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Philosophia Contributions to Philosophy Axiomatic Method
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Meixner,Uwe
Axiomatic MethodISBN: 978-3-88405-575-5 |
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€14.00
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Whether in mereology or any other area, applying the axiomatic method means to specify an axiomatic system S, consisting of the axioms of S and the logic of S. The axioms of S are certain basic statements, and the logic of S is a set of basic inferencerules which can be used to generate further statements from given statements (ultimately from the axioms). The specification of S must be effective, that is, it must be in every case decidable whether or not a given statement belongs to the axioms of S, and whether or not a given inference-rule belongs to the logic of S. Relative to the axiomatic system S – the axioms plus the logic – a notion of provability is recursively defined: (1) the axioms of S are provable in S; (2) if the premise(s) of an inference-rule of the logic of S are provable in S, then also the conclusion of that inference-rule is provable in S; (3) only statements that can be obtained by (1) and (2) are provable in S. Any statement that is provable in S but is not an axiom of S is called a theorem of S. Using the term ‘valid’ in the sense of ‘legitimate’, one can state six conditions that normally hold for axiom systems. (a) The language of an axiomatic system consists of (well-formed) statements that are ... |
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