Interpreting Intuitionistic Propositional Logic in Terms of Intuitionistic Protothetics

An algebra of sentences of the quite intuitionistic protothetics, that is, an intuitionistic propositional logic with quantifiers augmented by the negation of the excluded middle, is a faithful model of intuitionistic propositional logic.     

A short proof of Glivenko theorems for intermediate predicate logics

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic.     

Inhabitation of polymorphic and existential types

This paper shows that the inhabitation problem in the lambda calculus with negation, product, polymorphic, and existential types is decidable, where the inhabitation problem asks whether there exists some term that belongs to a given type. In order to do that, this paper proves the decidability of the provability in the logical system defined from the second-order natural deduction by removing implication and disjunction. This is proved by showing the quantifier elimination theorem and reducing the problem to the provability in propositional logic. The magic formulas are used for quantifier elimination such that they replace quantifiers. As a byproduct, this paper also shows the second-order witness theorem which states that a quantifier followed by negation can be replaced by a witness obtained only from the formula. As a corollary of the main results, this paper also shows Glivenko’s theorem, Double Negation Shift, and conservativity for antecedent-empty sequents between the logical system and its classical version.     

Glivenko like theorems in natural expansions of BCK‐logic

The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK‐logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK‐logic with negation by a family of connectives implicitly defined by equations and compatible with BCK‐congruences. Many of the logics in the current literature are natural expansions of BCK‐logic with negation. The validity of the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one‐variable formula in the language of BCK‐logic with negation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A complete many-valued logic with product-conjunction

A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Gödel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.This article was processed by the author using the LATEX style filepljorlm from Springer-Verlag.     

Iterated team semantics for a hierarchy of informational types

In this paper, we introduce and study a framework that is inspired by the team semantics for propositional dependence logic but deviates from it in several respects. Most importantly, instead of the two semantic layers used in dependence logic – possible worlds and teams – a whole hierarchy of contexts is introduced and different types of formulas are evaluated at different levels of this hierarchy. This leads to a rich stratification of informational types. In this framework, the dependence operator of dependence logic can be defined by the standard propositional connectives (negation, conjunction, disjunction and implication). We explore the formal aspects of this approach and apply it to a number of puzzling phenomena related to modalities and conditionals.     

An anytime deduction algorithm for the probabilistic logic and entailment problems

We study two basic problems of probabilistic reasoning: the probabilistic logic and the probabilistic entailment problems. The first one can be defined as follows. Given a set of logical sentences and probabilities that these sentences are true, the aim is to determine whether these probabilities are consistent or not. Given a consistent set of logical sentences and probabilities, the probabilistic entailment problem consists in determining the range of the possible values of the probability associated with additional sentences while maintaining a consistent set of sentences and probabilities.This paper proposes a general approach based on an anytime deduction method that allows the follow-up of the reasoning when checking consistency for the probabilistic logic problem or when determining the probability intervals for the probabilistic entailment problem. Considering a series of subsets of sentences and probabilities, the approach proceeds by computing increasingly narrow probability intervals that either show a contradiction or that contain the tightest entailed probability interval. Computational experience have been conducted to compare the proposed anytime deduction method, called ad-psat with an exact one, psatcol, using column generation techniques, both with respect to the range of the probability intervals and the computing times.     

Continuous L-domains in logical form

We introduce a framework of approximable disjunctive propositional logic, which is the logic that results from a disjunctive propositional logic by adding an additional connective. The Lindenbaum algebra of this logic is an approximable dD-algebra. We show that for any approximable dD-algebra, its approximable filters ordered by set inclusion form a continuous L-domain. Conversely, every continuous L-domain can be represented as an approximable dD-algebra. Moreover, we establish a categorical equivalence between the category of approximable dD-algebras with approximable dD-algebra morphisms and that of continuous L-domains with Scott-continuous functions. This extends Abramsky's Domain Theory in Logical Form to the world of continuous L-domains. As an application, we give an affirmative answer to an open problem of Chen and Jung.     

Ockham’s Calculus of Strict Implication

In his main work Summa Logicae written around 1323, William of Ockham developed a system of propositional modal logic which contains almost all theorems of a modern calculus of strict implication. This calculus is formally reconstructed here with the help of modern symbols for the operators of conjunction, disjunction, implication, negation, possibility, and necessity.     

A Shared Framework for Consequence Operations and Abstract Model Theory

In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically.     

经典命题逻辑中的近似推理与强近似推理

基于命题逻辑公式之间相似度的概念,在经典命题逻辑系统中提出了两种近似推理模式,得到了它们的一些基本性质。讨论了这两种推理模式与经典推理模式之间的关系。此外,当理论Γ有限时,本文分别给出Γ在α程度上近似推出公式A和理论Σ的充要条件。最后,我们给出公式A与有限理论Γ的结论集D(Γ)之间距离的计算公式。     

Bilattice logic of epistemic actions and knowledge

Baltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective.     

Hypersequent calculi for intuitionistic logic with classical atoms

We discuss a propositional logic which combines classical reasoning with constructive reasoning, i.e., intuitionistic logic augmented with a class of propositional variables for which we postulate the decidability property. We call it intuitionistic logic with classical atoms. We introduce two hypersequent calculi for this logic. Our main results presented here are cut-elimination with the subformula property for the calculi. As corollaries, we show decidability, an extended form of the disjunction property, the existence of embedding into an intuitionistic modal logic and a partial form of interpolation.     

Algorithms for generating arguments and counterarguments in propositional logic

A common assumption for logic-based argumentation is that an argument is a pair 〈Φ,α〉 where Φ is minimal subset of the knowledgebase such that Φ is consistent and Φ entails the claim α. Different logics provide different definitions for consistency and entailment and hence give us different options for formalising arguments and counterarguments. The expressivity of classical propositional logic allows for complicated knowledge to be represented but its computational cost is an issue. In previous work we have proposed addressing this problem using connection graphs and resolution in order to generate arguments for claims that are literals. Here we propose a development of this work to generate arguments for claims that are disjunctive clauses of more than one disjunct, and also to generate counteraguments in the form of canonical undercuts (i.e. arguments that with a claim that is the negation of the conjunction of the support of the argument being undercut).     

The Skolemization of prenex formulas in intermediate logics

The skolem class of a logic consists of the formulas for which the derivability of the formula is equivalent to the derivability of its Skolemization. In contrast to classical logic, the skolem classes of many intermediate logics do not contain all formulas. In this paper it is proven for certain classes of propositional formulas that any instance of them by (independent) predicate sentences in prenex normal form belongs to the skolem class of any intermediate logic complete with respect to a class of well-founded trees. In particular, all prenex sentences belong to the skolem class of these logics, and this result extends to the constant domain versions of these logics.     

A Sahlqvist theorem for distributive modal logic

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.     

A Square of Oppositions in Intuitionistic Logic with Strong Negation

In this paper, we introduce a Hilbert style axiomatic calculus for intutionistic logic with strong negation. This calculus is a preservative extension of intuitionistic logic, but it can express that some falsity are constructive. We show that the introduction of strong negation allows us to define a square of opposition based on quantification on possible worlds.     

Disturbing Fuzzy Propositional Logic and its Operators

In this paper, the concept of disturbing fuzzy propositional logic is introduced, and the operators of disturbing fuzzy propositions is defined. Then the 1-dimensional truth value of fuzzy logic operators is extended to be two-dimensional operators, which include disturbing fuzzy negation operators, implication operators, “and” and “or” operators and continuous operators. The properties of these logic operators are studied.     

Realisability for infinitary intuitionistic set theory

We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. Finally, we use a variant of our notion of realisability to show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.     

On a New Idiom in the Study of Entailment

This paper is an experiment in Leibnizian analysis. The reader will recall that Leibniz considered all true sentences to be analytically so. The difference, on his account, between necessary and contingent truths is that sentences reporting the former are finitely analytic; those reporting the latter require infinite analysis of which God alone is capable. On such a view at least two competing conceptions of entailment emerge. According to one, a sentence entails another when the set of atomic requirements for the first is included in the corresponding set for the other; according to the other conception, every atomic requirement of the entailed sentence is underwritten by an atomic constituent of the entailing one. The former conception is classical on the twentieth century understanding of the term; the latter is the one we explore here. Now if we restrict ourselves to the formal language of the propositional calculus, every sentence has a finite analysis into its conjunctive normal form. Semantically, then, every sentence of that language can be represented as a simple hypergraph, H, on the powerset of a universe of states. Entailment of the sort we wish to study can be represented as a known relation, subsumption between hypergraphs. Since the lattice of hypergraphs thus ordered is a DeMorgan lattice, the logic of entailment thus understood is the familiar system, FDE of first-degree entailment. We observe that, extensionalized, the relation of subsumption is itself a DeMorgan Lattice ordered by higher-order subsumption. Thus the semantic idiom that hypergraph-theory affords reveals a hierarchy of lattices capable of representing entailments of every finite degree.