This book is a collection of papers by Ruth Barcan Marcus, covering much ground in the development of her thought, and spanning from 1961 to 1990. Many of the papers deal with logical, semantic, metaphysical, and epistemological issues in intensional logic, and in particular, modalities. Some important themes that run through these papers are extensionality, the necessity of identity, the directly referential conception of proper names as “tags,” essentialism, substitutional quantification, and possibilia and possible worlds. What emerges from them is a robust defense of quantified modal logic in the light of a host of objections, particularly from Quine. Modalities also includes two papers on belief, which have consequences for epistemic logic and more widely for theories of rationality; two papers on ethical issues, which have consequences for deontic logic and practical reasoning; and finally, two papers on historical figures, Spinoza and Russell, dealing with the ontological proof of God’s existence, and the nature of particularity, identity, and individuation, respectively.
An Introduction to Symbolic Logic
Symbolic Logic is a relatively new subject, and the easiest methods of approach have not yet been determined. In point of arrangement, therefore, this Introduction has no predecessor. That is just why it was written: the need of some systematic guide, from the state of perfect innocence to a possible understanding of the classical literature, has become acute and commanding.
Chao Lun: The Treatises of Seng-chao
The Chao Lun (ca. 400 CE) is the main philosophical work of the Chinese Buddhist monk Sēngzhào, a student of Kumārajīva who first translated major Buddhist philosophical works like those of Nāgārjuna into Chinese. Sēngzhào develops a metaphysical conception of time according to which time is not real. What instead exists are an array of entirely static worlds – “time slices” in modern parlance (Sēngzhào’s views are an early example of a “B-theory” of time). Our selves are being moved from one world to the next, creating an illusion of progression and things like causation. For Sēngzhào, enlightenment can be found by simply remaining in the present, not moving on to the next state. Sēngzhào leans both on Buddhist and on Daoist influences, but also develops an epistemologically less radical view in chapter 3 that, unlike those of Nāgārjuna and Zhuangzi, cannot be associated with skepticism.
Aristotle’s Logic
Aristotle is generally credited with the invention of logic. More than two thousand years ago, he noticed that good, persuasive arguments have certain kinds of shapes, or structures. Logic was born when he began to study those structures, which he called syllogisms, identifying underlying patterns of ordinary human reasoning and then devising simple methods which can be used to determine whether someone is reasoning correctly. Aristotle was aware that this was a momentous discovery, and he himself thought that all scientific reasoning could be reduced to syllogistic arguments. There are hints that Aristotle saw his syllogistic logic as providing useful practice for participants in ancient debating contests, contests which he himself would have encountered as a student in Plato’s Academy. There are also hints that Aristotle might have envisioned his syllogistic logic as providing a way to catalogue facts about science. But Aristotle never fleshes out any such details. His Prior Analytics, the text in which he presents his syllogistic, focuses on the mechanics of the syllogistic far more than on its interpretation, and in fact the comparative lack of interpretive detail has sometimes led scholars to suppose that what we have today of Aristotle’s logic is his own sometimes scrappy lecture notes or his students’ notes, and not a polished, finished work. But in the end these are of course only guesses. In modern times, the main scholarly interest in Aristotle’s ancient logic has focused increasingly on the proper interpretation of the mechanical methods which Aristotle invented and on their relation to his wider philosophy.
The Limit of Language in Daoism
The paper is concerned with the development of the paradoxical theme of Daoism. Based on Chad Hansen’s interpretation of Daoism and Chinese philosophy in general, it traces the history of Daoism by following their treatment of the limit of language. The Daoists seem to have noticed that there is a limit to what language can do and that the limit of language is paradoxical. The ‘theoretical’ treatment of the paradox of the limit of language matures as Daoism develops. Yet the Daoists seem to have noticed that the limit of language and its paradoxical nature cannot be overcome. At the end, we are left with the paradoxes of the Daoists. In this paper, we jump into the abyss of the Daoists’ paradoxes from which there is no escape. But the Daoists’ paradoxes are fun!
Logic and Implication: An Introduction to the General Algebraic Study of Non-classical Logics
This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to (abstract) algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions.
Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics.
What is Mathematical Logic?
Mathematical logic has grown from an exotic branch of mathematics into an indispensable tool in computer science as well as other parts of mathematics. This concise book presents the subject of mathematical logic in a lively and approachable fashion although logic can be a formidably abstruse topic, even for mathematicians.
This second edition of What is Mathematical Logic?, originally published 50 years ago, deals with important ideas in modern mathematical logic, without the detialed mathematical work required of those with a professional interest in logic. The ideas are set forth simply and clearly in a pleasant style and, despite the book’s relative brevity, all the basic material is covered in these pages. Three new chapters have been added, coevering automatic theorem proving, logic beyond traditional first order logic, and other logics including intuitionistic, free, and modal logics.
Students of computer science and mathematical logic will find it a stimulating introduction and valuable supplement for courses, including current further reading suggestions in this lively area at the intersection of mathematics, philosophy, and computer science.
The Problem of Speaking for Others
As philosophers and social theorists we are authorized by virtue of our academic positions to develop theories that express and encompass the ideas, needs, and goals of others. However, we must begin to ask ourselves whether this is a legitimate authority. Is the discursive practice of speaking for others ever a valid practice, and, if so, what are the criteria for validity? In particular, is it ever valid to speak for others who are unlike me or who are less privileged than me?
Ever the Twain shall Meet? Chomsky and Wittgenstein on Linguistic Competence
It is a dominant view in the philosophical literature on the later Wittgenstein that Chomsky’s approach to the investigation of natural language stands in stark contrast to Wittgenstein’s, and that their respective conceptions of language and linguistic understanding are irreconcilable. The aim in this paper is to show that this view is largely incorrect and that the two approaches to language and its use are indeed compatible, notwithstanding their distinctive foci of interest. The author argues that there is a significant correspondence in at least five different areas of their work, and that once we pay attention to these there will be less temptation to see Wittgenstein and Chomsky as enemies.
The Autonomy of Grammar and Semantic Internalism
In his post-Tractatus work on natural language use, Wittgenstein defended the notion of what he dubbed the autonomy of grammar. According to this thought, grammar – or semantics, in a more recent idiom – is essentially autonomous from metaphysical considerations, and is not answerable to the nature of things. The argument has several related incarnations in Wittgenstein’s post-Tractatus writings, and has given rise to a number of important insights, both critical and constructive. In this paper I will argue for a potential connection between Wittgenstein’s autonomy argument and some more recent internalist arguments for the autonomy of semantics. My main motivation for establishing this connection comes from the fact that the later Wittgenstein’s comments on grammar and meaning stand in opposition to some of the core assumptions of semantic externalism.