Chapter 9 Lukasiewicz Logics

459

CHAPTER 9 LUKASIEWICZ LOGICS

The theory of Lukasiewicz-Moisil algebras originated in the need for an algebraic counterpart of the many-valued logics introduced by Lukasiewicz. In this chapter we present the three-valued Lukasiewicz logic in the Wajsberg axiomatization, the n-valued Lukasiewicz logics in the Cignoli axiomatization and a logic whose theorems are the propositions true for all than a fixed

k E I . The

i E I greater

Lindenbaum-Tarski algebras of these logics are the

3-valued Moisil algebras, a subclass of n-valued Moisil algebras and t h e dvalued LM-algebras, respectively. The latter logic is studied in more detail, including predicate calculus.

31. The Wajsberg axiomatization of the three-valued Lukasiewicz logic The first system of three-valued logic was constructed by J. Lukasiewicz in 1920 in connection with his investigations in modal logic (see Lukasiewicz [1920]). His idea was t o consider a third truth-value

between 0 (false-

hood) and 1(truth). In this way th e sentences of the three-valued logic are interpreted in

L3 =

{O,i,l}.

Lukasiewicz defined in L3 a unary operation

N (negation) and a binary operation Nz=l-z

--f

(implication):

and x + y = m i n ( l , l - ~ + y ) .

The other connectives were defined by Lukasiewicz in terms of N and +: z V y = (z+ y) + y

(disjunction)

z A y = N ( N x V Ny) (conjunction) . The first axiomatization of the three-valued logic was given by Wajsberg [1931] using axioms (Al)-(A4) presented in Definition 1.1below. This section is concerned with th e Wajsberg axiomatization of the three-valued

Lukasiewicz logics

460

Lukasiewicz logic. T h e proof of some syntactic properties of the three-valued propositional calculus

W is taken from Becchio [1972] and t h e proof of the

strong completeness theorem f r o m Goldberg, H. Leblanc and Weaver [1974]. Axioms (Al)-(A4)

induce o n the Lindenbaum-Tarski algebra of W a ca-

nonical structure of Wajsberg algebra. But t h e three-valued Moisil algebras and t he Wajsberg algebras are equivalent structures (Becchio [1978d]). Thus we can assert th a t th e three-valued Moisil algebras are algebraic models for the three-valued Lukasiewicz logic.

1.1. Defi nition. The sentences of the three-valued propositional calculus W are obtained from a countable set V of proposn'tional variables and t h e logical connecti-

ves N and +, according to th e following rules:

(i)

t he propositional variables are sentences;

(ii) if p , q are sentences then N p and p

---t

g are sentences;

(iii) every sentence is obtained by the above rules (i) and (ii)applied finitely many times.

E , is the (adjacent set of the) Peano algebra of type (2,l) generated by V . In other words, t h e set of sentences denoted by

W e use t he notation

for any sentence p E E and a fixed sentence p o E E . T h e axioms of W are th e sentences of t h e following forms:

W has modus ponens (m.p.) as rule of inference:

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic P,P

+

461

4

4

A proof o f a sentence p is a finite sequence p l , ...,p, = p o f sentences such that for any i 5 n we have one of the following possibilities: (a) p; is an axiom;

(b) there exists j , k

< i such that pk

is the sentence pj

-+

pi.

A sentence p is provable (I- p ) if there is a proof of it. If p E E and S C E then a proof of p from S is a finite sequence p1, ...,pn = p of sentences such that for any i 5 n we have one of the following possibilities: (a) pi is an axiom or a member of S;

(b) there exist j , k

< i such that pk

is the sentence

In this case, we say that p is provable f r o m

0 I- p

pj

+ pi.

S (S I- p ) .

is t he same as I- p . If the sentences s, p satisfy {s}

In particular

I- p we write

simply s I- p .

A set S of sentences if syntactically consistent if there is no p in W such that S I- p and S t- N p ; if not, S is syntactically inconsistent. A syntactically consistent set S is maximal consistent if S I- p for any sentence p such that S U { p } is syntactically consistent. Now we shall give, following Becchio [1972], some syntactic properties

of

w.

1.2. Lemma.

The following properties hold an W :

462

(1.5) (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) (1.12) (1.13) (1.14)

(1.15) (1.16) (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) (1.23) (1.24) (1.25) (1.26)

Lukasiewicz logics

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic

463

Proof. (1.1) and (1.2) follow by (Al), (A2) and m.p., while (1.3) follows by m.p. from (1.2). To obtain (1.4) use ( A l ) for p := r t p , q := ¶ and also for p := p , q := r , then apply (1.3). Now (1.5) follows from

(1.4). To obtain (1.6) use (A2), then again (A2) but for p := q 3 r , q := p --f r , r := s; finally apply (1.3). Now (1.7) follows by m.p. from (1.6), then (1.8) by m.p. from (1.7). To obtain (1.9) apply (1.1)for p := p , q := q --f N q , then (1.1)for p := ( q -+ N q ) --t p , q := p --t q, then (A3) for p := q , then (1.3) for p := p

+ q,

q := ( q + N q )

-+

q,

r := q; finally apply m.p. Further (1.10) follows by m.p. from (1.9) for p := p , q := r and (1.3) for p := p , q := p -+ r , r := r . Now

if t is a provable sentence then using ( A l ) for p := p , q := t , (1.6) for p := t , q := p , r := q , s := q , (1.3) for p := p , q := t -+ p , r := ( ( t t q ) + q ) t ( ( p t q ) --t q ) , (1.9) for p := t , q := q and (1.10) for p := (t + q ) t q , q := p , r := ( p q ) + q, we obtain (1.11).Further from (1.1)for p := q , q := r we obtain

via (1.2) for p

( ( q + r ) -+

r ) , r := p --t r ; then from (A2) for p := p , q := q --t r , r := r and (1.27) we get (1.12) via (1.3). Now (1.12) implies (1.13), then (A2) and (1.13) imply (1.14), while (1.14) implies (1.15). To obtain (1.16) apply (1.14) for p := r , q := p , r := q and (1.3). To obtain (1.17) apply (1.11) for p := q , q := r , then (1.15) for p := p , q := q , r := ( q -+ r ) -+ r. To obtain (1.18) apply (1.17) for p := p -+ q, q ._ .- q , r := r , then (1.13). To obtain := q , q := q

-+

(1.19) apply (1.3) t o ( A l ) for p := N p , q := N q and (A4) for p := q , q := p . Further (1.19) implies (1.20). To obtain (1.21) apply (1.3) t o (1.19) for p := N p , q := N q and (A4). Then (1.22) follows from (1.19) and (1.2) for p := N p , q := p + q, r := q. To obtain (1.23) apply (1.3) t o (1.21) for p := p , q := p + N p and (A3). Now (1.24) follows by m.p. from (1.23) for p := N p and (A4) for p := N N p , q := p . Further from (1.23) and (1.2) for p := N N p , q := p , r := q we get

Lukasiewicz logics

464

while from (1.24) for p T := N N q we obtain: (1.29)

:= q and (1.25) for p

:= N N p , q

:= q,

I- ( N N p -+ q ) -+ ( N N p -+ N N q ) ;

then (1.3) applied t o (1.28) and (1.29) yields

which together with (A4) for p := N p , q := N q , yields (1.25), again by 13 (1.3). Finally (1.26) follows from (1.24) and (1.23) by (1.3). 1.3. Lemma. T h e relation (1.31)

-

p-q

defined by iff

kp+q

and I - q - i p

is a n equivalence o n the set E of sentences of W . Proof. The relation is reflexive by (1.26), symmetric by definition and transitive by (1.3).

0

1.4. Theorem. Define the following operations

+, N ,

1 o n the quotient set E l -:

(1.32)

fi -+ 4 = p 3 q ,

(1.33)

Nfi = N^p

(1.34)

1 is the set of all provable sentences .

Then (E/

-,

,

N , 1) is a Wajsberg algebra, i.e. it satisfies conditions

3,

(W1)-(W 6 ) in Theorem 3.3.8. Proof. If p

-

-

p' and q q' then the sentences p + p', p' 4 p , q -+ q' and q' -+ q are provable. Then (1.15) and (1.2) imply t- ( p -+ q ) 4 ( p --+ q') and I- ( p -+ q') + (p' -+ q'), respectively. It follows by (1.3) that I- ( p -+ g ) -+ (p' -+ q') and similarly I- (p' 4 q') -+ ( p --+ q ) , therefore p + q p' -+ q'. This shows that the operation -+ is well defined. One proves similarly, using (1.25), that N is well defined. Further if t is provable and t t' then t' is

-

-

465

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic provable by m.p., while if t and t' are provable then t

-

t' by (1.1). This

shows that the set of provable sentences is actually an equivalence class. Finally axioms (Wl)-(W4)

are fulfilled in view of (Al)-(A4),

vely, (W5) reduces to the above remarked fact that I- t and

respecti-

I- t + t'

imply

I- t', while (W6) is straightforward from (1.31). 1.5. Remark. In accordance to Theorem 3.3.8, the Lindenbaurn-Tarski algebra E /

0

-

of

the three-valued propositional calculus is a three-valued Moisil algebra. This is a precise meaning of the assertion that the structure of three-valued Moisil

algebra is the algebraic counterpart of the Lukasiewicz three-valued logic. 1.6. Lemma.

The following properties hold in W : (1.35) (1.36)

(1.37) (1.38) (1.39) (1.40) (1.41) Proof. From Theorem 1.4 and Lemmas 3.3.9 and 3.3.10. 1.7. Lemma (Goldberg, Leblanc and Weaver [1974]).

Let S C E and p , q E E . (a)

If S I- p , then S' I- p for any superset S' of S .

(b) If S I- p, then there is a finite subset (c)

If S I- p and S I- p

+ q,

then S I- Q.

S'

of S such that S' I- p .

Lukasiewicz logics

466 (d) If S U { p } (e)

t- q t h e n S t- p

--t

( p + q).

If S i s syntactically inconsistent, t h e n S !- r foT a n y sentence r in E .

(f) S i s syntactically inconsistent iff S I- f. (g) If S U { p } is syntactically inconsistent, t h e n S t-

fi.

(h) If S U {fi} i s syntactically inconsistent, t h e n S t- p . Proof. (a)-(c):

Immediate from the definitions.

(d) Suppose p l , . . . , p n is a proof of q from S U { p } . We shall show, by induction on i, that S t- p t ( p t p i ) , 1 5 i 5 n. We shall distinguish the following cases:

(i) p; is an axiom or a member of S, hence S t- p i . Using twice (1.1) we get S I- p --t p i , then S I- p --f ( p + p i ) . (ii) p ; is p . Apply (Al).

(iii) pk and pk

+ pi

appear in the list p l , . . . , p i - l . Then

(1.42)

by t h e inductive hypothesis. Now apply twice (c) using (1.43) and

(1.35) for p := p , 4 := pk, (e) If S

t- p

and S I- N p for some p E

by(1.19), we get S

t- r

T

E

:= pi.

then since S

I- N p

+ ( p -+

r)

via (c).

(f) If S is syntactically inconsistent then S I- f by (e). Conversely, if S t- f then S I- N(p0 + PO) while S t- (PO --t P O ) by (1.26). (g) If S u { p } is syntactically inconsistent, then S U { p } I- N p by (e), therefore

S I- p + fi by (d), hence S t- fi via (1.40).

(h) If S U {fi} is syntactically inconsistent, then S Iand (c) yield S t- p .

5 by (g), hence (1.39) 0

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic

467

Now we turn t o semantics. We are going t o use the Lukasiewicz definition of the operations + and

N in L3:

Note that L3 endowed with the above operations is a Wajsberg algebra

1.8. Definition. An interpretation or a trzlth-value assignment of W is an arbitrary function ZJ : V + L3. Any interpretation v has a unique homomorphic extension V : E + L3, i.e. V J V = w and C(p 3 q ) = V ( p ) 3 i?(q), V ( N p ) = NV(p) for any p , q E E . We will also refer t o V as an interpretation and write simply

w instead of E.

A sentence p is valid (kp ) if w(p) = 1for any interpretation w. A set S of sentences is semanticalzy consistent if there is an interpretation 21 such that w(p) = 1for each p E S. If v(p) = 1for any interpretation v such that

v ( S ) = (1) then we shall wirte S 1.9. Proposition (Goldberg,

tactically consistent set S

p.

H. Leblanc and

Weaver [1974]). Every s y n -

c E is semantically consistent.

Proof. Assume S is syntactically consistent. As the set V of propositional variables is countable, one can prove that the whole set E of senten-

E = { p o , p l , ...}. Define by induction a sequence So = S, Sl, ... of syntactically consistent sets of sentences: Si = Si-l~{pi}if ces is also countable, say

Si-lu{pi} is syntactically consistent, otherwise Si = Si-1. Let

s=

00

U

i=O

Si.

S is syntactically consistent: 3 k p and S I- N p for some p would imply that S; I- p and Si t- N p for a sufficiently large i, a contradiction. Now we shall prove that S is maximal consistent. If Using Lemma 1.7 (b), it follows that

468

Lukasiewicz logics

it is not the case that S t- pi then pi 4 S , hence Si-l U { p i } is syntactically inconsistent. By Lemma 1.7 (f), Si-, U { p i } t- f, then S U { p i } I- f by Lemma 1.7 (a), hence S U { p i } is syntactically inconsistent. Let v be the following interpretation:

v(p) =

for any p E

1

I,

ifst-p

0,

i f SI- N p

1

otherwise

,

V . As a matter of fact we prove by induction that for any

sentence p :

(i) S I- p implies v ( p ) = I; (ii) S I- N p implies v(p) = 0; (iii) if neither

S I- p

nor

S I- N p , then v(p) = 3.

If p E V this holds by the previous definition. The inductive step comprises two cases. Case 1. p is N g , where g satisfies the inductive hypothesis.

(i)

If S I- N q then v(g) = 0, hence v ( p ) = N v ( g ) = 1.

(ii) If S I- N N q then S I- g by (1.23), hence v(g) = 1, therefore v(p) = N v ( q ) = 0. (iii) Assume neither S I- N q nor S I- N N g . But S t- q --t N N q by (1.24), hence it is not the case that S I- g. Since neither S t- g nor S I- N g we have v(g) = therefore v ( p ) = N v ( q ) =

i,

i.

Case 2. p is g -+ r , where q and r satisfy the inductive hypothesis.

(i) Suppose S I- g + r . Then the following cases are possible: S I- g 3 S I- r =+ v(r) = 1 + v(q + r ) = v(g) + w(r) = v(g) 1 = 1;

--t

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic

s I- N q * v(q) = 0 * v(q s I- r * v(r) = 1 =+ v(q

469

r ) = 0 -+ v(r) = 1; -+ r ) = 1; S I- N r (using S I- q --t r and (1.25)) S I- N q +-v(q + r ) = 1; none of th e sentences q, N q , r and N r is provable from S + v ( q ) = v(r)= v(q -+ r ) = -12 4 = 1. 4

*

3*

3

(ii) Suppose S I- N ( q -t r ) . Then using (1.36) and (1.37) we obtain S I- q and S I- N r , respectively, therefore v(q -+ r ) = v ( q ) + v(r) = 1+0=0.

(iii) Suppose neither S I- q -+ r nor S t- N ( q --t r ) . Notice that v(q) = 0 would imply S I- N q hence S I- q 4 r by (1.19), while v ( r ) = 1 would imply S I- r hence S I- q + T by ( A l ) . Thus v(q) # 0 and

# 1. S t- q. Then v ( r ) # 0 because v(r) = 0 would I- N r , hence S t- N ( q + r ) by (1.38). Thus w(r) = therefore imply v(q + r ) = 1-+ f = 12 ’ Case 1. v(q) = 1, hence

$, hence it is not the case that S I-

S is maximal consistent it follows that S U { q } is syntactically inconsistent, therefore S I- d by Lemma 1.7 (g). Then v(r) # f because v ( r ) = f would imply S Iby the above argument, hence S I- q -+ r by (1.41). Thus v(r) = 0 therefore w ( q -+ r ) = f + 0 = 1. 2 Case 2. v ( q ) =

It follows from (i) and S

q. Since

S that v ( p ) = 1for every p E S.

0

Proposition 1.9 has two important corollaries.

1.10. Theorem (The Strong Completeness Theorem).

I f S C E andpEE, thenSI-piffSbp. Proof. If S

+ p then SU{$} is semantically inconsistent because v ( S ) = (1)

implies v ( p ) = 1 therefore v(@) = v ( p ) -+ N v ( p ) = 1 + 0 = 0. It follows by Proposition 1.9 that SU {i;} is syntactically inconsistent, therefore S I- p

470

Lukasiewicz logics

by Lemma 1.7 (b). Conversely, if

S t- p

and o is an interpretation such that

v ( S ) = {l},take a proof pl, ...,pn= p of p from S and prove by induction that u(p;) = 1 for all i. 1.11. Corollary (The Wajsberg Completeness Theorem). I f p E E then I- p $7 p . 1.12. Remark. An algebraic proof of the Wajsberg Completeness Theorem can be found in Becchio

[1978d].

471

The Cignoli axiomatization of the n-valued Lukasiewicz logic $2. The Cignoli axiomatization of the n-valued Lukasiewicz logic

The Lukasiewicz n-valued logics ( n 2. 3) were introduced in 1922 (see the historical note of Malinowski in W6jcicki and Malinowski [1977]). Lukasiewicz defined th e following operations on the set L, = ( 0 ' 5 ,

+ y) ;

(2.1)

x + y = min(1,l-

(2.2)

NX = 1- X ;

(2.3)

x V y = m a x ( ~y) , = (X

(2.4)

2

2

..., 5, l}:

4

y)

4

y ;

A y = min(s, y) = N ( N z V Ny)

.

Recall t h a t L, has a canonical structure of n-valued Moisil algebra, namely (L,, A , V , N , O , l,cpl, ...,( 5

V y = max(z,y),

i +j 2

N(&)

~ ~ - (PI, 1 , ...,(Pn-

vi(&)

= -,

I),

where x A y = m i n ( ~ , y ) , is 0 if

i+j <

n and 1 if

n, (Pi = 1 - 'pi, cf. Example 3.1.20, where another notation has

been used.

It is easy t o see that LL = { 0 ,

We remark that, for n 2. 5,

&,s,

l} is a Moisil subalgebra of L,.

This example, given by A. Rose (see the introduction t o Cignoli [1969]) shows that, for n 2 5, the Lukasiewicz implication (2.1) cannot be expressed in an n-valued Moisil algebra in terms of t h e operators: V, A, N , cpl,...,cpn-l.

Consequently, for rz

2 5, the

n-valued Moisil algebras are

not adequate algebraic structures for the Lukasiewicz logics. For n = 3 and n = 4 the Lukasiewicz implication can be expressed in the language of Moisil

algebras (see (2.18) and (2.19)). A natural problem is whether the Lukasiewicz implication can be defined in a particular class of Moisil algebras.

The problem was solved by Cignoli in 1982. He defined the proper n-valued Moisil algebras and gave an axiomatization of the n-valued Lukasiewicz logic. The Cignoli axiomatization uses the characterization of Moisil algebras as Heyting algebras with some unary operators (see L. lturrioz [1977c]).

Lukasiewicz logics

472

The results presented in this section are taken from Cignoli [1980], [1982] and [1984]. 2.1. Notation. In any n-valued Moisil algebra set ~ o = x 0 and 9," = 1 for any x E L and define the unary operators J;, i = 0,1, ...,n - 1: (2.5)

J i ( x ) = cp,-;(x) A N v , - ; - ~ ( x ),

for any

2

EL

.

In particular note that in L,

(2.6)

Ji(-)

j

n-1

= 1 if j

= i and

J i ( 2 )

n-1

= O if j # i.

2.2. Lemma. I n any n-valued M o i s d algebra we have:

Proof. By induction on i. 2.3. Notation. We introduce the following sets:

S n = {(i,j)E N 213 5 i 5 n -2, 15 j 5 n - 4 , j < i} , if n 2 5 and S, = 0, if n < 5 ;

2.4. Definition (Cignoli [1980], [1982]).

A proper n-valued Moisil algebra is a structure (L, {F;j}fi,j)Esn), where L is an n-valued Moisil algebra and Fij,( i , j ) E S, are binary operations on L such that

The Cignoli axiomatization of the n-valued Lukasiewicz logic

(2.8)

Y) =

(PkF,j(%,

1

473

ifksi-j

O' Ji(x) A

foranyx,yEL,(i,j)ES,and

Jj(y)

k=1,

,

if k

>i - j

...,n - 1 .

2.5. Remark. For 2

= 0, therefore in this case any n-valued Moisil algebra is

5 n 5 4,S,

proper. 2.6. Remark. For any n

2 4 we

can extend the definition of Fij for any ( i , j ) E T n :

It is easy t o see that Fij satisfy conditions (2.8) for any ( i ,j ) E Tn. 2.7. Example. If x = 5,y = J n-1 are in L,, then we denote for ( i , j ) E S, (2.11)

F , j ( z , y)

=

n-1-i+j n-1

Fij(x,y) = 0 otherwise

if (r,s)= ( i , j ) and

.

We can prove t h a t F,, introduced by (2.11) verify (2.8). For example, if k > i - j and ( r , s ) = ( i , j ) then cpkF,.,(z,y) = (Pk (n-:1;+3) = 1, since k + ( n - 1 - r s) 2 n and Jr(x) A Jr(y) = 1 . In this way, L, has a

+

canonical structure of proper Moisil algebra. 2.8. Lemma.

Let ( L , {Ej}(i,j)E~n) be a proper n-valued Moisil algebra, x, y E L and a , b E C(L). Then the following properties hold

474

Lukasiewicz logics

Proof. (2.12): W e remark that

k > i - j iff k > ( n - 1 - j ) - (n - 1- i).

For

k > i - j we have:

For k

5 i-j, the previous equality is clear, hence, using the determination

principle, we get (2.12).

(2.13): W e have

and, similarly, J j ( y A b ) = Jj(y) A b. For le

> i - j , we can write

T h e Cignoli axiomatization of the n-valued Lukasiewicz logic

475

By the determination principle, we obtain (2.13). (2.14): Follows from (2.12) and (2.13). (2.15): By (2.13), F , j ( x , O ) = F , j ( x A 1,O A 0) = f i j ( ~ , O ) A 1 A 0 = 0, therefore, by (2.14), Fij(z, b) = F;j(x V 0,O V b) = Fij(z, 0) A N b = 0. 0

2.9. Lemma. Let ( L , { F ; j } ) ,(L’,{F;j}) be two proper n-valued h : L + L‘ is a morphism of Moisil algebras then (2.16)

h(F;j(z,y)) = F i j ( h ( ~ ) , h ( y ) ) for any (z,Y) E L and ( 4 j )E

Proof. If

Moisil algebras. If

T,,.

i - j < k 5 n - 1 then cpkh(F,j(s,y)) = h(cpkFij(s,y)) =

h(J;(z)A Jj(y)) = Ji(h(x)) A Jj (h(y)) = p&(h(x), h(y)). It is obvious t h a t this equality holds for any k = 1, ...,n - 1 and ( i , j ) E T,,hence (2.16) follows by the determination principle.

0

2.10. Remark. The precedent lemma shows that the category o f proper n-valued Moisil algebras is a full subcategory of th e category of n-valued Moisil algebras. 2.10’. Remark.

Let ( L , { F ; j } )be a proper n-valued Moisil algebra, F an n-filter of L and p t he congruence associated with F ,i.e. if (x,x’)E p and (y,y‘) E p then there exist a, b E F n C ( L ) such that x A a = x’A a and y A b = y’ A b. By (2.13) we have

F;j(x,y) A a A b = Fij(~A U , y A b ) = F;j(d A a, y’ A b ) = = F’~(z’,y’) A a A b for any ( i , j ) E

T,,.Since a A b E F

we get ( ~ j j ( z , y ) , ~ , j ( z ’ , y ’ ) )E p. It

follows that L / F = L / p has a canonical structure of proper Moisil algebra.

Lukasiewicz logics

476

2.11. Proposition. Any proper n-valued Moisil algebra is isomorphic to a subdirect product 5 f a family of szlbalgebras of L,. Proof. This is a consequence of Corollary 6.1.9 and Remark 2.10.

0

2.12. Notation. Let ( L ,{ F i j } ) be a proper n-valued Moisil algebra. For any z, y E L we set (2.17)

x

where

j

-+

y = (Z

+ y) V NXV

V

F,j(~,y)

( i , j ) E Tn

denotes the residuation defined and studied in $4.3.

2.13. Remark. It will be shown in Proposition 2.15 below that (2.17) is consistent with the notation (2.1) in L,. For a moment note that if n = 3 then from T3 = 0 we obtain (2.18)

z

-+

while if n = 4,

y = (z

+ y) V NX ,

T4= {(2,1)} implies

These relations show that for n = 3,4, the implication (2.17) can be defined in terms of operations of Moisil algebras. 2.14. Proposition. In every proper n-valued Moisil algebra ( L , {Fij}) the following properties hold:

(i)

'Pl@

(ii)

2

+

Y) = 'P&

+ y = ql(z -+

* Y); y) v y;

(iii) If a E C(L)then z + a = N z V a ;

477

The Cignoli axiomatization of the n-valued Lukasiewicz logic (iv)

If b E C ( L ) then b -+ x

(v)

1

(vi) x

--f

=Nbv x;

x =x;

-,y = 1 iff x 5 y.

Proof. For every ( k , j ) E

(i)

T, we

have 1

5 i -j ,

hence ' p l & , ( x , y ) = 0, by

(2.8). But x A q 1 N x = 0 5 y, therefore we get cpl.Nz 5 y 1 ( x + y) by Proposition 4.3.20(i), (ii). Consequently 'p1(x y) = 'pl(z +Y) v V l N X v v V l F i j ( X , Y> = ' p d " =+ Y>. -+

(&j)ETn

(ii) By (i),(4.3.3)and (4.3.4). (iii) In accordance t o (2.15) and Proposition 4.3.6 (ii) we have z a= u ) V N x = N'p,-lx V a V N x = c p l N x V N x V a = N x V a. (x --f

(iv) is obtained from

(2.15)and Proposition 4.3.6(i),while (v)

is a parti-

cular case of (iv). (vi) By (i) we have the equivalences x

cpl(x

j

y = 1 iff ' p ~ ( z-+ y) = 1 iff x + y = 1 iff x 5 y. -+

2.15. Proposition. In the proper MoisiE algebra L,, the operator

y) = 1 ifF 0

defined by (2.17)coincides with the Lukasiewicz implication introduced b y (2.1).

Proof. We shall prove that z

-+

y = min(1,l

x -+ y = 1 by Proposition 2.14 (vi) and 1 and let z =

-+

-x

5 1 -x

+ y). + y.

If x

5

y, then

Suppose y

<2

5, y = 5.Then F,j(x,y) = 0 if (i,j) # ( p , q ) and + y. We have also x + y = y. We shall distinguish two

Fpq(z,y) = 1-x cases.

E T, then by (2.17)and the previous remarks we obtain x -+ y = y v N x v (1 - x + y ) = max(y,l- x , 1 - x + y ) = 1 - x + y 5 1 .

(a) If ( p , q )

478

Lukasiewicz logics

(b) If ( p , g)

4 T, then z -+

y =y V

Nz = max(y, 1- z).

Assuming q = 0,

we have y = 0, therefore max(y, 1 - z) = 1 - z = 1- z consider t h e case g

# 0.

Since (p,q) @ Tn and g

< p, by the definition

15 g 5 n-3 cannot be simultaneously true. If it is not the case t h a t 2 5 p 5 n - 2 then it follows that p = n - 1; if 1 5 q 5 n - 3 do not hold, then we have q = n - 2, hence p = n - 1. In both cases p = n - 1, therefore 0 z = 1 and we obtain z + y = max(y,O) = y = 1 - z y _< 1.

of

T, it follows

that the inequalities 2

5p 5

+ y 5 1. Let us

n - 2,

+

In accordance with Proposition 2.15 t h e operator

-, defined by (2.17)

will be called the Lulcasiewicz implication. Now we shall give the syntactic construction of an n-valued propositional calculus Luk;.

2.16. Definition. Let V be a countable set of elementsp, q, T , ... called propositional variables. The set E of sentences of th e propositional calculus Luki is obtained as the adjacent set of th e Peano algebra:

( E ,A, V, =+,N , 9 1 , ...,Vn-1,

{Ej}(i,jleT,,)

of type ( 2 , 2 , 2 , 1 , { 1 } i ~ { i , . . . , n -(2}(i,j)E~,) ~~, freely generated by V .

We shall use the following abbreviations: A u B for ( A + B ) A ( B + A) and JiA for cp,-iA A NVn-j-lA, 1 5 i 5 n - 2. 2.17. Definition. The axioms of Luk: are the sentences of t h e following forms:

(Al)

WI

*

* wi)

( ~ 2

9

The Cignoli axiomatization of the n-valued Lukasiewicz logic

m

where

A w:

k=1

means (...((wi A w;) A w;>

The concept of proof in

A wk)

,

Luk: is defined in terms of the above axioms

and two rules of inference, modus ponens (m.p.) and (m.p.) W 1 , W =2 wz W

w2

...

479

;

(Tn)

(T,):

-Y1W

2.18. Definition. Let S E and w E E . By a proof of w from the hypotheses S is meant a finite sequence of sentences wl, wz, ...,wm such that w, = w and for every i E (1, ...,m} one of the following situations hold: (i)

wi is an axiom or w; E S;

Lukasiewicz logics

480 (ii) there exist j , k < i such that (iii) there exists j

Wj

= wk

+ Wi;

< i such that wi = v l w j .

We say that the proof wl,

...,w,

is o f length m and we denote its exis-

(S implies syntactically w). In particular if 0 t- w we write simply I- w and refer t o w as a theorem o f Luk',. tence by S I- w

2.19. Lemma.

Let S (i)

E and w, w', w1, w:, w2, wi E E . If S I- w1 M w{ and S I- w2 H wt then S I- wt. V 202 # wi V w:, S I- w1 A w2 w w{ A w i and S I- (w1 =$ w2) (j (w: + wi).

(ii) If S I- w w w' then S I- N w (iii) If S t w

e w' then S t-

cpiw

* Nw' 'piw' for i = 1, ...,n - 1.

(iv)

If S t 'piw + 9;w' for i = 1, ...,n - 1 then S I- w + w'.

(v)

If S I- N w then S t- w

(vi)

If S I- w (j w' then S I- Jiw

(vii) If S

+ w' for every sentence w' E E .

* Jiw' for i = 1,...,n - 2.

t- wl H w: and S I- w2

-S w;

then S t- 4j(w1,w2)

#

Ftj(wi, wi) for every ( i , j ) E Tn). Proof. We remark that S I- w

w w' iff S I- w

+ w' and S I- w' + w (this

follows i n one direction from (A6) and (A7) and i n the opposite direction

from ( A l ) and (A8)).

The proofs o f (i) and (ii) are straightforward: for example, (ii) follows from (AlO), (A14) and ( v n ) .

(iii) Assuming S I- w S I-

w' we have S I- cpl(w

+ w')

by ( v n ) ,therefore

n- 1

A

i=l

(piw

=$

'p;w') in accordance t o (A12).

S I- cpiw + cpiw' for every i = 1, ...,n - 1.

It follows that

The Cignoli axiomatization of the n-valued Luhiewicz logic

(iv)

If S I- 'p;w

+ (piw', 1 5 i 5 n - 1 then S I-

n-1

A ('piw

i= 1

481

+ 'piw'),

therefore S I- 'pl(w =+ w') by (A12) and, by applying (A14), we get

s I- w j . w'.

If S I- N w then S I- Nw'

(v)

+ Nw

by ( A l ) and map., hence S I-

'pl(Nw' + N w ) by (rn). Using (AlO), S t- 'pl(w 3 w'), then, by (A14), we obtain S I- w + w'.

(vi)

By (ii) and (iii).

(vii) Let us suppose that S t- w1

If ( i , j ) E T, and i - j

* w;, S I- w2 H w; and 1 5 k 5 n - 1.

< k In - 1, then,

in accordance with (A18)

and (vi) we have:

S I- cp&j(wl, w2) e 'pk&j(W;, w:). This property 1 5 k 5 i - j (see (A17) and (v)). By applying (iv)

By m.p. we get

is also true for

it follows that S I- &(w17 w2) -8 F,j(w;, wk).

-

0

-

2.20. Definition. Let be the equivalence relation on E defined by w w' iff I- w e w'. Denote t he equivalence class of a sentence w E E by 6.Let 1 be the class of all theorems. The quotient algebra

where 0 = N1 and

p; = N'pi, (i = 1,...,n - 1) is called the Lindenbaum-

Tarskd algebra of Luk:.

Lukasiewicz 1ogics

482 2.21. Proposition.

The Lindenbaum-Tarski algebra of Luki is a proper n-valued Moisil algebra with residuation +. Proof. The relation

N

is clearly an equivalence; moreover, it is a congruence

by Lemma 2.19, therefore the quotient algebra exists. If w1 and

-

wzare the-

w~ by (Al), while if w1 is a theorem and w1 w2 then since I- w1 + w1 by ( A l ) it follows from (A8) that I- wz A w l , hence 202 is a theorem by (A6). Therefore the class 1consisting of all theorems does exist. The next step is t o check axioms (pl)-(plo) of relatively pseudocomplemented lattices, given in Rasiowa [1974],Ch. 2, 52 and Ch. 4, $1. But orems then w~

(PI),

N

(pz) and (p5)-(p10) are the translations of axioms (Al)-(A8)

into the

til + G2 = tiz +- til = 1 then I- w1 + w2 and k w2 + wl, hence w1 w z or til = GZ,i.e. (p3) holds. Taking a theorem w' we get I- w + w' by (Al), i.e. ti + 1 = 1, which is (p4). Note

-

quotient-algebra language. If

further t h a t (2.21)

(w1 =+ w2) I- Nw2

+ Nu71

+ wz} implies syntactically w1 + 202, then.in turn 'pl(wl + WZ), 'pl(Nwz =+ N w l ) and Nwz + Null by ( r n ) , (A10) and (A14), rebecause {wl

spectively. In particular if obtain N1

5 NNG

t- w2

then k N w Z

+Nwl

= G by (A9). Therefore

El

complemented lattice with 0 = N 1 as zero, i.e.

To prove

-

El

and for w1 = N w we

-

is a relatively pseudois a Heyting algebra.

+ N(w1 A wz)and similarly k N W Z3 N(w1 A WZ),hence I- ( N w l V N w z ) + N(w1 A W Z )by (A5). For the converse implication we start from I- Nwl + ( N w l V N w z ) , which is (A3), and obtain I- N ( N w l V N w z ) + w1 by (2.21) and (A9); similarly I- N ( N w 1 V Nw2) + W Z . Applying (A8) we obtain I- N(Nw1 V N w Z )+ w1 A W Z ,hence I- N(wl A w2) + Nwl V N w z by (2.21). we notice that (A6) and (2.21) yield I- Nwl

It follows from (A9), (2.22) and Proposition 1.1.31 that N is an involutive dual endomorphism on E l -, therefore E l is also a De Morgan N

483

The Cignofi axiomatization of the n-valued Lukasiewicz logic

-

algebra. Moreover, properties (4.3.7), (4.3.14) and (4.3.16)- (4.3.19) hold by (A12), (A13), ( A l l ) and (A14)-(A16),

respectively. Therefore

El

is

an n-valued Moisil algebra in view of the lturrioz theorem quoted in Ch. 4,

$3. The algebra is proper by (A17) and (A18).

0

2.22. Definition. An interpretation of LukR in a proper n-valued Moisil algebra L is an arbitrary mapping v : V + L. For any interpretation v : V ---t L there exists a unique mapping V : E --t L such that Vlv = v and V preserves

N , (pi for i E (1,...,n - 1) and f i j for ( i ,j) E T,. A sentence w is valid in Luk; (+:L&:, v) if V(w) = 1for any interpretation v : V L,. A, V,

=$,

2.23. Proposition (The Completeness Theorem). For every sentence w of Luk; the following assertions are equivalent:

(i)

w is provable in Luk; (I- w);

(ii) Q = 1 in

El

-;

(iii) for every proper n-valued Maid algebra L and every interpretation v : V ---t L we have ~ ( w = ) 1; (iv) w i s valid

in Luk; (bLd:,w).

Proof. The implications (i) =+ (ii) =+ (iii) =+ (iv) are obvious and for (iv)

(i) we notice first t h a t (iv)

+ (iii) by Proposition 2.11,

by taking t h e interpretation v : V -+

El

+-

(iii) =+ (ii) follows -, v ( p ) = fi, while (ii) + (i) is

already proved.

0

2.24. Remark.

Jo, J1,..., JnV1are defined (in L,) usingonly the operations N and +. The unary operators H,, H I , ...,HnVl of L, are introduced by induction: In Rosser and Turquette [1952] th e operators

(2.23)

H o ( x )= N X ;

H k + l ( ~= ) x

+H

~ ( x.)

Lukasiewicz logics

484 Now we shall define Jo,

J,-l

J1,

...,Jn-1 using Ho, HI, ...,H,-1.

We start with

and Jo:

For 1 5

k 5 n -2

(n - 1)Hi(k)

let

(5)

i ( k ) be the greatest integer < -and r(k) =

*

The operators Jn-i,i = 2 , ...,n - 1 are defined by recurrence: if n r ( n - i), then (2.25) and if n (2.26)

Jn-i(X)

= Jn-l(Hi(n-i)(z) V 2)

- i < r ( n - i), Jn-i(z)

+

-i

=

(Hi(n-i)(x) A x)

then

= JT(n-i)

(Hi(n-i)(z))

*

One can prove (see Rosser and Turquette [1952], pp.

18-22) that

Jo, J17 ...,Jn-l coincide with the operators introduced by (2.5). By (2.7) it follows that (pl,...,(P,-~ can be defined in L, using only the operations N and +. In L, is also verified the equality:

By Proposition 2.14 (ii) we have in L,:

Consequently, by (2,3), (2,4) and by the previous remarks, the operations V,

A, *,91,...,(p,-l,

4jr( i , j ) E T,,can be expressed

in

L, in terms of N

and +. In Suchori

N and

[1974],th e operators (P~,...,(P,,-~ of L, are given in terms of

-+ with no use of

Jo,J1,...,J,-l.

2.25. Remarks.

V of propositional variables and using only the logical connectives -+ and N we can construct canonically the n- valued Lukasiewicz propositional calculus Luk,. An interpretation of Luk, is an arbitrary mapping v : V + L,. If E’ is the set of all sentences of Luk, Starting with a countable set

The Cignoli axiomatization of the n-valued Lukasiewicz logic

485

Luk, can be extended t o a unique mapping V : E' 4 L, t h a t preserves N and --1. A sentence w is valid in Luk, ( k ~ d w) , if V(w) = 1for every interpretation v of Luk,.

then every interpretation v of

If w is a sentence in Luk, we shall denote by w* the sentence of Luk: obtained by replacing the occurrences of + in w by the expression corresponding t o the relation (2.17). Conversely, with every sentence 20 of Luk: we can associate a sentence w o of Luk, by replacing t h e occurrences of V, A, +, cpi and Fij in w by the expressions corresponding to the relations (2.3), (2.4), (2.28), (2.7) and (2.27). By induction on the length of sentences one can prove for every sentence

w in Luk, that w is valid in Luk, iff W * is valid in Luk: and for every sentence w in Lukz that w is valid in Luk: iff wo is valid ain Luk,. This shows that the axioms (Al)-(A18) of Luk; give indeed an axiomatization o f the Lukasiewicz n-valued logic

Luk,.

2.26. Remark. Another axiomatization of the n-valued Lukasiewicz logic was given by Grigolia [1977] using the MV,-algebras

as algebraic models.

2.27. Remark.

The axioms (Al)-(A16) and the rules of inference m.p. and (r,) define an n-valued propositional calculus having as algebraic models the n-valued Moisil algebras. In accordance with Cignoli [1982], this logical system will

be called the n-valued Moisil propositional calculus. 2.28. Remark. Let us consider an n-valued Post algebra

( L , A , V, N,cpl, ..., ~ n - 1 j 0 , I r c 1 ,...,c,-2); cf. Ch. 4, $1,especially Corollary 4.1.9 and also Ch. 4, $2. Recall that cpicj = 1 if i + j

i

+j

< n. For every (i,j) E S,

(2.29)

Fij(Z, y)

and z , y E

= J ~ ( z A) Jj(y) A

An easy computation shows that

L

Cn-l-i+j

2n

and 9;cj = 0 if

let us define

-

Ej defined by (2.29)

verify (2.8) therefore

Lukasiewicz logics

486

every n-valued Post algebra has a canonical structure of proper Moisil algebra. If we replace th e axioms (A17) and ( A N ) by the axioms corresponding to the definition of the constants cl,

...,c,-~

we obtain an axiomatization of

the n-valued Post-logic (see Rasiowa [1974]). 2.29. Remark. An analysis of the predicate calculus for the n-valued Lukasiewicz logic can be found in Cignoli [1984]. The main tool in the proof of the completeness theorem for this logic is Cignoli’s theorem 4.5.12 which asserts that every nvalued Moisil algebra is completely chrysippian. Consequently the n-valued Moisil algebras provide a common algebraic framework for the treatment of

the n-valued logics of Lukasiewicz and Post. The above remarks 2.25, 2.27 and 2.28 establish the exact relations between the n-valued Moisil logic, the n-valued Lukasiewicz logic and the n-valued Post logic. See also Surma

[19751.

487

The 9-valued propositional calculus

53. The 29-valued propositional calculus In this section we introduce and examine a d-va,Jed propositional calculus with t he property that i t s theorems are the propositions true for all greater than a fixed

k E I. The

iEI

axiomatization of this calculus uses the

system of axioms of 6-valued calculus introduced by Boicescu (1973bl.The results of this section are taken from Filipoiu

3.1. Definition. Let V be an infinite

(19811.

set, whose elements will be called propositional varia-

bles. The proposation algebra of the d-valued propositional calculus on the set V is the free algebra Prop(V) on V in t h e class of all algebras of type = (2727 { l } i E ~ ,{ l } i E I ) .

The type T will be fixed throughout this section. The operations of any Talgebra will be denoted, without danger of confusion by A, V, cp,, @, (i E I).

3.2. Remark. Every LMd-algebra may be viewed as a .r-algebra. In particular ' L

is a

T-algebra.

3.3. Definition. If V is the set of propositional variables and A is a 7-algebra, then every mapping h : V + A will be called an interpretation. Now we introduce the concept of truth in the d-valued propositional calculus.

Let k E

I be fixed

in the sequel.

3.4. Definition. A valuation of Prop(V) is a .r-homomorphism w : Prop(V) -+ Lh'. We say that p E Prop(V) is k-true with respect t o w if w ( p ) ( k ) = 1 and k-false with respect t o v if w ( p ) ( k ) = 0.

Lukasiewicz logics

488 3.5. Remarks.

a) As Prop(V) is t h e free .r-algebra on V, there exists a bijection

Hom(Prop(V),

Lb4)

--t

Hornset(V, LL'), t o the effect that every valua-

tion is uniquely determined by its restriction t o the set of propositional variables and every mapping from V t o

Lh4 can be uniquely extended t o

a valuation.

b) If j E I, k

5 j , and p

j-true with respect t o

E Prop(V) is k-true with respect t o w , then p is

w.

3.6. Definition.

C Prop(V) and q E Prop(V). We say t h a t q is a k-consequence of F (or t h a t F semantically k-implies q ) if v ( q ) ( k )= 1for every valuation w such t h a t v ( p ) ( k ) = 1for all p E F . We shall write this F q.

a) Let F

+ k

b) We say t h a t p E Prop(V) is k-valid (or a k-tautology) and write /= p k

if

0

+ p , that is v(p)(lc)= 1for every valuation v. k

3.7. Remarks. a) For

F C Prop(V)

we denote by C o n k ( F ) = { p E Prop(V) 1 F

+ p}. k

Then Conk is a closure operation on Prop(V) (cf. Definition 1.1.14).

b) If j E

I, k 5 j

p then

and k

p. j

The &valued propositional calculus

489

'pk'pip) A ( P k ' p i p V (Pk'pjQiP). The computations use the fact that

homomorphism and the structure of

j

5k

@ ,

v is a

(cf. Example 3.1.3); thus e.g. if

then

Further we study the concept of proof in k-propositional calculus.

3.9. Definition.

-

For every p , 4 E Prop(V) we introduce the notation

P

(3.0)

k

q='PkPVvkq

p - 4 =

(

k

P-

--

4) A (4

k

k

P)

and call k - a x i o m , the propositions of the following forms:

(3.1)

P

(3.3)

PA4

(3.4)

PA4

k

(4

k

k

k

P

7

4

,

P)

7

r-

490

(3.8)

(P

-- -- q)

k

((r

k

k

Q)

k

(Pvr

k

Lukasiewicz logics 4))

,

(3.9)

* CpjP A Cpjq ,

for every j E

I

(3.10)

Cpj(p V q )

(3.11)

'PjP

* VicPjP

7

for every j , i E

I

,

(3.12)

'PjP

* CpiQjP

7

for every j , i E

I

,

(3.13)

Q j P e--t 'PiQjP

7

for every j 7 i E I

(3.14)

CpjP

(3.15)

ViP

k

k

k

k

t--t

k

+

k

CPiVjP > 'PjP

7

We denote t h e set o f Ic-axioms by

for every j , i E

I,

for every i , j E

I, i 5j

k-Axm.

In t h e next definition we introduce t h e logical system based o n axioms

(3.1)-(3.15)

and modus ponens as rule of inference:

3.10. Definition. Prop(V) and q E Prop(V). W e say t h a t q is a Ic-deduction f r o m F and write F t- q, if there exists a k-proof of q f r o m t h e assumption

a) Let F

k

F , i.e.,

E Prop(V) such t h a t pn = q and for each a E Gn, either p , E k-Axm U F or p , = pb + pa a finite sequence pl,p2, ...7pn of elements p ,

k

for some b, c

< a.

The &valued propositional calculus

491

b) We say that p E Prop(V) is a k-theorem and write I- p , provided k

0 I- p . We also use the k

for the property "if I-

k

notation

p l and

... and

I- p , then I- q". k

k

3.11. Remark. The k-axioms (3.1)-(3.8) form a set of axioms of positive logic with connectors V , A, -+; it follows that all of the theorems of positive logic are k-theorems; see Rasiowa [1974]. 3.12. Example. We shall write down some &theorems: (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22)

492 (3.23)

Lukasiewicz logics

[

((p

+p k q ) k

4

k

pkp),

where q is a k-theorem ,

(3.24)

(3.25)

(3.26) (3.27)

k k

( ( p k p + p k s ) +p ) , k k

where s is a k-theorem .

Let us check e.g. (3.19) (3.21) (3.26) and (3.27). But (3.19) means k ( p k p v y k y k p ) A ( p k v k p V V k p ) , which holds because the two factors k

of the conjunction are k-theorems and in positive logic any conjunction of

theorems is a theorem. For (3.21) we use in turn the equivalence theorem for positive logic in Rasiowa [1974], (3.9) (3.13), (3.12), (3.10):

493

The &valued propositional calculus Formula (3.26) means I- p k p k

v ( P k ( ( P k ( P k p v (Pkpkq) and

equivalence theorem it reduces t o Ik

using again the

(Pkpv(Pkpvpkq;t h e latter k-theorem

follows by applying modus ponens t o (3.16) and

k p k p v (Pkp -+ p k p v k

k

(pkp V pkq. Finally (3.27) can be written in the following equivalent forms:

the first factor of the conjunction is k-theorem (3.16), while the second is a k-theorem because (Pks. k

3.13. Notation. For every

F E Prop(V)

we set Dedk(F) = { p E Prop(V) I F Ik

p}.

3.14. Corollary.

Dedk i s a closure operator o n Prop(V), such that for every F E Prop(V) and every p E Dedk(F) there exists F’ F , F’ finite, such that p E Dedk( F’). In other words, Dedk is an algebraic closure operator on Prop(V). The proof is the same as in the bivalent case.

3.15. Proposition. L e t &, Vz be a n y two sets of propositional var&zbbs and f : Prop(&) + Prop(&) a .r-homomorphism. For a n y F C Prop(Vl) and p E Prop(V1) w e have

494

Lukasiewicz logics

Proof. a) Let pl,p2, . . . , p , be a k-proof of p from

-

F ; if pa E F

then f(pa) E f ( F ) .

Since f is a .r-homomorphism, if pa is a k-axiom in Prop(Vl), then f(pa) is a k-axiom in Prop(&); if p , = pb

<

p a , b,c

k

from

a, then f ( p , ) =

f(F).

b) Let v : Prop(V2) --t LLq be a valuation of Prop(V2) such that v ( f ( q ) ) ( k ) = 1 for each q E F . Then t h e composite mapping v

f

: Prop(V1) + Lh'

o

is a valuation of Prop(&), such t h a t ( v o f ) ( q ) ( k )=

1 for each q E F . Since F

p , we have (v o f ) ( p ) ( k ) = 1, i.e. k

Next we deal with consistency and completeness.

3.16. Lemma. Let F Prop(V) and p E Prop(V).

If F t- p then F I= p . k

k

v : Prop(V) -+ L'\ be a valuation such t h a t v ( q ) ( k )= 1 for every q E F. Let further p l , ...,pn = p be a k-proof of p from F . We will prove that v(p,)(k) = 1, a E 1 , . If p , E F U k-Azm then v(p,)(k) = 1, since for every q E k-Azm it is easy t o prove that q. If for b , c < a Proof. Let

+ k

495

The &valued propositional calculus we have p , = pb

+p a k

and v ( p c ) ( k )= v ( p b ) ( k ) = 1 then it follows from

(3.0) that

3.17. Proposition (The Deduction Theorem).

If F

E Prop(V), p , q E Prop(V) then F I- ( p k

+ q ) if and only k

if

Proof. By Remark 3.11 and th e well-known fact that t h e Deduction Theorem holds in positive logic; as a matter of fact axioms (3.1), (3.2) and (3.16) suffice (cf. e.g. Barnes and Mack [1975]).

0

3.18. Definition. Let F Prop(V). We say that F is k-consistent if p k p !$ D e d k ( F ) , for every k-theorem p ; otherwise F is said t o be k-inconsistent. F is called a mazimal k-consistent subset if it is k-consistent and maximal for inclusion

c

with this property.

3.19. Proposition. A set F Prop(V) is k-inconsistent if and only if there exists p E PrOp(V) such that p , qkp E D e d k ( F ) . Proof. If p , p k p E D e d k ( F ) then applying twice modus ponens t o (3.26) we obtain P k Q E D e d k ( F ) for every q. Conversely, if (Pkq E D e d k ( F ) for some k-theorem q, note that q E Dedk(0)

c Dedk(F).

0

3.20. Corollary.

The empty set Proof.

8 is k-consistent.

Otherwise p , p k p E Dedk(0) for some p E Prop(V), therefore

Lukasiewicz logics

496

p,Cpkp E Conk(@)by Lemma 3.16. This is a contradiction because v(Cpkp)(k) = Cpkv(p)(k) = v ( p ) ( k ) for every valuation v.

0

3.21. Lemma. T h e subset F C Prop(V) i s maximal k-consistent if and only if

# F , for

(3.28)

pkp

(3.29)

Dedk(F) = F ;

(3.30)

for every p E Prop(V), eather p E F

every k-theorem p ;

OT

pkp E F .

Proof. Let F be maximal k-consistent. For every k-theorem p , p k p # D e d k ( F ) I> F , therefore p k p # F . Since Dedk(Dedk(F)) = D e d k ( F ) , D e d k ( F ) is k- consistent and as F C D e d k ( F ) , we have F = D e d k ( F ) . Finally let p E Prop(V). If p

# F

then F U { p }

exists a k-theorem Q such that F U { p } l-

pkq, or

2

F l-

k

-

F , hence there (p

k

pkq) by

k

Proposition 3.17; using (3.23) we have p k ( p ) E D e d k ( F ) = F . If p E F then p E D e d k ( F ) hence p k p # D e d k ( F ) = F by (3.29) and Proposition 3.19. Now suppose F has properties (3.28)-(3.30). Then for every k-theorem p, pkp

#

F = D e d k ( F ) , i.e. F is k-consistent. If F1

’

#

F there exists

p E Fl such that p 6 F , then p k p E F . Thus p , p k p E I;; so that FI is not k-consistent by Proposition 3.19. 0

3.22. Remark.

for some k(3.29) & (3.30) + (3.28), since if we suppose that p k p E theorem p , then p $ F = D e d k ( F ) ; but p E Dedk(0) C D e d k ( F ) .

3.23. Lemma. If Dedk(8’) i s maximal k-consistent t h e n f o r every p, q E Prop(V)

The &valued propositional calculus p A q E Dedk(F)

(3.32)

497

++p E Dedk(F)

and q E Dedk(F) ;

Proof. (3.31) and (3.34) follow from (3.19) and (3.15) respectively. Suppose p , q E Dedk(F). Then q -+ (q

4

k

I-

k

q) by (3.16) and

((q

k

P)

7

((q

k

k

q)

p E Dedk(F) by (3.1), while

7

(q

k

(PQ))))

by (3.5) therefore p A g E Dedk(F). The converse implication of (3.32) follows from (3.4) and (3.5).

If p V q E Dedk(F) and p , q # Dedk(F) then @ k P , @ k q E Dedk(F) by Lemma 3.21, therefore pkp A pkq E Dedk(F) by (3.32), i.e. Cpk(p V q ) E Dedk(F) by the k-theorem dual to (3.21); this contradicts Proposition 3.19. The converse implication of (3.33) follows from (3.6) and (3.7). 0 3.24. Corollary. If F is maximal k-consistent then for every p , q E Prop(V): (3.35)

p EF

(3.36)

p A q E F H p E F and q E F ;

(3.37)

~ v ~ E F H o~ r .Eq EFF ;

(3.38)

(pip E

e (pkp E F

F

;

+ (pjp E F

for

Proof. From Lemmas 3.23 and 3.21.

i, j E I, i 5 j .

n

3.25. Proposition.

(i) Every k-consistent set is included in a maximal k-consistent set.

Lukasiewicz logics

498

(ii) FOTevery k-consistent set F there is a valuation w such that w ( p ) ( k )= 1 ~ O each T p E F. (iii) FOT every mazimal k-consistent set F there is a valuation w such that (3.39)

v(P)(~)= 1 u p i p E F

(Vp E Prop(V), Vi E I )

.

Proof.

(i) By a well-known argument using the Zorn lemma. (iii) Define f : V + Li4 by

(3.40)

f ( x > ( i )=

1

if pix E F

0

if pix

# F.

The mapping is well defined because (3.38) in Corollary 3.25 ensures that f ( x ) ( i ) 5 f ( x ) ( j ) for i 5 j . Let w : Prop(V) + Li' be the valuation extending f. We prove (3.39) by induction on the length of p . For p E V , (3.39) reduces to (3.40). The inductive step is based on Lemmas 3.21 and 3.23. For instance if p satisfies (3.39), so does p i p because taking also into account (3.14) and (3.13) we have

(ii) We use (i) and (iii): Let F*1 F be a maximal k-consistent set and w the valuation satisfying (3.39) with respect to F*. If p E F then p E F* = Dedk(F*) hence (pkp E Dedk(F') = F* by (3.31) in Lemma 0 3.23, therefore w ( p ) ( k )= 1.

499

The 29-valued propositional calculus 3.26. Theorem (completeness).

Let F E P r o p ( V ) , p E P r o p ( V ) . Then F

+ p if and only if F k

I-

p.

k

Comment. In other words Conk = Dedk. Proof. If F

t- p k

then

i=

F

k

p by Lemma 3.16.

Suppose

F

+ k

p.

We prove that F U { P k p } is k-inconsistent: if F U {pkp} is k-consistent, iL such that by Proposition 3.25 there is a valuation v : P r o p ( V ) 4 ' v ( q ) ( k ) = 1 for every q E F U { P k p } , i.e. v ( q ) ( k ) = 1 for every Q E F and 1 = v ( @ k p ) ( k )= ( P k V ( p ) ) ( k ) = v ( p ) ( k ) , which contradicts F

k

p.

Therefore F U { p k p } is not k-consistent, thus there exists a k-theorem s such that FLJ { p k p }

(PkS,

k

hence F

t- ( P k p

-+ p k S ) by t h e Deduc-

k

k

tion Theorem 3.17 and using (3.27) we obtain F

t- p .

0

k

3.27. Corollary.

FOTp E P r o p ( V ) we have

+ k

p if and only if I-

k

p.

In the sequel we construct the Lindenbaum-Tarski algebra of our &valued calculus. 3.28. Lemma.

If v

LLq is a valuation then the set Vk = { p E P r o p ( V ) I v ( p ) ( k ) = 1) is maximal k-consistent and every : Prop(V)

4

maximal k-consistent set i s of this form for a unique valuation. Proof. It is easy t o prove t h a t v k satisfies conditions (3.28)-(3.30) in Lemma

3.21 (for (3.29) use Theorem 3.26). Therefore v k is maximal k-consistent. If F P r o p ( V ) is maximal k-consistent then the valuation of Proposition

Lukasiewicz logics

500

Now let

be the set of k-theorems on the set Prop(V) and define the following relation: (3.42)

p

-

q ++ (pip

* cpiq E Tk , k

for every

iEI

This relation is an equivalence relation because the syllogism rule is valid in positive logic, hence in our calculus as well. 3.29. Lemma. T h e following assertions are equivalent:

(ii) w(p) = w ( q ) , for every valuation w.

-

-

3.30. Theorem. T h e relation is a congruence on Prop(V) and the r-afgebra Prop(V)/ is an LMd-algebra (called the Lindenbaum-Taraki algebra of the considered 19-calculus and denoted by P(V)).

The 6-valued propositional calculus Proof. The compatibility of prove: for example p

+

N

g

"N"

501

with the operations of

Prop(V)

+ w(p) = w(g), Vv valuation + .(pip)

is easy t o

-

= v(piq),

pig. It follows that P ( V ) = Prop(V)/ is a .r-algebra with operations defined by 01[2] = [ o ~ x ] , 0 2 [y] = [X 0 2 y], for

Vv valuation

pip

[XI

each unary operation o1 and binary operation

02.

Since all theorems of posi-

, V) is a distributive lattice tive logic are k-theorems it follows that ( P ( V ) A, (see e.g. Rasiowa [1974]). Moreover,

P ( V ) is bounded with 0 = [pkp] and

1 = [Cpkp],where p E Tk. Finally P(v)satisfies axioms (3.1.2)-(3.1.5) in Definition 3.1.1 of an LMd-algebra; for example

e

v(cp;p) = w(cpiq) Vi E I Vv valuation H

H cp;v(p) = cpiv(q) Vi E I Vv valuation

++

* cp;v(p)(j)= cpiw(q)(j)V i , j E I Vw valuation ++ e v ( p ) ( i )= w( q) ( i ) Vi E I Vw valuation

++

v(p) = v(q) Vv valuation

H

++ [p] = [q] .

0

Let L be an LM6-algebra. We introduce the operations + and k

as follows:

z * y = ( z + y ) A ( y - + ~ ) . k

k

k

3.31. Lemma. In every LMd-algebra L the following relations hold:

t+

k

Lukasiewicz logics

502

(3.46)

2

(3.48)

x + (y + Z ) = ( X A y )

+(y k

k

--t

k

z ) = (x + y ) + ( x + z ) ; k

k

k

-

k

z;

k

Proof. Straightforward .

0

3.32. Definition. Let L be an LMO-algebra. A subset S C L will be called a k-filter of L if s is a proper filter of the lattice L and x E s iff (Pkx E s. 3.33. Lemma. If S C_ L , t h e n the following assertions are equivalent: (i)

S

Proof.

(i)

a k-filter of L ;

23

+ (ii):

If x E S and x

+y E

S then (Pkx E S and (Pkx V (Pky E S

k

hence ( ~ kAx y k y = ' p k x A ( ( P k x V p k y )E S, which implies 'pky E S , therefore y E s. (ii) (i): If x , y E S then y --t (x --t y ) = 1 E S , z --t y E S;

+

since x

k

k

(z A y ) = x

-+

k

k

y it follows that x A y E

k

S. If x

E S , y E L,

503

The d-valued propositional calculus since x

--t

2

V y = 1, we have x V y

E S. Finally:

k

3.34. Remark. The set of k-filters of an LMd-algebra L is in bijective correspondence with the set of filters of the Boolean algebra

C(L).

3.35. Definition.

A proper subset F

#

Prop(V) will be called a k-deductive system of

formulas if

3.36. Proposition. The set of k-deductive systems of Prop(V) i s in bijective correspondence with the s e t of k-filters of the LMd-algebra P(V).

Prop(V)/ N= P ( V ) be the canonical map which is a homomorphism of .r-algebras. Let F C Prop(X) be a k-deductive system; we shall prove that g(F) is a k-filter of P(V). Since F it folh!s that 1 = g((pkp) E g(F), where p E Tk;if pk(Tk) c Tk g ( p ) , g ( p ) + g ( q ) E g ( F ) , then there exists pl,ql E F such that g(p) = Proof.

Let g

k

:

Prop(V)

--t

504

Lukasiewicz logics

p + q E F , hence q E F and finally g ( q ) E g ( F ) . k

q E g-'(g). Finally g(g-'(g)) = g for every k-filter

a of P(V)

since g is

& Prop(V); indeed if p E g-'(g(F)), then g ( p ) E g ( F ) , or g ( p ) = g(p'), p' E F , or p p', p' E F , hence p E F because F is a k-deductive system. surjective, and q-l ( g ( F ) ) = F for every k-deductive system F

N

3.37. Remark. 1) A subset F & Prop(V) is a k-deductive system iff F

# Prop(V)

and

F = Dedk(F). 2) Every k-deductive system is a k-consistent set. 3) A subset F

k-consistent .

Prop(V) is a maximal k-deductive system iff F is maximal

Andytic tableaux for the &valued propositional calculus

505

54. Analytic tableaux for the &valued propositional calculus In this section we introduce the method of analytic tableaux for the

6-

valued propositional calculus and obtain a completeness theorem using this method. For the classical calculus see Smullyan [1968]. The results of this section are taken from Filipoiu [1978].

Let v : P r o p ( V ) --t ' \L

be a valuation of P r o p ( V ) and set

then

4.1. Definition.

A family {vkl k E I } , where v k E P r o p ( V ) (Vk E (or truth-set) if conditions (4.1)-(4.5) hold.

I),will be called saturated

4.2. Remark.

If v : P r o p ( V ) -+ LLq is a map and v k ( k E I ) are defined by (4.0) then v is a valuation of P r o p ( V ) iff { v k I k E I } is a saturated family. Conversely, if { v k I k E I } , vk E P r o p ( V ) and we define v : P r o p ( V ) --t Li' by v ( p ) ( k ) =: 1 iff p E vk, then { v k 1 k E I } is a saturated family iff is a valuation of P r o p ( V ) in

L\'.

We consider the language of the 0-valued propositional calculus and for each k E I we introduce the symbols T k and T k .

Lukasiewicz logics

506 4.3. Definition.

An expression of the form T k p (or T k p ) , where p E Prop(V), will be called a signed proposition, or signed formula, of prefiz

T k(ofprefix

Tk).

Now we write down the following Gentzen-style (meta) formulas:

(4.9)

T k qj p

Tkpjp Tjp

Tjp

*

Formulas (4.9)-(4.9) suggest properties like "if p v q then p or q", "if not

( p V q ) then not p and not q" etc. This suggests further a classification of signed formulas into two classes according as they have "and" consequences or "or" consequences. Let us follow this suggestion in a formal definition. 4.4. Notation.

We denote by R any signed formula having one of the following forms: T k ( pv q ) 3 T k ( pA q ) 7 T k v j p7 T k p j p

(A)

and we denote by

any signed formula having one of the following forms:

Thus formulas (4.6)-(4.9) two: (4.10)

sz

A :-, 521

T k q j p7 T k q j p 7

can be succintly lumped into the following

B:-

522

r

rl I r 2

where in the case of signed formulas of type (A) of the form Tktpjp, Tktpjp,

T k + j p , T k p j p we consider Ri = R2 = T j p .

R1 = R2, for

example if

R

= T k p j p we set

Analytic tableaux for the 19-valued propositional calculus

507

4.5. Lemma.

Let S be a set of signed formulas and S; = { p E Prop(V) 1 T k p E S}, Si = { p E Prop(V) I T k p E S } . Suppose for every j , k E I, j 5 k we have Sr3 -C S; and Sj' 2 3;. Then {S; I k E I } is a saturated family i.f the

following conditions hold:

(i) for every k E I and every p E Prop(V) ezactly one of T k p , T k p belongs to S; (ii) 52 E S ($ R1 E S and (iii) r E S ($ rl E S

OT

R2 E S; E S.

Proof. If we define v : Prop(V) + L '\ (4.11)

V ( P ) ( k )=

{

by

1,

if T k p E S ,

0,

if T k p E S

,

v is well defined and v is a valuation iff conditions (i)-(iii) hold. Now apply Remark 4.2.

0

4.6. Definition. A set S of signed formulas is said t o be saturated if it satisfies conditions (i)-(iii) of Lemma 4.5.

We now describe the concept of tree. 4.7. Definition. An unordered tree is a triple A = (A,lev,s) where:

(i) A is a nonempty set, whose elements are called points.

(ii) lev : A + IN is a mapping, lev(x) is called the level of the point x E A. (iii) s E A x A such that if ( x l , y ) E s , (z2,y) E s then z1 = 5 2 . If (x,y) E s we say y is a direct successo~of x and x is a direct predecessor of y. More generally, if (z,y)E sn (i.e., there is a sequence

508

Lukasiewicz logics

...,(x,+~,y) E s),

(5,zl),(XI,zz),

while y is a successor of

we say t h a t z is a predecessor of y

x.

(iv) there exists a unique element 0 E A , such t h a t lev(0) = 1 and 0 has no predecessor.

(v)

if ( x , y ) E s then lev(y) = lev(z)

+ 1.

If x E A has no successor we say th a t z is an end point; if 2 E A has only one direct successor we say that x is a simple point, while if it has several direct successors we say th a t x is a branch point. A finite sequence (z1,22, ...,zn)E A” is called a path t o x, in the tree A if x1 = 0 and ( x 1 , ~ 2 ) , ( ~ 2 ,,..., ~ ) (zn-l,zn) E s. B y a mazimal p a t h or a brunch we mean a path

(21,

...,2,)

such th a t zn is an end point.

4.8. Definition. a) A tree

A is said to be finitely generated if each point has a finite number

of direct successors.

b) Let A = ( A ,lev, s) and A’ = (A’,lev’, 3’) be tree; A is called a subtree of

A’ if A G A’, lev

c) A tree is said t o be

= lev’lA, 0 = 0‘ and

S’IA~A

= s.

dyadic if each point has a t most two direct successors.

4.9. Definition. An analytic tableau for a signed formula x is a dyadic tree 7, whose points are signed formulas and there exists a finite sequence trees such t ha t

z,7z,..., 7, of dyadic

is th e tree w i th the single point z, 7, = 7 and for

E 1,n - 1, 7~~~ is a direct eztension of TA,t h a t is 7~is a subtree of 7 ~ + ~ and 7xtl is obtained fro m 7~by the application of one of the following t w o rules: a) choose an end point y of 7~and on th e path to y choose an $2, then adjoin either ill, or $2, as th e sole successor of y;

b) choose an end point y of 5 and on the p a t h to y choose a I?, then adjoin

rl and rzas the direct successors of y.

509

Analytic tableaux for the &valued propositional calculus 4.10. Definition.

a) A branch of an analytic tableau is called c h e d if it contains two points

of the form Tkp and T i p , for some k , j E

I , k 5 j and p E Prop(V). A

tableau is closed if all of its branches are closed.

b) A branch R of an analytic tableau is called complete, if for every point

R

which occurs in

which occurs in

R

R,both Q1 and n2occur in R a t least one of

called completed if every branch of

rl, I'2

and for every point

occurs in

R. A

tableau

I'

7 is

7 is either closed or complete.

For example an analaytic tableau for the signed formula X = rfi'[(ql(PkXV y3j(pkx)

A

('pl'pjrpkx V ( P k X ) ]

is given below:

This tableau i s closed. 4.11. Definition. a) Let

'u

: Prop(V) -+ Li4 be a valuation. A signed formula Tkp(or T k p )

is said t o be t r u e under

v if v(p)(lc)= 1(if v(p)(lc)= 0).

Lukasiewicz logics

510

b) A branch of an analytic tableau is said to be true under v , if all o f its points are true under v ; an analytic tableau is called true u n d e r v if it has at least a branch which is true under v. c) A set of signed formulas (a branch) is called satisfiable if there exists a valuation under which all the elements of the set (all t h e points of the branch) are true.

4.12. Definition. By a k-analytic proof of a formula p E Prop(V) is meant a closed tableau for T k p . 4.13. Proposition.

(i) If T k p has a closed analytic tableau, then p is a k-tautology. (ii) If Tkp has a closed analytic tableau, t h e n p is n o t k-satisfiable. Proof.

(i) If 7'is a direct extension of the analytic tableau 7, then 7' is true under every valuation v w i th the property t h a t 7 is true under v. Hence using Definition 4.1, it follows by induction t h a t every analytic tableau is true under a valuation v if i t s origin is true under this valuation. If 7

7 cannot be true under any valuation, because branch of 7 contains some Tkq,T j q , k , j E I , k 5 j and if such

is a closed tableau, then every

a branch is true under a valuation v , then v ( q ) ( k ) = 1, v ( q ) ( j )= 0, a contradiction. This implies th a t if T k p has a closed analytic tableau (i.e. p E Prop(V) has a k-analytic proof), then for every valuation v we have v ( p ) ( k ) = 1.

(ii) If T k p has a closed analytic tableau then v ( p ) ( k ) = 0 for every valuation 0 and so p is not k-satisfiable. 4.14. Remark. T he method of analytic tableaux is consistent, i.e. it is n o t possible t h a t

Analytic tableaux for the &valued propositional calculus

511

both T k pand T k p have closed analytic tableaux, where p E Prop(V). 4.15. Corollarv.

If a signed f o r m u l a y h a s a closed analytic tableau t h e n y i s n o t satisfiable. 4.16. Definition. By a Hintikka s e t we mean a set S of signed formulas which satisfies the following conditions:

(i) for every k , j E I , k 5 j, x E V it is not the case that Tka:E S and Tjx E S, where V is the set of propositional variables;

(ii) if R E S then

R1 E S and Q2

(iii) if r E S then

rl E S or rZ E S.

E S;

4.17. Lemma.

E v e r y Hintikka s e t i s satisfiable. Proof. Let vo : V -+

Lbq be the

interpretation of propositional variables

given by

{

~ o ( ~ ) (=i )

(4.12)

1,

0

,

if T k x E S for some k 5 i otherwise

,

.

It follows from Definition 4.16 (i) that vo(z) E ' \L

and v o ( z ) ( i ) = 0 whe-

PI E S. Furthermore there exists a valuation v : Prop(V) + L2 such that vlV = vo. It follows from (4.12) that every signed variable of S is true under v. As a matter of fact every element of S is true under v ; this is never

px

established easily by induction on the length of the formula, using conditions

(ii) and (iii) in Definition 4.16.

13

4.18. Corollary.

E v e r y complete n o t closed branch of a n y analytic tableau is satisfiable. Proof. If R is a complete not closed branch of the analytic tableau

7, then

the set of its points is a Hintikka set, hence it is satisfiable by Lemma 4.17. 0

Lukasiewicz logics

512

4.19. Theorern (corn pleteness). If p E Prop(V) is a k-tautology then every completed analytic tableau starting with T k p is closed. Proof. Let

7 be

a completed analytic tableau starting with T'p.

not closed then there is a complete branch Corollary 4.18 there is a valuation v : point of

R. It follows that v

be a k-tautology.

R

of

Prop(V)

If 7 is

7 which is not closed. By

--f

which satisfies every

satisfies T k p , i.e. v ( p ) ( k ) = 0, hence p cannot

The d-valued predicate calculus

513

$5. The d-valued predicate calculus In this section we introduce a &valued predicate calculus, a system of axioms for this calculus and we prove a completeness theorem. The construction is due t o Filipoiu [1981] and follows closely the model of Barnes and Mack [1975] for the classical calculus. 5.1. Definition. Let X be an infinite set whose elements are called individual variables, R a set whose elements are called predicate symbols and a map ar :

R

-+

N

called arity function. We will work with algebras of type T

= (2,2, { l } , ~ r ,{ l } i E ~ { , l } z E{ ~l, } z E their~ operations ); will be denoted

{ V Z } ~ € Xand { ~ X } ~ respectively. ~ X , By the f u l l by A, V, {Cpi}iEz, {&};El, first order algebra on (X,R) we mean the Peano algebra Prede(X,R) of type T freely generated by the set {r(zl,..., z,)Iz1, ...,z, E X , r E R,, n E N } , of atomic formulas, where r ( q ,...,z,) stands for (r,zl, ...,zn) and R, = { r E R I a r ( r ) = n } for n E N . The elements of Preds(X,R) are referred to as well-formed expressions or formulas.

5.2. Definition. The set of variables involved in w E Pred$(X, R ) is

(5.1)

X ( w ) = n { Y I Y C X , w E Preds(Y,R)}

5.3. Lemma.

FOTevery zl, ...,z,

z E X, r E

&, n E N ,

i E I

(5.5)

X(Vzw) = X ( 3 X W ) = X(w)

u {z} .

WI,2 0 2 , w

E Preds(X, R),

Lukasiewicz logics

514 Proof. Left t o the reader.

5.4. Corollary.

X ( w ) is finite for every w E Preds(X, R). Proof. Apply Lemma 5.3 t o a formative construction of w.

5.5. Definition. The depth of quantification of w E Prede(X, R ) is the number d(w) E

JV

defined recursively as follows:

(i)

d(r(x1, ...,x.,)

= 0,

(ii) d(wl A w2) = d(wl v w2>= max(d(wl),d(w2)), (iii) d(cpiw) = d(cpiw) = d(w), (iv) d(Vxw) = d(32w) = d(w)

+ 1.

5.6. Notation.

If w E Preds(X, R ) and x,y E X, we denote by w(x/y) the result of substituting y for x a t every occurrence of x in w, if any. In other words, w(z/y) is the image of w by the unique endomorphism h of Preds(X, R ) such that h ( x ) = y and h ( z ) = z for z E X - {x}. 5.7. Remarks. (i)

w(x/y) E Preds(X, R).

(ii) If 2 (iii) x

# X(w) then w(x/g)

= w.

4 x(w(x/y)).

The idea behind th e next definition is t h a t formulas Vxw and Vyw(x/y) express “the same thing” provided y g! X ( w ) - {x} (cf. Lemma 5.13). 5.8. Definition. w2 E Predfl(X,R), w1 M w2 i f For every wl,

515

The d-valued predicate calculus

(A) d(w1) = d(w2) = 0 and w1 = w2; or ( 6 ) d(w1) = d(w2)

> 0 and one of the following situations

hold:

(i)

Wh

= w i A w! ( h = 1,2) and w: x w; and wy x ws, or

(ii)

Wh

= wk V w; ( h = 1,2) and w i x w; and wy x w;, or

(iii) W h = cp;(wi) (i E I , h = 1,2) and w: (iv)

Wh

= ~p;(wL)( i E I , h = 1,2) and

M

w;, or

W: M

w:; or

(C) w1 = Vxw: and w2 = Vyw: and one of the following situations holds: (i) x = y and w: x w;, or

(ii) w{ x w and wk x w(z/y) where y

# X ( w ) ; or

(D) similar t o (C) but with 3 instead of V. 5.9. Lemma. (a) Suppose wl, w2 and w satasfy condition (C.ii) in Definition 5.8 and

take z E X , z # X(w1) U X(w2). T h e n there is w' E Preds(X,R) such that z # X(W') and ( C i ) hold3 f o r wl, w2 and w'.

(b)

If w1 x w2 t h e n d(wl)

(c)

~f wl x w2 and y # X(w1) u X(w2) t h e n wl(x/y)

(d) T h e relation

XI

= d(w2) M

w2(x/y).

is transitive.

Proof. Check simultaneously (a)-(d)

by induction on the depth of quantifi-

cation.

5.10. Proposition.

T h e relation x is a congruence of the algebra Preds(X, R). Proof. Reflexivity and consistency with the operations of Preds(X,R ) are immediate, while transitivity is Lemma 5.9 (d). The only delicate point in establishing symmetry is case (C.ii) of Definition 5.8. But w = w(z/y)(y/x) z

# X(w(z/y))

by Remark 5.7 (iii).

0

Lukasiewicz logics

516

5.11. Definition. The set F X ( w ) of free variab2es of a formula w E Prede(X,R) is defined recursively as follows:

(i)

~ ~ ( r ( z..., 1 2. ,))

= (51,...,G,},

(ii) FX(w1 A wz) = FX(w1V

WZ)

= FX(w1)U FX(w2),

(iii) FX(cpiw)= FX(cpiw) = FX(w), (iv)

FX(Vxw) = FX(3xw)= FX(w)- {x}.

The elements of the set

(5.6)

BX(w) = X(W)- F X ( w )

are referred t o as the bound varaables of w .

5.12. Lemma. If w1 M w 2 t h e n FX(wl)= FX(w2). Proof. By induction; left t o th e reader. 5.13. Lemma.

Let w E Preds(X, R) and x, y E X.

(i) If y

# X ( w ) - {x} t h e n Vxw

(ii) Ifx,y

# X(w)

M

Vyw(x/y).

t h e n Vxw M Vyw.

Proof.

(i) If y = x then Vyw(x/y)= Vxw. If y # X ( w ) the desired conclusion follows from case ( C i ) in Definition 5.8 with w{ = w and wi = w(x/y). (ii) From (i) and Remark 5.7 (ii).

0

5.14. Notation. Let Pd(X,R) or simply

Pd

stand for th e quotient algebra Preds(X,R)/

M.

The equivalence class containing w E Preds(X,R) will be denoted by

The 8-valued predicate calculus

517

[w]= E P $ ( X , R )or simply by [w] whenever there is no danger of confusion.

5.15. Remark. Using the simplified notation 5.14, condition (C) in Definition 5.8 reads: VXWl

=Vyw2

*

(i) x = y and w1 = w 2 ,or (ii) w2 = wl(s/y) where y $?! X(wl). 5.16. Lemma. Every element of Ps(X, R ) can be represented in t h e f o r m [w] where w E Preds(X, R) satisfies

(i) n o variable x E X appears in w more t h a n once in a quantifier 'dx OT 3 x and (ii) F X ( w ) n B X ( w ) = 0. Proof in two steps.

I) Let

be a formative construction (cf. Definition 1.5.b) of w E Preds(X, R).

6

Take a variable y X ( w ) and set wI, = Vyw*(x/y); then w i M Wh by Lemma 5.13 (i). Let further wy, ,..,w:-~, wI, be a formative construction o f w;L and let w;+~,...,w; be obtained from wl, ...,Wh-l, w i in the same way as W h + l , ..., wn were obtained from wl, ...,wh-1, Wh in (5.7). Then

is a formative construction of wk and since follows that w:M wt

(t = h+l,

M

is a congruence it also

...,n). Thus w; M w and the occurrences

of s in Vxw* have been removed from w;.

Lukasiewicz logics

518

II) Since X is infinite the above construction can be applied as many times as necessary, i.e. until we obtain a formula w' M w fulfilling properties (i) and (ii). 0 The next definition introduces an appropriate concept of interpretation. 5.17. Definition.

U be a non-empty set. By a 8-valued interpretation (or simply an interpretation) in U we mean a couple (f,G), where f : X --f U and Let

(5.9)

G : R

is such that

G(r)

-+

:

{gig : U"

U"'(')

--f

nEN}

--f

' i L

for every r E

R.

We will think of z E X as a name for the object f(z)E as a name for the &relation G ( r ) .

U

and of r

ER

5.18. Lemma. For every 8-interpretation (f,G)in U there is a unique f u n c t i o n v : Po 4 Lh' such that f o r every r(z1,...,I,) E Ps (r E R, 21,...,z, E X , n =

ar(r)), w1,w2,w,wo E

P79,; , j E 1:

(a) v(r(z~,*-*,zn))(j) = 1 @

G(r)(f(z~), * - . i f ( z n ) ) ( j )= 1,

(b) v(w1 A w ~ ) ( j )= 1 H w(wl)(j) = 1 and v(wz)(j) = 1, (c) v(w1

v w2)(j) = 1 @ v(w1)(j)

= 1 or v(wz)(j) = 1,

(f) condition (fn) holds f o r every n E N , (g) condition (g,) holds f o r every n E N ,

where conditions (fo), (go) are vacuously fulfilled, while f o r n > 0, (fn) if 20 = Vzwo and d(w) = n t h e n [v(w)(j) = 1 f o r every X' = X U { t } with t # X , every extension f' : X' + U o f f and every

The &valued predicate calculus (5.10)

V‘

I

519

: {w‘ E Pa(X‘,R) d(w’)<

that fu&l (a)-(e)

and (fk) f o r all k

TI} +

L2[Jl

< n, it follows that v’(w0(z/t))(j)=

117

(gn) similar to (fn) but with 3x instead of Vx and “there as a n extension”.

f‘ quantified by

Proof. Every element of the Peano algebra Preda(X,R) is uniquely repre-

sented in one and only one of the forms r(xl, ...,xn), w 1 A

cpiw,

Cpiw,

Vxw or 3x20; therefore conditions (a)-(g)

202,

w1 V w2,

determine uniquely

o(w) for each w E Preda(X,R). It remains t o show that if w1 M w2 then v ( w l ) = v ( w 2 ) E Liq. Suppose first d(wl)= 0. Then w1 = w 2 by Definition 5.8 (A). Clearly (a) implies that w(r(xl, ...,xn)) E LLq and it follows by induction via (a)-(e) t h a t v(wl) E LLq whenever d(wl) = 0. Next proceed by induction for d(wl) > 0. If w1 and w 2 satisfy Definition 5.8 (B) t he desired conclusion follows immediately from (b)-(e). Now suppose w1 = Vxw{ and

w2

= Vyw: satisfy Definition 5.8 (C). Then Lemma

5.9 (c) implies that in case (i) w{(x/t)M w i ( x / t ) = w t ( y / t ) , while in case (ii) w i ( x / t ) M w ( x / t ) and wk(y/t)M w ( s / y ) ( y / t ) = w ( x / t ) , therefore in both cases w : ( x / t )

w;(y/t),hence in condition (fn) we do have d ( w { ( x / t ) ) = d(w;(y/t)), therefore v(w1) = ~ ( 2 ~ 2 ) Condition . (fn) implies also

V(wl)

E

M

LL~.

0

5.19. Defi nitio n.

A quadruple (U, f,G , V ) satisfying the conditions of Lemma 5.18 will be called a d - d u e d valuation of Pa(X,R ) in the domain U .We also say simply that w is a valuation of Pa. As was done in 53, in the sequel we consider a fixed element k E

I.

5.20. Definition. Let H Pa and w E Pa. We say that H semantically k-implies w ,and we write

H

+ k

w ,if for every valuation v of Pa such that w(w’)(k) = 1for

520

Lukasiewicz logics

all w‘ E H it follows t h a t v ( w ) ( k ) = 1. We also use the notation (5.11)

Dedk(H) =

In particular if

0

{w E J‘s I H

20

k k

we say that

k

w1

w is a k-tautology and write simply

A “good” construction of a predicate calculus should “include” the corresponding propositional calculus. This is actually the case of the $-valued theory, as shown by Proposition 5.21 below, which is stated in semantical terms but is also valid for the corresponding syntactical concepts due t o the Completeness Theorems 3.26 and 5.36.

Let us extend th e notation Prop(V) introduced in Definition 3.1 t o the case of finite sets V. Preds(X,

If w’ E Prop({zl

R), we write simply w’(w1,

,...,5,))

...,w, E ...,z,/wn).

and w1,

...,w,)instead of w’(z1/w1,

5.21. Proposition.

Let n E and

IN - {0}, H U {w}

Prop({zl,

...,z,}),

w1,...,w, E Prede(X, R )

(i) If H semantically k-implies w i n the d-valued propositional calculus then H(w1, ...,w,) w(w1, ...,w,).

+ k

(ii) If w is a k-tautology in the d-valued propositional calculus then

I= k

w(w1,

...,%).

521

T h e &valued predicate calculus Proof.

(i) Let v be a valuation of Pd such that v(w")(k) = 1 for every w" E H(wl, ...,w,). Let h : Prop({z1,...,zn}) + Li' be the valuation of Prop((z1, ...,2,)) such t h a t h(xj) = v(wj) ( j = 1, ...,n). Then for ...,z,}) it follows that every w' E Prop( {q,

=

= w'(h(z,),...,h(z,)) =

...,z,))

h(w') = h(w'(z1,

...)v(wn))

W+(Wl),

In particular if 20' E H then w'(w1,

= v(w'(wl, ...,w,)) .

...,w,,)E H(w1, ...,w,) and

h ( w ' ) ( k )= v(w'(wl,..., wn))(k) = 1. This implies v(w(wl,

...,w,))(k) = h ( w ) ( k )= 1.

(ii) Immediate from (i).

0

The above elements of semantics will be related t o syntax, i.e. to the concept of proof which we introduce now. 5.22. Definition. The set k-Axm of axioms of the &valued predicate calculus is the set of those elements of P s ( X ,R)that have representatives of the following forms:

(5.13)

w'(w1,

...,wn), where ~ ' ( 2 1 ..., , z,)

is a k-axiom

of the &valued propositional calculus and

w1, ...,w, E Preds(X, R) (5.14')

Vx(w1

(5.15')

Vxw

(5.15")

w(x/y)

-+ w2) + k k

+ w(x/y) k --t

k

3zw

(wl

, + Vxw2) k

,

,

y

# BX(Vzw) ,

,

y

# BX(3xw) ,

x $? F X ( w , )

,

Lukasiewicz logics

522

(5.16“)

~~(VZW * ) V X C ~ ~, W i E

(5.17’)

Pi(3sw)

k

where + and k

H

k

*

Vx~piw,

I,

iEI,

are defined by (3.0).

k

We define by induction the concept of k-proof. 5.23. Definition. Let H C PS and w E Pa. A k-proof of length n of w from the hypotheses H is a sequence w1,w2, ...,w, of elements of Pd such that w, = w and wl, ...,w,-~ is a k-proof of length n - 1of w,-~ and (a) w,

E k - A x m U H , or

(b) wt = w,

--f

w, for some t,s

k

< n, or

t l , ...,t, E (1, ...,n } such that wtl, ...,wtm subset Ho H such that

(c) w, = Vxw’ and there exist is a k-proof of wt from a

x $! FX(H0) $ ! lJ{FX(w’)/w‘ E Ho} We denote by H I- w th e existence of a k-proof of w from H and we k

also say that

F syntactically k-implies w. In particular if 0 I-

k

t h a t w is a k-theorem and write simply I-

k

w.

w we say

The &valued predicate calculus

523

Note that by removing formulas involving cp; or

pi from axioms in Defi-

nition 5.22 we get the axioms of the bivalent propositional calculus. Also, the inference rules are the same as in the bivalent case: modus ponens and

generalization: W -

vxw

(6. Definition 5.23 (b) and (c), respectively). 5.24. Definition.

Let Pi = Po(Xl, R') and Pj = Pg(X2,R 2 ) . By a semi-morphism from to

Pj we

mean a couple (f,g) where

f

:

Pj

+

P$', g : XI

+

Pj

X2 and

the following conditions are fulfilled: (a)

g(X1) is infinite;

(b) f is a morphism with respect t o A, V, pi and pi (i f I);

Proof.

(i) As for the bivalent case; cf. Lemma 4.2 in Barnes and Mack [1975]. (ii) By induction. The only non-trivial case is w = Vzwo (or similarly w = 3zwo), where wg satisfies (ii). Using t h e infiniteness of g(X') and Lemma 5.13 (i) we may suppose t h a t g(z) # g(x), g(y) and z $!

X'(wo).

Then

Lukasiewicz logics

524

5.26. Theorem (The Substitution Theorem). L e t ( f , g ) be a semi-morphism from

Pi

Pj

to

and H G

Pi, w E Pi. If

w1,...,w, is a k-proof of w f r o m H , then f(wl), ,..,f(w,) is a k-proof of f(w)f r o m

f(W

Proof. Induction over n. For n = 1 note that f(k-Azm' U H ) = k-Asm2U

f ( H ) because one sees readily that the axioms of the forms (5.13), (5.16). (5.17) in Definition 5.22 are transformed into the corresponding axioms for

Pi,while for

the axioms of the forms (5.14) and (5.15) the same result is

obtained via Lemma 5.25; for (5.15) we have t o choose a representative of

f(w)such that g(y) # BX2(Vg(z)f(w))and similarly for 3. The inductive step amounts t o proving that f preserves modus ponens and generalization. The former is clear. To prove the latter suppose t h a t wn = Vxw and there is a k-proof w t l..., , wt, of w = w t mfrom a subset

HO

H such that x

# FX'(Ho).Then

by the inductive hypothesis

f(wt,), ...,f(w t m )is a k-proof o f f(w) from f ( H o ) . For each w' E Ho, from x # FXl(w') we infer g(x) @ FX2(f(w'))by Lemma 5.25 (i), therefore

is a k-proof from f ( H o ) .

0

5.27. Lemma.

For every H

5 PS and w

E Ps,af

H I- w k

Proof. Let w l ,...,w, = w be a k-proof o f such that v(w')(lc) = 1 for all w' E

then H

w. k

w from H and v

H. We will

a d-valuation

prove by induction over n

that v(w)(lc) = 1. For n = 1 we take w E k-Azm, say of the form (5.14') in Definition 5.22. Note first that

525

The &valued predicate calculus

w w' + w")(k)= 1

(5.18)

(

k

or w(w")(k)= 1 , therefore

2)(vx(w1

- w2)

k

k

(wl + Vxw2))(k)= 1 H k

w(wl)(k) = 0 or w(Vxwz)(k)= 1 -S

f' : X U { t } + U of f such that w'(w1(z/t))(k) = 1and w'(w2(x/t))(k)= 0) or w(wl)(k) = 0 or (for every extension f' : X U

H (there is an extension

{t}

-+

u, w'(wz(./t))(k)

= 1)

-

and the latter statement is clearly true. Similar proofs hold for the other k-axioms. For the inductive step suppose first that

w, = w t

k

wn, where

w(wd)(k) = w(wt)(k)= 1 by the inductive hypothesis; then (5.18) implies

w(w,)(k) = 1.

w = Vxwo and there exist t l , ...,t , E (1, ...,n} such that wtl, ...,wt, = w o is a k-proof of w ofrom HO C H , where x # F X ( H o ) . Take X' = X U { t } and w' as in condition (f,) from Lemma 5.18; the task Now suppose that

is t o prove w'(wo(x/t))(k) = 1. Reasoning as in Lemma 5.18, w' can be

extended t o a 6-valuation w" of Pa(X', R). On the other hand we claim that the following pair

(f,g)

is a semi-morphism:

X

f

: Pd(X, R) -+

Ps(X', R)

X' defined by g(x) = t and g(y) = y for y # z. Conditions (a)-(c) in Definition 5.24 are easily checked, while (d) becomes r(y/z)(z/t) = r(x/t)(g(y)/g(z)) and is also easily verified by considering the cases y # z # z, y # x = z and y = x # z . According t o Theorem 5.26, w t , ( x / t ) ,..., wt,(x/t) is a k-proof of wo(x/t) defined by f ( w ) = w ( x / t ) and g

:

--t

Lukasiewicz logics

526 from f ( H 0 ) . But f(H0) = HObecause z hypothesis implies

Ho

4 F X ( H o ) ,therefore the inductive

wo(z/t), hence t h e extension d' of v' satisfies k

v " ( w O ( z / t ) ) ( k= ) 1, which implies d(wo(x/t))(k)= 1.

0

5.28. Corollary.

~f I- w t h e n - ( Ik

(PkW).

k

Proof. As for Corollary 3.20.

D

5.29. Theorem (The Deduction Theorem).

If H

C_

P,q

and

H U { w } I-

w,w' E P,q then H

I-

(w

k

k

if and only i j

w')

w'.

k

Proof. Similar to that of Proposition 3.17, except that in the inductive step of the "if" part we must add the case when w is obtained by generalization: w' = Vzwo and

Ho I- wo, where Ho C H

U {w} and

k

z $! F X ( H 0 ) . If

w

4 HO then HO E H

followed by the k-axiom w' + (w k

from H . If w E Ho, from

HO- {w} I- w k

z

# FX(w).

k

k

k

w') yields a b-proof of w

Ho

+ w' k

Ho I- w o and the inductive hypothesis we obtain k

wo,therefore Ho - {w} I- Vz(w

Applying (5.14') we get HO -

H I- (w 4 w') k

--f

and a k-proof of w' from

k

{w} I-

k

w

-

--t

wo), where

k

k

Vxwo, hence 0

527

The $-valued predicate c d c u l u s

5.30. Definition. we say that H Pd is k-consistent if (PkW @ Dedk(H) for every k-theorem w ;otherwise H is said to be k-inconsistent. H is called a maximal kconsistent subset if it is k-consistent and maximal for inclusion with this property. 5.31. Remark. Proposition 3.19, Lemmas 3.21 and 3.23, Corollaries 3.20 and 3.24 and Remark 3.22 are extended to the &valued predicate calculus with the same proofs.

5.32. Lemma. Let H C Pd be k-consistent. If 3xw E H and t { w ( s / t ) } is k-consistent.

#

F X ( H ) then H U

Proof. Suppose there is a k-theorem w' such that H U { w ( z / t ) } I-

(PAW'.

k

Then H I- w ( z / t ) + (pkw' by Theorem 5.29, therefore k

k

H I- (pk(w(./t)) by (3.23), hence H k

we get

v@k(W(Z/t))

and using (5.17')

k

H I-

(pk3tW(T/t),

k

which yields H I-

(pk3SW

k

by Lemma 5.13 (i).

On the other hand from 3 s w E H and (3.19) we deduce H

I- cpk3sw, k

therefore H is k-inconsistent by Remark 5.31. 5.33. Lemma. F o r every k-consistent set H

P a ( X , R ) there ezist X' 2 X and H' c

PG(X*,R) such that (5.19)

H

(5.20)

H' is masimal Ic-consistent in Pd(X',R) ;

H' ;

528

Lukasiewicz logics

if 3 x w E H* t h e n w ( s / t ) E H' f o r s o m e t E X *

(5.21)

.

Proof. We are going to define four increasing families of sets X i , Hi, Hi, Pi such that Pj = P8(Xi,R ) and each Hi is maximal k- consistent in Pi (i E N ) . First we put X o = X and Ho = H ,then

where tL are new variables. We prove that if Hi is k-consistent, so is Hi+l. If not, there is a k-theorem wo such that @kw0 has a k-proof from H,!+l;let

A = {zf?;(x/t$,), ...,w~(s/t$h)} be the set of all elements of - Hi occurring in that k-proof. Then H j U A I- PkWO, which contradicts Lemma 5.32. Thus Hi+l is k-consistent k

and we take a maximal k-consistent set Hi+l 2 Hit1, which exists by the Zorn lemma. Now we take X * = U Xi and H* = U Hi. Then (5.19) holds and we prove (5.20) using Lemma 3.21 via Remark 5.31. First if (PkW E H* for some k-theorem w then (PkW E Hi for some i and this contradicts Lemma 3.21 for Hi; hence (Pkw $! H* for every k-theorem w. Further if w E Dedk(H*) take a k-proof w1, ...,wn-lr w, = w of w from H* in Pfl(X*,R);then this is also a k-proof of w from Hi in Pfl(Xj,R) for a sufficiently large i, hence w E Dedk(Hi) = Hi by Lemma 3.21 for Hi,therefore w E H'. This proves that Dedk(H*) = H*. One proves similarly (3.30) for H* by using (3.30) for

Hi for some i. Finally if w = 3sw' E H* take i such that w E Hi; then w'(z/tL) E

Hitl C H*.

0

5.34. Proposition. For every k-consistent set H c Pg(X, R ) there is v ( w ) ( k )= 1f o r every w E H . Proof. If H*

2H

satisfies conditions (5.19)-(5.21)

0

valuation v such that

and v* is a valuation

The &valued predicate calculus

529

of P8(X*,R)such that v*(w)(k) = 1 for every w E H*, then the restriction v = v*

I Pd(X,R ) is obviously a valuation

of P . ( X , R ) which satisfies

v(w)(k) = 1 for every w E H. Therefore we may suppose without loss of generality that (5.20) and (5.21) hold for

H . The desired valuation will

be a quadruple, as required in Definition 5.19, namely ( X , i d x , G , v ) , where --f {g I g : X" --t J$} is defined by

G : R

and v is obtained from the &interpretation (idx, G) in X by the construction in Lemma 5.18. The first point is the correctness of this construction: to prove that (idx, G) is actually an interpretation in X it remains to check that G(r)(xl,...,xn) E $. But if i , j E I,i I j and cp;r E H , then H I- cpjr k

by (3.15), therefore cpjr E H by Lemma 3.21 (3.29) via Remark 5.31. Now our theorem will be established if we succeed to prove that (5.23)

v ( w ) ( ~=) 1 ($ cpjw E H

(jE

I)

because if w E H then by applying Lemma 3.21 we obtain in turn (pkW $! H , (pkqkw E H and (PkW E H via (3.12), therefore v(w)(k) = 1by (5.23). Thus it remains to prove (5.23) by induction. For w = ~ ( $ 1 ,...,z,,),(5.23) follows from Lemma 5.18 (a) and (5.22). For the inductive step we use Properties (a)-(g) in Lemma 5.18 (6. Definition 5.19) and again Lemma 3.21.

If 201,

satisfy (5.23), so do wl A w2 and w1 V w2; e.g. for w1 A w2 we also use (3.36) and get 202

Now we suppose w satisfies (5.23) and prove that so do cpiw, p;w, Vxw

and 3xw. But

Lukasiewicz logics

530

by (3.11) and (3.29), while (3.30), (3,14), (3,13) and (3.29) imply

For Vxw suppose first yjVxw # H . Then cpjVxw E H as was just seen in (5.24), hence 3xcpjw E H by (5.17"), therefore (5.21) implies (cpjw)(x/t) E H for some t E X, hence (cpjcpjw)(z/t) E H by (3.13); but since cpjw satisfies also (5.23) we get v((cpjw)(x/t))(j) = 1, i.e. v(w(x/t))(j) = 0, therefore v(Vxw)(j) = 0. Conversely, suppose vjVxw E H. Then pjVxw # H by (5.24), hence 3xCpjw # H by (5.17"). Then it follows from (5.15") that for every t # BX(3xcpjw) = B X ( 3 x w ) we have (cpjw)(x/t) # H , hence (cpjw)(x/t) E H by (5.24), therefore v(w(x/t))(j) = 1. As t is arbitrary, the l a t t e r equality is equivalent to v(Vxw)(j) = 1. 0 The similar proof for 3 z w is left to the reader. 5.35. Theorem (The Completeness Theorem). Let H Pd(X,R) and w E Pd(X,R). T h e n H

k

w if and only if

H t w. k

Proof. As for Theorem 3.26 except that Lemma 3.16, Proposition 3.25 and Theorem 3.17 are replaced by Lemma 5.27, Proposition 5.34 and Theorem 5.29, respectively. 0 5.36. Corollary. For w E Pd(X, R ) we have

+ k

w if and only if I- w. k

Kripke-style semantics for $-valued predicate logics

531

$6. Kripke-style semantics for 8-valued predicate logics In this section we introduce the semantic 29-models for the 8-valued predicate calculus and establish the relationship with the algebraic models for

the case when 8 is an ordinal number. A similar semantics for the Post logic is defined in Maksimova and Vakarelov [1974].

6.1. Definition. Consider again the &valued predicate calculus Ps = P s ( X ,R ) . An algebraic 8-model is a triple M = ( L , U , S ) , where L is a completely chrysippian 8algebra, U a non-empty set and S : x U x --$ L is a map satisfying the following conditions:

w E Ps,z E X and for any interpretation f E U x exist in L : V{ws(f;) I c E U } , A {ws(f:) [ c E U},

for any there

where ws(f) = S ( w , f ) and

fay)

=

[

f(Y) c ,

7

if Y

#.

ify=x

6.2. Remark. Property (6.2) holds for any formula w .

,

for any y E X ;

Lukasiewicz logics

532 6.3. Definitiom. For every

H E

P8 and

w E

P8 let

H

b k

w mean that for any algebraic

hf = ( L , U , s ) , if Yk(W$(f)) = 1 for any w' E H and f E U x then V k ( W s ( f ) ) = 1. In particular, if 8 w then we say 20 is algebraically $model

k

k-valid and write simply

k

w.

6.4. Proposition.

Po and w E Ps then the following assertions are equivalent:

If H (a)

H t-

k

w;

Proof.

Let w1,w2,...,w , be a k-proof of w from H and M an algebraic &model such t h a t ( P k ( W $ ( f ) ) = 1 for any w' E H . We are .j (b):

(a)

going t o prove by induction on n that Y k ( w s ( f ) ) = 1. If n = 1 then

w E k - h m U H , therefore ( P k ( W S ( f ) ) = 1, because for any k-axiom s we have b s. Thus e.g. for the axioms (5.14') note first t h a t if z $ F X ( w ) k

then w s ( f ) = ws(f,")for every c E

U

by Remark 6.2, therefore

533

Kripke-style semantics for 19-valued predicate logics

The inductive step is performed as follows:

(1) if cpk(ws(f)) = 1 and (P~((w+ ~ ' ) ~ ( f=) 1 ) then k

(2) if wl,...,w, is a k-proof of w from Ho (Pk(wS(f)) (Pk(WS(f))

(b)

+ (c):

= 1 then

EH

# F X ( H o ) and {wS(ff) 1 c E u } =

where z

( P k ( ( v z W ) S ( f ) )= (Pk

A

= 1. Let us take

U"'(')

L = 'iL

and G :

R

+ {g

Ig

:

U" + Lh'}

Li'. We can consider the algebraic &model M = (Li4,U,S), where S : P8 x U x + LL4 is given by S(w,f) = v(f, G), where v(f, G) is the d-valuation associated with the interpretation (f, G). For this algebraic 19-model we have ( ~ k ( w ~ ( f ) = ) 1 iff v(f,g)(w)(k) = 1, such t h a t G ( r ) :

4

therefore we get the desired implication. (c)

+ (a): By the Completeness Theorem 5.35.

0

Lukasiewicz logics

534

Now we shall define the concept of semantic &model using the concepts of 6-structure and &space introduced in Definition

S

6.4.1and 6.4.6.If

( A , { A f } t E ~ , ; Eis ~ ) a &structure then we shall work below with a relation \I- C A x U x x Pd and we shall write x It- w instead of =

f

( X , f , W )E

It-.

6.5. Definition. A semantic 6-model (s.6-m.) is a triple 0 = (S,U,\I-) with S,

U ,IF as

above and satisfying the following conditions:

(6.7)

If x IF w and x 5 y then y [I-

(6.8)

For any atomic formula T and for any interpretations fi, f2 such that fl(ui) = f2(u:),i = 1, ...,rn we have

(6.9)

x It w v w ' e x [I- w or x It- w';

(6.10)

x [I- WAW'U x [I- w and x \I- w';

f

f

f

f

f

f

f

f

e there is c E U

(6.11)

x /I- 3zw

(6.12)

x It- Vzw u for any

CE

x

Itf

pjw

there is t E

and for any y E A:, y

(6.14)

x It- cpjw u there f

such that

x It- w ; ff

U , x It- w ff

f

(6.13)

w;

;

T such that x E At

It- w ; f

is t

E T such that z E A'

and there is y E A; such that y [If w f

.

535

Kripke-style semantics for 9-valued predicate logics

6.6. Remark. Property (6.8) holds for any formula w. 6.7. Definition. Let us consider a triple

0 = ( Y ,U,I I-),

where

Y

=

(Y, I , {di}iEi)

is a

&space. Such a triple is called a special semantic 9-model (s.s.9-m.) if it verifies axioms (6.7)-(6.12)

x IF Cpjw

(6.16)

f

in Definition 6.5 and

there is y

2 djx

such that y

Iy w . f

6.8. Remark. Using (6.7) we get the equivalence: d j x

Iy

that y

Iy f

w

e there

is y

2 djs

such

w.

f

6.9. Lemma. Any 8.8.8-m. is a 3.9-m. Proof. Since djx E A?] and y

2 djx for any y E A?], the following asserti-

ons are equivalent:

(i)

djx IF w;

(ii) y

1-

f

w, for any y E A?];

(iii) there is t E Y/ M , z E St and y

(iv)

2

IF 'pjw. f

Itf

w for any y E A;;

536

Lukasiewicz logics

6.10. Definition.

If 0 = (S, U,IF) is a s.6-m. we shall say that w E Pg is k-valid in 0 if for any

f E U x and for

any x E

A , x Ilf

(PkW.

A formula w is semantically

k-valid if it is k-valid in any s.6-m. If H G Ps and w E Po let H k

mean t h a t for any s.9-m.0, if every in

(s)w

w' E H is k-valid in 0 then w is k-valid

0. The following results establish the relationship between algebraic 9-models

and semantic 6-models. 6.11. Theorem.

Let S = ( A ,{ A f } t E ~ , jbe E ~a )9-structure and M ( S ) = ( B ( A ) ,n, U,8,A, {(pi}iEl, { @ i } i E ~ )the corresponding Moisilfield. If 0 = (S,U, I )-l as a semantic 6 - m o d e l define S by ws(f) = {t E

A

12

Ilf

w}f o r a n y w E Ps and f E U x . Then:

(1) M = ( M ( S ) , U , S )i s a n algebraic 6-model; ( 2 ) For any formula w,w is algebraically k-valid i n M iff w is semantically k-valid i n 0.

It is obvious that ws(f) E B ( A ) , for any w and f. The triple ( M ( S ) U, , S) satisfies conditions (6.1)-(6.6). We shall prove only (6.4)(6.6). In the 9-algebra M ( S ) we have, for w E Ps,f E U x ,i E I and any x E A: Proof.

Kripke-s tyle semm tics for &valued predicate logics

Iy

(for cp;w a similar proof with

5

E (3uw)s(f)

u

z

f

($5

537

and $Z),

II- 3uw H f

IF w for some c E U

f,"

++ z E

U ws(f,"),

CEU

(for Vuw a similar proof with "for all c"). For any formula w the following equivalences hold:

($for all 5 E

A,

5

\If

QkW

.

0

6.12. Theorem.

Let L be a completely chrysippian 8-algebra with card(I) _< X,, M = ( L , U , S ) a n algebraic 8-model and Q the countable set of the elements of L having the f o r m V ws(f:) OT A w~(f:), where u is a variable, CEU

CEU

w a formula and f a valuation. T h e n we can construct the 19-space S = (SpecL(Q),&,{d;};cl) and define the relation [I-c SpecL(Q) x U x x P8 by M It- w -Hws(f) E M , such that: f

(a)

0 = (S,U, It) is a special semantic d-model;

(b) FOTa n y formula w, w is k-valid in 0 ifl w is k-valid in M . Proof. The triple satisfies the conditions in the definition of a s.s.8-m.:

538

Lukasiewicz logics

* ws(f:) f

E M for some c E U u M II- w for some c E U ; f:

(Pkw fs ( ( P k w ) S ( f ) E

M

*

(Pk(wS(f))

E

M

*

T h e verification of the remaining conditions is easy. From the construction of 0 we obtain the equivalence of the following properties:

(i)

M II-

(ii)

((Pkw)S(f)

(iii)

((Pkw)S(f)=

f

(Pkw, for any

M E SpecL(Q)

and

f

E

Ux;

E M , for any M E SpecL(0) and f E U x ;

1.

0

6.13. Corollary. If 6 ds an ordinal number and card(I) 5 Xo f o r any w E F'd and H C Pa, then the following conditions are equivalent: (a)

H I- w ; k