Logic and Nonsense

Thomas Ferguson

Department of Philosophy
City University of New York, USA



"It is the basic idea of [Bertrand Russell’s] theory that the division of linguistic expressions into true and false is not sufficient, that a third categorymust be introduced which includes meaningless expressions. It seems to me that this is one of the deepest and soundest discoveries of modern logic." (Hans Reichenbach, Bertrand Russell’s Logic)

Overview: During the early 20th century, the problem of nonsense arose in a number of contexts. For example, Bertrand Russell solved the paradoxical sentences of naive class theory by dismissing them as meaningless, while positivists like Carl Hempel made similar assertions concerning any statements not empirically verifiable. Granted the existence of grammatical-yet-meaningless sentences, the question how a theory of deduction should approach such sentences becomes important. It is arguable that the classical Frege-Russell account of logic is ill-equipped to deal with such sentences. For example, there is a case to be made that such sentences are neither true nor false; should such sentences be grammatical while not possessing a truth value, then the semantics of classical logic cannot account for such sentences.

A number of responses to apparently nonsensical sentences within the bounds of classical logic have been proposed. Some—like Rudolph Carnap—have diagnosed all such sentences as artifacts of an “incorrectly constructed language,” suggesting that a category mistake such as “Socrates is an even number” would ever enter into a “logically perfect language.” Others—such as Willard Van Orman Quine—suggested that while such sentences are grammatical, classical logic is equipped to handle them by either treating them as “don’t cares” by assigning them truth values arbitrarily, or by uniformly assigning such sentences the value of falsehood. Each camp asserts that no revision of logic is necessitated, on the one hand by diagnosing nonsensical sentences as ill-formed and on the other by diagnosing these statements as irrelevant.

But just as many have argued that the existence of nonsensical sentences indeed demands a revision of the classical principles of inference, suggesting that a logic of nonsense provides a natural solution to many of the problems of early 20th century analytic philosophy. It is against this backdrop that logicians such as Dmitri Bochvar and Sören Halldén developed their programs of nonsense logics. Such programs aim to clear up the relationship between deduction and nonsense by examining how logic must be adapted in order to provide a theory of inference that acknowledges nonsensical sentences. Once the need for such a project is conceded, a new host of philosophical questions arise with respect to how to implement such a program: How should an operator “x is nonsense” behave? What should the appropriate generalization of semantic consequence? Is the negation of a nonsensical statement itself nonsensical? It is questions such as these that distinguish the major systems of nonsense logic, such as Bochvar’s Σ and Halldén’s C.

The aim of this tutorial is first to acquaint attendees with the primary philosophical problems giving rise to problems of logic and nonsense and the debates concerning their resolution. Then, we will introduce the most well-known logics of nonsense at the propositional level while examining the philosophical positions that motivate their definitions. Finally, we will review functional and first-order extensions of such systems, placing an emphasis on the relationship between Russell’s theory of types and the theories of predication posited by the proponents of nonsense logics.







Philosophical Topics Concerning Nonsense: We will discuss a number of the philosophical concerns that underlie the worry about the relationship between logic and nonsense. Emphasis will be placed on Russell’s comments on meaninglessness in the Principia Mathematica and on the empiricist theories of meaning forwarded by the logical positivists, as well as their respective resolutions to the apparent problem of grammatical-yet-nonsensical sentences. We will then proceed to examine the question of whether a revision of classical, bivalent logic is necessitated in light of such concerns.

Propositional Nonsense Logics: Assuming that a revisions of classical logic is needed in light of nonsensical sentences, we will first examine the philosophical matters that arise when formalizing logics of nonsense at the propositional level. We will discuss the philosophical differences between distinct schools of
nonsense logic with respect to challenges such as how to define semantic consequence in this context and how to generalize the propositional connectives of the Principia Mathematica. One the basis of these distinctions, we will first discuss the primary propositional logics of nonsense, i.e., Bochvar’s logic Σ and Halldén’s C, drawing on their commentators. Finally, we will consider some of the alternative accounts of logics of nonsense offered by Bochvar and Halldén’s successors.

First-Order Nonsense Logics: Many of the philosophical considerations with respect to nonsense lead naturally to questions concerning predicates, e.g., the notion of a category mistake is precisely the suggestion that some properties do not meaningfully apply to objects of certain types. In this session, we will review some of the accounts of predication and quantification formalized firstorder logics of nonsense. We will discuss Russell’s theory of types to provide some of the formal apparatus, before discussing the particular theses of Halldén, Goddard, and Routley on the matter of predicates and properties in logics of


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