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[[Aristotle]] was the first to formalize the practice of reasoning.  In particular, Aristotle developed a formalization of the [[syllogism]] and the Square of Opposition.  While Modus ponens and its complement [[Modus tollens]] were known to the Medievals, it is not clear when these central laws of deduction were first formalized.

[[Aristotle]] was the first to formalize the practice of reasoning.  In particular, Aristotle developed a formalization of the [[syllogism]] and the Square of Opposition.  While Modus ponens and its complement [[Modus tollens]] were known to the Medievals, 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]].  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 [[logicism|logicist]] project, which was shown to be inconsistent by [[Bertrand Russell]] in 1903 with what is famously called [[Russell's paradox]].

The next easily identifiable major development in logic comes from Boole in 1847 with the creation of [[Boolean algebra]].  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 [[logicism|logicist]] project, which was shown to be inconsistent by [[Bertrand Russell]] in 1903 with what is famously called [[Russell's paradox]].

By the 1950s, with the work of many logicians including [[Hilbert]], [[Emile Post]], [[Alfred Tarski]] and [[Kurt Goedel]], most of the major results in first-order logic had been proved, and in the 1960s [[Saul Kripke]] added a completeness proof for [[modal logic]].

By the 1950s, with the work of many logicians including [[Hilbert]], Emile Post, Alfred Tarski and [[Kurt Godel]], most of the major results in first-order logic had been proved, and in the 1960s Saul Kripke added a completeness proof for modal logic.

Most logical systems are [[bivalent]]; that is, they admit only two truth-values.  However, there is a fair amount of work done on non-bivalent systems of logic, especially [[intuitionist logic]] and [[relevance logic]].  In intuitionist logic, "true" and "false" are replaced with "proven true", "proven contradictory", and "not proven".  In relevance logic, "neither true nor false" and "both true and false" are added to the standard two truth values.  There is some debate about the value of these systems in philosophical circles, and [[Timothy Williamson]] claims to have a proof that any non-bivalent logic can be converted into a bivalent logic.

Most logical systems are bivalent; that is, they admit only two truth-values.  However, there is a fair amount of work done on non-bivalent systems of logic, especially intuitionist logic and relevance logic.  In intuitionist logic, "true" and "false" are replaced with "proven true", "proven contradictory", and "not proven".  In relevance logic, "neither true nor false" and "both true and false" are added to the standard two truth values.  There is some debate about the value of these systems in philosophical circles, and Timothy Williamson claims to have a proof that any non-bivalent logic can be converted into a bivalent logic.

Formal logic requires, since Frege, a distinction between the [[object language]] and the [[metalanguage]].  The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with a limited alphabet and syntax.

Formal logic requires, since Frege, a distinction between the [[object language]] and the [[metalanguage]].  The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with a limited alphabet and syntax.