scholarly journals CONDITIONAL BELIEFS: FROM NEIGHBOURHOOD SEMANTICS TO SEQUENT CALCULUS

2018 ◽  
Vol 11 (4) ◽  
pp. 736-779 ◽  
Author(s):  
MARIANNA GIRLANDO ◽  
SARA NEGRI ◽  
NICOLA OLIVETTI ◽  
VINCENT RISCH
Keyword(s):  
Decision Procedure ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Constructive Proof ◽  
Finite Model ◽  
Finite Model Property ◽  
Semantic Completeness ◽  
Multi Agent ◽  
Labelled Sequent Calculus ◽  
Direct Correspondence

AbstractThe logic of Conditional Beliefs (CDL) has been introduced by Board, Baltag, and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this article both the semantics and the proof theory for this logic are studied. First, a natural semantics forCDLis defined in terms of neighbourhood models, a multi-agent generalisation of Lewis’ spheres models, and it is shown that the axiomatization ofCDLis sound and complete with respect to this semantics. Second, it is shown that the neighbourhood semantics is equivalent to the original one defined in terms of plausibility models, by means of a direct correspondence between the two types of models. On the basis of neighbourhood semantics, a labelled sequent calculus forCDLis obtained. The calculus has strong proof-theoretic properties, in particular admissibility of contraction and cut, and it provides a decision procedure for the logic. Furthermore, its semantic completeness is used to obtain a constructive proof of the finite model property of the logic. Finally, it is shown that other doxastic operators can be easily captured within neighbourhood semantics. This fact provides further evidence of the naturalness of neighbourhood semantics for the analysis of epistemic/doxastic notions.

Logical questions concerning the μ-calculus: Interpolation, Lyndon and Łoś-Tarski

2000 ◽  
Vol 65 (1) ◽  
pp. 310-332 ◽  
Author(s):  
Giovanna D'agostino ◽  
Marco Hollenberg
Keyword(s):  
Decision Procedure ◽  
Sequent Calculus ◽  
Theoretical Perspective ◽  
Order Logic ◽  
Rite Of Passage ◽  
Finite Model ◽  
Finite Model Property ◽  
Exponential Time ◽  
Second Order Logic ◽  
Monadic Second Order Logic

The (modal) μ-calculus ([14]) is a very powerful extension of modal logic with least and greatest fixed point operators. It is of great interest to computer science for expressing properties of processes such as termination (every run is finite) and fairness (on every infinite run, no action is repeated infinitely often to the exclusion of all others).The power of the μ-calculus is also evident from a more theoretical perspective. The μ-calculus is a fragment of monadic second-order logic (MSO) containing only formulae that are invariant for bisimulation, in the sense that they cannot distinguish between bisimilar states. Janin and Walukiewicz prove the converse: any property which is invariant for bisimulation and MSO-expressible is already expressible in the μ-calculus ([13]). Yet the μ-calculus enjoys many desirable properties which MSO lacks, like a complete sequent-calculus ([29]), an exponential-time decision procedure, and the finite model property ([25]). Switching from MSO to its bisimulation-invariant fragment gives us these desirable properties.In this paper we take a classical logician's view of the μ-calculus. As far as we are concerned a new logic should not be allowed into the community of logics without at least considering the standard questions that any logic is bothered with. In this paper we perform this rite of passage for the μ-calculus. The questions we will be concerned with are the following.

scholarly journals On the Decidability of Intuitionistic Tense Logic without Disjunction

2020 ◽  
Author(s):  
Fei Liang ◽  
Zhe Lin
Keyword(s):  
Proof Theory ◽  
Tense Logic ◽  
Heyting Algebras ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Algebraic Models ◽  
Finite Axiomatizability ◽  
Finite Model Property ◽  
Modal Operators

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

scholarly journals A labelled sequent calculus for BBI: proof theory and proof search

2015 ◽  
Vol 28 (4) ◽  
pp. 809-872
Author(s):  
Zhé Hóu ◽  
Rajeev Goré ◽  
Alwen Tiu
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Proof Search ◽  
Labelled Sequent Calculus

The finite embeddability property for noncommutative knotted extensions of RL

2015 ◽  
Vol 25 (03) ◽  
pp. 349-379 ◽  
Author(s):  
R. Cardona ◽  
N. Galatos
Keyword(s):  
Classical Logic ◽  
Proof Theory ◽  
Mathematical Linguistics ◽  
Residuated Lattices ◽  
Automata Theory ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Embeddability Property

The finite embeddability property (FEP) for knotted extensions of residuated lattices holds under the assumption of commutativity, but fails in the general case. We identify weaker forms of the commutativity identity which ensure that the FEP holds. The results have applications outside of order algebra to non-classical logic, establishing the strong finite model property (SFMP) and the decidability for deductions, as well as to mathematical linguistics and automata theory, providing new conditions for recognizability of languages. Our proofs make use of residuated frames, developed in the context of algebraic proof theory.

Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics

1996 ◽  
Vol 61 (4) ◽  
pp. 1057-1120 ◽  
Author(s):  
D. M. Gabbay
Keyword(s):  
Proof Theory ◽  
Finite Model ◽  
Finite Model Property ◽  
The Past ◽  
Combined Logics ◽  
Intermediate Logics ◽  
Combination Of Logics ◽  
Combining Logics ◽  
New System ◽  
Intuitionistic Logics

AbstractThis is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems Li, i ∈ I, to form a new system LI. The methodology ‘fibres’ the semantics i of Li into a semantics for LI, and ‘weaves’ the proof theory (axiomatics) of Li into a proof system of LI. There are various ways of doing this, we distinguish by different names such as ‘fibring’, ‘dovetailing’ etc, yielding different systems, denoted by etc. Once the logics are ‘weaved’, further ‘interaction’ axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics.The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property.Some results on combining logics (normal modal extensions of K) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20–30 years. We hope our methodology will help organise the field systematically.

scholarly journals Semantical Proof of Subformula Property for the Modal Logics K 4.3, KD 4.3, and S4.3

2019 ◽  
Vol 48 (4) ◽  
Author(s):  
Daishi Yazaki
Keyword(s):  
Sequent Calculus ◽  
Modal Logics ◽  
Finite Model ◽  
Inference Rules ◽  
Sequent Calculi ◽  
Model Property ◽  
Finite Model Property ◽  
Subformula Property

The main purpose of this paper is to give alternative proofs of syntactical and semantical properties, i.e. the subformula property and the nite model property, of the sequent calculi for the modal logics K4.3, KD4.3, and S4.3. The application of the inference rules is said to be acceptable, if all the formulas in the upper sequents are subformula of the formulas in lower sequent. For some modal logics, Takano analyzed the relationships between the acceptable inference rules and semantical properties by constructing models. By using these relationships, he showed Kripke completeness and subformula property. However, his method is difficult to apply to inference rules for the sequent calculi for K4.3, KD4.3, and S4.3. Lookinglosely at Takano's proof, we nd that his method can be modied to construct nite models based on the sequent calculus for K4.3, if the calculus has (cut) and all the applications of the inference rules are acceptable. Similarly, we can apply our results to the calculi for KD4.3 and S4.3. This leads not only to Kripke completeness and subformula property, but also to finite model property of these logics simultaneously.

scholarly journals A Modified Subformula Property for the Modal Logic S4.2

2019 ◽  
Vol 48 (1) ◽  
Author(s):  
Mitio Takano
Keyword(s):  
Modal Logic ◽  
Sequent Calculus ◽  
Interpolation Property ◽  
Modal Logics ◽  
Finite Model ◽  
Model Property ◽  
Cut Rule ◽  
Finite Model Property ◽  
Axiom A ◽  
Subformula Property

The modal logic S4.2 is S4 with the additional axiom ◊□A ⊃ □◊A. In this article, the sequent calculus GS4.2 for this logic is presented, and by imposing an appropriate restriction on the application of the cut-rule, it is shown that, every GS4.2-provable sequent S has a GS4.2-proof such that every formula occurring in it is either a subformula of some formula in S, or the formula □¬□B or ¬□B, where □B occurs in the scope of some occurrence of □ in some formula of S. These are just the K5-subformulas of some formula in S which were introduced by us to show the modied subformula property for the modal logics K5 and K5D (Bull Sect Logic 30(2): 115–122, 2001). Some corollaries including the interpolation property for S4.2 follow from this. By slightly modifying the proof, the finite model property also follows.

Canonical rules

2009 ◽  
Vol 74 (4) ◽  
pp. 1171-1205 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Alternative Proof ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Finite Model Property ◽  
Admissible Rules ◽  
Superintuitionistic Logics ◽  
Canonical Formulas ◽  
Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

On the Modal μ-Calculus Over Finite Symmetric Graphs

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Giovanna D’Agostino ◽  
Giacomo Lenzi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Satisfiability Problem ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Symmetric Graphs ◽  
Trivial Consequence

AbstractIn this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

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