The general science of inference. Deductive logic, in which a conclusion follows from a set of premises, is distinguished from inductive logic, which studies the way in which premises may support a conclusion without entailing it. In deductive logic the conclusion cannot be false if the premises are true. The aim of a logic is to make explicit the rules by which inferences may be drawn, rather than to study the actual reasoning processes that people use, which may or may not conform to those rules. In the case of deductive logic, if we ask why we need to obey the rules, the most general form of answer is that if we do not we contradict ourselves (or, strictly speaking, we stand ready to contradict ourselves. Someone failing to draw a conclusion that follows from a set of premises need not be contradicting him or herself, but only failing to notice something. However, he or she is not defended against adding the contradictory conclusion to his or her set of beliefs). There is no equally simple answer in the case of inductive logic, which is in general a less robust subject, but the aim will be to find reasoning such that anyone failing to conform to it will have improbable beliefs. Aristotle is generally recognized as the first great logician, and Aristotelian logic or traditional logic (see syllogism) dominated the subject until the 19th century. It became increasingly recognized in the 20th century that fine work was done within that tradition, but syllogistic reasoning is now generally regarded as a limited special case of the forms of reasoning that can be represented within the propositional and predicate calculus. These form the heart of modern logic. Their central notions, of quantifiers, variables, and functions were the creation of the German mathematician Frege, who is recognized as the father of modern logic, although his treatment of a logical system as an abstract mathematical structure, or algebra, had been heralded by Boole (see Boolean algebra). Modern logic is thus called mathematical logic for two reasons: first, the logic itself is an object of mathematical study, but secondly, the forms introduced by Frege provided a language capable of representing all mathematical reasoning. This was something traditional logic had been quite incapable of tackling. The propositional and predicate calculus study ways of combining propositions with the connectives expressing truth functions, and of combining information about the quantity of times predicates are satisfied. These highly general operations can occur in any discourse, from mathematics to discussion of the football results. More specific logics study particular topics such as time, possibility, and obligation. Thus there exist deontic logics, modal logics, logics of tense, and so on. For other notions associated with the study of logic see interpretation, logical calculus, logical constants, logical form, model theory, proof theory, quantifier, truth function, variable.

http://www.nyu.edu/classes/velleman/blogic/Logic/ An online logic tutorial, with interactive exercises

http://www.dmoz.org/Society/Philosophy/Philosophy_of_Logic/ A list of internet resources on the philosophy of logic