Intuitionistic Negation Research Articles

Modalities combining two negations

Abstract In the context of positive logic expanded with the dual of intuitionistic negation, obtaining intuitionistic negation itself as a consequence, those two connectives allow to introduce modal concepts such as necessity and possibility. We study the resulting modal logic, paying attention to different extensions of it. We provide a syntactic approach and both algebraic and Kripke semantics.

Double Negation as Minimal Negation

N. Kamide introduced a pair of classical and constructive logics, each with a peculiar type of negation: its double negation behaves as classical and intuitionistic negation, respectively. A consequence of this is that the systems prove contradictions but are non-trivial. The present paper aims at giving insights into this phenomenon by investigating subsystems of Kamide’s logics, with a focus on a system in which the double negation behaves as the negation of minimal logic. We establish the negation inconsistency of the system and embeddability of contradictions from other systems. In addition, we attempt at an informational interpretation of the negation using the dimathematical framework of H. Wansing.

Paraconsistent da Costa Weakening of Intuitionistic Negation: What does it mean?

In this paper we consider the systems of weakening of intuitionistic negation logic mZ, introduced in [1], [2], which are developed in the spirit of da Costa's approach. We take a particular attention on the philosophical considerations of the paraconsistent mZ logic w.r.t. the constructive semantics of the intuitionistic logic, and we show that mZ is a subintuitionistic logic. Hence, we present the relationship between intuitionistic and paraconsistent subintuitionistic negation used in mZ. Then we present a significant number of examples for this subintuitionistic and paraconsistent mZ logics: Logic Programming with Fiting's fixpoint semantics for paraconsistent weakening of 3-valued Kleene's and 4-valued Belnap's logics. Moreover, we provide a canonical construction of infinitary-valued mZ logics and, in particular, the paraconsistent weakening of standard Zadeh's fuzzy logic and of the Godel-Dummet t-norm intermediate logics.

Equality and Apartness in Bi-intuitinistic Logic

In the present paper we argue that a symmetric picture of the relationships between equality and apartness can be attained by considering these notions on the background of bi-intuitionistic logic instead of the usual intuitionistic logic. In particular we show that, as the intuitionistic negation of a relation of apartness is an equality, the dualintuitionistic co-negation of an equality is a relation of apartness. At the same time, as the intuitionistic negation of equality is not an apartness, the co-intuitionistic negation of an apartness is not an equality.

On the relation of negations in Nelson algebras

The aim of this paper is to investigate the relation between the strong and the "weak" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.

A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3$$_{\text {DH}}$$

The logic DHb is the result of extending Sylvan and Plumwood’s minimal De Morgan logic B $$_{\text {M}}$$ with a dual intuitionistic negation of the type Sylvan defined for the extension CC $$\omega $$ of da Costa’s paraconsistent logic C $$\omega $$ . We provide Routley–Meyer ternary relational semantics with a set of designated points for DHb and a wealth of its extensions included in G3 $$_{\text {DH}}$$ , the expansion of G3 $$_{+}$$ with a dual intuitionistic negation of the kind considered by Sylvan (G3 $$_{+}$$ is the positive fragment of Godelian 3-valued logic G3). All logics in the paper are paraconsistent.

HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

Logics with Impossibility as the Negation and Regular Extensions of the Deontic Logic D2

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics (see [8] and [9]). In (Došen; [2]) a logic N has been investigated in the language with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation (see [3; p. 300]). In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21 (see [4] and [13]). The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+.

Paraconsistent Double Negations as Classical and Intuitionistic Negations

A classical paraconsistent logic (CP), which is regarded as a modified extension of first-degree entailment logic, is introduced as a Gentzen-type sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained for an intuitionistic paraconsistent logic (IP), and several versions of Glivenko and Godel-Gentzen translation theorems are proved for CP and IP.

Paraconsistent Logic, Evidence, and Justification

In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence (BLE) whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \(\square X\) can be read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense.

The attack as intuitionistic negation

We translate the argumentation networks ${\cal A}=(S, R)$ into a theory $D$ of intuitionistic logic, retaining $S$ as the domain and using intuitionistic negation to model the attack $R$ in ${\cal A}$: the attack is translated to $x\to\neg y$. The intuitionistic models of $D$ characterise the complete extensions of ${\cal A}$. The reduction of argumentation networks to intuitionistic logic yields, in addition to a representation theorem, some additional benefits: it allows us to give semantics to higher level attacks, where an attack $xRy$ can itself attack another attack $uRv$; one can make higher level meta-statements $W$ on $(S, R)$ and such meta-statements can attack and be attacked in the domain.

Substructural Negations

We present substructural negations, a family of negations (or negative modalities) classified in terms of structural rules of an extended kind of sequent calculus, display calculus. In considering the whole picture, we emphasize the duality of negation. Two types of negative modality, impossibility and unnecessity, are discussed and "self-dual" negations like Classical, De Morgan, or Ockham negation are redefined as the fusions of two negative modalities. We also consider how to identify, using intuitionistic and dual intuitionistic negations, two accessibility relations associated with impossibility and unnecessity.

A Modality Called ‘Negation’:

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for (the admissible) negations. The approach unifies in a philosophically motivated picture the following results: nothing can be called a negation properly if it does not satisfy (Minimal) Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds

Co-constructive Logics for Proofs and Refutations

Abstract This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov's logic of problems.

Some Modal Operators Over Intuitionistic Logic

We study a number of modal operators over intuitionistic logic defined as compositions of principal modal operator of logic in consideration with intuitionistic negation. We adopt an approach due to K. Dosen and M. Božic (see [1] and [3]), who introduced four systems of intuitionistic modal logics — HK , HK♦, HK ′ and HK♦′ — each dealing with one of the modal operators of necessity, possibility, un-necessity and impossibility, respectively, as a framework for investigating intuitionistic modal logics. By a composition we understand simply a sequence consisting of occurrences of one of four types of modal operators and intuitionistic negation and we mainly focus on basic compositions, that is compositions of the form ¬δ, δ¬ or ¬δ¬, where δ is the principal modal operator of the logic HKδ. It is well known that such compositions over classical logic can be regarded as natural definitions of four types of modal operators above by means of each other. We investigate which basic compositions yield modal operators of the same type over intuitionistic logic as over classical logic, which is a natural question, as there is no dualities between four types of modal operators over intuitionistic logic. It turns out that five basic compositions behave classically in that sense and for those five compositions we obtain theirs respective axiomatizations. Moreover, we show that logic KC is the smallest superintuitionistic logic over which all twelve basic compositions behave classically. We also investigate the so called Heyting-Kripke logicN∗ [2], which can be obtained by adding to intuitionistic non-modal base modal operator ∼, satisfying exactly the properties of negative operators ′ and ♦′ in logics HK ′ and HK♦′, respectively. We study some basic properties of N∗ such as finite model property, decidability and constructive properties. We then axiomatize two compositions, that is ¬ ∼ and ∼∼, in N∗ as necessity operators. We show that the former composition has an infinite axiomatization and the latter one is axiomatized in the form of logic, modal operator in which combines properties of positive operators and ♦ in logics HK and HK♦, respectively.

On classical behavior of intuitionistic modalities

We study connections between four types of modal operators – necessity, possibility, un-necessity and impossibility – over intuitionitstic logic in terms of compositions of these modal operators with intuitionistic negation. We investigate which basic compositions, i.e. compositions of the form ¬δ, δ¬ or ¬δ¬, yield modal operators of the same type over intuitionistic logic as over classical logic. We say that such compositions behave classically. We study which modal properties correspond to each basic compositions behaving classically over intuitionistic logic and also prove that KC constitutes the smallest superintuitionistic logic over which all basic compositions behave classically.

Composition of an intuitionistic negation and negative modalities as a necessity operator

The main objective of the paper is to axiomatize operators − ¬ and − □′ in logics N* and HK□′, respectively. The result is formulated in terms of normal extensions HKN□′ and HKNR of the logic HK□, which are embedded in corresponding logics via a natural translation. In addition, for the logic HKNR, the finite model property and decidability are established using a hybrid calculus.

Empirical Negation

An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.

Paraconsistent Logic and Weakening of Intuitionistic Negation

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. In an earlier paper [Notre Dame J. Form. Log. 49 (2008), 401–424], we developed the systems of weakening of intuitionistic negation logic, called and , in the spirit of da Costa's approach by preserving, differently from da Costa, the fundamental properties of negation: antitonicity, inversion and additivity for distributive lattices. Taking into account these results, we make some observations on the modified systems of and , and their paraconsistent properties.

Semantically closed intuitionistic abstract logics

Alfred Tarski called a language semantically closed if it contains its own truth predicate and technical means for (self-) reference. We consider the semantically closed non-Fregean logic ∈I, presented in (Lewitzka, 2009, Notre Dame Journal of Formal Logic, 50, 275–301), and introduce a predicate for validity by combining syntactical constructions with a modeltheoretic semantics. The resulting logic, called ∈I+ (Epsilon-I plus), is able to express its own finitary consequence relation. We show that every intuitionistic (or classical) abstract logic extends to a logic which has the semantic features of ∈I (of ∈I+), respectively, namely a truth predicate that satisfies the Tarski biconditionals, a predicate for falsity (as intuitionistic negation), means for propositional (self-) reference, and in the case of ∈I+ also predicates for validity and logical consequence. Applications of our construction to other non-classical abstract logics remain to be further investigated.