Sentential Falsehood Logic FL4 Sergey Pavlov

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 truthvalues 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 '(S_{1} ® S_{2}) ' stands for 'S_{1} implies S_{2}'. '(S_{1} ® S_{2})' is true iff 'S_{1}' is false or 'S_{2}' is true. '(S_{1} ® S_{2})' is false iff 'S_{1}' is true and 'S_{2}' is false. The notions of truth, falsehood and implication are interdependent. Falsehood and implication will be considered as basic logical connectives in FL4. The proposition about the truth of the sentence 'S' will stand for the proposition about falsehood of the negation of the sentence 'S' and we read it as follows "Sentence 'S' is true" (in symbols: '(S)', see D1.3). Language for FL4. Alphabet: s, s_{1}, s_{2}, ... are sentential variables;
Formation rules. i Every sentential variable is wff. ii If A, B are wff's, then  A, (A ® B) are wff's. Let A, B, ... be any wff’s. D1.1 0 =_{ df}  ( s ®  s) ('falsity' constant). D1.2 ~A =_{ df} (A ® 0) (negation). D1.3 A =_{ df}  ~A ( '' truth operator). D1.4 é A =_{ df}  (A ®  A) ( 'é ' strong truth operator). D1.5 (A É B) =_{ df} (é A ® é B) (deductive implication). It is convenient to define, inside of the class of wff's, the subclass of TFformulas that get only T or F as their truthvalue. iii If P is wff, then  P is TFformula (TFf). iv If P, Q are TFf's, then (P ® Q) is TFf. Let P, Q, R ... be any TFf’s. D2.1 (P Ù Q) =_{ df}  (P É  Q). D2.2 (P Ú Q) =_{ df} ( P É Q). D2.3 (P º Q) =_{ df} (P É Q) Ù (Q É P). Axiom schemata A1.1 (P É (Q É P)) A1.2 (P É (Q É R)) É ((P É Q) É (P É R)) A1.3 ( P É  Q) É (Q É P) Following axioms are added to classical logic axioms: A1.4 P º P A2.1 (A ® B) º  A Ú B A2.2  (A ® B) º A Ù  B Rule of Inference A, (A É B) / B Interpretation. For interpretation of the language of FL4 logic we use four truthvalues T, F, C, I, which means true and not false (strong true), false and not true (strong false), both true and false (contradictory), neither true nor false (indifference) respectively. The designated value is T. Note that truthvalues of the intended interpretation of FL4 are similar to those of Belnap's logic. In Belnap's logic: T  'tells the truth only', F  'tells the falsity only', B  'tells both the truth and the falsity', N  'tells neither the truth nor the falsity'. We used Belnap’s symbols in our truthvalue tables for initial and derivative operators and connectives.
Then characteristic matrix m_{FL4} is m_{FL4 }= <{T, F, B, N},  , ® , T > . Let's signify the algebra of characteristic matrix m_{FL4} as FA4algebra or as the falsehood algebra FA4 = <{T, F, B, N},  , ® > . Theorem about subalgebras. THEOREM 1. There exist only three subalgebras for the falsehood algebra FA4: FA3B = <{T, F, B },  , ® > , FA3N = <{T, F, N},  , ® > , FA2 = <{T , F},  , ® > . The operations of the algebra FA4 are unclosed for the other subsets of the set {T, F, B, N}. For FL4 metatheorems of consistency, deduction and completeness are fulfilled. Interrelations between FL4 and fourvalued Belnap’s logic and von Wright’s truth logic. In FL4 negation ~ correspond to negation ~ in 4valued Belnap’s logic. In FL4 conjunction & and disjunction V are defined in such a way that truthvalue tables for them correspond to those in 4valued logic of Belnap. D3.1 (A & B) =_{ df} ~(A ® ~B) D3.2 (A V B) =_{ df} (~A ® B) Due to Belnap, if A entails B, then the entailment preserves truth and preserves nonfalsity. We define implication for Belnap’s logic by such expression: D4. (A ® ^{B} B) =_{ df} (A ® B) Ù (  A ®   B) . Note that the truthvalue table for ® ^{B} coincides with the truthvalue table proposed by T.Smiley for the logic of tautological consequences E_{fde} . Among the truth logics, which were introduced by von Wright, the fourvalued logic T¢ ¢ LM is most similar to FL4. In FL4 truth operator , negation ~ and conjunction & are defined by such a way that truthvalue tables for them correspond to those in fourvalued truth logic T¢ ¢ LM . Difference between FL4 and truth logic T¢ ¢ LM is that deduction theorem is not fulfilled in last case. Subsystems of FL4. Let’s define three sublogics of FL4 that correspond to three subalgebras of FA4. D5.1 FL3N is threevalued logic which is obtained from FL4 by adding formula ( A Ú   A) as axiom. D5.2 FL3B is threevalued logic which is obtained from FL4 by adding formula (A Ú  A) as axiom. D5.3 FL2 is twovalued logic which is obtained from FL4 by adding formula (é A Ú é ~A) as axiom. THEOREM 2. FL2 is equivalent to classical logic Cl. Falsehood logics correlation is expressed by diagram: FL4 FL3N FL3B FL2(Cl) Let us compare Kleene strong connectives logic SK3 with FL3N logic. Kleene constructed a threevalued logic using regular tables for the connectives ~, &, Ú , ® which are considered in a strong sense. Kleene used three truthvalues: t ("truth"), f ("falsity"), u ("undefined"), or in the other interpretation "known truth", "known falsity", "unknown truth or falsity". Truthvalues and strong connectives of Kleene's logic correspond to the following truthvalues and connectives of FL3N:
A.S.Karpenko notes on functional equivalency of L ukasiewicz’s threevalued logic L_{3 }and FL3N logic. Lukasiewicz introduced the third truthvalue, argued that "there are propositions that are neither true nor false, but indifferent only", "indifferent propositions, which possibility corresponds, have third value". Indifferent propositions Lukasiewicz’s logic correspond to indifferent propositions FL3N logic. Implication for Lukasiewicz’s logic are defined in FL3N as: D6 (A ® L B) =_{df} (A ® B) V (A ® ^{B} B). This definition explains the sense of L ukasiewicz’s implication as disjunction of initial implication of Kleene’s logic (FL3N logic) and implication for Belnap’s logic. Truth operator  and negation falsehood operator ~  of FL3N logic are similar to operators of necessity N and possibility M of Lukasiewicz’s logic. Connectives, that similar to initial connectives of FL3N logic can be defined in Lukasiewicz’s logic: (A ® B) =_{df} ((~ A ®^{L} B) ®^{L} B);  A =_{df} ~ MA. Similarities that found, permit to proof next theorem. THEOREM 3. Lukasiewicz’s threevalued logic L_{3 }is functionally equivalent to FL3N logic. Since FL3N is sublogic of FL4 and functionally equivalent to L_{3}, it is possible to consider FL4 as logic that is functionally equivalent to fourvalued generalization of L_{3} and differs from Lukasiewicz’s generalization. Next theorems make the logic FL4 similar to the paraconsistent logics. THEOREM 4.1. It is not fulfilled  (A Ù  A). THEOREM 4.2. It is not fulfilled (A Ù  A) É B. Priest constructed a threevalued logic using tables for the connectives ~, Ù , Ú , ® and introducing the third truthvalue "paradoxicality" that is denoted as 2. Designated values are 1, 2. Truthvalues and connectives of Priest's logic correspond to the following truthvalues and connectives of FL3B:
There are two designated values in the interpretation of the Priest’s logic. For the interpretation FL3B, there is one designated truthvalue. Note that the formula A has value T if formula A takes T and B. The fact that formula A is valid in the paraconsistent Priest’s logic will be denoted as ¦ _{Pr} A. The following theorem holds. THEOREM 5. ¦ _{Pr} A Û ¦ _{FL3B} A. 
Notes (1) Pavlov S.A. Falsehood logic FL4 // Proceedings Scientific Seminar of the Logic Center, Institute of Philosophy RAS 1993. Moscow, 1994 (in Russian) ^{} 