|
Propositional calculus (also called propositional logic, sentential calculus, or sentential logic) is the branch of mathematical logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not” (negation) and "if" (but only when used to denote material conditional). The following is an example of a very simple inference within the scope of propositional logic: :Premise 1: If it's raining then it's cloudy. :Premise 2: It's raining. :Conclusion: It's cloudy. Both premises and the conclusions are propositions. The premises are taken for granted and then with the application of modus ponens (an inference rule) the conclusion follows. As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed anymore by logical connectives, this inference can be restated replacing those ''atomic'' statements with statement letters, which are interpreted as variables representing statements: :Premise 1: :Premise 2: :Conclusion: The same can be stated succinctly in the following way: : When is interpreted as “It's raining” and as “it's cloudy” the above symbolic expressions can be seen to exactly correspond with the original expression in natural language. Not only that, but they will also correspond with any other inference of this ''form'', which will be valid on the same basis that this inference is. Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a ''derivation'' or ''proof'' and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself. Usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of ''true'' or a truth value of ''false''. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic. ==History== (詳細はChrysippus in the 3rd century BC〔(Ancient Logic (Stanford Encyclopedia of Philosophy) )〕 and expanded by the Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood . Consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic. The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were reachieved by logicians like George Boole and Augustus De Morgan completely independent of Leibniz.〔(Leibniz's Influence on 19th Century Logic )〕 Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic was an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including Natural Deduction, Truth-Trees and Truth-Tables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. Truth-Trees were invented by Evert Willem Beth.〔Beth, Evert W.; "Semantic entailment and formal derivability", series: Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, Nieuwe Reeks, vol. 18, no. 13, Noord-Hollandsche Uitg. Mij., Amsterdam, 1955, pp. 309–42. Reprinted in Jaakko Intikka (ed.) ''The Philosophy of Mathematics'', Oxford University Press, 1969〕 The invention of truth-tables, however, is of controversial attribution. Within works by Frege〔(Truth in Frege )〕 and Bertrand Russell,〔(Russell's Use of Truth-Tables )〕 one finds ideas influential in bringing about the notion of truth tables. The actual 'tabular structure' (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently).〔 Besides Frege and Russell, others credited with having ideas preceding truth-tables include Philo, Boole, Charles Sanders Peirce, and Ernst Schröder. Others credited with the tabular structure include Łukasiewicz, Schröder, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis.〔 Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.".〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Propositional calculus」の詳細全文を読む スポンサード リンク
|