Sentential Falsehood Logic FL4 :: Philosophy Philosophical Logical Papers

Sentential Falsehood Logic FL4 ABSTRACT: In some philosophical conceptions, statements are valued as true, false, senseless (neither true nor false), or inconsistent. Falsehood logic FL4 makes it possible to operate correctly by such statements. Logic with falsehood operator FL4 is formulated. For FL4 metatheorems of consistency, deduction and completeness are fulfilled. Correlation between falsehood logic FL4 and four-valued Belnap’s logic and von Wright’s truth logic T"LM is considered. In FL4, the implication for Belnap’s logic is defined so that the truth-valued matrix of it is characterized for logic of tautological consequences Efde. Correlation between three-valued falsehood sublogic FL3N of FL4 and three-valued Kleene’s logic and Lukasiewicz’s logic is considered. Lukasiewicz’s three-valued logic is functionally equivalent to FL3N logic. Correlation between three-valued falsehood sublogic FL3B of FL4 and three-valued paraconsistent Priest’s logic is also con sidered. The construction of falsehood logic FL4 (1) and its analysis answer the question about the use of truth and falsehood notions. In some philosophical conceptions statements are valued as true, false, senseless (neither true nor false), inconsistent. Falsehood logic FL4 makes it possible to operate correctly by such statements. The main principles of falsehood logic FL4 are as follows: 1. The notion of falsehood will be considered as applied only to sentences of the following form: "Sentence 'S' is false" (in symbols: '(- S)' ). The proposition '(- S)' is a proposition about falsehood of the sentence 'S' and it is a proposition in a metalanguage related to the language in which a sentence 'S' is formulated. The set of propositions of language, metalanguage, metametalanguage and so on is considered as a whole. And one can operate with these propositions (viz. 'S', '(- S)', '(- S(- S))', ...) simultaneously in the language of FL4. 2. We shall consider the notion of falsehood as a primitive one which will be used as a logical operator in this formal system. 3. The sentence '(- S)' is always either true or false, while the sentence 'S' may have other truth-values than true or false. In other words, the laws of classical logic are valid for sentence '(- S)', but need not to be valid for sentence 'S'. 4. Sentences with the implication will be evaluated in standard way. Let '(S1 Â ® S2) ' stands for 'S1 implies S2'. '(S1 Â ® S2)' is true iff 'S1' is false or 'S2' is true. '(S1 Â ® S2)' is false iff 'S1' is true and 'S2' is false.