formal logic the science of correct reasoning, going back to Aristotle’s Prior Analytics, based upon the premise that the validity of an argument is a function of its structure or logical form. The modern embodiment of formal logic is symbolic (mathematical) logic. This is the study of valid inference in artificial, precisely formulated languages, the grammatical structure of whose sentences or well-formed formulas is intended to mirror, or be a regimentation of, the logical forms of their natural language counterparts. These formal languages can thus be viewed as (mathematical) models of fragments of natural language. Like models generally, these models are idealizations, typically leaving out of account such phenomena as vagueness, ambiguity, and tense. But the idea underlying symbolic logic is that to the extent that they reflect certain structural features of natural language arguments, the study of valid inference in formal languages can yield insight into the workings of those arguments. The standard course of study for anyone interested in symbolic logic begins with the (classical) propositional calculus (sentential calculus), or PC. Here one constructs a theory of valid inference for a formal language built up from a stock of propositional variables (sentence letters) and an expressively complete set of connectives. In the propositional calculus, one is therefore concerned with arguments whose validity turns upon the presence of (two-valued) truth-functional sentence-forming operators on sentences such as (classical) negation, conjunction, disjunction, and the like. The next step is the predicate calculus (lower functional calculus, first-order logic, elementary quantification theory), the study of valid inference in first-order languages. These are languages built up from an expressively complete set of connectives, first-order universal or existential quantifiers, individual variables, names, predicates (relational symbols), and perhaps function symbols. Further, and more specialized, work in symbolic logic might involve looking at fragments of the language of the propositional or predicate calculus, changing the semantics that the language is standardly given (e.g., by allowing truth-value gaps or more than two truth-values), further embellishing the language (e.g., by adding modal or other non-truth-functional connectives, or higher-order quantifiers), or liberalizing the grammar or syntax of the language (e.g., by permitting infinitely long well-formed formulas). In some of these cases, of course, symbolic logic remains only marginally connected with natural language arguments as the interest shades off into one in formal languages for their own sake, a mark of the most advanced work being done in formal logic today. See also DEONTIC LOGIC, EPISTEMIC LOGIC, FREE LOGIC , INFINITARY LOGIC , MANY -VAL – UED LOGIC , MATHEMATICAL INTUITIONISM , MODAL LOGIC , RELEVANCE LOGIC , SECOND – ORDER LOGI. G.F.S.