Interpolation in extensions of first-order logic

We prove a generalization of Maehara's lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig's interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.Comment: In this up-dated version of the paper a more general notion of singular geometric theory is provided allowing the extension of our interpolation results to further fundamental mathematical theorie

Interpolation in Extensions of First-Order Logic

We prove a generalization of Maehara\u2019s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig\u2019s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders

Equality in the Presence of Apartness: An Application of Structural Proof Analysis to Intuitionistic Axiomatics

The theories of apartness, equality, and n-stable equality are presented through contraction- and cut-free sequent calculi. By methods of proof analysis, a purely proof-theoretic characterization of the equality fragment of apartness is obtained

Intuitionistic Mereology

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import

Intuitionistic set theory

We describe the formal system of higher—order intuitionistic logic with power types and (impredicative) comprehension which provides the basis for our "set theory"; this is adapted from the system of FOURMAN (D.Phil. Thesis, Oxford 1974), and such theories are equivalent to the notion of a topos. We interpret the basic concepts of sets, relations and functions; we consider relations of equality, apartness and order, and develop some of the intuitionistic theory of complete Heyting algebras (cHa's). We give the semantical definitions which make sheaves over a cHa into a model for the formal system. We give a unified description of the Dedekind reals and Baire space as the spaces of models of geometric propositional theories, from which follows the usual characterisation of them as they appear in sheaf models. We investigate some notions from topology, adapting classical definitions and proofs where possible, and using sheaf models as counterexamples, especially for the Cauchy and Dedekind reals. Topologies are given by "open" sets as "closed" ones are unsuitable; strong forms of the basic separation principles arise in the presence of an apartness relation. Well—founded relations are those satisfying the principle of induction, and well—orderings act as the unique representatives of their "ranks". This class of well-orderings has good closure properties, but they need not be linearly ordered nor have Cantor normal forms; the Hartogs' number of an infinite set forms a regular limit well-ordering. We consider some notions of cardinality, in particular the many possible notions of finiteness. For sets with apartness different ones arise; we characterise strongly the sense in which real polynomials have "finitely many" roots

Intuitionistic Mereology

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import

Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)

The Workshop "Mathematical Logic: Proof Theory, Constructive Mathematics" focused on proofs both as formal derivations in deductive systems as well as on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory

Syntactic completeness of proper display calculi

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e. those calculi that support general and modular proof-strategies for cut elimination), and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analiticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e. those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (aka unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so that the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination and subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e. by showing that a (cut free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far this proof strategy for syntactic completeness has been implemented on a case-by-case base, and not in general. In this paper, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut free derivation can be effectively generated which has a specific shape, referred to as pre-normal form.Comment: arXiv admin note: text overlap with arXiv:1604.08822 by other author