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:"Laws of thought, traditionally, the three fundamental laws of logic: (1) the [[Law of Non-Contradiction|law of contradiction]], (2) the [[Law of the excluded middle|law of excluded middle]] (or third), and (3) the [[Law of identity|principle of identity]]. That is, (1) for all [[proposition]]s p, it is impossible for both p and not p to be true, or symbolically ∼(p · ∼p), in which ∼ means “not” and · means “and”; (2) either p or ∼p must be true, there being no third or middle true proposition between them, or symbolically p ∨ ∼p, in which ∨ means “or”; and (3) if a propositional function F is true of an individual variable x, then F is indeed true of x, or symbolically F(x) ⊃ F(x), in which ⊃ means “formally implies.” Another formulation of the principle of identity asserts that a thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x."<ref>[https://www.britannica.com/topic/laws-of-thought Laws of Thought], Encyclopedia Britannica</ref>

:"Laws of thought, traditionally, the three fundamental laws of logic: (1) the [[Law of Non-Contradiction|law of contradiction]], (2) the [[Law of the excluded middle|law of excluded middle]] (or third), and (3) the [[Law of identity|principle of identity]]. That is, (1) for all [[proposition]]s p, it is impossible for both p and not p to be true, or symbolically ∼(p · ∼p), in which ∼ means “not” and · means “and”; (2) either p or ∼p must be true, there being no third or middle true proposition between them, or symbolically p ∨ ∼p, in which ∨ means “or”; and (3) if a propositional function F is true of an individual variable x, then F is indeed true of x, or symbolically F(x) ⊃ F(x), in which ⊃ means “formally implies.” Another formulation of the principle of identity asserts that a thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x."<ref>[https://www.britannica.com/topic/laws-of-thought Laws of Thought], Encyclopedia Britannica</ref>

[[Aristotle]] was the first to formalize the practice of reasoning. In particular, Aristotle developed a formalization of the [[Syllogism|categorical syllogism]], the three laws of logic, and the Square of Opposition. While modus ponens and its complement modus tollens were known to the medieval philosophers, it is not clear when these central laws of deduction were first formalized.

[[Aristotle]] was the first to formalize the practice of [[Logical reasoning|reasoning]]. In particular, Aristotle developed a formalization of the [[Syllogism|categorical syllogism]], the three laws of logic, and the Square of Opposition. While modus ponens and its complement modus tollens were known to the medieval philosophers, it is not clear when these central laws of deduction were first formalized.

The next easily identifiable major development in logic comes from Boole in 1847 with the creation of [[Boolean algebra]] and Boolean logic. In the 1870s, Peirce introduces a logic of quantifiers, followed by [[Gottlob Frege]]'s 1879 ''Begriffsschrift'', which contains the first complete formalization of the propositional calculus. Frege's work was developed in service of his logicist project, which was shown to be inconsistent by [[Bertrand Russell]] in 1903 with what is famously called [[Russell's paradox]]. In 1913 Russell and Whitehead published ''Principia Mathematica'', a landmark work that sought to define mathematics with first-order logic.

The next easily identifiable major development in logic comes from Boole in 1847 with the creation of [[Boolean algebra]] and Boolean logic. In the 1870s, Peirce introduces a logic of quantifiers, followed by [[Gottlob Frege]]'s 1879 ''Begriffsschrift'', which contains the first complete formalization of the propositional calculus. Frege's work was developed in service of his logicist project, which was shown to be inconsistent by [[Bertrand Russell]] in 1903 with what is famously called [[Russell's paradox]]. In 1913 Russell and Whitehead published ''Principia Mathematica'', a landmark work that sought to define mathematics with first-order logic.