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Infinitary Logic
An infinitary logic is a logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ... that allows infinitely long statement (logic), statements and/or infinitely long Mathematical proof, proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be Compactness (logic), compact or Completeness (logic), complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses Hilbert system, Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Logical System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |