy of logic in its wider context. (shrink)

This special issue on general proof theory collects papers resulting from the conference on general proof theory held in November 2015 in Tübingen.

A textbook for modal and other intensional logics based on the Open Logic Project. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals.

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

This paper presents a simple but, by my lights, effective argument for a subclassical account of logic—an account according to which logical consequence is (properly) weaker than the standard, so‐called classical account. Alas, the vast bulk of the paper is setup. Because of the many conflicting uses of ‘logic’ the paper begins, following a disclaimer on logic and inference, by fixing the sense of ‘logic’ in question, and then proceeds to rehearse both the target subclassical account of logic and its (...) well‐known relative (viz., classical logic). With background in place the simple argument—which takes up less than five pages—is advanced. My hope is that the minimal presentation will help to get ‘the simple argument’ plainly on the table, and that subsequent debate can move us forward. (shrink)

Some logic students falter at the transition from the mechanical method of truth tables to the less-mechanical method of natural deduction. This short paper introduces a word game intended to ease that transition.

In Winter 2017, the first author piloted a course in formal logic in which we aimed to (a) improve student engagement and mastery of the content, and (b) reduce maths anxiety and its negative effects on student outcomes, by adopting student oriented teaching including peer instruction and classroom flipping techniques. The course implemented a partially flipped approach, and incorporated group-work and peer learning elements, while retaining some of the traditional lecture format. By doing this, a wide variety of student learning (...) preferences could be provided for. (shrink)

An introduction to informal logic, critical thinking and rhetoric utilizing actual public discourse .

In elementary logic textbooks, Venn diagrams are used to analyze and evaluate the validity of syllogistic arguments. Although the method of Venn diagrams is shown to be a powerful analytical tool in these textbooks, it still has limitations. On the one hand, such method fails to represent singular statements of the form, “a is F.” On other hand, it also fails to represent identity statements of the form, “a is b.” Because of this, it also fails to give an account (...) of the validity of some obviously valid arguments that contain these types of statements as constituents. In this paper, owing to the developments in the literature on Venn diagrams, we offer a way of supplementing the rules of the Venn diagram found in textbooks, and show how this retooled Venn diagram technique could account for the problem cases. (shrink)

This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood with examples (...) bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)

A thorough and practical introduction to inductive logic with a focus on arguments and the rules used for making inductive inferences.

The book contains an introduction to basic logical concepts and methods. It covers traditional logic of categorical judgment and syllogism, modern propositional logic, and introductory elements of predicate logic with corresponding methods (truth tables, natural deduction, truth trees).

It is made clear what a law of logic is and why the laws of classical logic are true.

Gottlob Frege (1848-1925) invented the discipline of mathematical logic. In this short work, it is clearly stated what Frege did and did not accomplish.

Understand Logic is a comprehensive introduction to this fascinating though sometimes challenging subject. As well as looking at logic in theoretical terms the book considers its everyday uses and demonstrates how it has genuine practical applications. It will take you step by step through the most difficult concepts and is packed with exercises to help you consolidate your learning at every stage. Covering everything from syllogistic logic to logical paradoxes and even looking at logic in Alice in Wonderland, this is (...) the only guide you will ever need. (shrink)

The aim of this article is to give students a small sense of what metamathematics is—that is, how one might use mathematics to study mathematics itself. School or college teachers could base a classroom exercise on the letter games I shall describe and use them as a springboard for further exploration. Since I shall presuppose no knowledge of formal logic, the games are less an introduction to Gödel's theorems than an introduction to an introduction to them. Nevertheless, they show, in (...) an accessible way, how metamathematics can be mathematically interesting. (shrink)

This book clearly explains what an infinite number is, how infinite numbers differ from finite numbers, and how infinite numbers differ from one another. The concept of recursivity is concisely but thoroughly covered, as are the concepts of cardinal and ordinal number. All of Cantor's key proofs are clearly stated, including his epoch-making diagonal proof, whereby he proved that that there are more reals than rationals and, more generally, that there are infinitely large, non-recursive classes. In the final section, Kurt (...) Gödel's celebrated Incompleteness Theorem is shown to be equivalent with a number of the various different theorems proved in this short but profound work. (shrink)