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Structural Co~Pleteness in Algebra and Logic

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Structural Co~Pleteness in Algebra and Logic COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 54, ALGEBRAIC LOGIC, BUDAPEST (HUNGARY), 1988 Structural Co~pleteness in Algebra and Logic CLIFFORD BERGMAN A number of authors, particularly those studying "intermediate" intu­ it\onistjc logic, have asked whether a given logic is structurally complete. See [14] and [12] for example. Roughly speaking, a logic is structurally com­ plete if every proper extension must contain new logical axioms (as opposed to nothing but new rules of inference). In particular, structural complete­ ness can be seen as a kind of maximality condition on a logic. Using techniques of algebraic logic, it is possible to study many log­ ics by finding an appropriate class of algebras in which various (algebraic) properties reflect the logical ones of interest. The property corresponding to structural completeness turns out to be quite interesting algebraically, and can be studied independently of any connections to logic. This has been done in a number of papers such as [7], [2] and [9]. In this paper we will briefly consider the notion of structural complete­ ness in its original logical context. We then go on to define and study the algebraic analog, as well as a hereditary version, called deductiveness. Unlike the situation in universal algebra, there is not yet any general agreement in algebraic logic on the definition of basic concepts (such as a logic!) or even on the motivating examples. Since our intent in section 1 is ' primarily motivational, we do not feel it is necessary to completely specify what we mean by a "logic". Instead we give what we hope is a minimal set of conditions that will hold in any reasonable definition of the term. For a rigorous treatment of the subject, the reader can consult [1], [3], or [5]. Research partially supported by National Science Foundation grant DMS-8701643 60 CLIFFORD BERGMAN The drawback of this approach is that we will not be able to establish any real connection between the definitions in .section 1 and the theorems ,, in section 2. We leave that to others with more expertise in the field. 1. The Logical Version Structural completeness arises naturally when considering the "deduc­ tive power" of a logic. Informally, one can think of a logic £, over a set Fm (the set of formulas), as being composed of a set !1 of logical axioms, and a set R of rules of inference. For any set r of formulas, the consequences of r are those formulas derivable from r together with all instances of formulas from !1 by means of the rules in R. In particular, a tautology is a formula derivable directly from !1 via R. We can write ,C = (!1, R) to denote the logic determined by those axioms and rules of inference, and r 1--c<p to indicate that <pis a consequence of r in £. For our purposes, it suffices to focus on the consequence operator Cc of ,C, defined by In fact, we identify a logic with its consequence operator, so that two logics ,C and £' are considered equal if Cc = CC'. The only assumptions we make of Cc is that it be an algebraic closure operator on the set Fm. That is, for all r,~ <;;Fm (cl) r <;;G(r) (c2) G(G(r)) <;;G(r) (c3) r <;;~ =* c(r) <;;c(~) (c4) G(r) <;;LJ { G(ro) : ro is a finite subset of r}. For the remainder .of this section, one can take (cl)-( c4) as the definition of a logic.1 However, the naive conception of a logic as a pair (!1,R) provides a more intuitive basis for the notions we will discuss. 1 Most modern conceptions of a logic diverge from this definition. For example, [3] requires that every set C .c(r) be closed under substitution. One can easily check that our results remain true under that particular assumption. On the other hand, in [1], condition (c4) defines the special case of compact logics. It is not clear that Proposition 1.2 holds in the failure of (c4). STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 61 We write £, < £', (and say that £,' is an extension of C) if £,' is a stronger logic in the sense that we can "derive more things". Furthermore, write £, =o £,' if £, and £,' have the same tautologies. In terms of the ,, consequence operator: ' c,:::;C' = (\7T <;;;Fm) Cc (r) <;;;Cc, (r) C =o C' = Cc (0) = Cc, (0). Definition 1. 1. A logic £, is structurally complete if for every logic £,', C < C' =* C.,to C'. In other words, £, is structurally complete if it is not possible to extend £, without creating new tautologies. Do structurally complete logics generally exist? The answer is 'yes' as long as we assume that (cl)-(c4) constitutes a definition of a consequence operator. Proposition 1.2. Every logic C, has a unique, structurally complete ex­ tension C', such that C, =o C'. Proof. Let x = {GM:£,:::; M & C, =o M }. x can be thought of as a subset of the monoid of self-maps of Sb(Fm) (the power set) under the operation of function-composition. Let x be the submonoid generated by x, Define a new operator D by D(r) = u{ G(r): CE x} for r <;;;Fm. We claim that DE x. The axioms(cl)-(c4) are all easily verified, with the exception of (c2). For this, we make the following two observations. First, every member ofx satisfies axiom (c4). Second, the family { G(r): CE x} is upper-directed by set-theoretic union. Now, if 'Y E D(D(r)), then for some C E x, 'Y E G(D(r)). Therefore, (by the first observation), there is ·1~" a finite subset A <;;;D(r) such that 'Y E C(A), and by upper-directedness, there is C' E x su_ch that A <;;;C'(r). Hence, 'Y E C(C'(r)) <;;;D(r), proving ( c2). Let £,' be the logic with consequence operator D. (In terms of logical axioms and rules of inference, those of £,' are obtained by taking the union 62 CLIFFORD BERGMAN over all of the logics referenced in x.) Then we clearly have £ :S £' =o .C, so that. D E x. To see that .C' is structurally complete, let £' :S M. 1° If £' =o M, then CM E x, from which it follows that M :S £'. The uniqueness is similar. I From the proposition, one sees that the structurally complete logics are precisely the, maximal members of the =a-equivalence classes. It is in this sense that structural completeness is a maximality condition. We now introduce the hereditary version of the same concept. Definition 1.3. A logic £ is deductive if every (not necessarily proper) extension of £ is structurally complete. Proposition 1.4. Let £ be a logic. The following are equivalent. (1) £ is deductive; (2) For each M c".£, and r <;;Fm, CM (r) = Cc (r u A), where A=CM(0) Proof. (1) * (2): First note that A<;; Cc (A) <;;CM (A)= A. Now define a consequence operator D by D(r) = Cc (AU r). That D satisfies (cl)-(c4) is straightforward. For any set r, (*) D(r) = Cc (Au r) <;;CM (Au r) <;; CM (Au CM (r)) = CM (CM (r)) = CM (r). In other words, if £' denotes the logic with consequence operator D, then £ :S £' :S M. Also, by taking r = 0 in (*), A = D(0) <;; CM (0) = A. Therefore,£' =o M. Now by the deductiveness of£, D = CM. (2) * (1): To show £ is deductive, let £ :S M :S N and assume M =o N. We must show M = N. Since M =o N, A= CM (0) = CN (0). Therefore CN (r) = C;:.(r U A)= CM (F), as desired. I According to the above proposition, dedcutiveness can be interpreted in the following way: Every new rule of inference can be replaced by a set of logical axioms. For if £ = (!1, R) is a deductive logic, and r is a new rule, let M = (!1,RU {r} ). The logic £' constructed in the proposition, is (!1U A, R), and th~ proof of 1.4 demonstrates that£'= M (more precisely, they have the same consequence operator). One of the goals of algebraic logic is to establish a correspondence be­ tween individual logics and classes of algebras. In the best of all possible STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 63 worlds, each logic£ would correspond to a quasivariety, £,Q, Under this correspondence £ :::;M {cc=} £q =2Mq ( t) £ =o M {cc=} V(£Q) = V(Mq). In other words, logical axioms correspond to identities ( on the associated algebra~) and rules of inference to quasi-identities. It is on this intuitive idea that the definitions of section 2 are based. Whether or not the relationships in ( t) are realized is a subject of debate in Algebraic Logic. In any case, the algebraic notions turn out to be quite interesting in their own right. Because it is not particularly natural from an algebraic standpoint, it is unlikely that the concept of a structurally ,: complete variety would ever have been considered if not for this intuitive I correspondence between algebras and logics. ' 2. The Algebraic Version By applying the idealized translation ( t) of section 1 to the notions of structural completeness and deductiveness, we produce corresponding notions for classes of algebraic structures (Definition 2.1, below). In order to proceed, we briefly review the relevant notions from universal algebra. A class ICof algebras is a quasivariety if it is defined by a set of quasi-identities, i.e. universal· sentences of the form 1 rn (f\o-i(x) cs Ti(x)) -->. µ(x) cs v(x) i=O in which n and tare natural numbers, and each <Ti, Ti,µ, v is a term in the variables x = xo,x1, ..
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