Axiomatic Extensions of IMT3 Logic

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ³ ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x³) ∨ x ≈ ?, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.     

Two notions of compactness in Gödel logics

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well.In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, Gödel_Δ, and Gödel_～ logics.There are several results (all for the countable set of propositional variables) concerning the compactness (based on satisfiability) of these logic by Cintula and Navara, and the question of compactness (based on entailment) for Gödel logic was fully answered by Baaz and Zach (see papers [3] and [2]).In this paper we give a nearly complete answer to the problem of compactness based on both concepts for all three logics and for an arbitrary cardinality of the set of propositional variables. Finally, we show a tight correspondence between these two concepts     

A Note on Algebras of Substitutions

We will study the class RSA of -dimensional representable substitution algebras. RSA is a sub-reduct of the class of representable cylindric: algebras, and it was an open problem in Andréka [1] that whether RSA can be finitely axiomatized. We will show, that the answer is positive. More concretely, we will prove, that RSA is a finitely axiomatizable quasi-variety. The generated variety is also described. We note that RSA is the algebraic counterpart of a certain proportional multimodal logic and it is related to a natural fragment of first order logic, as well.     

A Method of Generating Modal Logics Defining Jaśkowski’s Discussive Logic D<Subscript>2</Subscript>

Ja?kowski’s discussive logic D₂ was formulated with the help of the modal logic S5 as follows (see [7, 8]): \({A \in {D_{2}}}\) iff \({\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}\), where (–)^? is a translation of discussive formulae from For^d into the modal language. We say that a modal logic L defines D₂ iff \({{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}\). In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (?): have the same theses beginning with ‘\({\diamond}\)’ as S5. Of course, all logics fulfilling the above condition, define D₂. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D₂ is equivalent to having the property (?). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (?). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D₂.     

Smarandache BL-algebra

In this paper we define the Smarandache BL-algebra, Q-Smarandache ideal and Q-Smarandache implicative ideal, we obtain some related results. After that by considering the notions of these ideals we determine relationships between ideals in BL-algebra and Q-Smarandache (implicative) ideals of BL-algebra. Finally we construct quotient of Smarandache BL-algebras via MV-algebra and prove some theorems.     

Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains.     

An extension of the deontic calculus DSC1

The chief aim of the paper is to extend the calculusDSC ₁ (see [4]) in such a way as to satisfy all the requirements listed in [4] as well as a further stipulation — called ‘the principle of uninvolvement’ — to the effect that neither deontic compatibility nor deontic incompatibility of codes (see [2]) should be presupposed in deontic logic.     

Statistics of Intuitionistic versus Classical Logics

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences.     

Future determination of entities in Talmudic public announcement logic

Ordinary dynamic action logics deal with states and actions upon states. The actions can be deterministic or non-deterministic, but it is always assumed that the possible results of the actions are clear cut.Talmudic logic deals with actions (usually legally meaningful actions which can change the legal status of an entity) which depend on the future and therefore may be not clear cut at the present and need future clarifications.The clarification is modelled by public announcement which comes at a later time after the action has taken place.The model is further complicated by the need to know what is the status of formulas at a time before the results of the action is clarified, as we do not know at which state we are in. Talmudic logic treats such states much like the quantum superposition of states and when clarification is available we get a collapse onto a pure state.The Talmudic lack of clarity of actions arises from applying an action to entities defined using the future, like the statement of a dying man on his death bed:

Let the man who will win the jackpot in the lottery next week be the sole heir in my will now

We need to wait a week for the situation to clarify.There is also the problem of legal backwards causality, as this man, if indeed he exists, unaware of his possible good fortune, may have himself meanwhile donated all his property to a charity. Does his donation include this unknown inheritance?This paper will offer a model and a logic which can represent faithfully the Talmudic reasoning in these matters.We shall also see that we get new types of public announcement logics and (quantum-like) action logics. Ordinary public announcement logic deletes possible worlds after an announcements. Talmudic public announcement logic deletes accessibility links after an announcement. Technically these two approaches are similar but not equivalent.     

A Non-deterministic View on Non-classical Negations

We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ⁺, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is based on non-deterministic four-valued or three-valued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of non-determinism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ⁺ (with or without ff) are shown to be conservative over the underlying logic, and it is determined which of them are not.     

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL _ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. Presented by Heinrich Wansing     

Monadic Bounded Algebras

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA’s is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA’s as powerset algebras of certain directed graphs with a set of “marked” points.     

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p ₁, . . . , p _n , = (p ₁, . . . , p _n-1, p _n ) intuitively states that the value of p _n is determined by those of p ₁, . . . , p _n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Perfect MV-algebras are categorically equivalent to abelianl-groups

In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.Presented byDaniele Mundici     

Combining Possibilities and Negations

Combining non-classical (or sub-classical) logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will find that Kracht's results on the undecidability of classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic negation without corrupting the intuitionistic fragment of the logic.     

Bridging learning theory and dynamic epistemic logic

This paper discusses the possibility of modelling inductive inference (Gold 1967) in dynamic epistemic logic (see e.g. van Ditmarsch et al. 2007). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing (Lange et al. 1996) and iterated epistemic update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective.     

Three-Valued Temporal Logic <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">t</Emphasis></Subscript> and Future Contingents

Prior’s three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic Q _t and give its axiomatization and semantics. We also argue that Q _t provides a smooth solution to the problem of future contingents. Presented by Daniele Mundici