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Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical trivium of grammar, logic, and rhetoric. The word derives from Greek. Logic concerns the structure of statements and arguments, in formal systems of inference and natural language. Topics include validity, fallacies and paradoxes, reasoning using probability and arguments involving causality. Logic is also commonly used today in argumentation theory.
The syllogistic logic developed by Aristotle predominated until the mid-nineteenth century when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1879 Frege published Begriffsschrift which inaugurated modern logic with the invention of quantifier notation. In 1903 Alfred North Whitehead and Bertrand Russell published Principia Mathematica. on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931 Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues.
The development of logic since Frege, Russell and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments often as a compulsory discipline.
Nature of logic
Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic logic is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional Aristotelian syllogistic logic, which deals solely with categorical propositions.
- Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato[5] are a good example of informal logic.
- Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic, which were incorporated in the late nineteenth century into modern formal logic.[6] In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuance of natural language.)
- Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[4][7] Symbolic logic is often divided into two branches, propositional logic and predicate logic.
- Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.
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