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Hrafn Valtýr Oddsson

Publications and source records attributed to Hrafn Valtýr Oddsson.

5 recordsLinked to original sources
arXiv2026

Constructive proofs for the standard translation of many-sorted to unsorted predicate logic

It is well known that many-sorted logic can be reduced to unsorted first-order logic by adding predicates for each sort, relativizing quantifiers to these predicates, and adding appropriate axioms governing their behavior. Existing constructive proofs for the correctness of this translation break down when the many-sorted language includes equality and the unsorted target calculus includes the usual rules/axioms for equality. We give an elementary proof in the form of an effective procedure that closes this gap. As an application, we give a fully syntactic justification of van Dalen's translation of second-order logic into unsorted first-order logic. We also give a new proof for a claim made by Herbrand in his 1930 dissertation that, in the equality-free case, a sentence is derivable in many-sorted logic iff it is derivable in unsorted logic. Our proof avoids the heavy machinery of later proofs by Schmidt and Wang.

math.LO↗
arXiv2026

Cardinality in a paraconsistent and paracomplete set theory

This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that ``$x\in A$'' is neither true nor false for some $x$). We carefully analyze what it means for two potentially incomplete or inconsistent sets to have ``the same size'', construct the corresponding cardinal numbers, and develop the basic theory of cardinal arithmetic. A surprising result is that the cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, our cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals.

math.LO↗
arXiv2025

Proof-Theoretic Functional Completeness for the Connexive Logic C

We show the functional completeness for the connectives of the non-trivial negation inconsistent logic C by using a well-established method implementing purely proof-theoretic notions only. Firstly, given that C contains a strong negation, expressing a notion of direct refutation, the proof needs to be applied in a bilateralist way in that not only higher-order rule schemata for proofs but also for refutations need to be considered. Secondly, given that C is a connexive logic we need to take a connexive understanding of inference as a basis, leading to a different conception of (higher-order) refutation than is usually employed.

cs.LO↗
arXiv2023

Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell's paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ZFC to enable the development of interesting mathematics. We propose an axiomatic system BZFC, obtained by analysing the ZFC-axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set. Our theory is naturally bi-interpretable with ZFC, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel. Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory.

math.LO↗