Roughly speaking, logic is the study of prescriptive systems of reasoning, i.e. systems that people ought to use to reason deductively and inductively. How people actually reason is usually studied under other headings, including cognitive psychology. Logic is a branch of mathematics, and a branch of philosophy. As a science, logic defines the structure of statement and argument and devises formulae by which these are codified. Implicit in a study of logic is the understanding of what makes a good argument and what arguments are fallacious. Philosophical logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Following are more specific discussions of some systems of logic. Aristotelian logic Aristotelian logic was pioneered by Aristotle. Although it is possible that Aristotle was taught by someone else, the earliest study of reasoning can be attributed to Aristotle. Aristotle and his followers held that two of the most important principles of logic are the law of non-contradiction and the law of excluded middle. This kind of logic is now given various names to distinguish it from more recent systems of logic, e.g., Aristotelian logic or classical two-valued logic. The law of non-contradiction states that no proposition is both true and false and law of excluded middle states that a proposition must either be true or false. In combination, these laws require two truth values that are mutually exclusive. A proposition can be either true or false, but cannot be both at the same time. Formal logic Formal logic, also called symbolic logic, is concerned primarily with the structure of reasoning. Formal logic deals with the relationships between concepts and provides a way to compose proofs of statements. In formal logic, concepts are rigorously defined, and sentences are translated into a precise, compact, and unambiguous symbolic notation. Some examples of symbolic notation are: Lowercase letter p, q and r with italic font are conventionally used to denote propositions: p: 1 + 2 = 3 This statement defines p is 1 + 2 = 3 and that is true. Two propositions can be combined using conjunction, disjunction or conditional. They are called binary logical operators_. Such combined propositions are called compound propositions. For example, p: 1 + 1 = 2 and "logic is the study of reasoning." In this case, and is a conjunction. The two propositions can differ totally from each other. In mathematics and computer science, one may want to state a proposition depending on some variables: p: n is an odd integer. This proposition can be either true or false according to the variable n. A proposition with free variables is called propositional function with domain of discourse D. To form an actual proposition, one uses quantifiers. For every n, or for some n, can be specified by quantifiers: either the universal quantifier or the existential quantifier. For example, --

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