A four-valued modal logic arising from Monteiro's last algebras

The class of abstract logics projectively generated by the class of logics defined on tetravalent modal algebras by the family of their filters is studied. These logics are four-valued in the sense that they can be characterized by a generalized matrix on the four-element tetravalent modal algebra which generates this variety together with a family of homomorphisms. They can be called modal since this four-element algebra can be given a nice epistemic interpretation as an extension of Belnap's four-valued logic. The authors also characterize them by their abstract properties and prove a completeness theorem with respect to a sequent calculus suggested by the abstract version.<<ETX>>


INTRODUCTION
In this paper we are going to deal with algebras 21 = (.4, A, /, 0) of type (2,1,1,0), and we will follow some notational conventions: We always put 1 = ->0, a\/b = -i(-iflAii) for all a,b 6 A, and 21~= (^4,A,-i,0) denotes the I-less reduct of 21. We cali 97I4m the four-element algebra such that = 97í4 is the four-element De Morgan algebra (whose Hasse diagram is shown below) and whose unary operator / is defined in the following way: This algebra has two subalgebras, namely the two-element and three-element chains with the inherited unary operator I; we cali them 2^2m and 97Í3m respectively, and we also put 23^m = 2!$2 (the two-element Boolean algebra) and 97?^= 97l3 (the three-element Kleene algebra). Note that, aside from the presentation of the algebraic structure, 27?zm is the three-valued Lukasiewicz algebra.
Work partially supported by grant PB86-0269 of the Spanish DGICYT.
The algebra ÜDI4m generates the variety of the so-called Tetravalent Modal Algebras (TMAs). These algebras were considered by Antonio Monteiro motivated by Luiz Monteiro's independence proof [Mon] of the axiomatization of three-valued Lukasiewicz algebras. During his last stay in Lisbon between 1977 and 1979 he suggested the study of TMAs to Isabel Loureiro, who began to work under his advice and finally (two years after Monteiro's death) wrote her Ph.D. Dissertation [L3] and a number of papers on such topic. It seems (see [P, page xxxvi]) that she was one of Monteiro's last disciples and that TMAs were his last original "creation".
So far, TMAs have received little attention, and still only from the algebraic point of view. We believe, however, that they have a genuine interest from the point of view of logic, both the multiple-valued branch and the abstract branch, and we will try in this paper to present some facts and results to support this claim. Our goal is to use 97í4m as a (generalized) matrix to generate logics by the usual semantic methods and to clarify the status of TMAs among these logics. This will give us the means to characterize these logics in an abstract form and as a corollary to find a completeness theorem for them, by using an ad-hoc sequent calculus.
The four-element De Morgan algebra has been used by several authors to define a semantical entailment relation. In [Ma], for instance, we find a very elegant presentation, although no interpretation is given to the four valúes. It was in [Be] where for the first time such an "epistemic" interpretation was given, by reading the two intermediate valúes respectively as "neither true ñor false" (the well-known "undetermined" valué of classical three-valued logic) and "both true and false" (the new valué, sometimes called "overdetermined"). This seminal idea has lead to several developments, either in the study of question-answering data-bases and distributed logic programs dealing with information which might contain conflicts or gaps, either in some extensions of Kripke's theory of truth, where the truth predicate is partial.
The behaviour of WI4 has been generalized to the nice concept of bilattice. See [Fil], [Fi2] and [Vi] for more details and references.
The basic idea can be roughly explained as follows: If we admit that the informa¬ tion a Computer can access to answer the questions he is presented concerning some topic might contain gaps (impossibility to handle the question, for instance due to the structure of the data base itself) and conflicts (for instance if the information is drawn from several independent sources) then we have to admit that the computer's answer to a question of the form "is ... true ? " should inelude at least the four possibilities: Yes, No, Both (conflict) and None (gap). These four valúes can be ordered according to their "degree of truth", and the resulting ordered set is a lattice; together with the natural negation it becomes a model of 9TÍ4m : In this context we can consider adding a modal opera dor I of an epistemic character corresponding to questions of the type "can the Computer confirm that ... is true ?". It is clear that the answer is Yes just in the case where the answer to the first question is also Yes, while it is No in all other cases; this modal operator is thus truth-functional and gives us a model of the four-element TMA IUI4m .
We can extend the interpretation of the ordering relation of 97l4m in terms of "degrees (or modes) of truth" to find an appropriate way of definig a semantic entailment relation by taking valuations on 9JÍ4m. The basic principie should be that the valué of the conclusión should have a greater (or equal) degree of truth than that of the premisses, taken together (that is, of their conjunction). Looking at the Hasse diagram for and considering the two filters of the lattice: F\ = {1, a} and F2 = {1, b} we see we can implement the principie by asking that whenever the valué of the (conjunction of the) premisses falls in Ft then the valué of the conclu¬ sión should fall in Ft as well, for i = 1,2. The readers familiar with the methods of algébrale logic will recognize that this proposal is just to take the two logical matrices (9Jl4m,Fi) and (9H4m,F2) together, that is, to use the generalized matrix (9JÍ4m, {FuF2}) , and to projectively generate a logic from it by the family of all the homomorphisms (valuations).
We will generalize this construction by taking axbitrary families of homomor¬ phisms and by generating logics on any algebra of the type, obtaining a class of logics called Tetravalent Modal Logics (TMLs) because they are both modal (the operator I has an S5-type modal behaviour, see Proposition 3) and four-valued (by their own definition). We will see that certain logics canonically attached to TMAs have a distinguished position in this class of logics. This will allow us to obtain the abstract characterization of the original semantic operator and from it we will obtain the completeness result. For technical reasons the material in the paper follows a slightly different order: We begin by introducing the TMAs and some of their elementary properties, and then we present the class of TMLs and prove the main theorems about the relations linking TMAs and TMLs. After that we show the relation between TMLs and the generalized matrix on , while introducing the semantic operator, which receives the announced abstract characterization. Finally, after a change in the type of the algebras, we are able to define the syntactic operator and prove completeness.

Notation and Terminology
An abstract logic (briefly: a logic) is a pair L = (21, C) or L = (21 ,C), where 21 is an algebra, C is a closure operator over A, the carrier of 21, and C is a closure system on A. Given two closure operators Cj and C2 over the same set A, we say that Ci is weaker or smaller than C2 (in symbols Ci C2 ) if and only if VX C A, Ci(X) C C2(-A); it is equivalent to say that C2 C C\, that is, that C\ is finer than C2 • We also say that E_i is weaker or smaller than L2.
Given any two logics Li = (2li,Ci) and L2 = (2Í2,C2), we say that Li is projectively generated from L2 by a set Ti C Hom(2li, 2I2) if and only if C\ has the set { h-1(T) : T £ C2, h £ Ti } as a basis (where C2 can be substituted by any of its bases); it is equivalent to say that VX C Ai, Ci(X) = {a 6 Ai : h(a) £ C2(h(X)) Vh £ Ti]. In the case where Ti = [h] reduces to only one epimorphism h we say that h is a bilogical morphism between Li and L2. This is a central concept, as we see from the following properties. If h is a bilogical morphism between Li and í_2 then C\ = C2 as complete lattices, and we have both Ci = h~1oC2°h and C2 = LoCio/i-1. Given any logic L, to every 0 £ Con(21) one can associate the quotient logic of L by 0 by taking C/6 = {TC A/0 : 7r-1(T) £ C], where 7r : 21 -► 21/0 is the canonical projection, and putting L/0 = (21/0,C/0). On the other hand, we naturally have the equivalence relation 0(C) = { (a, b) £ A x A : C(a) = C(b) }, and we say that an equivalence relation 0 over A is an Lcongruence or a logical congruence whenever 0 £ Con(2l) and 0 C 0(C); in this case the canonical projection 7r becomes a bilogical morphism between L and L/0. Conversely, if h is a bilogical morphism between Li and L2 then the relation 9h = { (a, b) £ Ai x Ai : h(a) = h(b)} is an Li-congruence and by factorization we obtain a logical isomorphism between Li/0^and L2, that is, an algebraic isomorphism which is also an isomorphism between the closure systems; in other words, an "identification" of the logics. We thus see that in some sense bilogical morphisms play the role of epimorphisms in universal algebra.
In any lattice we usually denote by T the set of all its filters, and by V the set of all its prime filters. Our is a lattice, with V = and T = { {1}, Fi, F2, Mim} . The abstract logic determined by all the filters is denoted by ü_4m = (ÜX)?4mi F)! remark that since U7l4iri is a distributive lattice, V is a basis of F.

Tetravalent modal algebras
We are going to recall their definíton and some properties, together with some new ones we will need later. Our definítion will be slightly different from (although trivially equivalent to) that of Loureiro, to conform to the type of the algebras used here.
PROOF: Properties (1) to (9) are proved in [L3] (on the other hand, they are easy). To prove (10), from (1) we get -ia V -ij-ia = 1 Va 6 i, and the De Morgan rules for negation give a A I-'a = 0. Finally (11) comes directly from Definition 1. | We say that an element a € A is open iff a = la, and denote by B = {a € A : a = la} the set of all open elements, which by 2.9 is equal to I(A). We also say that X C A is open iff I(X) C X . Then the next proposition is the algebraic expression of "being a modal operator of S5 type"; this kind of modal operators can also be characterized using abstract logics, with the methods of [FV2], which will be applied to TMAs in [FR].
ship between open filters and prime filters; to express this relationship one needs to consider a transformation defined on the set of all prime filters, which sometimes (see Loureiro's and Monteiro's papers) is called the Birula-Rasiowa transformation : 5. DEFINITION. Let A be any set having a unary operation Then we denote by $ the function defined for all subsets of A in the following way: For any X C A, The definition given here is suitable for very general situations, but in the case of De Morgan algebras it is equivalent to the original one (which comes from the representation theory of these structures, see [Bi-Ra]) as shows the following result: 6. Proposition. If Qí is a De Morgan algebra (and, a fortiori, if it is a TMA), then Proposition. In any TMA the following properties hold: (1) If V QV is a family of prime filters closed under $, then Q P' E p+ ; (2) If F E F+ then V = {P E V : F C P} is a family of prime filters closed under $ and such that f]V = F ( 3 ) I f la E P and la £ 4>(P), then -i Ja E P and thus la A ->la =~>a A Ja = O E P, a contradiction. | According to Proposition 7 there is a correspondence between families of prime filters closed under $ and open filters. DifFerent such families, though, can have the same meet, and thus we remark that such correspondence is not one-to-one. It can be proved, however, that it is so if we restrb c ourselves to families of the form shown in 7.2; in terms of the closure systems generated by those families of prime filters, this is to restrict ourselves to algébrale closure systems. See [FR] for more details and for similar results in more general situations.

Tetravalent modal logics
We want to define these logics as the generalization of the logics determined on TMAs by the closure system of all filters. To this end we use some of the properties just proved, and Theorems 14 and 15 will confirm us that the choice made for the definition was right. As for the ñame given to them,it was so chosen to remark their relationship with TMAs, a term with a published tradition, although it seems somehow odd to use the latin term "tetravalent" in the place of "four-valued"; this last one would perhaps be linguistically better, but at the same time its meaning is too wide and ambiguous, and thus uninformative. 9. Definition. An abstract logic L = (51,C) is a quasi tetravalent modal logic (QTML) if and only if there is a basis £ of C such that every P E £ satisñes (1) P is an A-filter (that is, Va, b E A , a A b E P <=> a,b E P) ; (2) $(P) E £ and $2(P) = P ; (3) O^P ; (4) IaeP ifflae $(P) Va E A ; and (5) a E P O $(P) iff IaePH $(P) Va E A We say that L is a tetravalent modal logic (TML) iff it is a QTML and moreover C is algebraic (or finitary). (2) As a result of Proposition 8, for any TMA 21 we have one TML L = (21, T), where as usual T is the closure system of all its filters. As we have alreadv announced, these kind of TMLs will be the "prototypes" of TMLs, in the precise sense stated in Theorem 15. Previously in Theorem 14 we will also see that the properties shown in Proposition 7 characterize the QTMLs, although they are better characterized in Theorem 17.
The logics satisfying 9.1 and 9.2 are called De Morgan logics in [FV1] and [FV3], We are going to use some properties proved there for these logics, concerning mainly the non-modal part of our structures. For instance, we immediately have: 11. Proposition. Let L be a QTML. Then the following hold: (1) L satisñes the Property of Conjunction: C(aAb) = C(a,b) Va, b £ A; (2) Every P £ S is V-prime: a V b £ P a £ P or b £ P Va, 6 £ A; (3) L satisñes the Property of Disjunction: C(X, a V b) = C(X, a) fl C(X, b) Va, & £ A , VX C A. | 12. PROPOSITION. Each one of the conditions 9.4 and 9.5 is preserved nnder bilo-gicaJ morphisms, that is, if Li and L2 are two abstract logics and there is a bilogical morphism between them, then one of them satisñes condition 9.4 (resp. 9.5) if and only if the other one does.
PROOF: We must prove that each one of the conditions of Definition 9 is preserved under bilogical morphisms. Proposition 12 contains the proof for conditions 9.4 and 9.5; condition 9.3 is trivial; condition 9.2 is proved in Proposition 3 of [FV1] and condition 9.1 in Proposition 2.5 of [FV2]. Finally finitarity is dealt with for instance in a Corollary to Proposition 6 of [V]. | Recall that to any abstract logic L = (21, C) we naturally associate the equivalence relation #(C) = { (a, b) E A x A : C(a) = C(b) } . If no confusión is likely to arise (which is the normal situation) we will denote this relation simply as 9, and if necessaxy will attach to it the subor superscripts corresponding to the logic.
Also remark that if 8 is a basis of C, then (a, b) E #(C) if and only if for any P G f, aGPifffeGP.
14. THEOREM. Let L = (2t,C) be any abstract logic. Then the following conditions are equivalent: (1) L is a QTML ; (2) 9 G Con(21), 21/0 is a TMA, and C/9 is a closure system on A/9 having a basis of prime fílters closed imder $ ; and (3) There is a bilogical morphism between L and a logic L' = (21',C') where 21' is a TMA and C' is a closure system on A/9 having a basis of prime ñlters closed under $.
PROOF: (1) (2): In [FV2] it is proved that 9 E Con(2l~), that 21~/9 is a De Morgan lattice, and that C/9 has a basis of prime filters closed under $; it remains only to prove that 9 E Con(2t) and that 21¡9 is a TMA. Let (a, b) E 9 and supppose la £ P for some P £ S. Then by the assumption la £ P fl 4>(P) and also a G Pfl$(P); then from (a,6) € 0 we obtain b € Pn$ (P) and also Ib € Pfl$(P); a fortiori Ib £ P, therefore (la, Ib) £ 9 and this means that 8 £ Con ( 21). From now on we denote by a$ the equivalence class of a £ A in the quotient A/O. Condition 9.3 says that 0# is the infimum of the De Morgan lattice 21/0, therefore it is a De Morgan algebra. To prove that it is a TMA we must show that it satisfies: (1.1) las A -<as = 0 Va# £ 21 ¡8 ; and (1.2) as A -ias = as A -ilag Va# £ 21 /O To prove (1.1) we have that las A~>as < 0# iff VP £ £, la A~^a^P. lí la £ P then la £ P D $(P) and also a £ P D $(P), but a £ 4>(P) iff ->a^P, that is, la A -ia^P ] if la £ P then la A~>a^P either, because P is a filter.
To prove (1.2), note that a# A -na# < as A -<Ias iff a A ->a G P implies a A -Ta G P, VP G 5. If a A ->a G P and ->Ja^P, then Ja G $(P) and also la G Pfl$(P), that is,~'aAla £ P, a contradiction; thus we must have ->Ja G P and thus aA-'Ia £ P. Conversely, as A ->las < as A ->as iff a A -Ta G P implies a A ->a G P , VP G £.
If a A ->Ja G P and ->a^P , then a G $(P) and also a G P fl $(P) and thus la £ P C\ $(P) , a contradiction, because from ->Ja G P we get Ja^4>(P).
(3) =>■ (1): By Propositions 6 and 8, L' is a QTML, and thus by Proposition 12 so is L. | The preceding result tells us that QTMLs can all be generated from TMAs by taking families of prime filters closed under $ and then projectively generating logics by arbitrary epimorphisms from arbitrary algebras. Later on (Theorem 17) we will give another, more "constructive" method of finding all possible QTMLs, using the four-element algebra. First we give the parallel characterization of TMLs. (1) L is a TML; (2) 9 £ Con(S), 21 ¡9 is a TMA, and C¡9 is the set of all ñlters of 21/0; and (3) There is a bilogical morphism between L and a logic L' = (21', T') where 21' is a TMA and p' is the set of all its ñlters. PROOF: It consists in adding finitarity to Theorem 14; but this is independent of the modal paxt of the structure, and the proof given in [FV1] suits also here. | This Theorem offers a number of algebraic applications, among which we can quote the following results: It establishes a clear relationship between the categories of TMLs and TMAs. For every fixed 21, the correspondence 8 i->■ 8(C) establishes an isomorphism between the lattice of all closure operators C on 21 such that (21, C) is a TML and the lattice of all congruences 8 of 21 such that the quotient 21¡8 is a TMA. An algebra 21 is a TMA iff there is a closure operator C on A such that (21, C) is a TML and 8(C) = A. An abstract logic L = (21,C) is a simple TML (that is, a TML whose only logical congruence is A) iff 21 is a TMA and C is the set of all filters of 21. More details on such topics will appear in [FR], Four-valued LOGICS Now we want to characterize, given any algebra 21, the logic L4m(2l) semantically defined from l_4m = (9TÍ4m, T) by using the set of all homomorphisms. We will describe it as a distinguished member of the class of all TMLs over 2t. To this end we first need to consider logics defined on 21 by subsets of the set of all homomorphisms.
Although these logics have not as yet been given a clear logical significance, it is not difficult to imagine several reasons to consider restricting the class of valuations to a smaller class of "admissible" ones.
16. Proposition . Let L = (21, C) be a QTML, and let S be the basis of C mentioned in Deñnition 9. Then for every P (E £ there is an h £ Hom(%L, fAlim) such that P = or P = PROOF: For each P € £ we consider the logic Lp = (21, Cp) where Cp is the closure system generated by {P, $(P)}. Clearly this is a TML, and thus 21¡8p is a TMA, and it can have two, three, or four elements, depending on the relative positions of P and $(P). Its structure as a De Morgan algebra in the three possible cases is determined in Proposition 4 of [FV1], but this also determines its structure as a TMA, because it follows from the main results of [L4] that there can be at most one structure of TMA on a given De Morgan algebra (the same paper gives necessary and sufficient conditions for a De Morgan algebra to be a TMA). The three cases are the following: Case 1: P = $ (P). Then there are just two equivalence classes, and (21/8p)~= <B2, so 21 /8p = *82m • If we denote by irp the canonical projection and by i the embedding of 232m into D7t4m, then P = (í°7rp)_1(Fi).
Case 3: When P and $(P) are not comparable, the quotient has four elements and (21 /9P)~--DTÍ4, and thus 21 ¡6p -9Jt4m with P = np1(Fi). | 17. Theorem. An abstract logic L is a QTML if and only if L is projectively generated from L4m by some set Tí C Hom(21, 9Jt4m). PROOF: A ==¿>) Just take as Tí the family of homomorphisms found when applying Proposition 16 to all P E £, the basis of C mentioned in the definition of QTMLs.
PROOF: From Theorem 17 and the fact that by definition L4m(2l) is projectively generate i by the set of all homomorphisms. | Now we want to prove that L4m(2í) is also the fínest TML over 21. We need only prove that it is really a TML, that is, that it is finitary. If the algebra 21 is the formula algebra then this is usually called the compactness theorem, and is a well known result of the theory of logical matrices (see [Wo] for instance). But for an arbitrary 21 we need to complete the proof ourselves: 19. THEOREM. For any algebra 21, the logic L4m(2í) is finitary, and as a consequence L4m(2t) is the finest TML over 21.
PROOF: We know that L4m(2l) is a QTML, therefore by Theorem 14 there is a bilogical morphism h between it and a logic L' = (%L',C') where 21' is a TMA and C is a closure system of filters of 21 having a basis of prime filters closed under $. If L4m(2l) is not a TML then L' is not either, by Theorem 14, and thus it is strictly less fine than the logic of all filters L" = (21', X'), because this one is really a TML. Now the same epimorphism h projectively generates on 21 another TML L* = (21, C") from L", which will be strictly finer than L4m(2l); but this is against Corollary 18, since L* is a fortiori a QTML. |

COMPLETENESS THEOREMS
Sometimes, results in the form of Theorem 19 axe thought of as a kind of Abstract Completeness Theorems, for two main reasons: First, they characterize a semantic operator by means of its abstract properties. Second, because from these properties a syntactic operator can be defined on the algebra of formulas in such a way that the proof of the usual completeness theorem is rather straightforward. However, if we observe the conditions of Definition 9, we realize that all properties are expressed in terms of the closure system rather than in terms of the closure operator. To obtain an equivalent formulation allowing us to extract from it a syntactical definition we need to enrich our language by treating the connective V as primitive; in this new language we will fínd an equivalent concept of TML which will serve our purposes. 20. DEFINITION. Let 21' = (A, A, V, /, 0) be an algebra of type (2,2,1,1,0), and let L' = (2í',C) be an abstract logic over 21'. We say that L' is a (Q)TML when the following conditions are satisfied, for all a,b £ A and for all X C A: (1) C(a A b) = C(a, b) ; (2) C(X, a V b) = C(X, a) D C(X, b) ; (3) C(a) = C(-i-ia) ; (4) a £ C(6) ==>--ib £ C(~¡a) ; (5) C(0) = A ; (6) C(a) n C(-i/a) = C(0) ; (7) C(a, -<Ia) = C(a, -<a) ; and (8) C is finitary (only for TMLs).
21. THEOREM. Let 21 = {A, A, -1,7,0) be an algebra of type (2,1,1,0) and L = (21, C) be an abstract logic over 21. Then L is a (Q)TML in the sense of Definition 9 if and only if there is a binary operation V in A such that putting 21' = (A, A, V, 1,0) and L' = (21', C), L' is a (Q)TML in the sense of Definition 20. Moreover, in such a case it holds that C(a V b) = C(-'(-'a A -'b)).
PROOF: If L is a (Q)TML in the sense of Definition 9, then Theorem 14 tells us that # G Con(21) and that 21/# is a TMA. But 11.3 tells us that L satisfies 20.2 for aVfe = A~ife), and thus 9 6 Con(2!'). It is well-known that 20.1 is equivalent to 9.1, and then conditions 20.3 to 20.7 can be re-written as equations or implications which do hold in 21'/#, therefore they hold as stated in 21', because 21'/# is precisely the quotient of 21' by C (that is, the equality in 21'/# is exactly the C-equivalence in 21'). We conclude that L' is a (Q)TML in the sense of Definition 20. Note that, in view of 20.2, condition 20.6 is equivalent to C(a V -'la) = C(0).
For the converse part we are going to prove directly each one of the conditions 9 . 1 t o 9.5. In Proposition 1 of [FV3] it is proved that conditions 20.1 to 20.4 imply 9 . 1 a n d 9.2, where the basis S of C is taken to be the set of all V-prime A-filters of 21; it is also proved that C(a V b) = C(-i(-ia A~'b)), because they are equal in the quotient. Moreover 9.3 is equivalent to 20.5.
To prove 9.4 it is enough to prove that, for any P G £, la G P implies la G $(P), because <Í>2(P) = P. Suppose that la G P but la^$ (P), that is, -Ta G P; by (*) and 20.7 we would have -ia G P, that is, a^$(P). But $(P) G S and from 9 . 1 a n d 9.2 it follows that it is V-prime, and 20.6 implies that a V -<la G $(P); thus we must have ->Ia G $(P) which is the same as la^P, a contradiction.
Finally we prove 9.5: If la G P fl $(P) then (*) implies a G P D $ (P). To prove the converse, observe that 20.7 says that if a G P, then -<a G P iff -'la G P, that is, a G $(P) iff la G $(P); and using $2(P) = P it also says that if a G^(P), then a G P iff 7a G P. Therefore we conclude that if a G Pfl$(P) then la G Pfl$(P). | As a consequence of this result we will treat the two notions of (Q)TML as equivalent, and thus all the results proved until now for (Q)TMLs in the first setting can be used as if they were proved for the second one.
We are now ready to introduce a syntactic operator on the algebra of formulas 3" = (Form, A, V, □, _L), that is, the absolutely free algebra of the type (2,2,1,1,0). In this case the semantic operator ll_4m(3) is usually denoted by (3, |=) • The syntactic operator will be defined by means of a sequent calculus. Remark that our sequents are expressions of the form T H <p, where tp G Form and T is a finite (possibly empty) unordered set of formulas of Form.
22. Definition. Ls = (?, hs) is the logic defíned on 5" by: F I-5 if and only if there is a ñnite ro C T such that the sequent r0 b tp is derivable in the sequent calculus which has the following axiom and rules (for any A C Form, and for any a, /3,7 £ Form) : This definition has been chosen to make the proof of the following theorem rather straightforward: 23. THEOREM. The logic Ls = (í?, bs) is the least TML over J.
PROOF: Each of the conditions 20.1 to 20.8 can be proved for Ls from its definition; and given a proof of some sequent To b in the sequent calculus, and any other TML ($, C) over y, a routine inductive process tells us that the same "proof" can be "reproduced" for C to conclude that y> £ C(Fo), that is, bs^C. |