Rosalie Iemhoff - Academia.edu (original) (raw)

Uploads

Papers by Rosalie Iemhoff

Logic group preprint series, 2000

We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conj... more We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give s emantic criteria for admissibility.

Logic group preprint series, Apr 3, 2012

This paper contains a proof theoretic treatment of some aspects of unification in intermediate lo... more This paper contains a proof theoretic treatment of some aspects of unification in intermediate logics. It is shown that many existing results can be extended to fragments that at least contain implication and conjunction. For such fragments the connection between valuations and most general unifiers is clarified, and it is shown how from the closure of a formula under the Visser rules a proof of the formula under a projective unifier can be obtained. This implies that in the logics considered, for the n-unification type to be finitary it suffices that the m-th Visser rule is admissible for a sufficiently large m. At the end of the paper it is shown how these results imply several well-known results from the literature.

FLAP, 2017

It is shown that no intermediate predicate logic that is sound and complete with respect to a cla... more It is shown that no intermediate predicate logic that is sound and complete with respect to a class of frames, admits a strict alternative Skolemization method. In particular, this holds for intuitionistic predicate logic and several other well– known intermediate predicate logics. The result is proved by showing that the class of formulas without strong quantiﬁers as well as the class of formulas without weak quantiﬁers is sound and complete with respect to the class of constant domain Kripke models

Interpolation has been studied in a variety of settings since William Craig proved that classical... more Interpolation has been studied in a variety of settings since William Craig proved that classical predicate logic has interpolation in 1957. Interpolation is considered by many to be a “good ” property because it indicates a certain well-behavedness of the logic, vaguely reminiscent to analycity. In 1992 it was proved by Andrew Pitts that intuitionistic propositional logic IPC, which has interpola-tion, also satisfies the stronger property of uniform interpolation: given a formula ϕ and an atom p, there exist uniform interpolants ∀pϕ and ∃pϕ which are for-mulas (in the language of IPC) that do not contain p and such that for all ψ not containing p: ` ϕ → ψ ⇔ ` ∃pϕ → ψ ` ψ → ϕ ⇔ ` ψ → ∀pϕ. This is a strengething of interpolation in which the interpolant only depends on the premiss (in the case of ∃) or the conclusion (in the case of ∀) of the given implication. As the notation suggests, the fact that the uniform interpolants are definable in IPC also shows that the propositional quan...

For Dick de Jongh, on the occasion of his 65th birthday. If Visser’s rules are admissible for an ... more For Dick de Jongh, on the occasion of his 65th birthday. If Visser’s rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of Visser’s rules are admissible is not known. Here we study the situation for specific intermediate logics. We provide natural examples of logics for which Visser’s rule are derivable, admissible but non-derivable, or not admissible.

This note consists of a collection of observations on the notion of simplicity in the setting of ... more This note consists of a collection of observations on the notion of simplicity in the setting of proofs. It discusses its properties under formalization and its relation to the length of proofs, showing that in certain settings simplicity and brevity exclude each other. It is argued that when simplicity is interpreted as purity of method, different foundational standpoints may affect which proofs are considered to be simple and which are not.

Lecture Notes in Computer Science, 2011

Introduction to the constructive point of view in the foundations of mathematics, in particular i... more Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

This paper contains a proof-theoretic account of unification in (fragments of) transitive reflexi... more This paper contains a proof-theoretic account of unification in (fragments of) transitive reflexive modal logics, which means that the reasoning is syntactic and uses as little semantics as possible. New proofs of theorems on unification types are given and these results are extended to fragments. It is shown that transitive reflexive fragments that contain at least implication and conjunction have finitary unification, and that in the absence of disjunction they have unitary unification. The relation between classical valuations and projective unifiers is clarified, and it is shown that also in fragments the Visser rules form a basis for the admissible rules once they are admissible.

We prove a quadratic upper bound for the depth of cut free proofs in propositional intuitionistic... more We prove a quadratic upper bound for the depth of cut free proofs in propositional intuitionistic logic formalized with Gentzen's sequent calculus. We discuss bounds on the necessary number of reuses of left implication rules. We exhibit an example showing that this quadratic bound is optimal. As a corollary, this gives a new proof that propositional validity for intuitionistic logic is in PSPACE.

Logic Journal of the IGPL, Nov 11, 2013

arXiv (Cornell University), May 18, 2023

Annals of Pure and Applied Logic, Dec 1, 2021

Mathematical Logic Quarterly, May 1, 2003

In this paper we study the modal behavior of ¦-preservativity, an extension of provability which ... more In this paper we study the modal behavior of ¦-preservativity, an extension of provability which is equivalent to interpretability for classical superarithmetical theories. We explain the connection between the principles of this logic and some well-known properties of À , like the disjunction property and its admissible rules. We show that the intuitionistic modal logic given by the preservativity principles of À known so far, is complete with respect to a certain class of frames.

Annals of Pure and Applied Logic, Oct 1, 2006

In this paper an alternative Skolemization method is introduced that for a large class of formula... more In this paper an alternative Skolemization method is introduced that for a large class of formulas is sound and complete with respect to intuitionistic logic. This class extends the class of formulas for which standard Skolemization is sound and complete and includes all formulas in which all strong quantifiers are existential. The method makes use of an existence predicate first introduced by Dana Scott.

arXiv (Cornell University), Aug 9, 2022

arXiv (Cornell University), Aug 10, 2022

Logic group preprint series, 1999