Strong Completeness Theorem Research Articles

About Logically Probable Sentences

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered.

AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC

Abstract Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$ , which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$ . The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$ , we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$ , giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$ , and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

A probabilistic temporal epistemic logic: Strong completeness

Abstract The paper offers a formalization of reasoning about distributed multi-agent systems. The presented propositional probabilistic temporal epistemic logic $\textbf {PTEL}$ is developed in full detail: syntax, semantics, soundness and strong completeness theorems. As an example, we prove consistency of the blockchain protocol with respect to the given set of axioms expressed in the formal language of the logic. We explain how to extend $\textbf {PTEL}$ to axiomatize the corresponding first-order logic.

A Note on Calculi for Non-deterministic Many-valued Logics

We present two deductively equivalent calculi for non-deterministic many-valued logics. One is defined by axioms and the other – by rules of inference. The two calculi are obtained from the truth tables of the logic under consideration in a straightforward manner. We prove soundness and strong completeness theorems for both calculi and also prove the cut elimination theorem for the calculi defined by rules of inference.

An infinitary propositional probability logic

We introduce a logic for a class of probabilistic Kripke structures that we call type structures, as they are inspired by Harsanyi type spaces. The latter structures are used in theoretical economics and game theory. A strong completeness theorem for an associated infinitary propositional logic with probabilistic operators was proved by Meier. By simplifying Meier’s proof, we prove that our logic is strongly complete with respect to the class of type structures. In order to do that, we define a canonical model (in the sense of modal logics), which turns out to be a terminal object in a suitable category. Furthermore, we extend some standard model-theoretic constructions to type structures and we prove analogues of first-order results for those constructions.

S5-Style Non-Standard Modalities in a Hypersequent Framework

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

Performability of Actions

Action theory may be regarded as a theoretical foundation of AI, because it provides in a logically coherent way the principles of performing actions by agents. But, more importantly, action theory offers a formal ontology mainly based on set-theoretic constructs. This ontology isolates various types of actions as structured entities: atomic, sequential, compound, ordered, situational actions etc., and it is a solid and non-removable foundation of any rational activity. The paper is mainly concerned with a bunch of issues centered around the notion of performability of actions. It seems that the problem of performability of actions, though of basic importance for purely practical applications, has not been investigated in the literature in a systematic way thus far. This work, being a companion to the book as reported (Czelakowski in Freedom and enforcement in action. Elements of formal action theory, Springer 2015), elaborates the theory of performability of actions based on relational models and formal constructs borrowed from formal lingusistics. The discussion of performability of actions is encapsulated in the form of a strict logical system . This system is semantically defined in terms of its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since due to the nature of compound actions the system is not finitary, other semantic variants of are defined. The focus in on the system of performability of finite compound actions. An adequate axiom system for is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is highlighted. The relationship between performability of actions and dynamic logic is also discussed.

Ternary Relational Semantics for the Variants of BN4 and E4 which Contain Routley and Meyer's Logic B

Six interesting variants of the logics BN4 and E4—which can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively—were previously developed in the literature. All these systems are related to the family of relevant logics and contain Routley and Meyer's basic logic B, which is well-known to be specifically associated with the ternary relational semantics. The aim of this paper is to develop reduced general Routley-Meyer semantics for them. Strong soundness and completeness theorems are proved for each one of the logics.

Strategic coalitions in stochastic games

AbstractThe article compares two different approaches of incorporating probability into coalition logics. One is based on the semantics of games with stochastic transitions and the other on games with the stochastic failures. The work gives an example of a non-trivial property of coalition power for the first approach and a complete axiomatization for the second approach. It turns out that the logical properties of the coalition power modality under the second approach depend on whether the modal language allows the empty coalition. The main technical results for the games with stochastic failures are a strong completeness theorem for the logical system without the empty coalition and an incompleteness theorem which shows that there is no strongly complete logical system in the language with the empty coalition.

Calculi for Many-Valued Logics

We present a number of equivalent calculi for many-valued logics and prove soundness and strong completeness theorems. The calculi are obtained from the truth tables of the logic under consideration in a straightforward manner and there is a natural duality among these calculi. We also prove the cut elimination theorems for the sequent-like systems.

A Restricted Fragment of the Lambek Calculus with Iteration and Intersection Operations

The Lambek calculus (a variant of intuitionistic linear logic initially introduced for mathematical linguistics) enjoys natural interpretations over the algebra of formal languages (L-models) and over the algebra of binary relations which are subsets of a given transitive relation (R-models). For both classes of models there are completeness theorems (Andreka and Mikulas [J. Logic Lang. Inf., 3, No. 1, 1-37 (1994)]; Pentus [Ann. Pure Appl. Logic, 75, Nos. 1/2, 179-213 (1995); Fund. Prikl. Mat., 5, No. 1, 193- 219 (1999)]). The operations of the Lambek calculus include product and two divisions, left and right. We consider an extension of the Lambek calculus with intersection and iteration (Kleene star). It is proved that this extension is incomplete both w.r.t. L-models and w.r.t. R-models. We introduce a restricted fragment, in which iteration is allowed only in denominators of division operations. For this fragment we prove completeness w.r.t. R-models. We also prove completeness w.r.t. L-models for the subsystem without product. Both results are strong completeness theorems, i.e., they establish equivalence between derivability from sets of hypotheses (finite or infinite) and semantic entailment from sets of hypotheses on a given class of models. Finally, we prove $$ {\Pi}_1^0 $$ -completeness of the algorithmic problem of derivability in the restricted fragment in question.

Completeness theorems for σ–additive probabilistic semantics

We study propositional probabilistic logics (LPP–logics) with probability operators of the form P≥r (“the probability is at least r”) with σ–additive semantics. For regular infinite cardinals κ and λ, the probabilistic logic LPPκ,λ has λ propositional variables, allows conjunctions of <κ formulas, and allows iterations of probability operators. LPPκ,λ,2 denotes the fragment of LPPκ,λ where iterations of probability operators is not allowed. Besides the well known non-compactness of LPP–logics, we show that LPPκ,λ,2–logics are not countably compact for any λ≥ω1 and any κ, and that are not 2ℵ0+–compact for κ≥ω1 and any λ. We prove the equivalence of our adaptation of the Hoover's continuity rule (Rule (5) in [13]) and Goldblat's Countable Additivity Rule [9] and show their necessity for complete axiomatization with respect to the class of all σ–additive models. The main result is the strong completeness theorem for countable fragments LPPA and LPPA,2 of LPPω1,ω.

Borel functors, interpretations, and strong conceptual completeness for ℒ_{𝜔₁𝜔}

We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic L ω 1 ω \mathcal {L}_{\omega _1\omega } : every countable L ω 1 ω \mathcal {L}_{\omega _1\omega } -theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories ( L , T ) (\mathcal {L}, \mathcal {T}) and ( L ′ , T ′ ) (\mathcal {L}’, \mathcal {T}’) (in possibly different languages L , L ′ \mathcal {L}, \mathcal {L}’ ), every Borel functor Mod ( L ′ , T ′ ) → Mod ( L , T ) \text {Mod}(\mathcal {L}’, \mathcal {T}’) \to \text {Mod}(\mathcal {L}, \mathcal {T}) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L ω 1 ω ′ \mathcal {L}’_{\omega _1\omega } -interpretation of T \mathcal {T} in T ′ \mathcal {T}’ . This generalizes a recent result of Harrison-Trainor, Miller, and Montalbán in the ℵ 0 \aleph _0 -categorical case.

A Note on Strong Axiomatization of Gödel Justification Logic

Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators “t : ”, indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We replace this classical boolean base with Gödel logic, one of the three most prominent fuzzy logics, i.e. special instances of many-valued logics, taking values in the unit interval [0, 1], which are intended to model inference under vagueness. We extend the canonical possible-world semantics for justification logic to this fuzzy realm by considering fuzzy accessibility- and evaluation-functions evaluated over the minimum t-norm and establish strong completeness theorems for various fuzzy analogies of prominent extensions for basic justification logic.

Pure Sequent Calculi

Analyticity, also known as the subformula property, typically guarantees decidability of derivability in propositional sequent calculi. To utilize this fact, two substantial gaps have to be addressed: (i) What makes a sequent calculus analytic? and (ii) How do we obtain an efficient decision procedure for derivability in an analytic calculus? In the first part of this article, we answer these questions for pure calculi —a general family of fully structural propositional sequent calculi whose rules allow arbitrary context formulas. We provide a sufficient syntactic criterion for analyticity in these calculi, as well as a productive method to construct new analytic calculi from given ones. We further introduce a scalable decision procedure for derivability in analytic pure calculi by showing that it can be (uniformly) reduced to classical satisfiability. In the second part of the article, we study the extension of pure sequent calculi with modal operators. We show that such extensions preserve the analyticity of the calculus and identify certain restricted operators (which we call “Next” operators) that are also amenable for a general reduction of derivability to classical satisfiability. Our proofs are all semantic, utilizing several strong general soundness and completeness theorems with respect to non-deterministic semantic frameworks: bivaluations (for pure calculi) and Kripke models (for their extension with modal operators).

Deontology of Compound Actions

This paper, being a companion to the book [2] elaborates the deontology of sequential and compound actions based on relational models and formal constructs borrowed from formal linguistics. The semantic constructions presented in this paper emulate to some extent the content of [3] but are more involved. Although the present work should be regarded as a sequel of [3] it is self-contained and may be read independently. The issue of permission and obligation of actions is presented in the form of a logical system . This system is semantically defined by providing its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since the consequence relation is not finitary, other semantically defined variants of are defined. The focus is on the finitary system in which only finite compound actions are admissible. An adequate axiom system for it is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is emphasized.

A completeness theorem for continuous predicate modal logic

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

Strong standard completeness for continuous t-norms

This paper presents a proof of a strong completeness theorem for an extended axiomatic system of fuzzy logic BL with respect to all continuous t-norms. A finite strong standard completeness theorem for all continuous t-norms and their residua, the basic fuzzy logic, was proved across two papers Hájek (1998) and Cignoli et al. (2000). In Montagna (2007), the language of BL is extended by an additional connective and the axiomatic system includes an infinitary rule to achieve strong completeness result. In this paper we provide a proof of strong completeness for BL with a different infinitary inference rule but without extending the language of BL. We will also prove strong completeness for the Łukasiewicz and product t-norms using this extended axiomatic system.

RETRACTED ARTICLE: A completeness theorem for continuous predicate modal logic

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

The Axiomatization of Propositional Logic

Summary This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α), (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)), (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented.