Constructible Falsity Research Articles

Constructive Logic is Connexive and Contradictory

It is widely accepted that there is a clear sense in which the first-order paraconsistent constructive logic with strong negation of Almukdad and Nelson, QN4, is more constructive than intuitionistic first-order logic, QInt. While QInt and QN4 both possess the disjunction property and the existence property as characteristics of constructiveness (or constructivity), QInt lacks certain features of constructiveness enjoyed by QN4, namely the constructible falsity property and the dual of the existence property. This paper deals with the constructiveness of the contra-classical, connexive, paraconsistent, and contradictory non-trivial first-order logic QC, which is a connexive variant of QN4. It is shown that there is a sense in which QC is even more constructive than QN4. The argument focuses on a problem that is mirror-inverted to Raymond Smullyan’s drinker paradox, namely the invalidity of what will be called the drinker truism and its dual in QN4 (and QInt), and on a version of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations that treats proofs and disproofs on a par. The validity of the drinker truism and its dual together with the greater constructiveness of QC in comparison to QN4 may serve as further motivation for the study of connexive logics and suggests that constructive logic is connexive and contradictory (the latter understood as being negation inconsistent).

Dual counterpart intuitionistic logic

Abstract We introduce dual counterpart intuitionistic logic (or DCInt): a constructive logic that is a conservative extension of intuitionistic logic, a sublogic of bi-intuitionistic logic, has the logical duality property of classical logic, and also retains the modal character of its interpretation of the connective dual to intuitionistic implication. We define its Kripke semantics along with the corresponding notion of a bisimulation, and then prove that it has both the disjunction property and (its dual) the constructible falsity property. Also, for any class $ {\mathcal{C}}$ of Kripke frames from our semantics, we identify a condition such that $ {\mathcal{C}}$ will have the disjunction property if it satisfies the condition. This provides a method for generating extensions of DCInt that retain the disjunction property.

On the Provable Contradictions of the Connexive Logics C and C3

Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive connexive logic C, which is obtained by a simple modification of a system of constructible falsity, namely N4, as well as its non-constructive extension C3. For these systems, various observations concerning provable contradictions are made, using mainly a proof-theoretic approach. The topics covered in this paper include: how new contradictions are found from parts of provable contradictions; how to characterise provable contradictions in C3 that are constructive; how contradictions can be seen from the relative viewpoint of strong implication; and as an appendix an attempt at generating provable contradictions in C3.

Type Theory with Opposite Types: A Paraconsistent Type Theory

Abstract A version of intuitionistic type theory is extended with opposite types, allowing a different formalization of negation and obtaining a paraconsistent type theory ($\textsf{PTT} $). The rules for opposite types in $\textsf{PTT} $ are based on the rules of the so-called constructible falsity. A propositions-as-types correspondence between the many-sorted paraconsistent logic $\textsf{PL}_\textsf{S} $ (a many-sorted extension of López-Escobar’s refutability calculus presented in natural deduction format) and $\textsf{PTT} $ is proven. Moreover, a translation of $\textsf{PTT} $ into intuitionistic type theory is presented and some properties of $\textsf{PTT} $ are discussed.

Constructive truth and falsity in Peano arithmetic

Abstract Artemov (2019, The provability of consistency) offered the notion of constructive truth and falsity of arithmetical sentences in the spirit of Brouwer–Heyting–Kolmogorov semantics and its formalization, the logic of proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano arithmetic. For this purpose, we generalize the constructive falsity to $n$-constructive falsity in Peano arithmetic where $n$ is any positive natural number. Based on this generalization, we also analyse the logical status of well-known Gödelean sentences: consistency assertions for extensions of PA, the local reflection principles, the ‘constructive’ liar sentences and Rosser sentences. Finally, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor $n$-constructively false for any $n$.

Explicit analyses of proof/refutation interaction for constructible falsity and Heyting–Brouwer logic

Abstract Nelson’s logic of constructible falsity $\textsf{N}$ and Rauszer’s Heyting–Brouwer logic $\textsf{HB}$ are well-known cases of extensions of intuitionistic logic $\textsf{Int}$ enriched with novel connectives. Wansing has suggested that Gödel’s provability interpretation of $\textsf{Int}$ can be extended to these systems by pairing the category of formal proofs with a distinct category of formal refutations. In this paper, we extend the framework of Artemov’s justification logic to provide explicit analyses of $\textsf{N}$ and $\textsf{HB}$ (and the dual-intuitionistic logic $\textsf{DualInt}$) that respect a distinction between proofs and refutations. The application distinguishes the categories by reinterpreting the agents of multiple-agent justification logic as devices that operate exclusively on one or the other category. The analyses reveal that differences between $\textsf{N}$ and $\textsf{HB}$ can be reduced to competing interaction principles characterizing the coordination between proofs and refutations. We conclude by reappraising some of the unusual features of $\textsf{HB}$ in light of the explicit analysis of $\textsf{HB}$.

A bilateral Hilbert-style investigation of 2-intuitionistic logic

AbstractWe develop a bilateral Hilbert-style calculus for 2-intuitionistic logic of Heinrich Wansing. This calculus is defined over signed formulas of two types: formulas signed with plus correspond to assertions, while formulas signed with minus correspond to rejections. In this way, the provided system is a Hilbert-style calculus, which does take rejection seriously by considering it a primitive notion on par with assertion. We show that this presentation is not trivial and provide two equivalent axiomatizations obtained by extending intuitionistic and dual intuitionistic logics, respectively. Finally, we show that 2-intuitionistic logic is in some sense definitionally equivalent to a variant of Nelson’s logic with constructible falsity.

Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯ , the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson’s paraconsistent N4, this also supplements Nelson’s own proof of the Post consistency of N4♯ . Insofar as the present technique allows infinite models, this resolves Nelson’s concern that N4♯ is of interest only to those accepting that there are finitely many natural numbers.

A decidable paraconsistent relevant logic: Gentzen system and Routley‐Meyer semantics

In this paper, the positive fragment of the logic of contraction‐less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four‐valued logic . This extended relevant logic is called , and it has the property of constructible falsity which is known to be a characteristic property of . A Gentzen‐type sequent calculus for is introduced, and the cut‐elimination and decidability theorems for are proved. Two extended Routley‐Meyer semantics are introduced for , and the completeness theorems with respect to these semantics are proved.

Representation of interlaced trilattices

Trilattices are algebraic structures introduced ten years ago into logic with the aim to provide a uniform framework for the notions of constructive truth and constructive falsity. In more recent years, trilattices have been used to introduce a number of many-valued systems that generalize the Belnap–Dunn logic of first-degree entailment, proposed as logics of how several computers connected together in a network should think in order to deal with incomplete and possibly contradictory information. The aim of the present work is to develop a first purely algebraic study of trilattices, focusing in particular on the problem of representing certain subclasses of trilattices as special products of bilattices. This approach allows to extend the known representation results for interlaced bilattices to the setting of trilattices and to reduce many algebraic problems concerning these new structures to the better-known framework of lattice theory.

Negative facts and constructible falsity

We formalise the notion of negative facts due to Russell in terms of the logic of constructible falsity. We generalise Fraassen's theory of facts so that we can deal with a positive fact as a true proposition and a negative fact as a false proposition in a constructive manner. We provide some adequacy results for the revised van Fraassen semantics with respect to several versions of Kripke semantics for constructible falsity.

Modal Operators over Constructive Logic

The article deals with modal extensions of Nelson's constructive logic with strong negation N4. Different logics are constructed with respect to the notions of semantical and formal dualities introduced in Odintsov and Wansing (2004, First-Order Logic Revisited, 269–286) and to a new notion of negative semantical duality. Soundness and completeness are proved for each logic. It is also shown that all the logics are conservative extensions of N4, and they possess the disjunction property and the constructible falsity property.

Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.

The Trilattice of Constructive Truth Values

We introduce an abstract algebraic structure – a lattice defined on a generalized truth value space of constructive logic. For background one can refer to the idea of ‘under‐determined’ and ‘over‐determined’ valuations (Dunn), a ‘useful four‐valued logic’ (Belnap), and the notion of a bilattice (Ginsberg). We consider within one general framework the notions of constructive truth and constructive falsity, as well as the notions of non‐constructive truth and non‐constructive falsity. All possible combinations of the basic truth values give rise to an interesting ‘16‐valued logic’. It appears that these 16 truth values constitute what we call a trilattice – a natural mathematical structure with three partial orderings that represent respectively an increase in information, truth and constructivity. The presentation of the paper is essentially conceptual: the stress is laid on introducing new concepts and structures as well as on their general interpretation.

Partiality and Its Dual

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including Łukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".

Partial Up and Down Logic

This paper presents logics for reasoning about extension and reduction of partial information states. This enterprise amounts to nonpersistent variations of certain constructive logics, in particular the so-called logic of constructible falsity of Nelson. We provide simple semantics, sequential calculi, completeness and decidability proofs.

A Note on the Proof Method for Constructive Falsity

A Note on the Proof Method for Constructive Falsity

On the Proof Method for Constructive Falsity

On the Proof Method for Constructive Falsity

Refutability and elementary number theory

It would thus appear that there is not much difference in the case of elementary number theory whether one uses for the underlying logic intuitionism, constructible falsity or refutability. When the domain of discourse is the species of natural numbers, mathematical induction is a very powerful principle which overcomes any deficencies which may be present. Thus if one wishes to obtain a separation of the concepts of intuitionism, constructible falsity and refutability then either one has to give up mathematical induction or else consider some species other than the natural numbers. The latter would probably be more interesting.

On constructible falsity in the constructive logic with strong negation

On constructible falsity in the constructive logic with strong negation