Subminimal negation

  title={Subminimal negation},
  author={Almudena Colacito and Dick de Jongh and Ana Lucia Vargas},
  journal={Soft Computing},
Minimal logic, i.e., intuitionistic logic without the ex falso principle, is investigated in its original form with a negation symbol instead of a symbol denoting the contradiction. A Kripke semantics is developed for minimal logic and its sublogics with a still weaker negation by introducing a function on the upward closed sets of the models. The basic logic is a logic in which the negation has no properties but the one of being a unary operator. A number of extensions is studied of which the… 
Contrapositionally Complemented Pseudo-Boolean Algebras and Intuitionistic Logic with Minimal Negation
Two kinds of relational semantics for ILM and ILM-${\vee}$ are obtained, and an inter-translation between the two semantics is provided, and relationships are established between the different algebraic and interpersonal semantics for the logics defined in the work.
A Kuroda-style j-translation
It is shown that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda's negative translation, where the key is to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well.
A Study of Subminimal Logics of Negation and Their Modal Companions
It is shown that there are uncountably many subminimal logics and model-theoretic and algebraic definitions of filtration for minimal logic are dual to each other and these constructions ensure that the propositional minimal logic has the finite model property.
Proof Theory for Positive Logic with Weak Negation
Cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property, which is used to conclude that the considered logical systems are PSPACE-complete.
Subminimal Logics in Light of Vakarelov's Logic
A new infinite class of subsystems of minimal logics is formulated, using the framework of subminimal logics by A. Colacito, D. de Jongh and A. Vargas to investigate a subsystem of minimal logic related to D. Vakarelov's logic.
A note on Humberstone's constant Ω
The main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics.
Universal Models for the Positive Fragment of Intuitionistic Logic
An alternative proof of Jankov's theorem is given stating that the intermediate logic $$\mathsf {KC}$$, the logic of the weak law of excluded middle, is the greatest intermediate logic extending 𝕂(1,2,3,4) that proves exactly the same positive formulas as £1.
The logical challenge of negative theology
  • P. Urbańczyk
  • Philosophy
    Studies in Logic, Grammar and Rhetoric
  • 2018
Abstract In this paper I present four interpretations of so-called negative theology and provide a number of attempts to model this theory within a formal system. Unfortunately, all of them fail in
Subminimal Negation on the Australian Plan
Frame semantics for negation on the Australian Plan accommodates many different negations, but it falls short on accommodating subminimal negation when the language contains conjunction and


Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation
This paper generalizes constructive logic with Nelson negation weakening maximally the underlying intuitionistic negation, and shows 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.
Negation in the Light of Modal Logic
This is a summary of some things that can be said about negation understood as an impossibility operator. To model negation one may use possible-worlds models in the style of Kripke that have an
An algebraic approach to intuitionistic connectives
Abstract. It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to
Substructural fuzzy logics
Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy and completeness is proved for the logic extended with Takeuti and Titani's density rule.
The Universal Model for the negation-free fragment of IPC
We identify the universal n-model of the negation-free fragment of the intuitionistic propositional calculus IPC. We denote it by U(n) and show that it is isomorphic to a generated submodel of the
From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931
The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded
Unification and admissible rules for paraconsistent minimal Johanssons' logic J and positive intuitionistic logic IPC+
On the Principle of Excluded Middle
I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is
Non-Classical Negation in the Works of Helena Rasiowa and Their Impact on the Theory of Negation
The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation,–intuitionistic negation and some of its generalizations: minimal negation of Johansson and semi-negation.
Normal Monomodal Logics Can Simulate All Others
Many old and new results in modal logic can be derived in a straightforward way by simulations of polymodal normal logics, sheding new light on the power of normal monomodal logic.