Syntax and semantics of simple paracomplete logics Текст научной статьи по специальности «Математика»

Syntax and semantics of simple

abstract. For an arbitrary fixed element 3 in {1, 2, both a sequent calculus and a natural deduction calculus which axiomatise simple paracomplete logic are built. Additionally, a valuation semantic which is adequate to logic is constructed. For an arbitrary fixed element y in {1, 2, 3,... } a cortege semantic which is adequate to logic I2,7 is described. A number of results obtainable with the axiomatisations and semantics in question are formulated.

Keywords: paracomplete logic, paraconsistent logic, cortege semantics, valuation semantics, sequent calculus, natural deduction calculus

We study logics ¡2,1, I2>2,12,3, • • • I2,w presented in [8]. These logics are paracomplete counterparts of paraconsistent logics I1,1, I1>2, I1>3, ...I1;W from [7]. In the paper, (a) simple paracomplete logics I2>1, I2,2, I2,3, ... I2,w are defined (see [8]); these logics form (in the order indicated above) a strictly decreasing (in terms of the set-theoretic inclusion) sequence of logics, (b) for any j in {0,1,2,3,... w} both a sequent calculus GI2,j (see [10]) and a natural deduction calculus NI2,j which axiomatise logic I2,j are formulated, (c) for any j in {1,2, 3, ...w}, we propose a valuation semantics for logic I2,j (see [9]), (d) for any j in {1,2, 3,... }, we propose a cortege semantics for logic I2,j (see [9]). Below there are some results obtained with the semantics and calculi in question.

The language L of each logic in the paper is a standard propositional language with the following alphabet: {&, V,

xThe paper is supported by Russian Foundation for Humanities, project №10-03-00570a and project №13-03-00088a (both authors).

Vladimir M. Popov, Vasiliy O. Shangin

D, —, (, ),p1,p2,p3, • • • }• As it is expected, V, D are binary logical connectives in L, - is a unary logical connective in L, brackets (, ) are technical symbols in L and p1,p2,p3, • • • are propositional variables in L. A definition of L-formula is as usual. Below, we say 'formula' instead of 'L-formula' only and adopt the convention on omitting brackets as in [4]. A formula is said to be quasi-elemental iff no logical connective in L other than - occurs in it. A length of a formula A is, traditionally, said to be the number of all occurrences of the logical connectives in L in A. We denote the rule of modus ponens in L by MP and the rule of substitution of a formula into a formula instead of a propositional variable in L by Sub. A logic is said to be a non-empty set of formulas closed under MP and Sub. A theory for logic L is said to be a set of formulas including logic L and closed under MP. It is understood that the set of all formulas is both a logic and a theory for any logic. The set of all formulas is said to be a trivial theory. A complete theory for logic L is said to be a theory T for logic L such that, for some formula A, A € T or —A € T. A paracomplete theory for logic L is said to be a theory T for logic L such that T is not a complete theory and any complete theory for logic L, which includes T, is a trivial theory. A paracomplete logic is said to be a logic L such that there exists a paracomplete theory for logic L. Simple paracomplete logic is said to be a paracomplete logic L such that for any paracomplete theory T for logic L holds true: there exists a quasi-elemental formula A such that neither A, nor —A belongs to T.

Let us agree that anywhere in the paper: a is an arbitrary element in {0,1,2,3, • • • w}, / is an arbitrary element in {1,2,3, • • • w}, y is an arbitrary element in {1,2,3, • • • }. We define calculus HI2,a. This calculus is Hilbert-type calculi, the language of HI2a is L. HI2a has MP as the only rule of inference. The notion of a derivation in HI2a (of a proof in HI2a, in particular) is defined as usual; and for HI2a, both notion of a formula derivable from the set of formulas in this calculus and a notion of a formula provable in this calculus are defined as usual. Now we only need to define the set of axioms of

HI2,a-

A formula belongs to the set of axioms of calculus HI2a iff it is one of the following forms (hereafter, A, B, C denote formulas):

(I) (A D B) D ((B D C) D (A D C)), (II) A D (A V B), (III) B D (A V B), (IV) (A D C) D ((B D C) D ((A V B) D C)), (V) (A&B) D A, (VI) (A&B) D B, (VII) (C D A) D ((C D B) D (C D (A&B))), (VIII) (A D (B D C)) D ((A&B) D C), (IX) ((A&B) D C) D (A D (B D C)), (X) ((A D B) D A) D A, (XI, a) (E D —(B D B)) D —E, where E is formula which is not a quasi-elemental formula of a length less than a, (XII) —A D (A D B).

Let us agree that, for any j in {0,1,2,3,... w}, I2,j is the set of formulas provable in HI2,j.

The following theorems 1 and 2 are shown.

Theorem 1. Sets I2,0, h,1, I2,2, I2,3, ■■■h,u are logics, and, for any k and l in {0,1, 2, 3,... w}, if k < l, then I2,i C I2 k■

Theorem 2. Logic I2,0 is the set of the classical tautologies in L■

Let us establish connections between logics I2>1, I2,2, I2,3, ... I2)W and logic I2,0 (that is, the classical propositional logic in L).

Let y be a mapping of the set of all formulas into itself satisfying the following conditions: (1) y(p) is not a quasi-elemental formula, for any propositional variable p in L, (2) for any propositional variable p in L, formulas p D y(p) and y(p) D p belong to logic I2,0, (3) y(B o C) = y(B) o y(C), for any formulas B, C and for any binary logical connective o in L, (4) y(—B) = —y(B), for any formula B.

Following these conditions, theorem 3 is shown.

Theorem 3. For any j in {1, 2, 3,... w} and for any formula A: A e I2,o iff <fi(A) e I2,j ■

Let now ^ be such a mapping the set of all formulas into itself satisfying the following conditions: (1) ^(p) = p, for any propositional variable p in L, (2) ^(B o C) = ^(B) o ^(C), for any formulas B, C and for any binary logical connective o in L, (3) ^(—B) = ^(B) D —(p1 D p1), for any formula B. Following these conditions, theorem 4 is shown.

Theorem 4. For any j in {1, 2, 3,... w} and for any formula A: A e 12,0 iff 1>(A) e h,j ■

Let us now show a method to build up a sequent calculus GI2,^ which axiomatises logic I2,@. Calculus GI2)/g (see [10]) is a Gentzen-

type sequent calculus. Sequents are of the form r ^ A (hereafter, r, A, £ and 0 denote finite sequences of formulas). The set of basic sequents of GI2)/g is the set of all sequents of the form A ^ A. The only rules of GI2are the rules R1-R15, R16(/), R17 listed below.

r, A, B, A ^ © r, B, A, A ^ © r ©,A,A

R1,

r ^ ©,A r^ A, a

R4

r ^ A, A, B, © r ^ A, B, A, ©

r ©

R2,

A, r B, E^©

©

■ R5,

A, A, r ^ © A, r ^ ©

r ©

R3,

A D B, r, E ^ A, ©

R7,

a, r

r

©, B

©, A R8,

R6,

A, r ^ © A&B, r ^ ©

r ^ ©, A r ^ ©, A V B r ©,A

■ R9,

R12,

a, r ■

r ^ ©, A D B © „ r —> ©, a

B&A, r ^ ©

r ^ ©, A r ©,B\Z A

R10,

r ~^©,B

R13,

r ^ ©, A&B A, r ^ © B, r ^ © a v B, r ©

■ R11,

R14,

-A, r ^ © E, r ^ © r ^ ©, -E

r A, a

R15,

R16(^), where E is a formula which is not a quasi-elemental

formula of a length less than 3,

a, ^ ^ ©

r, E A, ©

R17 (cut rule)

A derivation in calculus GI2,^ is defined in a standard sequent calculus fashion. The definition of a sequent provable in GI2,^ is as usual. The cut-elimination theorem is shown (by Gentzen's method presented in [3]) to be valid in GI2,^. The following theorem 5 is shown.

Theorem 5. For any j in {1,2,3, • • • w} and for any formula A: A € I2,j iff a sequent ^ A is provable in GI2,j.

Let us now show a method to build up a Fitch-style natural deduction calculus NI2)/g which axiomatises logic I2,@.

The set of NI2)/g-rules is as follows, where [A]C denotes a derivation of a formula C from a formula A.

C&C\ C

&el1

cfecj cl

&el2

C, Cl C&Cl

C V ci, [c]c2 [cijc, C w ci ..

' Vel „ „ Vini-

c2 l c V ci c V ci

c D ci, c [c]ci [a D b]a

D el —---— Din -:-DP

Ci C D Ci A

[E] —(C D C)

\ni(p), where E is a formula which is not a quasi-

E

elemental formula of a length less than

—Ci, Ci

■ —in 2

C

A derivation in NI2| is defined in a standard natural deduction calculus fashion.

The following theorem 6 is shown.

Theorem 6. For any j in {1,2,3,... w} and for any formula A : A € I2,j iff A is provable in NI2,j.

The proof search procedures which were proposed to the classical and a variety of non-classical logics are applicable [1, 2].

Let us construct I2,|-valuation semantics for I2|. By Q| we denote the set of all quasi-elemental formulas of a length less or equal to /. By I2)/g-valuation we mean any mapping v set Q| into the set {0, 1} such that, for any quasi-elemental formula e of a length less than if v(e) = 1, then v(—e) = 0. Let Form denote the set of all formulas and let Val2,| denote the set of all I2,|-valuations. It can be shown there exists a unique mapping (denoted by ) satisfying the following six conditions: (1) £2| is a mapping a Cartesian product Form x Val2,| into the set {1, 0}, (2) for any quasi-elemental formula Y in Q| and any I2,|-valuation v: (Y,v) = v(Y), (3) for any formulas A, B and any I2,|-valuation v: (A&B,v) = 1 iff £2,i3 (A) = 1 and (B) = 1, (4) for any formulas A, B and any I2|-valuation v: (A VB,v) = 1 iff (A) = 1 or (B) = 1, (5) for any formulas A, B and any I2,|-valuation v: (A d B,v) = 1 iff £2,l (A) = 0 or {2|(B) = 1, (6) for any formula A which is not a quasi-elemental formula of a length less than /, and for any I2,|-valuation v: (—A,v) = 1 iff (A,v) = 0. A formula A is said to be I2,|-valid iff for any I2,|-valuation v, (A,v) = 1.

The following theorems 7 and 8 are shown.

Theorem 7. For any j in {1,2,3,... w}, for any formula A, for any set r of formulas: formula A is derivable from Y in HI2,j iff for

any I2,j-valuation v, if for any formula B in r, {2 ,j (B,v) = 1, then 6 j (A,'v) = 1.

Theorem 8. For any j in {1,2,3, • • • w} and for any formula A, A € I2,j iff formula A is I2,j-valid.

It should be noted that the proposed I2 ,@-valuation semantics is consistent to the requirements, which, in our point of view, N.A. Vasiliev considers to be necessary in [11]: (1) no proposition cannot be true and false at once, (2) in general case, a value of the proposition that is a negation of a proposition P, is not determined by the value of P.

Let us construct I2,7-cortege semantics for \2,y. By \2,y-cortege we mean an ordered 7 + 1-tuplet of elements of the set {1, 0} such that for any two neighboring members of this ordered 7 + 1-tuplet, at least one of them is 0. By a designated I2>7-cortege we mean I2,y-cortege, where the first member is 1. By S2>7 we denote the set of all I2)7-corteges and by D2>7 we denote the set of all designated I2,y-corteges. By a normal I2>7-cortege we mean I2>7-cortege such that any two neighboring members of this I2>7-cortege are different. By a single I2>7-cortege we mean a normal I2>7-cortege such that the first member of it is 1. By a zero I2>7-cortege we mean a normal I2,y-cortege such that the first member of it is 0.

It is clear that there exists a unique single I2>7-cortege (denoted by 1Y) and there exists a unique zero I2>7-cortege (denoted by 0Y). It can be shown that there exists a unique binary operation on S2>7 (denoted by &2>7) satisfying the following condition, for any X, Y in S2>7: if the first member of I2>7-cortege X is 1 and the first member of I2, Y-cortege Y is 1 then X&2,yY is 1y; ^terw^ X&2,yY is 0y. It can be shown that there exists a unique binary operation on S2>7 (denoted by V2>7) satisfying the following condition, for any X and Y in S2>7: if the first member of I2>7-cortege X is 1 or the first member of I2)7-cortege Y is 1 then X V2,Y Y is 1Y; otherwise, X V2,Y Y is 0Y. It can be shown that there exists a unique binary operation on S2>7 (denoted by D2)7) satisfying the following condition, for any X and Y in S2>7: if the first member of I2>7-cortege X is 0 or the first member of I2>7-cortege Y is 1 then X d2>7 Y is 1Y; otherwise, X d2>7 Y is 0Y. It can be shown that there exists a unique unary

operation on S2,7 (denoted by -2,7) satisfying the following condition, for any I2)7-cortege < x1,x2, • • • ,x7, x7+1 >: if x7+1 is 1 then -2)7(< x1,x2,...,x7, x7+1 >) =< x2,...,x7 ,x7+1,0 > and if, if x7+1 is 0, then -2,7(< x1,x2,...,x7, x7+1 >) = < x2, • • •, x7, x7+1,1 >.

It is clear that < S2,7, D2,7, &2,7, V2,7, D2,7, -2)7 > is a logical matrix. This logical matrix (denoted by M2,7) is said to be I2)7-matrix. M2>7-valuation is said to be a mapping the set of all propositional variables in L into S2,7. The set of all M2,7-valuations is denoted by ValM2,7. It can be shown that there exists a unique mapping (denoted by £M2,7) satisfying the following conditions: (1) £M2,7 is a mapping a Cartesian product Form x ValM2>7 into the set S2,7, (2) for any propositional variable p in L and for any M2,7-valuation w, £M2,7(p,w) = w(p), (3) for any formulas A, B and for any M2,7-valuation w, £M2,7(A&B,w) =

Y(A,w)&2,y£M2,y(B,W), (4) for any formulas A B and for any

M2,7-valuation w, £M2,7(A V B, w) = £M2,7(A, w) V2,7 £M2,7(B, w), (5) for any formulas A, B and for any M2,7-valuation w, £M2,7(A d B,w) = £M2,7(A,w) d2,7 £M2,7(B,w), (6) for any formula A and for any M2,7-valuation w, £M2,7(-A,w) = -2,7£M2,7(A, w).

A formula A is said to be M2,7-valid iff for any M2,7-valuation w, £M2,7(A, w) € D2,7.

The following theorems 9-11 are shown.

Theorem 9. For any j in {1,2,3, • • • }, for any formula A and for any set r of formulas, formula A is derivable from r in HI2,j iff for any M2,j-valuation w, if for any formula B from r, £M2,j (B,w) € Dhj then £M2j (A,w) € D2j.

Theorem 10. For any j in {1, 2,3, • • • } and for any formula A, A € I2,j iff A is M2,j-valid.

Theorem 11. For any j in {1,2,3,...} and for any formula A, A is M2,j-valid iff for any M2,j-valuation w, £M1j (A,w) € 1j.

The following theorems 12-19 are shown with the help of the axiomatisations and semantics presented in the paper.

Theorem 12. Logics I2>1, I2,2, I2,3, ■ ■ ■ I2,u are simple paracomplete logics■

Theorem 13. For any j and k in {1,2, 3,...w}, if j = k then I2,j = I2,k ■

Theorem 14. For any j in {1,2,3,... w}, the positive fragment of logic I2,j is equal to the positive fragment of logic I2)0■

Theorem 15. For any j in {1,2, 3,... w}, logic I2,j is decidable■

Theorem 16. For any j in {1, 2, 3,... }, logic I2,j is finitely-valued■

Theorem 17. Logic I2;W is not finitely-valued^

Theorem 18. Logic I2;W is equal to the intersection of logics I2>1, I2,2, I2,3, ■ ■ ■

Theorem 19. There is a continuum of logics which include I2>U) and are included in I2,1^

References

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