Online research in philosophy

This paper considers the question of whether Mill's account of the nature and justificatory foundations of deductive logic is psychologistic. Logical psychologism asserts the dependency of logic on psychology. Frequently, this dependency arises as a result of a metaphysical thesis asserting the psychological nature of the subject matter of logic. A study of Mill's System of Logic and his Examination reveals that Mill held an equivocal view of the subject matter of logic, sometimes treating it as a set of psychological processes and at other times as the objects of those processes. The consequences of each of these views upon the justificatory foundations of logic are explored. The paper concludes that, despite his providing logic with a prescriptive function, and despite his avoidance of conceptualism, Mill's theory fails to provide deductive logic with a justificatory foundation that is independent of psychology.

The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnemed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct rea soning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism. The purpose of this article is to begin the articulation of this relationship, pointing out its sources and its limitations.

What is moral reasoning? For that matter, what is any sort of reasoning? Let me begin by making a few distinctions. First, there is a distinction between reasoning as something that that people do and the abstract structures of proof or “argument” that are the subject matter of formal logic. I will be mainly concerned with reasoning in the first sense, reasoning that people do. Second, there is a distinction between moral reasoning with other people and moral reasoning by and for yourself . Moral reasoning with others may involve discussion with them, bargaining with them, and possibly arguing with them.

What is the relationship between logic and reasoning? How do logical norms guide inferential performance? This paper agrees with Gilbert Harman and most of the psychologists that logic is not directly relevant to reasoning. It argues, however, that the mental model theory of logical reasoning allows us to harmonise the basic principles of deductive reasoning and inferential perfomances, and that there is a strong connexion between our inferential norms and actual reasoning, along the lines of Peacocke’s conception of inferential role.

Any discussion of the concept of “formal science” must acknowledge that the term is used in different ways, for different purposes, by different people. For some, the formal sciences are defined by the exclusive use of deductive methods for discovering, or reasoning about, the properties of formal, abstract systems. On this view, the formal sciences are synonymous with mathematics, formal logic, and certain branches of linguistics and computer science that emphasize the study of formal languages. For others, “formal science” means something like “exact science”, or “formalized science”. On this view, any scientific discipline that places heavy emphasis on mathematical or logical formalization of key theoretical concepts and theories, could be described as a formal science. This latter conception of formal science is much more liberal than the former, and would include all of physics, much of chemistry, and some parts of biology, ecology, psychology and economics, as well as newer computationoriented disciplines like artificial life and artificial intelligence that do not fit easily within the traditional classification of the sciences.

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.

The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable intimations concerning the nature and constitution of the human mind” [2, p. 1]. Logic is normative, not descriptive. For, the laws of thought do not “manifest their presence otherwise than by merely prescribing the conditions of formal inference” [2, p. 419]. They are, “properly speaking, the laws of right reasoning only” [2, p. 408]. So they “form but a part of the system of laws by which the actual processes of reasoning, whether right or wrong, are governed” [2, p. 409]. Boole’s idea that logic is the study of the laws of thought was taken over by Hilbert. According to him logic is “a discipline which expresses the structure of all our thought” [31, p..

I begin by summarizing the first two chapters of (Harman 1986). The first chapter stresses the importance of not confusing inference with implication and of not confusing reasoning with the sort of argument studied in deductive logic. Inference and reasoning are psychological events or processes that can be done more or less well. The sort of implication and argument studied in deductive logic have to do with relations among propositions and with structures of propositions distinguished into premises, intermediate steps, and conclusion. Deductive logic is not a particular psychological subject and is not a particularly normative subject, although one might attempt to develop a logic of belief or a deontic logic, for example.

Logic brings elementary logic out of the academic darkness into the light of day. Paul Tomassi makes logic fully accessible for anyone trying to come to grips with the complexities of this challenging subject. This book is written in a patient and user-friendly way which makes both the nature and value of formal logic crystal clear. This textbook proceeds from a frank, informal introduction to fundamental logical notions to a system of formal logic rooted in the best of our natural deductive reasoning in daily life. The book includes plenty of exercise to put the students' reading to test, summay boxes of key points, a glossary and many illustrations. This book will be useful to any student who needs a patient and comprehensible introduction to what otherwise can be a daunting subject.

The view that logic is true independently of a subject matter is criticized—enlarging on Quine's criticisms and adding further ones. It is then argued apriori that full reflective understanding of logic and deductive reasoning requires substantial commitment to mathematical entities. It is emphasized that the objectively apriori connections between deductive reasoning and commitment to mathematics need not be accepted by or even comprehensible to a given deductive reasoner. The relevant connections emerged only slowly in the history of logic. But they can be recognized retrospectively as implicit in logic and deductive reasoning. The paper concludes with discussion of the relevance of its main argument to Kant's question—how is apriori knowledge of a subject matter possible?