Wim Ruitenburg

Basic Propositional Calculus I

We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If B is a formula such that (T B) B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula (T B) B is derivable exactly when B is provably equivalent to a formula of the form ((T A) A) (T A).

Kripke submodels and universal sentences

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including ⊤ and ⊥) using ∧, ∨, ∀, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model-consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Period of Sequences

Dans le calcul propositionnel classique pour toute proposition A(p) on a: A(p)↔A 3 (p). On etudie ce qui reste de ceci dans le cas intruitionniste. On montre que pour toute proposition A(p) on a: il y a un n∈N tel que A n (p)↔A n+2 (p)

(A^n(p))

Abstract. Let ℒ and ? be propositional languages over Basic Propositional Calculus, and ℳ = ℒ∩?. Weprove two different but interrelated interpolation theorems. First, suppose that Π is a sequent theory over ℒ, and Σ∪ {C⇒C′} is a set of sequents over ?, such that Π,Σ⊢C⇒C′. Then there is a sequent theory Φ over ℳ such that Π⊢Φ and Φ, Σ⊢C⇒C′. Second, let A be a formula over ℒ, and C1, C2 be formulas over ?, such that A∧C1⊢C2. Then there exists a formula B over ℳ such that A⊢B and B∧C1⊢C2.

in Intuitionistic Propositional Calculus

We characterize the first-order formulas with one free variable that are preserved under bisimulation and persistence or strong persistence over the class of Kripke models with transitive frames and unary persistent predicates.

Basic Propositional Calculus II. Interpolation

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω

Basic Logic, K4, and Persistence

From classical, Fraisse-homogeneous, (≤ ω)-categorical theories over finite relational languages (which we refer to as JRS theories), we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. The technique we use considers Kripke models as functors from a small category to the category of L-structures with morphisms, rather than the usual interpretation wherein the frame of a Kripke model is a partial order. While one can always “unravel” a functor Kripke model to obtain a partial order Kripke model with the same intuitionistic theory, our technique is perhaps an easier way to consider a Kripke model that includes a single classical node structure and all of the endomorphisms of that classical JRS structure. We also determine the intuitionistic universal fragments of these theories, in accordance with the hierarchy of intuitionistic formulas put forth in [9] and expounded on by Fleischmann in [11]. This portion of the thesis (Chapter 1) is the result of joint work with Ben Ellison, Jonathan Fleischmann, and Wim Ruitenburg, as published (up to minor structural changes) in [10]. Given a classical JRS theory, we determine axiomatizations of the corresponding intuitionistic theory in Chapter 2. We first do so by axiomatizing properties apparent from the behavior of the model, and discuss improvements to that axiom system. We then present another axiomatization, this time by axiomatizing the properties of quantifier elimination. We discuss improvements to this system, and show how this system and various subsystems thereof are equivalent to our first axiomatization and corresponding subsystems thereof.

Notions of Relative Ubiquity for Invariant Sets of Relational Structures

We generalize the double negation construction of Boolean algebras in Heyting algebras, to a double negation construction of the same in Visser algebras (also known as basic algebras). This result allows us to generalize Glivenko’s Theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras. Mathematics Subject Classification: Primary: 03G05, 03G25, 06D20, secondary: 03B20.

QUANTIFIER ELIMINATION FOR A CLASS OF INTUITIONISTIC THEORIES

Let k be a discrete field. A k-algebra is a ring A that is also a vector space over k, satisfying λ(ab) = (λa)b = a(λb) for each λ in k and a, b in A. If A and B are k-algebras, then a homomorphism from A to B is a ring homomorphism that is also a k-linear transformation. The term finite dimensional, when applied to a structure S that is a vector space over k, like a k-algebra, signifies that S is a finite-dimensional vector space over k.

Boolean Algebras in Visser Algebras

We present an overview of the unintended interpretations of intuitionistic logic that arose after Heyting formalized the “observed regularities” in the use of formal parts of language, in particular, first-order logic and Heyting Arithmetic. We include unintended interpretations of some mild variations on “official” intuitionism, such as intuitionistic type theories with full comprehension and higher order logic without choice principles or not satisfying the right choice sequence properties. We conclude with remarks on the quest for a correct interpretation of intuitionistic logic.