Constructing Natural Extensions of Propositional Logics

The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to ?o? and Suszko. However, it was recently observed by Cintula and Noguera that both of these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the ?o?–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality \({\kappa}\) may have more than one conservative extension of cardinality \({\kappa}\) by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana.     

Free-Variable Tableaux for Propositional Modal Logics

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.     

A Modal View on Resource-Bounded Propositional Logics

Studia Logica - Classical propositional logic plays a prominent role in industrial applications, and yet the complexity of this logic is presumed to be non-feasible. Tractable systems such as...     

Uniform Interpolation and Propositional Quantifiers in Modal Logics

We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible. We shall present such a proof of the uniform interpolation theorem for normal modal logics K and T. It provides an explicit algorithm constructing the interpolants. Presented by Heinrich Wansing     

Expressive Power and Incompleteness of Propositional Logics

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this.     

On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics

By using algebraic-categorical tools, we establish four criteria in order to disprove canonicity, strong completeness, w-canonicity and strong w-completeness, respectively, of an intermediate propositional logic. We then apply the second criterion in order to get the following result: all the logics defined by extra-intuitionistic one-variable schemata, except four of them, are not strongly complete. We also apply the fourth criterion in order to prove that the Gabbay-de Jongh logic D₁ is not strongly w-complete. This revised version was published online in June 2006 with corrections to the Cover Date.     

Propositional Epistemic Logics with Quantification Over Agents of Knowledge

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal (epistemic) operators or over agents of knowledge and extended by predicate symbols that take modal (epistemic) operators (or agents) as arguments. Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There exist epistemic logics whose languages have the above mentioned properties (see, for example Corsi and Orlandelli in Stud Log 101:1159–1183, 2013; Fitting et al. in Stud Log 69:133–169, 2001; Grove in Artif Intell 74(2):311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science (LNCS), vol 1193, pp 71–85, 1996). But these logics are obtained from first-order modal logics, while a logic of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal (epistemic) operators and predicate symbols that take modal (epistemic) operators as arguments. Among the logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re (between ‘knowing that’ and ‘knowing of’). We show the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) with the help of the loosely guarded fragment (LGF) of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal (epistemic) operators. The family of this logics coincides with \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols (see Grove and Halpern in J Log Comput 3(4):345–378, 1993). Some logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as counterparts of logics defined in Grove and Halpern (J Log Comput 3(4):345–378, 1993). We prove that the satisfiability problem for these logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) is Pspace-complete using their counterparts in Grove and Halpern (J Log Comput 3(4):345–378, 1993).     

Order-Dual Relational Semantics for Non-distributive Propositional Logics: A General Framework

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures (frames) and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic (the logic of bounded lattices) and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as (non-distributive) Generalized Galois Logics (GGL’s).     

一类命题逻辑的一般弱框架择类语义

命题逻辑的一般弱框架择类语义是相干邻域语义的变形,其特点是：采用择类运算来刻画逻辑常项;语义运算与逻辑联结词之间有清晰的对应关系,可以从整体上处理一类逻辑,具有普适性。本文将这种语义用于一类B、C、K、W命题逻辑,包括相干逻辑R及其线性片段、直觉主义逻辑及其BCK片段等,并借助典范框架和典范赋值,证明了这些逻辑系统的可靠性和完全性。     

Quantification in Some Non-normal Modal Logics

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems by Stolpe (Logic Journal of the IGPL 11(5), 557–575, 2003), but unfortunately that work used only models and not frames.     

Some Metacomplete Relevant Modal Logics

A logic is called metacomplete if formulas that are true in a certain preferred interpretation of that logic are theorems in its metalogic. In the area of relevant logics, metacompleteness is used to prove primeness, consistency, the admissibility of γ and so on. This paper discusses metacompleteness and its applications to a wider class of modal logics based on contractionless relevant logics and their neighbours using Slaney’s metavaluational technique.     

Propositional Epistemic Logics with Quantification Over Agents of Knowledge (An Alternative Approach)

Studia Logica - In the previous paper with a similar title (see Shtakser in Stud Log 106(2):311–344, 2018), we presented a family of propositional epistemic logics whose languages are...     

Some Logics of Iterated Belief Change

The problems that surround iterated contractions and expansions of beliefs are approached by studying hypertheories, a generalisation of Adam Grove's notion of systems of spheres. By using a language with dynamic and doxastic operators different ideas about the basic nature of belief change are axiomatised. It is shown that by imposing quite natural constraints on how hypertheories may change, the basic logics for belief change can be strengthened considerably to bring one closer to a theory of iterated belief change. It is then argued that the logic of expansion, in particular, cannot without loss of generality be strengthened any further to allow for a full logic of iterated belief change. To remedy this situation a notion of directed expansion is introduced that allows for a full logic of iterated belief change. The new operation is given an axiomatisation that is complete for linear hypertheories.     

On the Existence of Continua of Logics Between Some Intermediate Predicate Logics

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.     

Measuring Inconsistency in Some Logics with Modal Operators

Studia Logica - The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The...     

Sequent Calculi for Some Strict Implication Logics

We introduce various sequent systems for propositional logicshaving strict implication, and prove the completeness theoremsand the finite model properties of these systems.The cut-eliminationtheorems or the (modified) subformula properties are provedsemantically.     

Some Remarks on Ultrafilter and Normality Logics

The paper presents the main ideas of Ultrafilter Logic (UL), as introduced by Veloso and others. A new proposal, Normality Logic (NL), is outlined for expanding the expressive power of UL. The system NL appears to offer a simpler solution to the problem of expressive power than the sorting strategy of Carnielli and Veloso. Interpretations of NL are discussed and an important point of contact to Hansson's notion of non-prioritized belief revision is observed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Minimally Abnormal Models In Some Adaptive Logics

In an adaptive logic APL, based on a (monotonic) non-standardlogic PL the consequences of can be defined in terms ofa selection of the PL-models of . An important property ofthe adaptive logics ACLuN1, ACLuN2, ACLuNs1, andACLuNs2 logics is proved: whenever a model is not selected, this isjustified in terms of a selected model (Strong Reassurance). Theproperty fails for Priest's LP ^m because its way of measuring thedegree of abnormality of a model is incoherent – correcting thisdelivers the property.     

A Loop-Free Decision Procedure for Modal Propositional Logics K4, S4 and S5

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.