Online research in philosophy

Thus far the logic out of which mathematics has developed has been First-order Predicate Calculus with Identity, that is the logic of the sentential functors, ¬, →, ∧, ∨, etc., together with identity and the existential and universal quotifiers restricted to quotify- ing only over individuals, and not anything else, such as qualities or quotities themselves. Some philosophers—among them Quine— have held that this, First-order Logic, as it is often called, con- stitutes the whole of logic. But that is a mistake. It leaves out Second-order Logic, which we need if we are to characterize the natural numbers precisely, and pays scant attention to the logic of relations, especially transitive relations, which is the key to much of modern mathematics. Quine’s argument for restricting logic to First-order Logice was based on the grounds that only First- order logical theories display “Law and Order” and himself regards modal logic as belonging with witchcraft and superstition.1 Pred- icates are ontologically more suspect than individuals, and have a different logic, which is liable to give rise to paradox and inconsis- tency. Moreover, Second-order Logic lacks the completeness that First-order Logice has, which provides a pleasing parallel between syntactic and semantic notions, and argues for the analyticity of deductive logic.

Modern thinkers regard logic as a purely formal discipline like number theory, and not to be confused with any empirical discipline such as cognitive psychology, which may seek to characterize how people actually reason. Opposed to this is the traditional view that even a formal logic can be cognitively veridical – descriptive of procedures people actually follow in arriving at their deductive judgments (logic as Laws of Thought). In a cognitively veridical logic, any formal proof that a deductive judgment, intuitively arrived at, is valid should ideally conform to the method the reasoning subject has used to arrive at that judgment. More specifically, it should reveal the actual reckoning process that the reasoning subject more or less consciously carries out when they make a deductive inference. That the common logical words used in everyday reasoning – words such as 'and', 'if,''some', 'is''not,' and 'all'– have fixed positive and negative charges has escaped the notice of modern logic. The present paper shows how, by unconsciously recognizing 'not' and 'all' as 'minus-words', while recognizing 'and', 'some', and 'is' as 'plus words', a child can intuitively reckon, for example, 'not (−) all (−) dogs are (+) friendly' as equivalent to 'some (+) dogs aren't (−) friendly': −(−D+F) = +D−F.

This introduction to the basic forms of deductive inference as evaluated by methods of modern symbolic logic is de­signed for sophomore-junior-level stu­dents ready to specialize in the study of deductive logic. It can be used also for an introductory logic course. The inde­pendence of many sections allows the instructor utmost flexibility. The text consists of eight chapters, the first six of which are designed to intro­duce the student to basic topics of sen­tence and predicate logic. The last two chapters extend the procedures of the first six to alethic modal logic, the logic of imperatives, and deontic logic. Throughout the text there is an attempt to relate symbolic techniques to issues in the philosophy of logic.

The argument of this paper rests on the distinction between two types of what are, loosely speaking, logical claims: A general (speaker-independent) claim that some favoured principle of inference is both truth-preserving, and consistent with certain others. A claim by a particular speaker that he/she has reasonable deductive grounds for concluding that some particular statement is true. The first is a matter of pure logic—a question of what (allegedly) follows from what. The second is a matter of epistemic logic—a question of whether someone has, or more generally, whether there are, reasonable deductive grounds for concluding that something is the case. I shall argue that this distinction has a crucial bearing on the disagreement between classical logicians and non-classical logicians, which is essentially a disagreement about inferential behaviour. The argument is laid out in a manner designed to maximise the chances of any errors being detected. The paper concludes with some considerations of the relevance of relevant logic to the psychologist investigating inference behaviour.

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.

A philosophical argument in ordinary language is made the basis for a series of deductive logic exercises. Problems of translating the reasoning and alternative symbolizations are discussed to help guide students toward accurate charitable formalizations. Finally, the inference is critically evaluated in light of its deductive validity.

Classic deductive logic entails that once a conclusion is sustained by a valid argument, the argument can never be invalidated, no matter how many new premises are added. This derived property of deductive reasoning is known as monotonicity. Monotonicity is thought to conflict with the defeasibility of reasoning in natural language, where the discovery of new information often leads us to reject conclusions that we once accepted. This perceived failure of monotonic reasoning to observe the defeasibility of natural-language arguments has led some philosophers to abandon deduction itself (!), often in favor of new, non-monotonic systems of inference known as `default logics'. But these radical logics (e.g., Ray Reiter's default logic) introduce their desired defeasibility at the expense of other, equally important intuitions about natural-language reasoning. And, as a matter of fact, if we recognize that monotonicity is a property of the form of a deductive argument and not its content (i.e., the claims in the premise(s) and conclusion), we can see how the common-sense notion of defeasibility can actually be captured by a purely deductive system.

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.

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.

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.