Related papers: Following all the rules: Intuitionistic completene…
We study abstract intermediate justification logics, that is arbitrary intermediate propositional logics extended with a subset of specific axioms of (classical) justification logics. For these, we introduce various semantics by combining…
Following G. Mints(Kluwer 2000 and draft 2013), we present terminating and bicomplete proof searches in multi-succedent sequent calculi for intuitionistic propositional logic, fragments of intuitionistic predicate logic and full…
This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a…
We give a calculus for reasoning about the first-order fragment of classical logic that is adequate for giving the truth conditions of intuitionistic Kripke frames, and outline a proof-theoretic soundness and completeness proof, which we…
In Feferman's work, explicit mathematics and theories of generalized inductive definitions play a central role. One objective of this article is to describe the connections with Martin-Lof type theory and constructive Zermelo-Fraenkel set…
We introduce a method of verifying termination of logic programs with respect to concrete queries (instead of abstract query patterns). A necessary and sufficient condition is established and an algorithm for automatic verification is…
Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for…
We obtain modal completeness of the interpretability logics ILP_0 and ILR w.r.t. generalized Veltman semantics. Our proofs are based on the notion of smart (full) labels. We also give shorter proofs of completeness w.r.t. generalized…
Positive logic is a generalisation of full first-order logic that does not have negation built in. Still, many model-theoretic ideas, tools and techniques work perfectly fine in positive logic. Importantly, there is a compactness theorem.…
Predicate intuitionistic logic is a well established fragment of dependent types. According to the Curry-Howard isomorphism proof construction in the logic corresponds well to synthesis of a program the type of which is a given formula. We…
It has been argued that reduction procedures are closely connected to the question about identity of proofs and that accepting certain reductions would lead to a trivialization of identity of proofs in the sense that every derivation of the…
Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a…
In the framework of propositional {\L}ukasiewicz logic, a suitable notion of implicit definability, tailored to the intended real-valued semantics and referring to the elements of its domain, is introduced. Several variants of implicitly…
Einstein's general theory of relativity is the standard theory of gravity, especially where the needs of astronomy, astrophysics, cosmology and fundamental physics are concerned. As such, this theory is used for many practical purposes…
It is shown that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain topological property reminiscent of compactness, the Tarski union property and preservation under…
Research in information systems includes a wide range of approaches which make a contribution in terms of knowledge, understanding, or practical developments. The measure of any research is, ultimately, its validity: are its finding true,…
Generic absoluteness is the phenomenon that certain truths in the set-theoretic universe remain stable under forcing expansions. A classical result by Kripke asserts that every complete Boolean algebra completely embeds into a countably…
Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to…
These lecture notes survey the emerging area of Universal Proof Theory, which investigates general questions about the existence, equivalence, and characterization of good proof systems for broad classes of logics. In particular, the notes…
The concept of informal mathematical proof considered in intuitionism is apparently vulnerable to a version of the liar paradox. However, a careful reevaluation of this concept reveals a subtle error whose correction blocks the…