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 considered.

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 S2) ' stands for 'S1 implies S2'.

'(S1 S2)' is true iff 'S1' is false or 'S2' is true.

'(S1 S2)' is false iff 'S1' is true and 'S2' 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, s1, s2, ... are sentential variables;

- , are logical constants; ( , ) are technical symbols.

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 A = df - (|A - A) ( ' ' strong truth operator).

D1.5 (A B) = df ( A B) (deductive implication).

It is convenient to define, inside of the class of wff's, the subclass of TF-formulas that get only T or F as their truth-value.

iii If P is wff, then - P is TF-formula (TF-f).

iv If P, Q are TF-f's, then (P Q) is TF-f.

Let P, Q, R ... be any TF-f’s.

D2.1 (P Q) = df - (P - Q). D2.2 (P Q) = df (- P Q).

D2.3 (P Q) = df (P Q) (Q P).

Axiom schemata

A1.1 (P (Q P))

A1.2 (P (Q R)) ((P Q) (P R))

A1.3 (- P - Q) (Q P)

Following axioms are added to classical logic axioms:

A1.4 |P P

A2.1 |(A B) - A |B

A2.2 - (A B) |A - B

Rule of Inference A, (A B) / B

Interpretation. For interpretation of the language of FL4 logic we use four truth-values 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 truth-values 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 truth-value tables for initial and derivative operators and connectives.

  A -A ~A |A A   T F B N   T F B N
  T F F T T   T T F B N   T T F F F
  F T T F F   F T T T T   F T T T T
  B T B T F   B T B B T   B T T T T
  N F N F F   N T N T N   N T T T T

Then characteristic matrix mFL4 is

mFL4 = <{T, F, B, N}, - , , T > .

Let's signify the algebra of characteristic matrix mFL4 as FA4-algebra 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 four-valued Belnap’s logic and von Wright’s truth logic.

In FL4 negation ~ correspond to negation ~ in 4-valued Belnap’s logic.

In FL4 conjunction & and disjunction V are defined in such a way that truth-value tables for them correspond to those in 4-valued 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 non-falsity.

We define implication for Belnap’s logic by such expression:

D4. (A B B) = df (|A |B) (- - A - - B) .

Note that the truth-value table for B coincides with the truth-value table proposed by T.Smiley for the logic of tautological consequences Efde .

Among the truth logics, which were introduced by von Wright, the four-valued logic T LM is most similar to FL4. In FL4 truth operator |, negation ~ and conjunction & are defined by such a way that truth-value tables for them correspond to those in four-valued 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 three-valued logic which is obtained from FL4 by adding formula (- |A - - A) as axiom.

D5.2 FL3B is three-valued logic which is obtained from FL4 by adding formula (|A - A) as axiom.

D5.3 FL2 is two-valued logic which is obtained from FL4 by adding formula ( A ~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 three-valued logic using regular tables for the connectives ~, &, , which are considered in a strong sense. Kleene used three truth-values: t ("truth"), f ("falsity"), u ("undefined"), or in the other interpretation "known truth", "known falsity", "unknown truth or falsity". Truth-values and strong connectives of Kleene's logic correspond to the following truth-values and connectives of FL3N:

Kleene's logic SK3 Logic FL3N
t, f, u correspond to T, F, N respectively.
~, &, , correspond to ~, &, V, ?? respectively.

A.S.Karpenko notes on functional equivalency of L ukasiewicz’s three-valued logic L3 and FL3N logic.

Lukasiewicz introduced the third truth-value, 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 three-valued logic L3 is functionally equivalent to FL3N logic.

Since FL3N is sublogic of FL4 and functionally equivalent to L3, it is possible to consider FL4 as logic that is functionally equivalent to four-valued generalization of L3 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 - A).

THEOREM 4.2. It is not fulfilled (|A - A) B.

Priest constructed a three-valued logic using tables for the connectives ~, , , and introducing the third truth-value "paradoxicality" that is denoted as 2. Designated values are 1, 2.

Truth-values and connectives of Priest's logic correspond to the following truth-values and connectives of FL3B:

Priest's logic Pr3 Logic FL3B
0, 1, 2 correspond to F, T, B respectively.
~, , , correspond to ~, &, V, respectively.

There are two designated values in the interpretation of the Priest’s logic. For the interpretation FL3B, there is one designated truth-value. 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 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)