Abstract. The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to metavariables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation

This paper is an investigation into the logic of information flow. The basic

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We present well-ordering proofs for Martin-Lof's type theory with W-type and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 -CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...

We investigate the defining power of stratified and hierarchical logic programs. As an example for the treatment of negative information in the context of these structured programs we also introduce a stratified and hierarchical closed-world assumption. Our analysis tries to relate the defining power of stratified and hierarchical programs (with and without an appropriate closed-world assumption) very precisely to notions and hierarchies in classical definability theory. Stratified and hierarchical logic programs are two well-known and typical candidates of what one may more generally denote as structured programs. In both cases we have to deal with normal logic programs which satisfy certain syntactic conditions with respect to the occurrence of negative literals. Recently they have gained a lot of importance in connection with the search for nice declarative semantics for logic programs and the treatment of negative information in logic programming (e.g., Lloyd [10]). Stratified programs were introduced into logic programming by Apt, Blair, and Walker [2] and van Gelder [17] not long ago. In mathematical logic, however, theories of this kind have been studied for more than 20 years under the general theme of iterated inductive definability. Indeed, stratified programs can be understood as systems for (finitely) iterated inductive definitions where the definition clauses are of very low logical complexity. The notion of hierarchical program (e.g., Clark [6], Shepherdson [15]), on the other hand, is motivated by database theory and tries to reflect the idea of iterated explicit definability by simple principles. From a conceptual point of view we are interested in the relationship between logic programming, inductive definability and equational definability. By making u...

Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.

We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rst-order part is a boolean combination of existential formulas, and (ii) over nite successor structures, the Kolaitis-Thakur hierarchy of minimization problems collapses completely and the Kolaitis-Thakur hierarchy of maximization problems collapses partially. The normal form theorem for 1 2 fails if 1 2 is replaced with 1 1 or if innite structures are allowed. 1 Introduction We consider second-order logic with equality (unless otherwise stated explicitly) and without function symbols of positive arity. Predicates are denoted by capitals and individual variables by lower case letters; a bold face version of a letter denotes a tuple of corresponding symbols. For brevity, we say that a formula reduces t...

(2), W (2) and I (3). To make it easier to remember the meaning of the symbols, we give the following hints: r is the (unique) element of an identity type I; n k is the nth element of the finite type N k with k elements, C the Casedistinction for this type; O is the zero, S the Successor, P Primitive recursion or induction over the natural numbers N ; i stands for left inclusion, j for right inclusion, D is the choice in the type A +B of disjoint union of A and B; p 0 and p 1 are the projections, p the pairing for the #-typ

. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the so-called Anti-Foundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...

We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-Lof type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-Lof type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.