Bergmann, Merrie, and . An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems

2008, Cambridge University Press.

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.

Blanchette, Patricia, and . Models and Modality

2000, Synthese 124(1): 45-72.

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.

Blanchette, Patricia, and . Logical Consequence

2001, Lou Goble (Ed). Blackwell Guide to Philosophical Logic. Wiley-Blackwell: 115-135.

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.

Cauman, Leigh S., and . First Order Logic: An Introduction

1998, Walter de Gruyter & Co.

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.

Dutilh Novaes, Catarina, and . Formal Languages in Logic: A Philosophical and Cognitive Analysis

2012, Cambridge: Cambridge University Press

Formal languages are widely regarded as being above all mathematical objects and as producing a greater level of precision and technical complexity in logical investigations because of this. Yet defining formal languages exclusively in this way offers only a partial and limited explanation of the impact which their use (and the uses of formalisms more generally elsewhere) actually has. In this book, Catarina Dutilh Novaes adopts a much wider conception of formal languages so as to investigate more broadly what exactly is going on when theorists put these tools to use. She looks at the history and philosophy of formal languages and focuses on the cognitive impact of formal languages on human reasoning, drawing on their historical development, psychology, cognitive science and philosophy. Her wide-ranging study will be valuable for both students and researchers in philosophy, logic, psychology and cognitive and computer science.

Comment: This book addresses important questions about formal languages: why formalization works and the limitations of formalization. The questions are answered from cognitive, historical and logical points of view. It is a good introductory material for teaching on formal language and psychology of reasoning.

Eriksson, Lina, and Alan Hájek. What are Degrees of Belief?

2007, Studia Logica 86(2): 185-215.

Probabilism is committed to two theses:
1) Opinion comes in degrees – call them degrees of belief, or credences.
2) The degrees of belief of a rational agent obey the probability calculus.

Correspondingly, a natural way to argue for probabilism is:
i) to give an account of what degrees of belief are, and then
ii) to show that those things should be probabilities, on pain of irrationality.

Most of the action in the literature concerns stage ii). Assuming that stage i) has been adequately discharged, various authors move on to stage ii) with varied and ingenious arguments. But an unsatisfactory response at stage i) clearly undermines any gains that might be accrued at stage ii) as far as probabilism is concerned: if those things are not degrees of belief, then it is irrelevant to probabilism whether they should be probabilities or not.

In this paper, the authors scrutinize the state of play regarding stage i). We critically examine several of the leading accounts of degrees of belief: reducing them to corresponding betting behavior (de Finetti); measuring them by that behavior (Jeffrey); and analyzing them in terms of preferences and their role in decision-making more generally (Ramsey, Lewis, Maher). We argue that the accounts fail, and so they are unfit to subserve arguments for probabilism. We conclude more positively: “degree of belief” should be taken as a primitive concept that forms the basis of our best theory of rational belief and decision: probabilism.

Comment: This paper is accessible to an advanced undergraduate audience in a formal philosophy course, since it provides an overview of the different accounts of the notion of degrees of belief. However, it's most adequate for graduate level, where it could be used in a formal epistemology course or in a course on the philosophy of probability.

Fisher, Jennifer, and . On the Philosophy of Logic

2007, Cengage Learning.

Publisher’s Note: Jennifer Fisher’s On the Philosophy of Logic explores questions about logic often overlooked by philosophers. Which of the many different logics available to us is right? How would we know? What makes a logic right in the first place? Is logic really a good guide to human reasoning? An ideal companion text for any course in symbolic logic, this lively and accessible book explains important logical concepts, introduces classical logic and its problems and alternatives, and reveals the rich and interesting philosophical issues that arise in exploring the fundamentals of logic.

Comment: This book provides an introduction to some traditional questions within philosophy of logic. Moreover, it presents some non-classical logics. It includes an introduction to formal classical logic, so no previous technical knowledge is required. Adequate for a first course on philosophy of logic, either as main or further reading.

Grover, Dorothy, Joseph Kamp and Nuel Belnap. A Prosentential Theory of Truth

1975, Philosophical Studies 27(1): 73-125.

Summary: Classic presentation of the prosentential theory of truth: an important, though minority, deflationist account of truth. Prosententialists take ‘It is true that’ to be a prosentence forming operator that anaphorically picks out content from claims made further back in the anaphoric chain (in the same way that pronouns such as ‘he’, ‘she’ and ‘it’ anaphorically pick out referents from nouns further back in the anaphoric chain).

Comment: Good as a primary reading on a course on truth, philosophy of language, or on deflationism more generally. Any course that treats deflationary accounts of truth in any detail would deal with the prosentential theory of truth, and this is one of the most historically important presentations of that theory. Would be best used in advanced undergraduate or graduate courses.

Grover, Dorothy, and . Inheritors and Paradox

1977, Journal of Philosophy 74(10): 590-604

Summary: Classic account of the way in which the prosentential theory of truth handles the liar paradox. Prosententialists take ‘It is true that’ to be a prosentence forming operator that anaphorically picks out content from claims made further back in the anaphoric chain (in the same way that pronouns such as ‘he’, ‘she’ and ‘it’ anaphorically pick out referents from nouns further back in the anaphoric chain). Liar sentences have no proposition-stating antecedents in the anaphoric chain. As a result, the problem of the liar does not arise.

Comment: Good as a primary reading on a course on truth, paradox, philosophy of language, or on deflationism more generally. Any course that treats deflationary accounts of truth in any detail would deal with the prosentential theory of truth, and this is one of the most historically important presentations of that theory. This is particularly useful in courses on paradox, as it is a rare articulation of the idea that the liar paradox is not "deep" and does not require large revisions to classical logic. Would be best used in advanced undergraduate or graduate courses.

Grover, Dorothy, and . How Significant is the Liar?

2008, In J. C. Beall & Bradley Armour-Garb (eds.), Deflationism and Paradox. OUP Oxford.

Summary: Grover argues that one should be unconcerned about the liar paradox. In formal languages there are uniform ties between syntax and semantics: a term, in all its occurrences, carries a fixed meaning; and sequences of sentences that are (syntactically) proofs are always (semantically) inferences. These two features do not hold of natural languages. Grover makes use of this claim to argue that there are no arguments to contradictions from liar sentences in natural languages, as the relevant syntactic ‘moves’ do not come with relevant semantic ‘moves’.

Comment: This would be useful in a course on truth, the philosophy of language or paradoxes. It provides a very up to date account of the prosentential theory of truth and how it may be able to deal with semantic paradoxes. Not as technical as some literature on the topic.