## Logic

Logic is the systematic study of patterns of inference and is intended to clarify the underlying structure of “good” arguments. When we call an argument valid, we are not judging the truth or falsity of its premises or conclusion. We are evaluating its structure. An argument is said to be sound if it is both valid in argumentative structure and its premises are in fact true. This principle was articulated by Aristotle (384-322 BC) as the syllogism, which may be seen as an early and durable basis for logic but is more accurately considered a structure for argument and rhetoric (see “What Philosophy Professors Want from Student Writers” in Chapter 5) because it falls far short of creating a fully descriptive system of logic. That is, syllogistic logic does not address such areas as propositional calculus (also called propositional logic), predicate calculus (also called predicate logic), modal logic, and other, newer logics. Thus, logic is a discrete branch of philosophy of great importance and (often) technical complexity.

## Note

Most introductory philosophy courses and many traditional or classical philosophy courses do not touch upon modern logics. Indeed, the technical nature of these logical disciplines may be outside the course curriculum for the general philosophy student. Some of the content in this chapter, therefore, may be advanced significantly beyond both the interests and (at this point) competence of novice philosophy students. Nevertheless, even such students may encounter advanced material and the terms associated with it. For this reason, we include a compact yet comprehensive introductory treatment of major modern logics.

A key issue in logic is whether the various formal languages developed to study patterns of inference are primarily descriptive or prescriptive—that is, whether they are to be thought of as merely making explicit the underlying logical structure of natural language arguments that we already make, or as constructing alternatives to our natural language arguments to which we should aspire. For example, there is a range of different conversational and conventional implications associated with the English connectives “and” and “but,” although both are formalized in terms of the same truth tables. Similarly, a conditional statement of the form “if P, then Q” is usually rendered in terms of the material conditional (P ⊃ Q), which is evaluated as true whenever the antecedent P is false, or the conclusion Q is true. It is not clear, however, that this captures what we mean when we use such a construction. There is a question, then, as to whether these differences are something that a logical language will inevitably fail to capture or superfluous idiosyncrasies of natural language that a logical language manages to avoid.

A second issue concerns the competing desiderata involved in constructing our formal languages. On the one hand, we want our logical languages to be exhaustive insofar as they can capture as many of our (valid) patterns of inference as possible. On the other hand, however, we want these languages to be conservative in the sense that they do not add anything or introduce arbitrary new patterns of inference merely as a consequence of their technical construction. These desiderata can pull apart, and there are now a range of fully developed nonclassical logics that are argued to provide a better compromise between these competing demands. Some of these languages, in turn, involve further philosophical commitments. Thus, in the same way that we might take the theoretical virtues of a scientific theory (such as simplicity and predictive power) as an argument for accepting its ontological commitments, so some philosophers argue that the technical virtues of a logical language (such qualities as completeness and harmonious proof system) provide arguments for different metaphysical conclusions.

The existence of different logical languages also raises questions about the universality of logic: whether there is one “correct” logic that somehow captures the underlying structure of reality, or whether different logics can be thought of as merely useful tools for describing different domains of inquiry.