A classical-logic view of a paraconsistent logic
aa r X i v : . [ c s . L O ] A ug A Classical-Logic View of a Paraconsistent Logic
C.A. Middelburg
Informatics Institute, Faculty of Science, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, the Netherlands
Abstract.
This paper is concerned with the first-order paraconsistentlogic LPQ ⊃ , F . A sequent-style natural deduction proof system for thislogic is given and, for this proof system, both a model-theoretic justifica-tion and a logical justification by means of an embedding into first-orderclassical logic is presented. For no logic that is essentially the same asLPQ ⊃ , F , a natural deduction proof system is currently available in theliterature. The presented embedding provides both a classical-logic ex-planation of this logic and a logical justification of its proof system. Keywords: paraconsistent logic, classical logic, natural deduction, em-bedding, three-valued logic.
Mathematics Subject Classification (2010):
A set of formulas is contradictory if there exists a formula such that both thatformula and the negation of that formula can be deduced from it. In classicallogic, every formula can be deduced from every contradictory set of formulas. Aparaconsistent logic is a logic in which not every formula can be deduced fromevery contradictory set of formulas.In [10], Priest proposed the paraconsistent propositional logic LP (Logic ofParadox) and its first-order extension LPQ. The paraconsistent logic consideredin this paper, called LPQ ⊃ , F , is LPQ enriched with a falsity constant and an im-plication connective for which the standard deduction theorem holds. A sequent-style natural deduction proof system for LPQ ⊃ , F is presented. In addition to theusual model-theoretic justification of the proof system, a logical justification bymeans of an embedding into FOCL (First-Order Classical Logic) is given. Clas-sical logic is used meta-logically here: the embedding provides a classical-logicexplanation of LPQ ⊃ , F .LPQ ⊃ , F is essentially the same as CLuNs [1], LFI1 ∗ [4], QLFI1 ◦ [5], J ∗ [6],and LP ◦ [9]. The proof systems for these logics available in the literature areHilbert systems for the first four logics and a Gentzen-style sequent systemfor the last one. To fill this gap, a natural deduction proof system is given forLPQ ⊃ , F in this paper. An important reason to present a justification of this proofsystem by means of an embedding into classical logic is to draw attention to theviewpoint that, although it may be convenient to use a paraconsistent logic likePQ ⊃ , F if contradictory formulas have to be dealt with, classical logic is theultima ratio of formal reasoning.The only difference between CLuNs and LPQ ⊃ , F is that the former has a bi-implication connective and the latter does not have that connective. However,bi-implication is definable in LPQ ⊃ , F . LFI1 ∗ , QLFI1 ◦ , J ∗ , and LP ◦ do nothave the falsity constant of LPQ ⊃ , F and J ∗ and LP ◦ also do not have theimplication connective of LPQ ⊃ , F . Instead, each of LFI1 ∗ , QLFI1 ◦ , J ∗ , andLP ◦ has a connective that is foreign to classical logic. However, the constantsand connectives of LPQ ⊃ , F are definable in terms of those of each of these logicsand vice versa. That is why it is said that LPQ ⊃ , F is essentially the same as theselogics. I prefer LPQ ⊃ , F because it does not have a connective that is foreign toclassical logic.The structure of this paper is as follows. First, the language of the paracon-sistent logic LPQ ⊃ , F is defined (Section 2). Next, a sequent-style natural deduc-tion proof system for LPQ ⊃ , F is given (Section 3). After that, a model-theoreticjustification of this proof system is given (Section 4). Then, a justification ofthis proof system by means of an embedding into FOCL is given (Section 5).Following this, selected points related to the preceding sections are discussed(Section 6). Finally, some concluding remarks are made (Section 7). ⊃ , F In this section the language of the paraconsistent logic LPQ ⊃ , F is described.First, the assumptions which are made about function and predicate symbols aregiven and the notion of a signature is introduced. Next, the terms and formulasof LPQ ⊃ , F are defined for a fixed but arbitrary signature. Thereafter, notationalconventions and abbreviations are presented and some remarks about free vari-ables and substitution are made. In coming sections, the proof system of LPQ ⊃ , F and the interpretation of the terms and formulas of LPQ ⊃ , F are defined for afixed but arbitrary signature. It is assumed that the following has been given: (a) a countably infinite set V of variable symbols , (b) a countably infinite set C of constant symbols , (c) for each n ∈ N , a countably infinite set F n of function symbols of arity n , and, (d) foreach n ∈ N , a countably infinite set P n of predicate symbols of arity n . It is alsoassumed that all these sets and { = } are mutually disjoint.We write S ym for V ∪ C ∪ S {F n | n ∈ N } ∪ S {P n | n ∈ N } . The notation w ≡ w ′ , where w, w ′ ∈ S ym , is used to indicate that w and w ′ are identical.A signature Σ is a subset of C ∪ S {F n | n ∈ N } ∪ S {P n | n ∈ N } .We write S ig for the set of all signatures. We write C( Σ ), F n ( Σ ), and P n ( Σ ),where Σ ∈ S ig and n ∈ N , for Σ ∩ C , Σ ∩ F n , and Σ ∩ P n , respectively.The language of LPQ ⊃ , F will be defined for a fixed but arbitrary signature Σ .This language will be called the language of LPQ ⊃ , F over Σ or shortly the lan-2uage of LPQ ⊃ , F ( Σ ). The corresponding proof system and interpretation will becalled the proof system of LPQ ⊃ , F ( Σ ) and the interpretation of LPQ ⊃ , F ( Σ ). The language of LPQ ⊃ , F ( Σ ) contains terms and formulas. They are constructedaccording to the formation rules given below.The set of all terms of LPQ ⊃ , F ( Σ ), written T LPQ ⊃ , F ( Σ ), is inductively definedby the following formation rules: – if x ∈ V , then x ∈ T LPQ ⊃ , F ( Σ ); – if c ∈ C( Σ ), then c ∈ T LPQ ⊃ , F ( Σ ); – if f ∈ F n ( Σ ) and t , . . . , t n ∈ T LPQ ⊃ , F ( Σ ), then f ( t , . . . , t n ) ∈ T LPQ ⊃ , F ( Σ ).The set of all formulas of LPQ ⊃ , F ( Σ ), written F LPQ ⊃ , F ( Σ ), is inductively definedby the following formation rules: – F ∈ F LPQ ⊃ , F ( Σ ); – if t , t ∈ T LPQ ⊃ , F ( Σ ), then t = t ∈ F LPQ ⊃ , F ( Σ ); – if P ∈ P n ( Σ ) and t , . . . , t n ∈ T LPQ ⊃ , F ( Σ ), then P ( t , . . . , t n ) ∈ F LPQ ⊃ , F ( Σ ); – if A ∈ F LPQ ⊃ , F ( Σ ), then ¬ A ∈ F LPQ ⊃ , F ( Σ ); – if A , A ∈ F LPQ ⊃ , F ( Σ ), then A ∧ A , A ∨ A , A ⊃ A ∈ F LPQ ⊃ , F ( Σ ); – if x ∈ V and A ∈ F LPQ ⊃ , F ( Σ ), then ∀ x • A, ∃ x • A ∈ F LPQ ⊃ , F ( Σ ).For the connectives ¬ , ∧ , ∨ , and ⊃ and the quantifiers ∀ and ∃ , the classicaltruth-conditions and falsehood-conditions are retained. Except for implications,a formula is classified as both-true-and-false exactly when it cannot be classifiedas true or false by these conditions. In the sequel, some notational conventions and abbreviations will be used.The following will sometimes be used without mentioning (with or withoutsubscripts): x as a syntactic variable ranging over all variable symbols from V , t as a syntactic variable ranging over all terms from T LPQ ⊃ , F ( Σ ), A as a syntacticvariable ranging over all formulas from F LPQ ⊃ , F ( Σ ), and Γ as a syntactic variableranging over all finite sets of formulas from F LPQ ⊃ , F ( Σ ).The string representation of terms and formulas suggested by the formationrules given above can lead to syntactic ambiguities. Parentheses are used toavoid such ambiguities. The need to use parentheses is reduced by ranking theprecedence of the logical connectives ¬ , ∧ , ∨ , ⊃ . The enumeration presents thisorder from the highest precedence to the lowest precedence. Moreover, the scopeof the quantifiers extends as far as possible to the right and ∀ x • · · · ∀ x n • A isusually written as ∀ x , . . . , x n • A .Non-equality, truth, and bi-implication are defined as abbreviations: t = t stands for ¬ ( t = t ), T stands for ¬ F , A ≡ A stands for ( A ⊃ A ) ∧ ( A ⊃ A ).3 .4 Free variables and substitution Free variables of a term or formula and substitution for variables in a term orformula are defined in the usual way.We write free ( e ), where e is a term from T LPQ ⊃ , F ( Σ ) or a formula from F LPQ ⊃ , F ( Σ ), for the set of free variables of e . We write free ( Γ ), where Γ is afinite set of formulas from F LPQ ⊃ , F ( Σ ), for S { free ( A ) | A ∈ Γ } .Let x be a variable symbol from V , t be a term from T LPQ ⊃ , F ( Σ ), and e be aterm from T LPQ ⊃ , F ( Σ ) or a formula from F LPQ ⊃ , F ( Σ ). Then [ x := t ] e is the resultof replacing the free occurrences of the variable symbol x in e by the term t ,avoiding — by means of renaming of bound variables — free variables becomingbound in t . ⊃ , F ( Σ ) The proof system of LPQ ⊃ , F ( Σ ) is formulated as a sequent-style natural deduc-tion proof system. This means that the inference rules have sequents as premisesand conclusions. First, the notion of a sequent is introduced. Next, the inferencerules of the proof system of LPQ ⊃ , F ( Σ ) are presented. Then, the notion of aderivation of a sequent from a set of sequents and the notion of a proof of asequent are introduced. An extension of the proof system of LPQ ⊃ , F ( Σ ) whichcan serve as a proof system for FOCL( Σ ) is also described. In LPQ ⊃ , F ( Σ ), a sequent is an expression of the form Γ ⊢ A , where Γ is a finiteset of formulas from F LPQ ⊃ , F ( Σ ) and A is a formula from F LPQ ⊃ , F ( Σ ). We write ⊢ A instead of ∅ ⊢ A . Moreover, we write Γ, Γ ′ for Γ ∪ Γ ′ and A for { A } on theleft-hand side of a sequent.The intended meaning of the sequent Γ ⊢ A is that the formula A is a logicalconsequence of the formulas Γ . There are several sensible notions of logicalconsequence in the case where formulas can be classified as both-true-and-false.The notion underlying LPQ ⊃ , F is precisely defined in Section 4. It corresponds tothe intuitive idea that one can draw conclusions that are not false from premisesthat are not false. Sequents are proved by (natural deduction) proofs obtainedby using the rules of inference given below. The sequent-style natural deduction proof system of LPQ ⊃ , F ( Σ ) consists of theinference rules given in Table 1. In this table, x is a syntactic variable rangingover all variable symbols from V , t , t , and t are syntactic variables rangingover all terms from T LPQ ⊃ , F ( Σ ), and A , A , A , and A are syntactic variablesranging over all formulas from F LPQ ⊃ , F ( Σ ). Double lines indicate a two-wayinference rule. 4 able 1. Natural deduction proof system of LPQ ⊃ , F ( Σ )I Γ, A ⊢ A T -I Γ ⊢ ¬ F ∧ -I Γ ⊢ A Γ ⊢ A Γ ⊢ A ∧ A ∨ -I Γ ⊢ A i Γ ⊢ A ∨ A for i = 1 , ⊃ -I Γ, A ⊢ A Γ ⊢ A ⊃ A ∀ -I Γ ⊢ AΓ ⊢ ∀ x • A †∃ -I Γ ⊢ [ x := t ] AΓ ⊢ ∃ x • A =-I Γ ⊢ t = t ¬ -M Γ ⊢ ¬¬ AΓ ⊢ A ∨ -M Γ ⊢ ¬ ( A ∨ A ) Γ ⊢ ¬ A ∧ ¬ A ∀ -M Γ ⊢ ¬∀ x • AΓ ⊢ ∃ x • ¬ A EM Γ ⊢ A ∨ ¬ A F -E Γ ⊢ F Γ ⊢ A ∧ -E Γ ⊢ A ∧ A Γ ⊢ A i for i = 1 , ∨ -E Γ ⊢ A ∨ A Γ, A ⊢ A Γ, A ⊢ A Γ ⊢ A ⊃ -E Γ ⊢ A ⊃ A Γ ⊢ A Γ ⊢ A ∀ -E Γ ⊢ ∀ x • AΓ ⊢ [ x := t ] A ∃ -E Γ ⊢ ∃ x • A Γ, A ⊢ A Γ ⊢ A ‡ =-E Γ ⊢ t = t Γ ⊢ [ x := t ] AΓ ⊢ [ x := t ] A ∧ -M Γ ⊢ ¬ ( A ∧ A ) Γ ⊢ ¬ A ∨ ¬ A ⊃ -M Γ ⊢ ¬ ( A ⊃ A ) Γ ⊢ A ∧ ¬ A ∃ -M Γ ⊢ ¬∃ x • AΓ ⊢ ∀ x • ¬ A † restriction on rule ∀ -I: x / ∈ free ( Γ ); ‡ restriction on rule ∃ -E: x / ∈ free ( Γ ∪ { A } ). In LPQ ⊃ , F ( Σ ), a derivation of a sequent Γ ⊢ A from a finite set of sequents H is a finite sequence h s , . . . , s n i of sequents such that s n equals Γ ⊢ A and, foreach i ∈ { , . . . , n } , one of the f ollowing conditions holds: – s i ∈ H ; – s i is the conclusion of an instance of some inference rule from the proofsystem of LPQ ⊃ , F ( Σ ) whose premises are among s , . . . , s i − .A proof of a sequent Γ ⊢ A is a derivation of Γ ⊢ A from the empty set ofsequents. A sequent Γ ⊢ A is said to be provable if there exists a proof of Γ ⊢ A .An inference rule that does not belong to the inference rules of some proofsystem is called a derived inference rule if there exists a derivation of the conclu-5ion from the premises, using the inference rules of that proof system, for eachinstance of the rule.The difference between CLuNs and LPQ ⊃ , F is that bi-implication is a logicalconnective in CLuNs and must be defined as an abbreviation in LPQ ⊃ , F . In [1],a proof system of CLuNs is presented which is formulated as a Hilbert system.Removing the axiom schemas A ≡
1, A ≡
2, and A ≡ A ≡ A in this proof system as abbreviations yieldsa proof system of LPQ ⊃ , F formulated as a Hilbert system. Henceforth, this proofsystem will be referred to as the H proof system of LPQ ⊃ , F and the proof systempresented in Section 3.2 will be referred to as the ND proof system of LPQ ⊃ , F . Σ ) In FOCL, the same assumptions about symbols are made as in LPQ ⊃ , F and thenotion of a signature is defined as in LPQ ⊃ , F . The languages of FOCL( Σ ) andLPQ ⊃ , F ( Σ ) are the same. A natural deduction proof system of FOCL( Σ ) canbe obtained by adding the following inference rule to the ND proof system ofLPQ ⊃ , F ( Σ ): C Γ ⊢ A Γ ⊢ ¬ A Γ ⊢ A .This proof system is known to be sound and complete. There exist better knownalternatives to it, but this proof system is arguably the most appropriate one inthis paper.In Section 5, the sequents of LPQ ⊃ , F ( Σ ) will be translated to sequents ofFOCL( Σ ′ ) ( Σ ′ is a particular signature related to Σ ). The translation concernedhas the property that what can be derived remains the same after translation.This implies that the inference rules of the proof system of LPQ ⊃ , F ( Σ ) becomederived inference rules of the above-mentioned proof system of FOCL( Σ ′ ) aftertranslation. Thus, the translation provides a logical justification for the inferencerules of LPQ ⊃ , F ( Σ ). A model-theoretic justification is afforded by the interpre-tation given in Section 4. ⊃ , F ( Σ ) The proof system of LPQ ⊃ , F is based on the interpretation of the terms andformulas of LPQ ⊃ , F ( Σ ) presented below: the inference rules preserve validityunder this interpretation. The interpretation is given relative to a structure andan assignment. First, the notion of a structure and the notion of an assignmentare introduced. Next, the interpretation of the terms and formulas of LPQ ⊃ , F ( Σ )is presented. The terms from T LPQ ⊃ , F ( Σ ) and the formulas from F LPQ ⊃ , F ( Σ ) are interpretedin structures which consist of a non-empty domain of individuals and an in-6erpretation of every symbol in the signature Σ and the equality symbol. Thedomain of truth values consists of three values: t ( true ), f ( false ), and b ( bothtrue and false ).A structure A of LPQ ⊃ , F ( Σ ) consists of: – a set U A , the domain of A , such that U A = ∅ and U A ∩ { t , f , b } = ∅ ; – for each c ∈ C( Σ ),an element c A ∈ U A ; – for each n ∈ N , for each f ∈ F n ( Σ ),a function f A : U A × · · · × U A | {z } n times → U A ; – for each n ∈ N , for each P ∈ P n ( Σ ),a function P A : U A × · · · × U A | {z } n times → { t , f , b } ; – a function = A : U A × U A → { t , f , b } such that, for each d ∈ U A ,= A ( d, d ) = t or = A ( d, d ) = b .Instead of w A we write w when it is clear from the context that the interpretationof symbol w in structure A is meant. An assignment in a structure A of LPQ ⊃ , F ( Σ ) assigns elements from U A to thevariable symbols from V . The interpretation of the terms from T LPQ ⊃ , F ( Σ ) andthe formulas from F LPQ ⊃ , F ( Σ ) in A is given with respect to an assignment α in A . Let A be a structure of LPQ ⊃ , F ( Σ ). Then an assignment in A is a function α : V → U A . For every assignment α in A , variable symbol x ∈ V and element d ∈ U A , we write α ( x → d ) for the assignment α ′ in A such that α ′ ( x ) = d and α ′ ( y ) = α ( y ) if y x . The interpretation of the terms from T LPQ ⊃ , F ( Σ ) is given by a function mappingterm t , structure A and assignment α in A to the element of U A that is thevalue of t in A under assignment α . Similarly, the interpretation of the formulasfrom F LPQ ⊃ , F ( Σ ) is given by a function mapping formula A , structure A andassignment α in A to the element of { t , f , b } that is the truth value of A in A under assignment α . We write [[ t ]] A α and [[ A ]] A α for these interpretations.The interpretation functions for the terms from T LPQ ⊃ , F ( Σ ) and the formulasfrom F LPQ ⊃ , F ( Σ ) are inductively defined in Table 2. In this table, x is a syntacticvariable ranging over all variable symbols from V , c is a syntactic variable rangingover all constant symbols from C( Σ ), f is a syntactic variable ranging over allfunction symbols from F n ( Σ ) (where n is understood from the context), t , . . . ,7 able 2. Interpretation of the language of LPQ ⊃ , F ( Σ )[[ x ]] A α = α ( x ) , [[ c ]] A α = c A , [[ f ( t , . . . , t n )]] A α = f A ([[ t ]] A α , . . . , [[ t n ]] A α )[[ F ]] A α = f , [[ t = t ]] A α = = A ([[ t ]] A α , [[ t ]] A α ) , [[ P ( t , . . . , t n )]] A α = P A ([[ t ]] A α , . . . , [[ t n ]] A α ) , [[ ¬ A ]] A α = t if [[ A ]] A α = ff if [[ A ]] A α = tb otherwise , [[ A ∧ A ]] A α = t if [[ A ]] A α = t and [[ A ]] A α = tf if [[ A ]] A α = f or [[ A ]] A α = fb otherwise , [[ A ∨ A ]] A α = t if [[ A ]] A α = t or [[ A ]] A α = tf if [[ A ]] A α = f and [[ A ]] A α = fb otherwise , [[ A ⊃ A ]] A α = t if [[ A ]] A α = f or [[ A ]] A α = tf if [[ A ]] A α = f and [[ A ]] A α = fb otherwise , [[ ∀ x • A ]] A α = t if , for all d ∈ U A , [[ A ]] A α ( x → d ) = tf if , for some d ∈ U A , [[ A ]] A α ( x → d ) = fb otherwise . [[ ∃ x • A ]] A α = t if , for some d ∈ U A , [[ A ]] A α ( x → d ) = tf if , for all d ∈ U A , [[ A ]] A α ( x → d ) = fb otherwise . t n are syntactic variables ranging over all terms from T LPQ ⊃ , F ( Σ ), P is a syntacticvariable ranging over all predicate symbols from P n ( Σ ) (where n is understoodfrom the context), and A , A , and A are syntactic variables ranging over allformulas from F LPQ ⊃ , F ( Σ ),The logical consequence relation of LPQ ⊃ , F ( Σ ) is based on the idea that aformula A holds in a structure A under an assignment α in A if [[ A ]] A α ∈ { t , b } .Let Γ be a finite set of formulas from F LPQ ⊃ , F ( Σ ) and A be a formula from F LPQ ⊃ , F ( Σ ). Then A is a logical consequence of Γ , written Γ | = A , iff for allstructures A of LPQ ⊃ , F ( Σ ), for all assignments α in A , [[ A ′ ]] A α = f for some A ′ ∈ Γ or [[ A ]] A α ∈ { t , b } .As mentioned before, the difference between CLuNs and LPQ ⊃ , F is that bi-implication is a logical connective in CLuNs and must be defined as an abbrevi-ation in LPQ ⊃ , F . In [1], an interpretation of the formulas of CLuNs is presentedwhose restriction to formulas without occurrences of the bi-implication connec-8ive is essentially the same as the interpretation of the formulas of LPQ ⊃ , F givenabove. The soundness and completeness properties for the Hilbert proof systemof CLuNs proved in [1] directly carry over to LPQ ⊃ , F . Theorem 1.
The ND proof system of
LPQ ⊃ , F ( Σ ) presented in Section 3.2 issound and complete, i.e., for each finite set Γ of formulas from F LPQ ⊃ , F ( Σ ) andeach formula A from F LPQ ⊃ , F ( Σ ) , Γ ⊢ A is provable in the ND proof system of LPQ ⊃ , F ( Σ ) iff Γ | = A .Proof. Because it is known from [1] that these properties hold for the H proofsystem of LPQ ⊃ , F , it is sufficient to prove that, for each finite set Γ of formulasfrom F LPQ ⊃ , F ( Σ ) and each formula A from F LPQ ⊃ , F ( Σ ), Γ ⊢ A is provable in theH system of LPQ ⊃ , F ( Σ ) iff Γ ⊢ A is provable in the ND system of LPQ ⊃ , F ( Σ ).The only if part is straightforwardly proved by induction on the length ofthe proof of Γ ⊢ A in the H system, using that (a) for each axiom A ′ of the Hsystem, ⊢ A ′ can be proved in the ND system and (b) for each inference rule ofthe H system, a corresponding derived inference rule of the ND system can befound.The if part is straightforwardly proved by induction on length of the proof of Γ ⊢ A in the ND system, using that (a) the standard deduction theorem holdsfor the H system, (b) for each inference rule of the ND system different from I, ⊃ -E, ∀ -I, and ∃ -E, there exists a corresponding axiom of the H system, (c) foreach of the inference rules ⊃ -E, ∀ -I, and ∃ -E, a corresponding derived inferencerule of the H system can be found, and (d) ⊢ A ⊃ A can be proved in the Hsystem. ⊓⊔ In addition to the notion of logical consequence, the notions of logical equiva-lence and consistency are semantic notions that are relevant for a paraconsistentlogic. The logical equivalence relation is semantically defined as it is semanti-cally defined in classical logic. Let A and A be formulas from F LPQ ⊃ , F ( Σ ).Then A is logically equivalent to A , written A ⇔ A , iff for all structures A of LPQ ⊃ , F ( Σ ), for all assignments α in A , [[ A ]] A α = [[ A ]] A α . The consistencyproperty is not semantically definable in classical logic. Let A and A be formu-las from F LPQ ⊃ , F ( Σ ). Then A is consistent iff for all structures A of LPQ ⊃ , F ( Σ ),for all assignments α in A , [[ A ]] A α = b .Unlike in classical logic, it does not hold that A ⇔ A iff ⊢ A ≡ A , butit holds that A ⇔ A iff ⊢ ( A ≡ A ) ∧ ( ¬ A ≡ ¬ A ). Moreover, it holds that A is consistent iff ⊢ ( A ⊃ F ) ∧ ( ¬ A ⊃ F ). In other words, the notions of logicalequivalence and consistency can both be internalized in LPQ ⊃ , F ( Σ ). ⊃ , F ( Σ ) into FOCL( Σ ) To give a classical-logic view of LPQ ⊃ , F , the terms, formulas and sequents ofLPQ ⊃ , F ( Σ ) are translated in this section to terms, formulas and sequents, re-spectively, of FOCL( Σ ). The mappings concerned provide a uniform embed-ding of LPQ ⊃ , F ( Σ ) into FOCL( Σ ). What can be proved remains the same after9ranslation. Thus, the mappings provide both a classical-logic explanation ofLPQ ⊃ , F ( Σ ) and a logical justification of its proof system. In the translation, a canonical mapping from symbols of LPQ ⊃ , F ( Σ ) to symbolsof FOCL( Σ ) is assumed. For each w ∈ S ym , we write w for the symbol to which w is mapped and, for each W ⊆ S ym , we write W for the set { w | w ∈ W } .The mapping concerned is further assumed to be injective and such that: – each x ∈ V is mapped to an x ∈ V , – each c ∈ C( Σ ) is mapped to a c ∈ C( Σ ) , – each f ∈ F n ( Σ ) is mapped to an f ∈ F n ( Σ ) , – each P ∈ P n ( Σ ) is mapped to a P ∈ P n ( Σ ) .It is also assumed that U , B ∈ P , true , false , both ∈ C , X ∈ V , and, for each i ∈ N , X i ∈ V .For the translation of terms from T LPQ ⊃ , F ( Σ ), one translation function isused: ([ ]) : T LPQ ⊃ , F ( Σ ) → T FOCL ( Σ )and for the translation of formulas from F LPQ ⊃ , F ( Σ ), three translation functionsare used: ([ ]) t : F LPQ ⊃ , F ( Σ ) → F FOCL ( Σ ) , ([ ]) f : F LPQ ⊃ , F ( Σ ) → F FOCL ( Σ ) , ([ ]) b : F LPQ ⊃ , F ( Σ ) → F FOCL ( Σ ) . For a formula A from F LPQ ⊃ , F ( Σ ), there are three translations of A to FOCL( Σ ).([ A ]) t is a formula of FOCL( Σ ) stating that the formula A of LPQ ⊃ , F ( Σ ) is truein LPQ ⊃ , F ( Σ ). Likewise, ([ A ]) f is a formula of FOCL( Σ ) stating that the formula A of LPQ ⊃ , F ( Σ ) is false in LPQ ⊃ , F ( Σ ) and ([ A ]) b is a formula of FOCL( Σ ) statingthat the formula A of LPQ ⊃ , F ( Σ ) is both true and false in LPQ ⊃ , F ( Σ ).The translation functions for the terms from T LPQ ⊃ , F ( Σ ) and the formulasfrom F LPQ ⊃ , F ( Σ ) are inductively defined in Table 3. In this table, x is a syntacticvariable ranging over all variable symbols from V , c is a syntactic variable rangingover all constant symbols from C( Σ ), f is a syntactic variable ranging over allfunction symbols from F n ( Σ ) (where n is understood from the context), t , . . . , t n are syntactic variables ranging over all terms from T LPQ ⊃ , F ( Σ ), P is a syntacticvariable ranging over all predicate symbols from P n ( Σ ) (where n is understoodfrom the context), and A , A , and A are syntactic variables ranging over allformulas from F LPQ ⊃ , F ( Σ ).The translation rules strongly resemble the interpretation rules of LPQ ⊃ , F ( Σ )that are given in Section 4: the rules for the mapping ([ ]) t correspond to thetruth-conditions and the rules for the mapping ([ ]) f correspond to the falsehood-conditions.A translation for sequents of LPQ ⊃ , F ( Σ ) can also be devised:([ Γ ⊢ A ]) = Ax( Σ, Γ ∪ { A } ) ∪ { ([ A ′ ]) t ∨ ([ A ′ ]) b | A ′ ∈ Γ } ⊢ ([ A ]) t ∨ ([ A ]) b , able 3. Translation of the language of LPQ ⊃ , F ( Σ )([ x ]) = x , ([ c ]) = c , ([ f ( t , . . . , t n )]) = f (([ t ]) , . . . , ([ t n ])) . ([ F ]) t = F , ([ t = t ]) t = eq (([ t ]) , ([ t ])) = true , ([ P ( t , . . . , t n )]) t = P (([ t ]) , . . . , ([ t n ])) = true , ([ ¬ A ]) t = ([ A ]) f , ([ A ∧ A ]) t = ([ A ]) t ∧ ([ A ]) t , ([ A ∨ A ]) t = ([ A ]) t ∨ ([ A ]) t , ([ A ⊃ A ]) t = ([ A ]) f ∨ ([ A ]) t , ([ ∀ x • A ]) t = ∀ x • U ( x ) ⊃ ([ A ]) t , ([ ∃ x • A ]) t = ∃ x • U ( x ) ∧ ([ A ]) t , ([ F ]) f = T , ([ t = t ]) f = eq (([ t ]) , ([ t ])) = false , ([ P ( t , . . . , t n )]) f = P (([ t ]) , . . . , ([ t n ])) = false , ([ ¬ A ]) f = ([ A ]) t , ([ A ∧ A ]) f = ([ A ]) f ∨ ([ A ]) f , ([ A ∨ A ]) f = ([ A ]) f ∧ ([ A ]) f , ([ A ⊃ A ]) f = ¬ ([ A ]) f ∧ ([ A ]) f , ([ ∀ x • A ]) f = ∃ x • U ( x ) ∧ ([ A ]) f , ([ ∃ x • A ]) f = ∀ x • U ( x ) ⊃ ([ A ]) f , ([ A ]) b = ¬ (([ A ]) t ∨ ([ A ]) f ) . where Ax( Σ, Γ ∪ { A } ) consists of the following formulas: – true = false ∧ true = both ∧ false = both ; – ∀ X • B ( X ) ≡ ( X = true ∨ X = false ∨ X = both ); – ∃ X • U ( X ); – ∀ X • ¬ ( U ( X ) ≡ B ( X )); – ∀ X , . . . , X n • U ( X ) ∧ . . . ∧ U ( X n ) ⊃ U ( f ( X , . . . , X n )) for each f ∈ F n ( Σ ),for each n ∈ N ; – ∀ X , . . . , X n • U ( X ) ∧ . . . ∧ U ( X n ) ⊃ B ( P ( X , . . . , X n )) for each P ∈ P n ( Σ ),for each n ∈ N ; – ∀ X , X • U ( X ) ∧ U ( X ) ⊃ B ( eq ( X , X )); – ∀ X • eq ( X, X ) = true ∨ eq ( X, X ) = both ; – U ( x ) for each x ∈ free ( Γ ∪ { A } ).Ax( Σ, Γ ∪ { A } ) contains formulas asserting that the domain of truth valuescontains exactly three elements, the domain of individuals contains at least one11lement and is disjoint from the domain of truth values, application of a func-tion yields an element from the domain of individuals, application of a predicate,including the equality predicate, yields an element from the domain of truth val-ues, application of the equality predicate does not yield false if the arguments areidentical, and free variables are always elements from the domain of individuals. An important property of the translation of sequents of LPQ ⊃ , F ( Σ ) to sequents ofFOCL( Σ ∪{ true , false , both , B , U , eq } ) presented above is that what can be provedremains the same after translation. This means that the translation provides auniform embedding of LPQ ⊃ , F ( Σ ) into FOCL( Σ ∪ { true , false , both , B , U , eq } ). Theorem 2.
For each finite set Γ of formulas from F LPQ ⊃ , F ( Σ ) and each for-mula A from F LPQ ⊃ , F ( Σ ) , Γ ⊢ A is provable in LPQ ⊃ , F ( Σ ) iff ([ Γ ⊢ A ]) isprovable in FOCL( Σ ∪ { true , false , both , B , U , eq } ) .Proof. The only if part is easily proved by induction over the length of a proof of Γ ⊢ A and case distinction on the last inference rule applied, using that the NDproof system for FOCL( Σ ∪ { true , false , both , B , U , eq } ) described in Section 3.4contains all inference rules of LPQ ⊃ , F ( Σ ).The if part is proved making use of Theorem 1. Let A be a structure ofLPQ ⊃ , F ( Σ ). Then A can be transformed in a natural way into a structure A ∗ ofFOCL( Σ ∪ { true , false , both , B , U , eq } ) with the following properties: [[ A ]] A α = t iff [[([ A ]) t ]] A ∗ α = t , [[ A ]] A α = f iff [[([ A ]) f ]] A ∗ α = t , and [[ A ]] A α = b iff [[([ A ]) b ]] A ∗ α = t (for all assignments α in A ). Now assume that A is a counter-model for Γ ⊢ A .Then, for its above-mentioned properties, A ∗ is a counter-model for ([ Γ ⊢ A ]).From this, by Theorem 1, the if part follows immediately. ⊓⊔ The translation of sequents extends to inference rules in the obvious way.
Corollary 1.
The translation of the inference rules of the presented proof sys-tem of
LPQ ⊃ , F ( Σ ) are derived inference rules of the proof system of FOCL( Σ ∪ { true , false , both , B , U , eq } ) described in Section 3.4. In this section, properties the propositional fragment of LPQ ⊃ , F and the differentways in which LPQ ⊃ , F is close to classical logic are briefly discussed. ⊃ , F In [8], the propositional fragment of LPQ ⊃ , F , called LP ⊃ , F , is presented. In thatpaper, it is among other things shown that LP ⊃ , F is the only three-valued para-consistent propositional logic whose logical consequence relation has all prop-erties proposed as desirable for such logics in the literature and whose logical12quivalence relation satisfies the identity, annihilation, idempotent, and commu-tative laws for conjunction and disjunction, the double negation law for negation,and two laws that uniquely characterize its implication connective.Because closeness to classical logic is generally considered important, theabove-mentioned properties of the logical equivalence relation concerning con-junction, disjunction, and negation should arguably also be taken as desirablefor paraconsistent propositional logics. Unlike in classical logic, it does not holdin three-valued paraconsistent logics that logical equivalence is the same as log-ical consequence and its inverse. This necessitates taking desirable properties ofthe logical equivalence relation explicitly into account. If the above-mentionedproperties concerning conjunction, disjunction, and negation are taken into ac-count, then the number of ‘ideal’ three-valued paraconsistent propositional logicsreduces from 8192 to 16, among which LP ⊃ , F (cf. [8]).In [2], an application of LP ⊃ , F can be found. The properties of the logicalequivalence relation that are essential for that application, among which most,but not all, properties considered desirable above, reduces the number of idealthree-valued paraconsistent propositional logics that are applicable even to 1,namely LP ⊃ , F . This strengthens the impression that among the paraconsistentlogics that deserves most attention are LP ⊃ , F and its first-order extensions. How-ever, the question arises whether a paraconsistent logic is really needed to dealwith contradictory sets of formulas. The embedding of LPQ ⊃ , F into FOCL givenin this paper shows that it can be dealt with in classical logic but in a much lessconvenient way. ⊃ , F is close to classical logic LPQ ⊃ , F is a paraconsistent logic whose logical consequence relation and logicalequivalence relation have all properties proposed as desirable for such logics.These properties are all related to closeness to classical logic. Moreover, LPQ ⊃ , F has no connective that is foreign to classical logic and the inference rules of itsnatural deduction proof system are all known from classical logic: – except for the inference rules concerning the negation connective, the in-ference rules are the ones found in all natural deduction proof systems forclassical logic; – the inference rules concerning the negation connective are a rule that corre-sponds to the law of the excluded middle and rules that correspond to thede Morgan’s laws for all connectives and quantifiers; – the rule corresponding to the law of the excluded middle is also found in nat-ural deduction proof systems for classical logic and the rules correspondingto the de Morgan’s laws are well-known derived rules of natural deductionproof systems for classical logic.This means that natural deduction reasoning in the setting of LPQ ⊃ , F differs fromclassical natural deduction reasoning only by slightly different, but classicallyjustifiable, reasoning about negations. 13 Concluding Remarks
The paraconsistent logic LPQ ⊃ , F has been presented. A sequent-style naturaldeduction proof system has been given for this logic. A natural deduction proofsystem has not been given before in the literature for one of the logics that areessentially the same as LPQ ⊃ , F . In addition to the model-theoretic justification ofthe proof system, a logical justification by means of an embedding into classicallogic has been given. Thus, a classical-logic view of LPQ ⊃ , F has been provided.It appears that a classical-logic view of a paraconsistent logic has not been givenbefore.A classical-logic view of a paracomplete logic has been given before in [7], fol-lowing the same approach as in the current paper. It is likely that this approachworks for all truth-functional finitely-valued logics. The approach concerned isreminiscent of the method, described in [3], to reduce the many-valued interpre-tation of the formulas of a truth-functional finitely-valued logic to a two-valuedinterpretation. References
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