Showing papers in "Journal of Symbolic Logic in 1985"

On the logic of theory change: Partial meet contraction and revision functions

Abstract: This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate “partial meet contraction functions”, which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gardenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are “relational” and “transitively relational”, are studied in detail, and their connections with certain “supplementary postulates” of Gardenfors investigated, with a further representation theorem established.

Logics Without the Contraction Rule

Abstract: We will study syntactical and semantical properties of propositional logics weaker than the intuitionistic, in which the contraction rule (or, the exchange rule or the weakening rule, in some cases) does not hold. Here, the contraction rule means the rule of inference of the form if we formulate our logics in a Gentzen-type formal system. Some syntactical properties of these logics have been studied firstly by the second author in [11], in connection with the study of BCK-algebras (for information on BCK-algebras, see [9]). There, it turned out that such a syntactical method is a powerful and promising tool in studying BCK-algebras. Using this method, considerable progress has been made since then (see, e.g., [8], [18], [27]). In this paper, we will study these logics more comprehensively. We notice here that the distributive law does not hold necessarily in these logics. By adding some axioms (or initial sequents) and rules of inference to these basic logics, we can obtain a lot of interesting nonclassical logics such as Łukasiewicz's many-valued logics, relevant logics, the intuitionistic logic and logics related to BCK-algebras, which have been studied separately until now. Thus, our approach will give a uniform way of dealing with these logics. One of our two main tools in doing so is Gentzen-type formulation of logics in syntax, and the other is semantics defined by using partially ordered monoids.

Fibered Categories and the Foundations of Naive Category Theory

Abstract: Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related. (0.1) Noncontradiction: Namely, to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent, or at least relatively consistent with a well-established and “safe” theory, e.g. Zermelo-Frankel (ZF). (0.2) Adequacy, in the following, nontechnical sense: (i) The basic notions must be simple enough to make transparent the syntactic structures involved. (ii) The translation between the formal language and the usual language must be, or very quickly become, obvious. This implies in particular that the terminology and notations in the formal system should be identical, or very similar, to the current ones. Although this may seem minor, it is in fact very important. (iii) “Foundations” can only be “foundations of a given domain at a given moment”, therefore the frame should be easily adaptable to extensions or generalizations of the domain, and, even better, in view of (i), it should suggest how to find meaningful generalizations. (iv) Sometimes (ii) and (iii) can be incompatible because the current notations are not adapted to a more general situation. A compromise is then necessary. Usually when the tradition is very strong (ii) is predominant, but this causes some incoherence for the notations in the more general case (e.g. the notation f(x) for the value of a function f at x obliges one, in category theory, to denote the composition of arrows (f, g) → g∘f, and all attempts to change this notation have, so far, failed).

Cuts, consistency statements and interpretations

Abstract: Interpretability in reflexive theories, especially in PA, has been studied in many papers; see e.g. [3], [6], [7], [10], [11], [15], [26]. It has been shown that reflexive theories exhibit many nice properties, e.g. (1) if T, S are recursively enumerable reflexive, then T is interpretable in S iff every Π1 sentence provable in T is provable in S; and (2) if S is reflexive, T is recursively enumerable and locally interpretable in S (i.e. every finite part of T is interpretable in S), then T is globally interpretable in S (Orey's theorem, cf. [3]).In this paper we want to study such statements for nonreflexive theories, especially for finitely axiomatizable theories (which are never reflexive). These theories behave differently, although they may be quite close to reflexive theories, as e.g. GB to ZF. An important fact is that in such theories one can define proper cuts. By a cut we mean a formula with one free variable which defines a nonempty initial segment of natural numbers closed under the successor function. The importance of cuts for interpretations in GB was realized already by Vopěnka and Hajek in [30]. Pioneering work was done by Solovay in [24]. There he developed the method of “shortening of cuts”. Using this method it is possible to replace any cut by a cut which is contained in it and has some desirable additional properties; in particular it can be closed under + and ·. This introduces ambiguity in the concept of arithmetic in theories which admit proper cuts, namely, which cut (closed under + and ·) should be called the arithmetic of the theory? Cuts played the crucial role also in [20].

Strongly Majorizable Functionals of Finite Type: A Model for Barrecursion Containing Discontinuous Functionals

Abstract: In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howard's majorizability notion; see [T, p. 456].

Second-order languages and mathematical practice

Abstract: There are well-known theorems in mathematical logic that indicate rather profound differences between the logic of first-order languages and the logic of second-order languages. In the first-order case, for example, there is Godel's completeness theorem: every consistent set of sentences (vis-a-vis a standard axiomatization) has a model. As a corollary, first-order logic is compact: if a set of formulas is not satisfiable, then it has a finite subset which also is not satisfiable. The downward Lowenheim-Skolem theorem is that every set of satisfiable first-order sentences has a model whose cardinality is at most countable (or the cardinality of the set of sentences, whichever is greater), and the upward Lowenheim-Skolem theorem is that if a set of first-order sentences has, for each natural number n, a model whose cardinality is at least n, then it has, for each infinite cardinal κ (greater than or equal to the cardinality of the set of sentences), a model of cardinality κ. It follows, of course, that no set of first-order sentences that has an infinite model can be categorical. Second-order logic, on the other hand, is inherently incomplete in the sense that no recursive, sound axiomatization of it is complete. It is not compact, and there are many well-known categorical sets of second-order sentences (with infinite models). Thus, there are no straightforward analogues to the Lowenheim-Skolem theorems for second-order languages and logic.There has been some controversy in recent years as to whether “second-order logic” should be considered a part of logic, but this boundary issue does not concern me directly, at least not here.

Logics containing K 4. Part II 1

Abstract: There are two main lacunae in recent work on modal logic: a lack of general results and a lack of negative results. This or that logic is shown to have such and such a desirable property, but very little is known about the scope or bounds of the property. Thus there are numerous particular results on completeness, decidability, finite model property, compactness, etc., but very few general or negative results. In these papers I hope to help fill these lacunae. This first part contains a very general completeness result. Let I n be the axiom that says there are at most n incomparable points related to a given point. Then the result is that any logic containing K4 and I n is complete. The first three sections provide background material for the rest of the papers. The fourth section shows that certain models contain no infinite ascending chains, and the fifth section shows how certain elements can be dropped from the canonical model. The sixth section brings the previous results together to establish completeness, and the seventh and last section establishes compactness, though of a weak kind. All of the results apply to the corresponding intermediate logics.

Adjoining dominating functions

Abstract: If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number (and hence the distributivity number ) remains at ω 1.

Sequent-systems for modal logic

Abstract: The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. ?0. Introduction. The aim of this work is to present sequent formulations of the modal logics S5 and S4 based on sequents of higher levels. Sequents of level 1 have collections of formulae of a given formal language on the left and right of the turnstile, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants will involve sequents of level 2, whereas rules for other customary logical constants of first-order logic (with identity) will involve only sequents of level 1. We shall show how a restriction on Thinning of level 2, which when applied to Thinning of level 1 produces intuitionistic out of classical logic, produces in this case S4 out of S5. Both in passing from classical to intuitionistic logic and in passing from S5 to S4, only Thinning is changed-all the other assumptions are unchanged. In particular, this means that S5 and S4 will be formulated with identical assumptions for the necessity operator. We shall also show in what sense our characterization of the necessity operator is Received January 5, 1982; revised December 3, 1983. 1980 Mathematics Subject Classification. Primary 03B45, 03F99. (? 1985, Association for Symbolic Logic 0022-4812/85/5001-001 5/$03.00

Jumps of Quasi-Minimal Enumeration Degrees

Abstract: Enumeration reducibility is a reducibility between sets of natural numbers defined as follows: A is enumeration reducible to B if there is some effective operation on enumerations which when given any enumeration of B will produce an enumeration of A. One reason for interest in this reducibility is that it presents us with a natural reducibility between partial functions whose degree structure can be seen to extend the structure of the Turing degrees of unsolvability. In [7] Friedberg and Rogers gave a precise definition of enumeration reducibility, and in [12] Rogers presented a theorem of Medvedev [10] on the existence of what Case [1] was to call quasi-minimal degrees. Myhill [11] also defined this reducibility and proved that the class of quasi-minimal degrees is of second category in the usual topology. As Gutteridge [8] has shown that there are no minimal enumeration degrees (see Cooper [3]), the quasi-minimal degrees are very much of interest in the study of the structure of the enumeration degrees. In this paper we define a jump operator on the enumeration degrees which was introduced by Cooper [4], and show that every complete enumeration degree is the jump of a quasi-minimal degree. We also extend the notion of a high Turing degree to the enumeration degrees and construct a “high” quasi-minimal enumeration degree—a result which contrasts with Cooper's result in [2] that a high Turing degree cannot be minimal. Finally, we use the Sacks' Jump Theorem to characterise the jumps of the co-r.e. enumeration degrees.

On minimal pairs of enumeration degrees

Abstract: For sets of natural numbers A and B, A is enumeration reducible to B if there is some effective algorithm which when given any enumeration of B will produce an enumeration of A. Gutteridge [5] has shown that in the upper semilattice of the enumeration degrees there are no minimal degrees (see Cooper [3]), and in this paper we study those pairs of degrees with gib 0. Case [1] constructed a minimal pair. This minimal pair construction can be relativised to any gib, and following a suggestion of Jockusch we can also fix one of the degrees and still construct the pair. These methods yield an easier proof of Case's exact pair theorem for countable ideals. 0″ is an upper bound for the minimal pair constructed in §1, and in §2 we improve this bound to any Σ2-high Δ2 degree. In contrast to this we show that every low degree c bounds a degree a which is not in any minimal pair bounded by c. The structure of the co-r.e. e-degrees is isomorphic to that of the r.e. Turing degrees, and Gutteridge has constructed co-r.e. degrees which form a minimal pair in the e-degrees. In §3 we show that if a, b is any minimal pair of co-r.e. degrees such that a is low then a, b is a minimal pair in the e-degrees (and so Gutteridge's result follows). As a corollary of this we can embed any countable distributive lattice and the two nondistributive five-element lattices in the e-degrees below 0′. However the lowness assumption is necessary, as we also prove that there is a minimal pair of (high) r.e. degrees which is not a minimal pair in the e-degrees (under the isomorphism). In §4 we present more concise proofs of some unpublished work of Lagemann on bounding incomparable pairs and embedding partial orderings.As usual, {Wi}i ∈ ω is the standard listing of the recursively enumerable sets, Du is the finite set with canonical index u and {‹ m, n ›}m, n ∈ ω is a recursive, one-to-one coding of the pairs of numbers onto the numbers. Capital italic letters will be variables over sets of natural numbers, and lower case boldface letters from the beginning of the alphabet will vary over degrees.

The Decision Problem for Branching Time Logic

Abstract: First we describe a certain branching time logic. We borrow this logic from Prior (1967) and call it BTL. It is a propositional logic whose formulas are built from propositional symbols by means of usual boolean connectives and additional unary connectives PAST, FUTURE, NECESSARY.

Syntactic translations and provably recursive functions

Abstract: Syntactic translations of classical logic C into intuitionistic logic I are well known (see [Kol25], [Gli29], [God32], [Kre58b], [M063], [Cel69] and [Lei71]). Harvey Friedman [Fri78] used a translation of a similar nature, from I into itself, to reprove a theorem of Kreisel [Kre58a] that various theories based on I are closed under Markov's rule: if ¬¬∃x.α is a theorem, where x is a numeric variable and α is a primitive recursive relation, then ∃x.α is a theorem. Composing this with Godel's translation from classical to intuitionistic theories, it follows that the functions provably recursive in the classical version of the theories considered are provably recursive already in their intuitionistic version. This conservation result is important in that it guarantees that no information about the convergence of recursive functions is lost when proofs are restricted to constructive logic, thus removing a potential objection to the use of constructive logic in reasoning about programs (see [C078] for example). Conversely, no objection can be raised by intuitionists to proofs of formulas that use classical reasoning, because such proofs can be converted to constructive proofs (this has been exploited extensively; see [Smo82]).Proofs of closure under Markov's rule had required, until Friedman's proof, a relatively sophisticated mathematical apparatus. The chief method used Godel's “Dialectica” interpretation (see [Tro73, §3]). Other proofs used cut-elimination, provable reflection for subsystems [Gir73], and Kripke models [Smo73]. Moreover, adapting these proofs to new theories had required that the underlying meta-mathematical techniques be adapted first, not always a trivial step.

On the Role of the Baire Category Theorem and Dependent Choice in the Foundations of Logic

Abstract: The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Cech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The RasiowaSikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces. The relevance of the Baire Category Theorem to the fundamental metalogical principle of deductive completeness has long been known. Rasiowa and Sikorski [1950], in their Boolean-algebraic proof of Gddel's completeness theorem for firstorder logic, applied the Baire Category Theorem to the compact Hausdorff Stone space of a Boolean algebra to obtain their celebrated lemma about the existence of ultrafilters respecting countably many meets. Grzegorczyk, Mostowski, and RyllNardzewski [1961] later adapted this approach to obtain the completeness theorem for co-logic, by applying the Baire theorem to the complete metric space of co-models of an w-complete theory. A similar argument may also be developed for omittingtypes theorems. The aim of this paper is to isolate that part of the Rasiowa-Sikorski Lemma that does not depend on the Ultrafilter Theorem. A result about the existence of certain filters is obtained that is dubbed "Tarski's Lemma" since it is closely allied to Tarski's algebraic proof of the Rasiowa-Sikorski Lemma, as reported by Feferman [1952]. It will be shown that in set theory without choice, Tarski's Lemma is equivalent to each of (i) the Baire Category Theorem for Cech-complete spaces, (ii) the Baire Category Theorem for complete metric spaces, (iii) the Principle of Dependent Choice, (iv) the existence theorem for generic sets of forcing conditions, and (v) a proof-theoretic principle which abstracts the technique introduced by Henkin [1949] for proving completeness proofs. From this follows a proof that the Rasiowa-Sikorski Lemma is equivalent to the conjunction of the Ultrafilter Theorem with the Baire Category Theorem for compact Hausdorff spaces. Throughout this paper, assertions that one statement implies another, or that two statements are equivalent (imply each other) will mean that the implications Received November 1, 1983; revised February 23, 1984. ? 1985, Association for Symbolic Logic 0022-48 12/85/5002-0016/$02. 1 0

The geometry of weakly minimal types

Abstract: Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p(M). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl(X) ∩ D, Y = acl(Y) ∩ D and X ∩ Y ≠ ∅,Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp(a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1.Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H∖acl(A), b ∈ G∖acl(A) there area′ ∈ H, b′ ∈ G such that a′ ∈ acl(abb′A)∖acl(aA). Similarly when H and G are the realizations of complete types or strong types over A.

Nonsplitting Subset of $\mathscr{P}_\kappa(\kappa^+)$

Abstract: Assuming the existence of a supercompact cardinal, we construct a model of ZFC + (There exists a nonsplitting stationary subset of ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes stationary

The Extensions of the Modal Logic $K5$

Abstract: Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K 5. We associate with each logic extending K 5 a finitary index , in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K 5 and an abstract characterization of the lattice of such extensions. This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10. By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A ↔ B infer □ A ↔□ B ) and normal if it is classical and contains □ ⊤ and □ ( P → q ) → (□ p → □ q ). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □ A ).

A Direct Proof of the Finite Developments Theorem

Abstract: Let M be a term of the type free λ -calculus and let be a set of occurrences of redexes in M . A reduction sequence from M which first contracts a member of and afterwards only residuals of is called a development (of M with respect to ). The finite developments theorem says that developments are always finite. There are several proofs of this theorem in the literature. A plausible strategy is to define some kind of measure for pairs ( M, ), which—if M ′ results from M by contracting a redex occurrence in and ′ is the set of residuals of in M ′— decreases in passing from ( M , ) to ( M ′, ′). This procedure is followed as a matter of fact in the proofs in Hyland [4] and in Barendregt [1] (both are covered in Klop [5]). If, as in the latter proof, the natural numbers are used as measures, then the measure of ( M , ) will actually denote an upper bound of the number of reduction steps in a development of M with respect to . In the present proof we straightforwardly define for each pair ( M , ) a natural number, which can easily be seen to indicate the exact number of reduction steps in a development of maximal length of M with respect to .

Universal recursion theoretic properties of r.e. preordered structures

Abstract: When dealing with axiomatic theories from a recursion-theoretic point of view, the notion of r.e. preordering naturally arises. We agree that an r.e. preorder is a pair = 〈 P , ≤ P 〉 such that P is an r.e. subset of the set of natural numbers (denoted by ω ), ≤ P is a preordering on P and the set {〈; x, y 〉: x ≤ P y } is r.e.. Indeed, if is an axiomatic theory, the provable implication of yields a preordering on the class of (Godel numbers of) formulas of . Of course, if ≤ P is a preordering on P , then it yields an equivalence relation ~ P on P , by simply letting x ~ P y iff x ≤ P y and y ≤ P x . Hence, in the case of P = ω , any preordering yields an equivalence relation on ω and consequently a numeration in the sense of [4]. It is also clear that any equivalence relation on ω (hence any numeration) can be regarded as a preordering on ω . In view of this connection, we sometimes apply to the theory of preorders some of the concepts from the theory of numerations (see also Ersov [6]). Our main concern will be in applications of these concepts to logic, in particular as regards sufficiently strong axiomatic theories (essentially the ones in which recursive functions are representable). From this point of view it seems to be of some interest to study some remarkable prelattices and Boolean prealgebras which arise from such theories. It turns out that these structures enjoy some rather surprising lattice-theoretic and universal recursion-theoretic properties. After making our main definitions in §1, we examine universal recursion-theoretic properties of some r.e. prelattices in §2.

Cylindric-Relativised Set Algebras have Strong Amalgamation

Abstract: In algebra, the properties of having the (strong) amalgamation property and epis being surjective are well investigated; see the survey [10]. In algebraic logic it is shown that to these algebraic properties there correspond interesting logical properties, see e.g. [15], [12], [4], and [8, p. 311, Problem 10 and the remark below it]. In the present paper we show that the varieties Crsa (of cylindric-relativised set algebras) and Bo. (of Boolean algebras with operators) have the strong amalgamation property. These contrast to the following result proved in Pigozzi [E15]: No class K with Gs. c K c CA_ has amalgamation property. Note that Gs. c Crs_ c Bo. and CA_ C Boa. For related results see [3], [1], [16], [11]. For more connections with logic and abstract model theory see [14] and ?4.3 of [9]. BA denotes the class of all Boolean algebras. Let a be any ordinal. From now on, throughout in the paper, a is an arbitrary but fixed ordinal. Recall from [7, p. 430, Definition 2.7.1] that an a-dimensional BA with operators, a Boa, is an algebra W = ijej of the same similarity type as CAa's such that 319 = is a BA and the operations c' (i E x) are additive, i.e., W # ci (x + y) = cix + cjy for all i E a. If W E Bo. then 019 is called the Boolean reduct of 9. Note that BA = Boo. A Bo, is said to be normal if {cj0 = 0: i E o} is valid in it, and a Bo, is said to be extensive if {x < cix: i E o} is valid in it. Boa's were introduced in [17]. The class Crs, of all cylindric-relativised set algebras is defined in Definition I.1.1(iii) of [8, p. 4]. We give a definition in the present paper, too-see Definition 5 below. It is shown in [13] that ICrs, is a variety. Our main result is (i) of Theorem 1 below, but we obtain (ii)-(vi), too, as a byproduct from the proof. THEOREM 1. Let o be any ordinal. Then (i)-(vi) below hold. (i) Crsa has the strong amalgamation property (SAP). (ii) Bo, has SAP. (iii) The class of all normal Boa's has SAP. (iv) The class of all extensive Boa's has SAP. (v) The class of all normal and extensive Boa's has SAP.

Automorphisms of supermaximal subspaces

Abstract: An infinite-dimensional vector space V ∞ over a recursive field F is called fully effective if V ∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V ∞ has a dependence algorithm , that is a uniformly effective procedure which when applied to x , a 1 ,…, a n , ∈ V ∞ determines whether or not x is an element of { a 1 ,…, a n }* (the subspace generated by { a 1 ,…, a n }). The study of V ∞ , and of its lattice of r.e. subspaces L ( V ∞ ), was introduced in Metakides and Nerode [15]. Since then both V ∞ and L ( V ∞ ) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology. In [15] Metakides and Nerode observed that a study of L ( V ∞ ) may in some ways be modelled upon a study of L ( ω ), the lattice of r.e. sets. For example, they showed how an e -state construction could be modified to produce an r.e. maximal subspace, where M ∈ L ( V ∞ ) is maximal if dim( V ∞ / M ) = ∞ and, for all W ∈ L ( V ∞ ), if W ⊃ M then either dim( W / M ) V ∞ / W ) However, some of the most interesting features of L ( V ∞ ) are those which do not have analogues in L ( ω ). Our concern here, which is probably one of the most striking characteristics of L ( V ∞ ), falls into this category. We say M ∈ L ( V ∞ ) is supermaximal if dim( V ∞ / M ) = ∞ and for all W ∈ L ( V ∞ ), if W ⊃ M then dim( W / M ) W = V ∞ . These subspaces were discovered by Kalantari and Retzlaff [13].

3088 varieties: A solution to the Ackermann constant problem

Abstract: It is shown that there are exactly six normal DeMorgan monoids generated by the idntity element alone. The free DeMorgan monoid with no generators but the identity is characterised and shown to have exactly three thousand and eighty-eight elements. This result solves the “Ackermann constant problem” of describing the structure of sentential constants in the logic R .

The Completeness of a Predicate-Functor Logic

Abstract: Predicate-functor logic, as founded by W. V. Quine ([1960], [1971], [1976], [1981]), is first-order predicate logic without individual variables. Instead, adverbs or predicate functors make explicit the permutations and replications of argumentplaces familiarly indicated by shifting variables about. For the history of this approach, see Quine [1971, 309ff.]. With the evaporation of variables, individual constants naturally assimilate to singleton predicates or adverbs, leaving no logical subjects whatever of type 0. The orphaned "predicates" may then be taken simply as terms in the sense of traditional logic: class and relational terms on model-theoretic semantics, schematic terms on Quine's denotational or truth-of semantics. Predicate-functor logic thus stands forth as the pre-eminent first-order term logic, as distinct from propositional-quantificational logic. By the same token, it might with some justification qualify as "first-order combinatory logic", with allowance for some categorization of the sort eschewed in general combinatory logic, the ultimate term logic. Over the years, Quine has put forward various choices of primitive predicate functors for first-order logic with or without the full theory of identity. Moreover, he has provided translations of quantificational into predicate-functor notation and vice versa ([1971, 312f.], [1981, 651]). Such a translation does not of itself establish semantic completeness, however, in the absence of a proof that it preserves deducibility. An axiomatization of predicate-functor logic was first published by Kuhn [1980], using primitives rather like Quine's. As Kuhn noted, "The axioms and rules have been chosen to facilitate the completeness proof" [1980, 153]. While this expedient simplifies the proof, however, it limits the depth of analysis afforded by the axioms and rules. Mindful of this problem, Kuhn ([1981] and [1983]) boils his axiom system down considerably, correcting certain minor slips in the original paper. At about the same time, apparently, Egli developed Gentzen and Hilbert-style formulations of predicate-functor logic [1979], which were proved complete by Kndpfler [1979] and Zimmermann [1979] respectively. An interesting feature of Egli and Kndpfler's treatment is their representation of singular terms by one-place predicate functors. This approach is extended to operation symbols by Grfinberg

Church-Rosser theorem for typed functional systems

Abstract: Introduction. This paper contains a new proof of the Church-Rosser theorem for the typed λ-calculus, which also applies to systems with infinitely long terms. The ordinary proof of the Church-Rosser theorem for the general untyped calculus goes as follows (see [1]). If is the binary reduction relation between the terms we define the one-step reduction 1 in such a way that the following lemma is valid. Lemma. For all terms a and b we have: a b if and only if there is a sequence a = a 0 , …, a n = b, n ≥ 0, such that a i i a i + 1 for 0 ≤ i n . We then prove the Church-Rosser property for the relation 1 by induction on the length of the reductions. And by combining this result with the above lemma we obtain the Church-Rosser theorem for the relation . Unfortunately when we come to infinite terms the above lemma is not valid anymore. The difficulty is that, assuming the hypothesis for the infinitely many premises of the infinite rule, there may not exist an upper bound for the lengths n of the sequences a i = a 0 , …, a n = b i ( i α ); cf. the infinite rule (iv) in §6. A completely new idea in the case of the typed λ -calculus would be to exploit the type structure in the way Tait did in order to prove the normalization theorem. In this we succeed by defining a suitable predicate, the monovaluedness predicate, defined over the type structure and having some nice properties. The key notion permitting to define this predicate is the notion of I-form term (see below). This Tait-type proof has a merit, namely that it can be extended immediately to the case of infinite terms.

Decision Problems Concerning $S$-Arithmetic Groups

Abstract: This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work. We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”. (1) Isomorphism problems . Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z -generated modules over a fixed finitely generated ring}. Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q -algebras.

The <-ordering on normal ultrafilters

Abstract: consistent that K has the maximum possible number of normal ultrafilters. Starting with assumptions stronger than measurability, Mitchell [Mi-i] filled in the gap by constructing models of ZFC + GCH satisfying "there are exactly i normal ultrafilters over K", where i could be K+ or K++ (measured in the model), or anything < K. Whether or not Mitchell's results can be obtained by starting only with a measurable cardinal in the ground model and defining a forcing extension is unknown.

Why some people are excited by Vaught's conjecture

Abstract: §I . In 1961, R. L. Vaught ([V]) asked if one could prove, without the continuum hypothesis, that there exists a countable complete theory with exactly ℵ 1 isomorphism types of countable models. The following statement is known as Vaught conjecture: Let T be a countable theory. If T has uncountably many countable models, then T has countable models . More than twenty years later, this question is still open. Many papers have been written on the question: see for example [HM], [M1], [M2] and [St]. In the opinion of many people, it is a major problem in model theory. Of course, I cannot say what Vaught had in mind when he asked the question. I just want to explain here what meaning I personally see to this problem. In particular, I will not speak about the topological Vaught conjecture, which is quite another issue. I suppose that the first question I shall have to face is the following: “Why on earth are you interested in the number of countable models—particularly since the whole question disappears if we assume the continuum hypothesis?” The answer is simply that I am not interested in the number of countable models, nor in the number of models in any cardinality, as a matter of fact. An explanation is due here; it will be a little technical and it will rest upon two names: Scott (sentences) and Morley (theorem).