Modal Logic and Universal Algebra I. Modal axiomatizations of structures. 3 Valentin Goranko1 and Dimiter Vakarelov2 1
Department of Mathematics, Rand Afrikaans University PO Box 524, Auckland Park 2006, Johannesburg, South Africa E-mail:
[email protected], URL: http://www.rau.ac.za/staff/goranko vf Department of Mathematical Logic Faculty of Mathematics and Computer Science, So a University blvd James Bouchier 5, 1126 So a, Bulgaria E-mail:
[email protected] 2
We study the general problem of axiomatizing structures in the framework of modal logic and present a uniform method for complete axiomatization of the modal logics determined by a large family of classes of structures of any signature. abstract.
Keywords:
complex algebras of structures, modal logics, axiomatizations, completeness, de nability.
Introduction
By a structure we mean a non-empty universe with a collection of functions and predicates of nite arity. Every such a structure can be regarded as a generalized Kripke frame with a collection of accessibility relations corresponding to the principal predicates and the graphs of the principal 3 The rst author was partly supported by a research grant GUN 2034353 of the Foundation for Research Development of South Africa. The second author was partly supported by the Bulgarian Ministry of Science and Education
1
functions. Given a class of structures C of a signature (-structures), we consider the induced multi-modal language M with modalities corresponding to these accessibility relations, and the multi-modal logic M LC in that language which captures the modal formulae valid in C. On the other hand, with every class of -structures we associate the class of their complex algebras (power structures in Brink 1993) which provides the algebraic semantics for that logic. In this paper we give uniform schemes for complete axiomatization of the modal logics for a broad family of elementary classes (incl. all 502 classes) of -structures, assuming that the di erence operator is present in the corresponding modal language. That operator is de nable in many important classes of structures, such as all expansions of implicative lattices (e.g. Heyting and Boolean algebras) and all expansions of groups (e.g. rings and modules). In other cases, such as arbitrary lattices and semigroups, it has to be added to the language. The di erence operator has recently been widely studied and used as a useful tool both for boosting the expressiveness of modal languages and for axiomatizing extended modal logics, combined with using speci c non-Hilbert style context dependent rules, often referred to as \Gabbay-style" rules. For some related results see Goranko 1990, Gargov and Goranko 1993, de Rijke 1992, de Rijke 1993, Venema 1993, Goranko 1998. This paper, in particular, contributes further in that direction, by presenting a general completeness proof which does not assume versatility of the languages, as in Venema 1993. It is based on a modi ed canonical construction, very close to the Henkin completeness method for the rst-order logic. The idea of the construction is taken from Passy and Tinchev 1991 where similar construction has been applied to PDL with so called \data constants". Besides the various speci c applications for particular classes of structures, the obtained results can be appreciated from a few general viewpoints. First, the modal logic M LC corresponding to a class of structures C encapsulates the universal fragment of the monadic second-order theory of C, and thus provides an axiomatization for that fragment. Second, that modal logic can be regarded as an axiomatization of the class of corresponding complex algebras (see Goldblatt 1989) of the structures in C. Third, from viewpoint of modal logic, the structures in C can be regarded as Kripke frames in which the universe of possible worlds has a speci c algebraic structure, besides the traditional relational component. In cases when suitable representation theorems exist, (e.g. for groups, lattices, Boolean algebras etc.) that algebraic structure can accordingly 2
re ect some internal structure of the possible worlds themselves. A numbers of recent papers contain particular studies of such \structured" Kripke frames. For instance in many-dimensional modal logic Venema 1992 possible worlds have an internal structure of n-tuples; in some arrow logics Vakarelov 1992, Vakarelov 1996 possible worlds have a structure of the arrows of a multigraph; in the interval tense logic Venema 1990 they have a structure of intervals of some ordered set. We should also mention Orlowska's work Orlowska 1996 where relational framework for modalization of semigroups and various extensions is developed. The main precursor of the present paper is Goranko and Vakarelov 1998 where the method presented here is applied to axiomatize the modal logic of Boolean algebras, i.e. the class of frames (called there hyperboolean algebras) the worlds of which are elements of a Boolean algebra, and the frame is endowed with the usual Boolean operations. The structure of the paper is as follows. In section 1 we introduce structures and complex -algebras, as well as the syntax and semantics of the corresponding multimodal languages. We also discuss the role of the di erence operator and de ne a translation of the universal rst-order formulae on -structures to identities in complex -algebras. Section 2 introduces an axiomatic system for the minimal multimodal logic of the class of all -structures and presents a detailed proof of its completeness. Section 3 considers modal logics for classes of -structures. It gives a uniform axiomatization, by means of additional axioms, of the modal logic of any universal rst-order class of -structures, which is then extended to a uniform axiomatization, by means of additional rules, of the modal logic of any universal-existential class. We also brie y discuss second-order de nability on structures and second-order quanti cation over substructures or congruences, de nable in the modal languages and their expressiveness as universal fragments of monadic second-order languages. The paper ends with some concluding remarks and questions arising from the present study. The reader is assumed to have a technical knowledge on modal logics and some background in abstract algebra. 1
Preliminaries
1.1 -structures and complex -algebras. Let be an arbitrary signature. For notational convenience we shall assume that it consists of nitely many functional, constant and predicate symbols resp. f1; : : : fk ; c1 ; : : : cl ; r1; : : : rm , and let be the arity function.
3
A -structure W is any non-empty set with designated functions, constants and relations corresponding to the symbols in : W
= hW ; F1; : : : Fk ; C1; : : : Cl ; R1 ; : : : Rm i
With every signature we associate an algebraic (purely functional) signature 3 which extends the signature of Boolean algebras, containing the usual Boolean symbols ?; :; ^ (and >; _; ); , accordingly de nable), with a set of operators fhf1i; : : : hfk i; hc1i; : : : hcl i; hr1i; : : : ; hrm ig of arities respectively (fi ); 0, and (ri ) 01 corresponding to the functional, constant, and predicate symbols. Further, with every -structure W we associate a 3 -structure called here its complex -algebra: P
(W ) = hP (W ); ;; 0; \; hF1i; : : : hFk i; hC1 i; : : : hCl i; hR1 i; : : : hRm ii
which is the Boolean algebra of sets over W endowed with the corresponding power operations, respectively de ned as follows: hFi i( X1 ; : : : ; Xn ) = fFi (x1 ; : : : ; xn )jx1 2 X 1 ; : : : ; xn 2 Xn g, hCi i = fCi g, and hRi i(X1 ; : : : ; Xn) = fx 2 W j(9 x1 2 X1 ) : : : (9xn 2 X n )( Ri (x; x1 ; : : : ; xn ))g. In particular, for unary relational symbols, hRi i = fx 2 W jRi (x)g. For every non-constant symbol s in we introduce a \box" operator, dual to the \diamond" operator: [s](A1 ; : : : ; An ) = :hsi(:A1 ; : : : ; :An). Given a complex algebra P (W ) we further extend it to a di erentiated complex algebra by adding to its signature a di erence operator h6=i de ned as follows: h6=iX = def fx 2 W j9y 2 X and x = 6 y g. We denote its dual by [= 6 ]. Note that di erentiated complex algebras are Boolean algebras with (normal and additive) operators in the sense of Jonsson and Tarski (Jonsson and Tarski 1951). It turns out that for a large family of important structures the di erence operator is de nable by means of the other operators in the complex algebras. Two important general cases are given below.
Implicative lattices and expansions.
As shown in Goranko and Vakarelov 1998, the di erence operator can be de ned in complex Boolean algebras (called there \hyperboolean algebras") as follows: A= 4
:A ^ ((A < 1 >) >)) _ A A) ^ : < 1 >) < 1 >) >)
(
((((
This de nition obviously works in all implicative lattices, hence all pseudo-Boolean (Heyting) algebras, as well as all structures expanding pseudo-Boolean or Boolean algebras with additional operators, such as Post algebras, modal, relation, cylindric algebras, etc.
Groups and expansions.
The di erence operator can be de ned for complexes of groups, too:1
h6=iA = (:Ahi(AhiAh i )) _ (((AhiAh i ) ^ :hei)hi>: 0
where
0
hi; h0i, and hei are the power operators of the group operations
and identity. Thus,
h6=i
is de nable in all expansions of groups, such as ordered
groups, rings, modules, vector spaces, (ordered) elds etc. On the other hand, the di erence operator seems not de nable in other important classes of structures such as lattices (even distributive lattices with top and bottom) and semigroups (even monoids). Henceforth, we shall assume that the di erence operator is part of the signature of any complex algebra, unless otherwise indicated.
1.2 Modal -languages and -frames. Syntax and semantics of multimodal -logics.
M VAR = fp1; p2; :: :g of propositional variables, the usual Boolean symbols ?; :; ^ (and >; _; ); , accordingly de nable) and a set of modalities fh6=i ; hf 1 i; :: : hf k i; hc1 i; :: : hcl i, hr1 i; :: :; h rm ig corresponding to the algebraic operators of 3 . For technical convenience we identify the language M with the algebraic language for complex -algebras Every signature
determines a propositional multi-modal language
containing a denumerable set
and use the same symbols in both languages. The notion of a formula of
M is de ned as usual:
besides the classical formulae, hci is a formula for c, and if A1 ;: ::; An are formulae and h i is an nary modality, then h i(A 1 ; :: :;A n ) is a formula, too. [ ](A1 ;: :: ;An ) is an abbreviation of:h i(:A1 ;: :: ; :An ) as usual. We adopt the standard omission of parentheses. The set of formulae of M will be denoted by every constant symbol
FOR . 1
A similar formula was independently proposed by Yde Venema 5
According to the notational convention above, every modal formula of
M can also be regarded as an algebraic term for complex -algebras. Complex -algebras provide an algebraic semantics for the language M in the usual way: by a valuation in a complex -algebra P (W ) we mean any function v from VAR into P (W ). Each valuation v is then extended to arbitrary formulae by a straightforward induction:
v(?) = ;, v(A ^ B ) = v(A) \ v(B ) v(:A) = W 0 v (A) v(h6=iA) = h6=i(v (A)), v(hsi i) = hSi i for any constant or unary relational symbol si , v(hfi i(A1 ;: :: ;An )) = hFi i(v (A1);: ::; v(An)) for any n-ary functional symbol f i , v(hri i(A1 ; ::: ;An)) = hRi i(v (A1 ); :: :;v (An )) for any n+1-ary predicate symbol ri . A formula A is valid in the complex algebra P (W ) if v ( A) = W for any valuation v . Given a class C of -structures, by C3 we denote the class of complex algebras of the structures from A formula of from
M
C. C if it is valid in every complex algebra
is valid in
C . A formula of M is valid if it is valid in every complex -algebra. 3
The semantics given above can be regarded as a Kripke-style semantics
-structures by considering every complex -algebra as a Kripke frame, called here a -frame: on
FW = hW ; 6=;GF ; :: :GFk ; fC1g; :: :; fCl g;R1; :: :Rm i 1
where
GF
is the graph of the function
argument rst:
GF
=
F , though taken with the last
f(F (x1 ;: :: ;xn );x1; :: :; xn )jx1 ; ::: ;xn 2 W g
(just
in order to comply with the customary notation in modal logic).
Then the truth de nitions above become standard truth de nitions of
n-ary modalities. They x j=v A: \the formula A is true at x under the valuation v ", de ned as follows: x j=v A i x 2 v (A). This relation can be de ned independently by induction on A as usual. A pair (FW ;v ) for a -structure W and a valuation v in W is called a model (over W ). A formula A is valid in a model (FW ;v ) if for any x 2 W , x j=v A. A is valid in FW if it is valid in all models over W . Clearly, any formula is valid in a -frame FW if and only if it is valid in the complex algebra of W . the Kripke semantics respectively generalized for
can be reformulated in terms of the satisfaction relation
6
Finally, a formula is algebra.
valid in a -structure W if it is valid in its complex
Hereafter, an arbitrary nitary signature indicated.
1.3
is xed, unless otherwise
Using the di erence operator
The presence of the di erence operator in the complex algebras of a class of structures renders the modal language very expressive (see e.g. Gargov and Goranko 1993, de Rijke 1992, de Rijke 1993).
Here we shall
only mention some algebraic consequences which are of importance for the axiomatization of the corresponding modal logic.
h6 i
The following are de nable by means of the di erence operator = :
the universal modalities:
6 ]X , U ]X =def :hU i:X = X ^ [= and the \Only" operator: O X =def X ^ [6=]:X .
hU iX =def X _ h6=iX
and [
Remark 1.1 Taking the universal modality and the primitives we can de ne the di erence operator:
Only operator as
h6=iX =def hU iX ^ :O X .
This makes possible to base our investigations not on di erence but on the universal modality and the Only operator. In some sense this will be more natural, because in the applications of di erence we mainly use the universal modality and the Only operator. We prefer however to preserve the tradition, because di erence is better known and more popular in the area of modal logic.
Remark 1.2 Using the universal modality we can de ne a discriminator term t(x; y;z ) in the complex algebras:
t(x; y;z ) = ([U ](x , y ) ^ z ) _ (:[U ](x , y ) ^ x): The existence of such a term has a signi cant impact on the algebraic properties of the complex algebras and the variety generated by them (see e.g.
Burris and Sankappanavar 1981).
In particular, it allows for
reduction of all universal formulae of the language of complex to identities.
-algebras
U ] and the operator O we can of all universal formulae of the rst-order language L of -structures into identities in the rst-order language L3 of their complex -algebras. Moreover, using the universal modality [
construct a uniform translation
7
For simplicity and further applications we assume that the atomic for-
L are the following: x = xj , x = ci , x = fi (x1 ;:: :; xn), ri (x;x1 ;:: :; xn), where x; x1; ::: ;xn are individual variables of L . We
mulae of
may do that without loss of generality since we consider languages with
equality, hence every term can be \unnested" into a conjunction of such atomic formulae. (That assumption, however, is not essential for the existence of the translation.) Also, for notational convenience we use the same
L and L3 . The translation is de ned as follows (x; x1 ;: :: ;xn are
variables for the languages
De nition 1.3
variables): For atomic formulae: (x = xj ) = hU i(Ox ^ Oxj ), (x = ci ) = hU i(Ox ^ hci i) for any constant symbol ci , (x = fi(x1 ;: :: ;xn)) = hU i(O x ^ hfi i(Ox1 ;: ::; Oxn)) for any n-ary functional symbol fi , (ri (x;x1; :: :; xn ) = hU i(Ox ^ hri i(O x; Ox1; :: :; O xn )) for any relational symbol ri . For open formulae: (:') = : ('), (' ^ ) = (') ^ ( ). For universal formulae: Let = (8x1 ) : :: (8xn )'(x1 ;: :: ;xn) where '(x1; ::: ;xn ) is an open formula. Then ( ) = hU i(O x1 ) ^ : :: ^ hU i(Oxn ) ) ('(x1 ;: :: ;xn )). Lemma 1.4 (Modal de nability lemma for universal classes) A closed universal formula is valid in a -structure W i the corresponding identity ( ) = > holds in P(W ). Proof. Straightforward. |
2 The minimal modal logic for the class of all -structures. -logics. ML consisting of the valid formulae of the class of all complex algebras of -structures for any signature , and prove completeness of that axiomatic system.
In this section we axiomatize the minimal modal logic
8
2.1 Axiomatic system for
2.1.1 Axioms:
ML .
I. Enough propositional tautologies. II. Axioms of the minimal multimodal logic:
K [ ]) [ ](A1 ;: ::; Ai01; B ) C; Ai+1 ;: :: ;An ) ) ([ ](A 1 ; :: :;A i01 ;B;Ai+1 ; :: :; An ) ) [ ](A1 ;: ::; Ai01 ; C; Ai+1 ;:: :; An )),
(
and
h i(A1 ;: :: ;An ) , :[ ](:A1 ;: ::; :An) n-ary operator [ ] and i 2 f1;: ::; ng. III. Axioms for [= 6 ]: (D1) A _ [= 6 ]:[= 6 ]A, (D2) h6=ih6= iA ) ( A _ h6=iA ), (Dual)
for every
IV. Axioms for [U]:
U ]A ) [ ](A1 ;: ::; Ai01; A;Ai+1; :: :; An),
(U)
[
V. Axioms for the constant and functional modalities: (c) For every constant symbol
hU iOhci, hci ) Ohci,
(c1) (c2) (f)
f
For every
( 1)
f
( 2)
n-ary functional symbol f from ,
hU iOA1 ^ :: : ^ hU iOAn ) ) hU ihf i(OA1 ;: ::; OAn )
(
and
c from ,
hf i(OA1 ;: :: ; OAn) ) Ohf i(O A1; :: :; OAn).
2.1.2 Rules of inference:
Uniform substitution (SUB): where
sub(A)
A sub(A)
is the result of application of any uniform substitution of
formulae for variables in
Modus Ponens (MP):
A.
Necessitation for [U ] (N[U]):
A; A ) B B A UA
[ ]
Witness rule schema: 9
(WIT):
A ) ([ ](A1; :: :;A 01 ; O p ) B;A +1 ; ::: ;A )); for all p 2 VAR A ) ([ ](A1 ;: ::; A 01; B; A +1 ;: :: ;A )) for any n-ary modality [ ] and i 2 f1; ::: ;ng. n
i
i
i
i
n
Let us note that, as usual, the Witness rules can be replaced by nitary rules assuming the proviso not for all
in A; B; A1 ;: ::; A
n.
p 2VAR but for some p not occurring
The Witness rule used here shares the idea of Gabbay's Irre exivity
rule in Gabbay 1981 and originates from Passy & Tinchev's COV rule (see Passy and Tinchev 1991, Gargov and Goranko 1993).
Such rules and
have been used in a number of recent publications, e.g.
Venema 1992,
Venema 1993, Gargov and Goranko 1993, de Rijke 1992, de Rijke 1993, Balbiani et al. 1997.
Remark 2.1 As we have mentioned earlier, the di erence operator h6=i
U O: < U > A ^ :OA. So, the logic MLC can be alternatively axiomatized using only the operators [U ] and O . Then, instead of the axioms for and its dual can be de ned by the universal modality [ ] and
h6 iA h6 i =
=
= (D1 and D2), we can use the following list of axiom schemes: (K[U])
U A ) B ) ) ([U ]A ) [U ]B ) U ]A ) A, A _ [U ]:[U ]A, [U ]A ) [U ][U ]A, [U ](A , B ) ^ O A ) O B , OA ) A, OA ) OOA, A^ < U > OA ) OA, < U > A ^ O(A _ B ) ) O A.
(S5[U])
O O O O O
(
1)
(
2)
(
3)
(
4)
(
5)
[ ]( [
6
The rules are as before, only for the rule (WIT) with the de nition of [=].
= [= 6 ] we use
For the proof of completeness of the axiomatic system for
ML
we need
some preparatory work, which will be done in the following subsections.
2.2 By a
-logics. The sublogic
ML=. 6
-logic we mean any set L of formulae closed under substitution of
propositional variables, containing the axioms and closed under the rules
-logic L is consistent if ? is not a theorem of L. ML is based on the axioms from the groups I-IV, Modus Ponens and Necessitation for [U ] (witness rules are dropped).
of the logic
ML
. A
An important sublogic of
10
It is the minimal normal polymodal logic with di erence modality and will be denoted by
ML6= . All the axioms of this logic are in a Sahlqvist form, so
it is complete in the semantics, de nable by its axioms. The semantics for the di erence modality now is not standard. If we denote the accessibility
h6 i R6= xR6= y ! yR6=x xR6=y yR6= z R x;x1; :: :; x
, we see that it satis es the following conditions:
relation for = by
,
and
If
If
(
n
x = z or xR=6 z . ) then xR x for each i = 1; :: :; n. then
U
i
U R then xR y i x = y or xR=6 y . It is easy to see that R is an equivalence relation containing R6= and hence [U ] is an S5 modality.
If we denote the accessibility relation of the modality [ ] by we have:
U
U
U
We will use this fact later on saying \by S5". The semantics of
O
now is also non-standard:
x j= OA i x j= A and (8y )(xR=6 y ) y j6 = A).
Later on we make use of the completeness theorem of
ML=6 with respect
to the above nonstandard semantics instead of making formal derivations
from the axioms. We will refer to that as: \by the nonstandard semantics of
ML6= ".
Lemma 2.2 The following rules are derivable in ML=6 :
(i) Necessitation for arbitrary modality [ ] (N [ ]): A [ ]A; (ii) Monotonicity for [ ] and h i: A)B [ ]( P1 ;: :: ;P 01 ; A;: ::; P ) ) [ ](P1 ; ::: ;P 01 ; B; :: :; P ); and A)B < > (P1 ;: ::; P 01 ;A; :: :; P ) )< > (P1; :: :; P 01 ;B;: ::; P ); (iiia) A ) [U ]B A)B (iiib) A)B A ) [U ]B i
n
i
n
n
i
i
11
n
Proof.
U N [U ]) and axiom IV.
(i) follows from the Necessitation for [ ] ( (ii) follows from (i) and axiom II. (iii) follows by S5.
|
Lemma 2.3 The following variations of the Witness rule are derivable in the logic ML . (i) Basic Witness rule (WIT0 ): Op ) A; for all p 2 VAR A (ii) Extended Witness rule (WIT1): A ) [U ](B ) ([ ](A1 ;:: :; A 01; Op ) C;: :: ;A ))); for all p 2 VAR A ) [U ](B ) ([ ](A1 ;: :: ;A 01 ;C; :: :; A ))) for any n-ary modality [ ] and i 2 f1; :: :; ng,
n
i
n
i
Proof. (i) Easily obtained from (WIT) by taking = U; i = 1;B = A, and A = >, using the Necessitation for [U ] and the ML -theorem [U ]A ) A.
(ii) Suppose
ML
for all
ML
for all
ML
`A) U B ) p2 ` hU iA ) B ) p2 ` hU iA ^ B ) [
A1; :: :; A 01 ; Op ) C;A +1 ;: :: ;A
](
([ ](
i
i
n )))
VAR. Then, by the rule 2.2(iiia),
A ;: :: ;A 01 ; Op ) C; A +1 ;:: :; A
(
([ ]( 1
i
i
n )))
VAR, hence
(
)
A ; ::: ;A 01 ; Op ) C; A +1 ;: ::; A
([ ]( 1
i
i
:
n )))
Then, by (WIT),
` hU iA ^ B )
([ ]( 1
` hU iA ) B )
([ ]( 1
ML so
ML
(
)
(
A ;: ::; A 01; C; A +1 ;:: :; A i
i
n )))
A ;: :: ;A 01 ;C; A +1 ;: :: ;A i
i
n )))
hence, by the rule 2.2(iiib),
|
ML
`A) U B) [
](
A ; :: :; A 01; C;A +1; :: :; A
([ ]( 1
12
i
i
:
n )))
Note that every instance of all three Witness rule schemata can be represented in a uniform way:
'(Op ) A);
for a suitable formula
'(Op ) A)='(A).
'(q).
for all
'(A)
p 2 VAR
We shall refer to such a representation as
Note also that the antecedent of any instance of (WIT) or (WIT1 ) can
be
>
, and therefore omitted.
In the next lemma we list some technical theorems of be of later use.
ML=6 , which will
Lemma 2.4 The following formulae are theorems of ML=6 :
(i) < U > (O p ^ A) ) [U ](O p ) A), (ii) < U > Op ) (< U > (Op ^ A) , [U ](Op ) A)), (iii) < U > (OA ^ OC )^ < U > (OB ^ OC ) )< U > (OA ^ OB ), (iv) < U > Op ) (< U > (Op ^:A) , : < U > (O p ^ A)), (v) < U > (Op ^ (A _ B )) , (< U > (Op ^ A)_ < U > (Op ^ B )), (vi) < U > (O p^ < > (Oq1 ^ A1 ;: :: ; Oq ^ A )) , < U > (Op^ < > (Oq1; :: :; Oq ))^ < U > (Oq1 ^ A1) ^ :: : ^ < U > ( O q ^ A ), (vii) < U > O p ) (< U > O p^h6=i(Oq ^A)) , : < U > (Op ^Oq)^ < U > (Oq ^ A))). Proof. Use the nonstandard semantics for ML=6 . The following is an important lemma for ML6= , which is based only on n
n
n
n
n
the properties of the universal modality and axiom IV.
Lemma 2.5 (Strong Replacement Theorem) Let '(p) be a formula in which the variable p has unique occurrence and let '(A) be the result of the replacement of p by A. Then the following formula is a theorem of ML6= : U A , B ) ) ('(A) , '(B )):
[ ](
Proof.
The proof can be done by induction on the complexity of
nontrivial case is not in
true.
P1 ;: ::; P
n
'(p) = [ ](P1 ;: ::; P 01 ; (p); P +1 ;: :: ;P i
i
'(p).
The
n ), where
and by the induction hypothesis the assertion for
p is p
( ) is
The following is a sketch of the proof of this case. For notational con-
U ](A , B ) ) ( (A) , (B )) { by the induction hypothesis,
venience we will assume that [ ] is a one-place modality. 1. [
13
U U A , B ) ) [U ]( (A) , (B )) { from 1 by the monotonicity
2. [ ][ ](
U
of [ ],
U ](A , B ) ) [U ]( (A) , (B )) { from 2 by S5, U ]( (A) , (B )) ) [ ](( (A) , (B )) { by axiom IV. 5. [ ](( (A) , (B )) ) ([ ] (A) , [ ] (B )) { by the minimal modal logic K , 6. [U ](A , B )) ) ([ ] (A ) , [ ] (B )) { by the propositional logic 3. [
4. [
from 3, 4 and 5.
2.3
|
Theories in -logics
L to be a xed consistent -logic. theory in L if it satis es the following
Throughout this section we consider A set of formulae 0 is called a conditions: (t1) All theorems of
L are contained in 0,
(t2) 0 is closed under Modus Ponens, i.e., if
A;A ) B 2 0 then B 2 0,
(t3) 0 is closed under the rules (WIT), (WIT0 ) and (WIT1 ) i.e., whenever all premises of an instance of any of these rules are in 0 then its conclusion is in 0.
L is a theory of L. consistent if ? 2= 0. 0 is said to be a maximal theory if it is consistent and for all A: either A 2 0 or :A 2 0. Obviously the set of all theorems of
A theory 0 is
Note that every maximal theory is consistent.
We will use without explicit mentioning the following properties of a maximal theory 0:
:A 2 A^B 2 A_B 2
A 2= 0, 0 i A 2 0 and B 2 0, 0 i A 2 0 or B 2 0. Let 0 be a set of formulae and A be a formula. De ne: 0 + A = fB 2 FOR jA ) B 2 0g Lemma 2.6 (Deduction lemma for theories) Let 0 be a theory in L. Then: (i) 0 + A is the smallest theory, containing 0 and A, (ii) 0 + A is consistent i :A 2= 0. 0 i
14
Proof.
Standard. Note that
A ) '(Op ) B )=A ) '(B ) is tautologically '(Op ) B )='(B ) is
equivalent to an instance of a Witness rule whenever such an instance.
|
Lemma 2.7 (Lindenbaum Lemma)
(i) Any consistent theory 0 can be extended to a maximal theory 1. (ii) If 0 is a theory and A 2= 0 then there exists a maximal theory 1 such that 0 1 and A 2= 1. (iii) If A is not a theorem of L then there exists a maximal theory 0 such that A 2= 0. Proof. (i). Let A0; A1 ;: :: be an enumeration of all formulae. We de ne inductively a sequence of consistent theories 00 ; 01 ;: :: in the following way. De ne 00 = 0 and suppose that 00 ; :: :; 0 are de ned and consistent. n
For 0n+1 we consider two cases.
Case 1: If 0 Case 2: Let
+
n
A
n
0n +
is consistent, then put 0n+1 = 0n +
A
n
be inconsistent.
there are nitely many representations of Witness rules
'1 (B1 );: :: ;' (B k
k ).
A
Then n
:A 2 n
A
n.
0n .
Note that
as a conclusion of any of the
;: :: ; 0 ;: :: ; i < k
We de ne the nite sequence 00 n
0 inductively as follows. Let 00 n = 0n and suppose that 0n
0in ,
k n
p such that 0 + :' (Op ) B ) is consistent. For, suppose the contrary, i.e. for any p, 0 + :' (Op ) B ) is inconsistent. Then ' (Op ) B ) 2 0 for any variable p and by the corresponding Witness rule we obtain that ' (B ) = A 2 0 . But :A also belongs to 0 , which implies that 0 is inconsistent | a contradiction. Let p be the rst variable such that 0 + :' (O p ) B ) is consistent. Then de ne 0 +1 = 0 + :' (O p ) B ). are de ned. Then there exists a propositional variable i
i
i
i
i n
i n
n
i
Finally 0n+1 = 0k n.
; ;: ::.
00 01
Now put 1 =
S1
n=0
i n
i n
i
i
i
i
i n
n
i
i
i n
i
i
That completes the de nition of the sequence
0n .
It is straightforward to show that 1 is a
maximal theory containing 0. Conditions (ii) and (iii) follow from (i).
W
i n
i
i
i n i n
|
L. Lemma 2.8 (Witness Lemma) For any 0 2 W there exists a variable p such that Op 2 0. Proof. Suppose, for the sake of contradiction, that 0 2 W and for any variable p we have O p 2 = 0. Then, by the maximality of 0 for any variable p we have that Op ) ? 2 0 and hence, by the Basic Witness rule, ? 2 0 We denote by
L
the set of all maximal theories of L
L
| a contradiction.
|
15
2.4 Canonical models for -logics Throughout this section we will assume that L is a xed consistent -logic and 0 is a maximal theory of L. Following an idea very similar to the clas-
sical Henkin's construction of canonical models in rst-order logic we will construct a canonical -structure (W 0; f10; : : : ; fk0 ; c01 ; : : : ; c0l ; r10 ; : : : ; rm0 ), and a canonical valuation on it, related to 0. First we de ne 60 = fp 2 V ARj < U > Op 2 0g. 6 ;. We de ne the following relation in By the witness lemma 60 = 60 : p q i < U > (O p ^ Oq) 2 0. It is easy to see that is a re exive and symmetric relation in 60. By 2.4(iii) we obtain that is also a transitive relation. Hence is an equivalence relation in 60 . Then de ne: jpj = fq 2 60 jp qg, W 0 = fjpj j p qg. 2.4.1 De nition of relations:
Let r be a relational symbol of arity n + 1 from the signature . For jpj; jq1j; : : : ; jqn j 2 W 0 de ne r0(jpj; jq1j; : : : ; jqn j) i < U > (Op^ < r > (Oq1 ; : : : ; Oqn)) 2 0. The correctness of this de nition follows from the next lemma. Lemma 2.9 Let r be an n + 1-place relational symbol from the signature and let for i = 1; : : : ; n the following hold: 1. 2.
p; p ; qi ; qi 2 60 , p p , qi q i , 0
0
0
0
and 3.
< U > (O p^ < r > (Oq1; : : : ; O qn )) 2 0.
Then
< U > (Op ^ < r > (Oq1; : : : ; Oqn ) 2 0. 0
0
0
Suppose that the conditions (1){(3) are ful lled. Then by (1) and (2) we obtain: < U > (Op ^ Op ) 2 0, < U > (Oqi ^ Oqi) 2 0, i = 1; : : : ; n. Then, applying 2.4(i), we obtain [U ](Op , O p ) 2 0, [U ](O qi , O qi ) 2 0, i = 1; : : : ; n. From this, applying n + 1 times the strong replacement lemma to (3), we obtain the result. |
Proof.
0
0
0
0
16
2.4.2 De nition of functions:
Let f be a functional symbol of arity n. Before de ning the function f 0 we de ne an (n + 1)-ary relation Rf as in the de nition of relations. Namely, for jpj; jq1 j; : : : ; jqnj 2 W 0 we put Rf ((jpj; jq1j; : : : ; jqn j) i < U > (Op^ < f > (Oq1; : : : ; O qn )) 2 0:
The correctness of this de nition follows from 2.9. In the next lemma we will show that Rf is a total functional relation with respect to jq1j; : : : ; jqn j. Let f be an n-ary functional symbol from the signature . Then for every jq1 j; : : : ; jqn j 2 W 0 there exists a unique jpj 2 W 0 such that Rf (jpj; jq1 j; : : : ; jqn j).
Lemma 2.10
(Existence) Let jq1 j; : : : ; jqnj 2 W 0 . Then < U > Oq1 ; : : : ; < U > O qn 2 0. From this and axiom (f 1) we obtain (1) < U >< f > (Oq1; : : : ; O qn ) 2 0. Now we will show that there exists p 2 V AR such that (2) < U > (O p^ < f > (Oq1 ; : : : ; O qn )) 2 0. Suppose the contrary, namely that for every p 2 V AR we have < U > (Op^ < f > (Oq1 ; : : : ; Oqn)) 2= 0. From this, by the maximality of 0 we get that for every p 2 V AR we have [U ](O p ) : < f > (Oq1; : : : ; qn )) 2 0. Then by the Extended Witness rule for 0 we get [U ](: < f > (Oq1 ; : : : ; qn) 2 0 and from here : < U >< f > (Oq1 ; : : : ; qn ) 2 0, which contradicts (1). Using monotonicity of < U > it follows from (2) that < U > Op 2 0, jpj 2 W 0 and Rf (jpj; jq1j; : : : ; jqn j) | the existence part is proved. (Uniqueness) Suppose Rf (jpj; jq1 j; : : : ; jqnj) and Rf (jp j; jq1j; : : : ; jqn j). By the de nition of Rf we obtain (3) < U > (Op^ < f > (Oq1 ; : : : ; Oqn )) 2 0 and (4) < U > (Op ^ < f > (Oq1 ; : : : ; Oqn)) 2 0 Applying the monotonicity of < U > and axiom (f 2) to (3) and (4) we obtain (3') < U > (Op ^ O < f > (Oq1; : : : ; O qn )) 2 0 and (4') < U > (Op ^ O < f > (O q1 ; : : : ; Oqn)) 2 0. Now from (3'), (4') and Thm 2.4(iii) we obtain < U > (Op ^ Op ) 2 0, which shows that p p and consequently jpj = jp j. | Now the de nition of f 0 is the following: we put jpj = f 0(jq1j; : : : ; jqn j) i Rf (jpj; jq1 j; : : : ; jqnj).
Proof.
0
0
0
0
0
0
17
2.4.3 De nition of constants
Let c be a constant symbol from the signature . If we treat c as zero-place function we can repeat the construction for the functions given above. We will only formulate the relevant lemma. Lemma 2.11 Let c be a constant symbol from the signature . Then there exists unique jpj 2 W 0 such that < U > (Op^ < c >) 2 0.
Similar to the proof of 2.10, using axioms (c1) and (c2). | Now we put c0 = jpj, where jpj is the unique element of W 0 from 2.11. We have de ned the canonical -structure. To de ne the canonical model it remains to de ne the canonical valuation v 0 .
Proof.
2.4.4 De nition of the canonical valuation:
For arbitrary q 2 V AR we put v0 (q) = fjpj 2 W 0 j < U > (Op ^ q) 2 0g. The correctness of this de nition is obtained from the following lemma. Lemma 2.12 1. 2.
Let
p p and < U > (O p ^ q) 2 0. 0
Then
< U > (Op
0
^ q) 2 0.
From (1) we obtain < U > (Op ^ Op ) 2 0. Applying 2.4(i) we obtain [U ](O p , Op ) 2 0. Now from this, (2), and strong replacement lemma we obtain the result.
Proof.
0
0
|
2.5 The Truth Lemma
The following theorem, often called \truth lemma", states that the canonical valuation has the same form for arbitrary formulae. (Truth Lemma)
Let v = v0 be the canonical valuation. The following equivalence is true for any formula Q and jpj 2 W 0 : jpj 2 v (Q) i < U > (Op ^ Q) 2 0
Theorem 2.13
Proof.
The proof will be done by induction on the complexity of Q. 18
Q 2 V AR. This case is true by the de nition of v 0. Suppose Q has one of the forms :A, A _B , h6=iA, < r > (A1; : : : ; An), < f > (A1 ; : : : ; An ), < c > and for A; B; A1 ; : : : ; An the assertion is
true (induction hypothesis, IH). Q = :A: jpj 2 v (:A) i jpj 2 = v(A) i (by IH) < U > (Op ^ A) 2= 0 i (by the maximality of 0) : < U > (Op ^ A) 2 0 i (by 2.4(iv)) < U > (O p ^ :A) 2 0. Q = A _ B: jpj 2 v(A _ B ) i jpj 2 v(A) or jpj 2 v (B ) i (by IH) < U > (Op ^ A) 2 0 or < U > (Op ^ B ) 2 0 i (by the maximality of 0) < U > (Op ^ A)_ < U > (Op ^ B ) 2 0 i (by 2.4(v)) < U > (Op ^ (A _ B )) 2 0. Q =< r > (A1 ; : : : ; An ); Q =< f > (A1; : : : ; An); Q =< c >; Q = h6=iA. The proofs of these cases follow from the next four lemmas.
Lemma 2.14 (i) (ii)
The following are equivalent:
< U > (O p^ < > (A1; : : : ; Ai 1 ; Ai; Ai+1 ; : : : ; An )) 2 0, There exists qi 2 V AR such that < U > (O p^ < > (A1; : : : ; Ai 1 ; (Oqi ^ Ai ); Ai+1 ; : : : ; An )) 2 0. There exist jqi j 2 W 0 , i = 1; : : : ; n such that < U > (O p^ < > (Oq1 ^ A1 ; : : : ; Oqn ^ An)) 2 0, (For the case = r.) There exist jqij 2 W 0, i = 1; : : : ; n such that r0 (jpj; jq1 j; : : : ; jqnj) and < U > (Op ^ Ai ) 2 0 for each i = 1; : : : ; n. 0
0
(iii)
(iv)
(i) ! (ii): Suppose (i) and, for the sake of contradiction, that (ii) is not ful lled. Then by the maximality of 0 we obtain that for any qi 2 V AR the following is true: [U ](Op ) [ ](:A1 ; : : : ; :Ai 1 ; Oqi ) :Ai ; :Ai+1; : : : ; :An )) 2 0. From this, by (WIT1) we obtain [U ](Op ) [ ](:A1 ; : : : ; :Ai 1 ; :Ai ; :Ai+1; : : : ; :An )) 2 0 which is equivalent to : < U > (Op^ < > (A1 ; : : : ; Ai 1 ; Ai ; Ai+1 ; : : : ; An )) 2 0 which contradicts (i). Let us note that qi , the existence of which is claimed in (ii), belongs to 60 . This follows from the monotonicity of < U > and < >, and axiom (U ).
Proof.
0
0
0
19
(ii) ! (i): This implication follows by the monotonicity of < U > and < >. Thus we have obtained the equivalence (i) $ (ii). Using the above note and applying this equivalence n-times for i = 1; : : : ; n we obtain the equivalence (i) $ (iii). The equivalence (iii) $ (iv ) follows by 2.4(v) and the de nition of r0.
|
Lemma 2.15
The following are equivalent:
< U > (Op^ < f > (A1 ; : : : ; An )) 2 0, there exist jqij 2 W 0, i = 1; : : : ; n, such that jpj = f 0(jq1j; : : : ; jqn ) and < U > (O p ^ Ai ) 2 0 for any i = 1; : : : ; n. (i) (ii)
Proof.
The proof is similar to that of 2.14 by using the de nition of f 0 . As a particular case we obtain:
Lemma 2.16 (i) (ii)
|
The following conditions are equivalent:
< U > (Op^ < c >) 2 0, jpj = c0.
Lemma 2.17
The following conditions are equivalent: (i) < U > (O p ^ h6=iA) 2 0, (ii) There exists jqj 2 W 0 such that jpj = 6 jqj and < U
> (O q ^ A) 2 0.
By 2.14 condition (i) is equivalent to the following one: (i') There exists q 2 V AR such that < U > (Op ^ h6=i(Oq ^ A)) 2 0. For this q we easily obtain that jqj 2 W 0 , hence, by 2.4(vii), (i') is equivalent to: (i") There exists jqj 2 W 0 such that : < U > (Op ^ Oq)^ < U > (Oq ^ A) 2 0. Finally, (i") is obviously equivalent to (ii). | The proof of the Truth Lemma is completed. |
Proof.
2.6
The canonical model lemma
(Canonical Model Lemma) Let L be an arbitrary consistent -logic. Then the following conditions are equivalent for any formula A: (i) A is a theorem of L, (ii) A is true in all canonical models of L.
Lemma 2.18
(i) ! (ii) Suppose that A is a theorem of L and, for the sake of contradiction, that A is not true in some canonical model of L determined by some maximal theory 0. Then there is a propositional variable p such
Proof.
20
that jpj 2 W 0 and that jpj 2= v0 (A). So jpj 2 v 0(:A). By the Truth Lemma (2.13) we obtain that < U > (Op ^ :A) 2 0. By the monotonicity of < U > we get that < U > :A 2 0 and hence that :[U ]A 2 0. But since A is a theorem of L then [U ]A is a theorem of L, too, and hence [U ]A 2 0 | a contradiction. (ii) ! (i). Suppose that A is not a theorem of L. Then by the Lindenbaum Lemma (2.7(iii)) there exists a maximal theory 0 such that A 2= 0, so :A 2 0. We shall show that A is falsi ed in the canonical model determined by 0. By the Witness Lemma there is a propositional variable p such that Op 2 0. Then Op ^ :A 2 0 and by S5 that < U > (Op ^ :A) 2 0. By the Truth Lemma we obtain that jpj 2 v 0(:A), hence jpj 2= v 0(A), which shows that A is not true in the canonical model determined by 0. |
ML
. 2.7 Soundness and completeness of Theorem 2.19 The logic ML is sound with respect to its algebraic and Kripke-style semantics.
It is straightforward to check that all axioms are valid and that all rules preserve validity. (For the soundness of the Witness rule, see similar results e.g. in Passy and Tinchev 1991, Gargov and Goranko 1993, Goranko 1998.) |
Proof.
Theorem 2.20
The following are equivalent for any formula (i) A is a theorem of ML . (ii) A is valid in all -structures. (iii) A is valid in all complex -algebras.
A in ML :
(i) ! (ii): | this is the soundness theorem for ML . (ii) ! (i): Suppose A is true in all -structures. Then A is true in all models over all -structures and consequently A is true in all canonical models of ML . Then by the Canonical Model Lemma A is a theorem of ML . Finally, (ii) and (iii) are equivalent by de nition. |
Proof.
3 Modal logics for classes of -structures. In this section we introduce uniform methods for complete axiomatization of the modal logic MLC consisting of the valid formulae of a class C of -structures for a large family of elementary classes.
21
3.1 Canonical de nability
Theorem 3.1
(Canonical de nability of universal formulae) Let 0 be a maximal theory in a -logic L and let W 0 be the canonical structure determined by 0. Let '(x0 ; : : : ; xn) be an open formula in the rst-order language of the signature and be the translation de ned by 1.3. Then: (i) For any jp0j; : : : ; jpn j 2 W 0 , '(jp0 j; : : : ; jpn j) holds in W 0 i ('(p0 ; : : : ; pn)) 2 0. (ii) Let be the closed formula (8x0 ) : : : (8xn )'(x0 ; : : : ; xn). Then ( ) is a theorem of L i holds in all canonical structures of L.
Proof.
(i) We will proceed by induction on the complexity of The case for atomic formulae:
'(x0 ; : : : ; xn).
If '( x0 ; xj ) is x0 = xj then: p0 j = jpj j i p0 pj i < U > (Op0 ^ Opj ) 2 0 i (p0 = pj ) 2 0. If '( x0 ; x1 ; : : : ; xn ) is r (x0 ; x1 ; : : : ; xn ) then: r0(jp0j; jp1j; : : : ; jpnj) is true in W 0 i < U > (Op0 ^ < r > (Op1 ; : : : ; Opn ) 2 0 i (r(p0; p1 ; : : : ; pn )) 2 0. '(x0; x1; : : : ; xn ) is x0 = f (x1; : : : ; xn ). Then we have: 0 jp0 j = f (jp1 j; : : : ; jpn j) i < U > (O p^ < f > (O p1 ; : : : ; O pn )) 2 0 (p0 = f (p1 ; : : : ; pn)) 2 0. '(x0) is x0 = c. Then we have: 0 i < U > (Op^ < c >) 2 0 i (p = c) 2 0. jp0 j = c 0 j
The case of arbitrary open formulae:
'(x0; : : : ; xn ) has a form : (x0; : : : ; xn ) or
(x0 ; : : : ; xn ) _ (x0 ; : : : ; xn )
and by induction hypothesis the assertion is true for are standard. (ii) Recall that (!) Suppose that
(
(
i
and
.
Both cases
< U > Op0 ^ : : : ^ < U > Opn ) '(p0; : : : ; pn). ) is a theorem of L but, for the sake of contradiction, ) is
is not true in some canonical structure over some maximal theory
0. Then for some
p0 j; : : : ; jpn j
j
2
W0
not hold. By (i) we obtain
we have that
'(jp0 j; : : : ; jpn j)
does
('(p0 ; : : : ; pn )) 2= 0. 0 we obtain jp0 j; : : : ; jpn j 2 W (2) < U > Opi 2 0, i = 0; : : : ; n. But (1) and (2) imply that ( ) 2 = 0, which contradicts the assumption that ( ) is a theorem of L. ( ) Suppose that ( ) is not theorem of L. We shall show that is not true in some canonical structure of L. Since ( ) is not theorem of L then by the Lindenbaum Lemma (2.7(iii)) there exists a maximal theory 0 such that ( ) 2 = 0. From here we obtain (1)
From
22
< U > Opi 2 0, i = 0; : : : ; n, and ('(p0 ; : : : ; pn ) 2= 0.
(3) (4)
From (3) we get
p0j; : : : ; jpn j 2 W 0
(5)
j
and from (4) we obtain
'(jp0 j; : : : ; jpn j) is not true in the canonical structure determined
(6) by 0.
From (5) and (6) we obtain that
of
L.
is not true in all canonical models
|
3.2
The modal logic of a universal class of structures.
Given a universal class
C of -structures, axiomatized by a set of univerM LC of all valid formulae in C by
sal sentences 1, we obtain the logic extending (
ML
with the following additional group of axioms:
Axioms for the class C3 :
3)
(
),
for every formula
from 1.
Theorem 3.2 For any universal class C the following are equivalent for any formula A of M : (i) A is a theorem of M LC , (ii) A is valid in all -structures of the class C. (ii) A is valid in all complex -algebras from C3 . Proof. (i) ! (ii):
this is the soundness theorem for
the soundness theorem for (ii)
!
and lemma 1.4.
M LC
which follows from
(i ): Note that by the Canonical De nability Theorem, all canon-
ical structures of all
M L
-structures
MLC
are in the class
from the class
C.
C.
Then
Suppose now that
A
A
is true in
is true in all models over all
-structures from the class C and consequently A is true in all canonical M LC . Then by the Canonical Model Lemma A is a theorem of M LC .
models of
Again, (ii) and (iii) are equivalent by de nition.
|
This theorem covers a large family of important classes of structures, such as:
implicative lattices, pseudo-Boolean algebras, Boolean algebras (axiomatized in Goranko and Vakarelov 1998 using a method which is generalized here), Post algebras, N-lattices, modal algebras, dynamic algebras, (representable) relation algebras, (representable) cylindric algebras, etc.
23
groups, abelian groups, groups of exponent
p,
torsion-free groups;
ordered groups, etc.
rings, commutative rings, integral domains, etc.
3.3 The modal logic of a
502 class of structures.
We shall extend the result above by providing a uniform complete axiomatization of any 50 2 class of structures, by means of additional
rules of the
type of Witness.
C of -structures axiomatized by a set of 502 sentences 1, we obtain the logic M L C of all valid formulae in C by extending M L Given a 50 2 class
with the following additional group of rules:
Rules for the class C3 :
For every formula
y 1 ; : : : ; ym
=
x y (x; y)
8 9
from 1, where x =
are disjoint strings of variables and
lowing rule to the axiomatic system ML :
x1; : : : ; xk ; y =
is open, we add the fol-
(WIT ):
n-ary modality [ ] and i 2 f1; : : : ; ng, if A ) [ ](A1 ; : : : ; Ai01 ; (hU iO y ^ ( (x; y))) ) B; Ai+1 ; : : : ; An ) for all y1 ; : : : ; y m 2 VAR, where hU iO y = hU iO y1 ^ : : : ^ hU iO ym , then A ) [ ](A1; : : : ; Ai01; hU iOx ) B; Ai+1 ; : : : ; An ) where hU iO x = h U iO x1 ^ : : : ^ hU iO xk . For any
`
Theorem 3.3
For any 502 class C the following are equivalent for any formula A of M : (i) A is a theorem of M LC , (ii) A is valid in all -structures of the class C. (ii) A is valid in all complex -algebras from C3 .
Proof.
We shall outline the major steps in the proof.
For more details of
similar results, see Goranko 1998. (i)
!
(ii): We only need to show the soundness of each of the rules
(WIT ). Let
=
x y ( x; y), hU iOx and hU iOy be as above
8 9
and suppose
M LC 6j= A ) ([ ](A1 ; : : : ; Ai01; (hU iOx) ) B; Ai+1 : : : ; An)): Then, for some structure S 6j=
S 2
C,
A ) ([ ](A1 ; : : : ; Ai01 ; (hU iO x) ) B; Ai+1 ; : : : ; An ));
i.e. for some valuation
v
and
a 2 S, 24
S
; v; a 6j= A ) ([ ](A1; : : : ; Ai01 ; (hU iOx) ) B; Ai+1; : : : ; An)):
; v; a j= A and S ; v; aj 6j= Aj , S ; v; ai j6 = (hU iOx) ) B for some such that R ( a; a1 ; : : : ; an ) where R is the relation in S corresponding to . Then S ; v; ai j= hU iO x and S ; v; ai j6 = B . Therefore, there are elements s1 ; : : : ; s k 2 S such that v ( x1 ) = fs 1 g; : : : ; v (xk ) = fs k g. For these s1; : : : ; sk there exist t1; : : : ; tm 2 S such that Then
S
a1 ; : : : ; a n
2 S
S j=
(s1 ; : : : ; sk ; t1 ; : : : ; tm ):
y1 ; : : : ; ym be variables not occurring in A ) ([ ](A1; : : : ; Ai01 ; (hU iOx) ) B; Ai+1; : : : ; An)) and v 0 be a valuation in S modifying v on y 1 ; : : : ; ym 0 ft1 g; : : : ; v (ym ) = ftm g. Then
Let
S
by
v 0(y1)
=
; v 0 j= (hU iOy ^ ( ( x; y)))
and S
; v 0 ; ai j6 = B
hence S
; v 0 ; a j6 = A ) ([ ](A1 ; : : : ; Ai01; (hU iOy ^ ( ( x; y))) ) B; : : : ; An )):
Therefore,
M LC 6j= A ) ([ ](A1 ; : : : ; Ai01; (hU iOy ^ ( ( x; y))) ) B; : : : ; An )); whence the soundness of the rule (WIT ). (ii)
!
(i): First, note that analogues of the Basic Witness and the
Extended Witness rules are likewise derivable for each of the rules (WIT ). We now modify the completeness proof for
ML by building the canon-
ical model from theories, additionally closed under all (basic, standard and extended versions of) rules (WIT ). All lemmas apply accordingly, hence, as in the previous theorem, it suces to show that all canonical
M LC are in C. We shall demonstrate that for every canonical structure W 0 and axiom = 8x 9y ( x; y) as above, W 0 j= . In0 deed, let jx1 j; : : : ; jxk j 2 W , hence hU iO x1 ; : : : ; hU iO xk 2 0, and suppose W 0 6j= (jx1 j; : : : ; jxk j; jq1 j; : : : ; jqm j) for any jq1 j; : : : ; jqm j 2 W 0. Then for any y1 ; : : : ; y m 2 VAR, hU iO y ^ ( ( x; y)) 2 6 0, hence ( hU iO y1 ^ : : : ^ h U iO ym ^ ( ( x; y))) ) ? 2 0. Then, by closedness under (WIT ), h U iO x ) ? 2 0, whence ? 2 0 | a contradiction. | structures of
25
3.4 Axioms vs rules. Since every universal formula is a 50 2 formula, we have a choice to axiomatize a universal class either by means of axioms or rules, and it is quite easy to derive the axiom from the corresponding rule.
Same choice exists for various modally de nable 50 2 properties. Here is
a representative sample of some algebraically important 50 2 classes which can be modally axiomatized either way.
Atomless Boolean algebras can be modally axiomatized by extending the axiomatic system for Boolean algebras with the additional axiom: (O A
U i(Ah^i> ^ :A ^ :h0i);
^ :h0i) ) h
Rings with division
can be modally axiomatized by extending the
axiomatic system for rings with the additional axiom: (O A
^ :h0i) )
[U ](Ah2i> ^ >h2iA)
Division rings are then axiomatized as ring with division without zero divisors (a universal condition).
Fields are modally axiomatized over integral domains with the additional axiom: (O A
^ :h0i) )
[U ](h1i
)
Ah2i>)
The modal logic of elds is easily extended further to elds of nite characteristics, or elds with characteristics 0, ordered elds, etc. Note however, that some modal formulae (such as
determines Church-Rosser's 50 2 property
32p
xyz 9t(Rxy
)
Rxz
23p which
! ( Ryt ^ Rzt))) although canonical in modal logics without rules, cease to be such in 8
^
the stronger sense relevant to logics with Witness type rules (where there are fewer maximal consistent sets), and hence added as axioms to such logics present a problem, while the corresponding rules co-operate smoothly. This suggests that using rules for axiomatization is generally preferable, at least what concerns proving completeness. Moreover, we argue that rules behave better than axioms in the formal derivations, too, and this will be discussed in more detail elsewhere.
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3.5 Some notes on the second-order expressiveness of the modal languages.
The modal languages for complex algebras, being essentially of a secondorder nature, naturally allow for universal monadic second-order quanti cation over important types of subsets of structures, such as substructures. For instance, universal quanti cation over subgroups can be effected by using a modally de nable predicate for a subgroup: s(X ) = h U i X ^ [U ](X h2iX h i ) X ), and then s(X ) ) 9( X ) is valid in a group i all subgroups of the group have the property 9. This pattern obviously generalizes to substructures of an arbitrary structure. Finally, the congruences in many classes of structures (such as Boolean algebras, groups, rings) etc. can be determined by speci c subsets (resp. lters, normal subgroups, ideals), which sometimes are modally de nable and thus enable universal quanti cation over congruences, hence over homomorphic images, too. For instance, a modally de nable predicate for a lter in a Boolean algebra is: f (X ) = [U ]((h1i ) X ) ^ (X ) :h0i) ^ (X h^iX ) X ) ^ (X h_i> ) X )), and for an ultra lter: u(X ) = f (X ) ^ [U ](h0iX , :X ). 0
4 Concluding remarks This study raises a number of interesting questions of logical or algebraic importance, which will be addressed in a subsequent paper. We shall brie y mention some. Logical issues. Traditional logical problems arise around the modal languages and logics for complex algebras. Some general questions are: Expressiveness. The modal languages, via their standard translation a la van Benthem, cover a fragment of the universal monadic second-order extension of the rst-order theory of the underlying class of structures. Can that fragment be characterized in a coherent model-theoretic fashion extending Goldblatt & Thomason's results from classical modal logic? Modal axiomatizability. The axiomatization results presented here can be further extended in various ways. Some of them are based on appropriate generalizations of Sahlqvist-type syntactic forms (see Venema 1993 for a discussion and some results) or of semantic persistence conditions on formulae (see Goranko 1998) which would ensure preservation of their validity from canonical structures (in the stronger sense used in this paper) to the underlying frames, i.e. their behavior like canonical formulae in modal logics without additional 27
rules, and hence the successful co-operation of such axioms with the Witness-type rules employed by the axiomatic systems. Others call upon extension of well developed model-theoretic techniques (such as ltration, unraveling etc.) for proving completeness of non-canonical modal logics to the general framework studied here. A number of more speci c logical questions arise regarding the axiomatizations proposed in this paper, one of them being: when is the Witness rule schema redundant, or at least replaceable by ( nitely many) axiom schemata? This question calls upon a general study of the proof theory of Witness-type rules. Another one is about axiomatizing the nite structures of a given class. For instance, on partially ordered structures, Grzegorczyk's formula
2(2(p ! 2p) ! p) ! p
where the modality 2 corresponds to the partial ordering, de nes the class of structures with no in nite descending chains, hence on linear orderings or some nite partial orderings in which a bound on the width can be established (such as Boolean algebras) it de nes the class of nite structures. The question arises when Grzegorczyk's formula added to the modal logic of a class of partially ordered structures axiomatizes the nite structures of that class. Decidability. When is the modal logic of the complexes of a class of structures decidable? This question is closely related to the previous one. In general, the (un)decidability of the rst-order theory does not imply (un)decidability of the modal logic as the two languages are incomparable, so a ner analysis of the expressiveness of the latter is neccessary in order to adapt known techniques such as e.g. tiling, or word problems. On the other hand, since the full monadic secondorder theory covers the modal language, decidability of the former implies decidability of the latter, so in particular, Rabin's theorem guarantees decidability of the modal logic of any class of structures interpretable in trees with at most countably many successors. Complexity. How does the complexity change when passing from the (universal) rst-order theory of structures to the modal logic of their complexes? The rst-order theory of complex algebras. The full rst-order language for complex -algebras encapsulates, via the translation , the 28
full monadic second-order theory of -structures. What are the logical implications of that translation? A challenging program in that respect is to study the full rst-order theory of complex -algebras. Algebraic issues. The main one is: to what extent is the proposed modal treatment of arbitrary structures interesting from algebraic perspective and how does it contribute to the study of intrinsically algebraic properties of these structures? More speci c questions include: What is the precise algebraic characteristic of the (non)de nability of the di erence operator in a class of -structures? What are the algebraic properties of the complex -algebras and how does the algebraic theory of the underlying -structures determine them? In particular, when do their substructures (up to isomorphism) form a variety? One general approach to this question comes from Goldblatt's study of varieties of complex algebras Goldblatt 1989. What algebraically important, essentially second-order properties of speci c -structures are de nable in the modal language of their complex -algebras? And how far reaching are the consequences of the capacity of the modal languages to quantify over substructures and congruences for the formal modal analyzing of the algebraic theory of structures?
References Balbiani, Ph., L. Fari nas del Cerro, T. Tinchev, and D. Vakarelov. 1997. Modal Logics for Incidence Geometries. J. Logic Computat. 7(1):59{78. Brink, C. 1993. Power structures. Algebra Universalis 30:177{216. Burris, S., and H. Sankappanavar. 1981. A course in universal algebra. SpringerVerlag. de Rijke, M. 1992. The modal Logic of inequality. J. Symb. Logic 57:566{584. de Rijke, M. 1993. Extending Modal Logic. Doctoral dissertation, ILLC, University of Amsterdam. Gabbay, D. 1981. An Irre exivity Lemma with Applications to Axiomatizations of Conditions on Tense Frames. In Aspects of Philosophical Logic, ed. U. Monnich. 67{89. Reidel, Dordrecht. Gargov, G., and V. Goranko. 1993. Modal Logics with Names. J. Philos. Logic 22(6):607{636.
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Goldblatt, R. 1989. Varieties of complex algebras. Ann. Pure and Appl. Logic 44:173{242. Goranko, V. 1990. Modal De nability in Enriched Languages. Notre Dame J. of Formal Logic 31:81{105. Goranko, V. 1998. Axiomatizations with Context Rules of Inference in Modal Logic. Studia Logica 61:179{197. Goranko, V., and D. Vakarelov. 1998. Hyperboolean Algebras and Hyperboolean Modal Logic. J. Appl. Non-classical Logics. Jonsson, B., and A. Tarski. 1951. Boolean algebras with operators, Parts I-II. American J. Math. 73:891{993, 74:127{162. Orlowska, E. 1996. Relational Environment for Semigroup Logics. In Logic, Action and Information, ed. A. Fuhrmann and H. Rott. 351{391. Walter de Gruyter, Berlin-New York. Passy, S., and T. Tinchev. 1991. An Essay in Combinatory Dynamic Logic. Information and Computation 93:263{332. Vakarelov, D. 1992. A modal theory of arrows. Arrow logic I. In Logics in AI, European Workshop JELIA'92, ed. D. Pearce and G. Wagner, 1{24. SpringerVerlag. Vakarelov, D. 1996. Many-dimensional Arrow Structures. Arrow logics II. In Arrow Logic and Multi-Modal Logic, ed. L. Polos M. Marx and M. Masuch. 141{187. CSLI Publications, Stanford. Venema, Y. 1990. Expressiveness and completeness in an interval tense logic. Notre Dame J. Formal Logic 31:529{547. Venema, Y. 1992. Many-Dimensional Modal Logic. Doctoral dissertation, Department of Mathematics and Computer science, University of Amsterdam. Venema, Y. 1993. Derivation rules as anti-axioms in modal logic. J. Symb. Logic 58:1003{1034.
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