Logics of belief change without linearity

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of $${\int}_{\Omega} f(x,u,\nabla u)\,dx $$ with f subject to the general structural conditions $$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$ where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ Lt(Ω) for some t > n/p−, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Holder-continuous.

The idealizations resulting from the use of Kripke semantics in Epistemic Logic are inherited by formalizations of group epistemic notions. For example, distributed knowledge is often taken to reflect the potential knowledge of a group: what agents would know if they had unbounded means of communication and deductive ability. However, this does not specify whether/how this potential can be actualized, especially since real people are not unbounded reasoners. Inspired by experiments on group reasoning, we identify two dimensions of actualizing distributed knowledge: communication and inference. We build a dynamic framework with effortful actions accounting for both, combining impossible-worlds semantics and action models, and we provide a method for extracting a sound and complete axiomatization.

We start from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and adapt the case of first-order formulas by employing certain rings equipped with infinitary operations. This paper defines the notion of M-ring, a kind of polynomial ring that can be naturally associated to each first-order structure and each first-order theory, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences can be seen as a legitimate algebraic semantics for first-order logic, an alternative to cylindric and polyadic algebras and closer to the primordial forms of algebraization of logic. We suggest how the method and its generalization could be lifted successfully to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.

We introduce a general notion of semi-implication which generalizes both the implications used in classical, intuitionistic, and many other logics, as well as those used in relevance logics. It is mainly based on the relevant deduction property (RDP)—a weak form of the classical-intuitionistic deduction theorem which has motivated the design of the intensional fragments of the relevance logic R. However, CL ↔, the pure equivalential fragment of classical logic, also enjoys the RDP with respect to ↔. We show that in the language of → this is the only exception. This observation leads to an adequate definition of semi-implication, according to which a finitary logic L has a semi-implication → iff L has a strongly sound and complete Hilbert-type system which is an extension by axiom schemas of HR → (the standard Hilbert-type system for the implicational fragment of R). We also show that in the presence of a conjunction, or a disjunction, or an implication, a connective → of a logic L is a semi-implication iff it is an implication (i.e. it satisfies the classical-intuitionistic deduction theorem), and the same is true if L is induced by a matrix which has a single designated value or a single non-designated value.

The study of belief change has been an active area in philosophy and AI. In recent years two special cases of belief change, belief revision and belief update, have been studied in detail. In a companion paper (Friedman & Halpern, 1997), we introduce a new framework to model belief change. This framework combines temporal and epistemic modalities with a notion of plausibility, allowing us to examine the change of beliefs over time. In this paper, we show how belief revision and belief update can be captured in our framework. This allows us to compare the assumptions made by each method, and to better understand the principles underlying them. In particular, it shows that Katsuno and Mendelzon's notion of belief update (Katsuno & Mendelzon, 1991a) depends on several strong assumptions that may limit its applicability in artificial intelligence. Finally, our analysis allow us to identify a notion of minimal change that underlies a broad range of belief change operations including revision and update.

Decision making alternatives are sometimes incomparable with each other and violate the completeness axiom of the rational behavior axiom of von Neumann-Morgenstern. Lattice order theory has therefore been used to deal with the incomparability issue among alternatives. This paper proves that the incomparable relation of the set of alternatives in same level under lattice order structure is equal to equivalence relation. The alternatives set with lattice ordered structure is classified as subsets of alternatives in terms of their corresponding location layers. The subsets make up a chain structure under the preference relations of equivalence classes, which realizes the chain-forming of preference structure, and transforms the lattice order structure to total order or asymmetric weak order structure. A new decision making method based on lattice-ordered preference structure is proposed and the proposed method is efficient to make decisions for the incomparable paired alternatives in the decision making alternatives set.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

In this paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner.

I propose a new semantics for intuitionistic logic, which is a cross between the construction-oriented semantics of Brouwer-Heyting-Kolmogorov and the condition-oriented semantics of Kripke. The new semantics shows how there might be a common semantical underpinning for intuitionistic and classical logic and how intuitionistic logic might thereby be tied to a realist conception of the relationship between language and the world.

The purpose of the study is to list the characteristics of organizational culture as an object of change and an influential tool for implementing strategic decisions, justifying its role and place in implementing planned and unplanned changes. Materials and methods. The research used such general scientific techniques and methods as structural-logical analysis (at the stage of construction of general logic and structure of research), synthesis (in the analysis of existing publications), induction and deduction, comparisons and key components. Research results. Summarizing the results of the study, we should note the role and importance of organizational culture of the enterprise in the process of introducing changes in modern conditions. It is emphasized that organizational culture is not only a unique managerial, social and psychological phenomenon, but also an effective tool for implementing tactical and strategic decisions in the economy. The processes of introduction of changes taking place in Ukraine today cover the majority of domestic business and business structures. The main reasons are: formation of high-tech market relations; integration of economies and globalization of socio-economic processes; increased competition in international markets and the like. All these and other factors are forcing businesses to respond promptly to changes in the economy and society. Managers implementing new approaches and the latest strategy are faced with the fact that the primary object of change in the enterprise is the organizational culture itself, a culture that was formed during the command and administrative system and which for a long time influenced the economic activity of the enterprise. It is a culture dominated by high levels of power distance, authoritarian management style, inflexible rules and management procedures. Therefore, managing the overall processes of change in enterprises require a reorganization in the current culture. The main directions of such changes are: optimization of organizational and personnel structure, enhancement of the general and managerial culture of managers and main specialists; overcoming the boundaries of the uneven distribution of power between the personnel of the enterprise and others. Given that organizational culture is a powerful means of regulating the behavioral activity of employees, it is necessary to take into account the historical experience, its national peculiarities and focus on those areas that most effectively influence the results of transformational changes in the organization. Organizational culture is not only the object of change, it is a powerful driving force and an influential tool for implementing and implementing a series for other changes (organizational, innovative, financial, technological, human resources, etc.) in the enterprise, which will eventually lead to the expected positive economic result. Conclusions. Organizational culture, therefore, is an important tool for the economic and psychological unity of enterprise staff to implement transformational change in a changing environment. Studies show that changes aremade faster in an organization and that a real economic effect isobtained where the level of organizational culture is highest.

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus (IPC) turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC.

Recently, there has been a growing interest in constructive reinterpretations of description logics. This has been motivated by the need to model in the DLs setting problems that have a consolidate tradition in constructive logics. In this paper we introduce a constructive description logic for the language of ALC based on the Kripke semantics for Intuitionistic Logic. Moreover we give a tableau calculus and we show that it is sound, complete and terminating.

In this paper we first introduce a new neighbourhood semantics for propositional intuitionistic logic. We then naturally extend this semantics to first-order intuitionistic logic. We also study bisimulation between neighbourhood models and prove some of their basic properties for both propositional and first-order intuitionistic logic.

As pointed out by Hans Rott and others, there are two ways of reading AGM, the theory of Alchourrón, Gärdenfors and Makinson presented in a classic paper that has come to be regarded as the fons et origo of today’s formal study of belief revision [Alchourrón et al., 1985]. Under the “iterative” reading there is room for multiple sequential belief change, but under the “one-shot” reading the perspective is limited to one change. Roughly and informally, the “one-shot” reading recognizes doxastic situations of two kinds. First, there is status quo, the (generic but unique) anterior situation. Then a belief change takes place, landing the agent in a posterior situation. And this is the entire scenario! There is no inkling that there could be further belief changes and corresponding post-posterior situations. Iterative theories are richer and more interesting. But as they are more difficult to devise, one might begin by studying one-shot theories (to which from now on we will refer without scare quotes). In this paper, we suggest how one-shot AGM can be rendered within dynamic doxastic logic (DDL). This author’s first effort in this area was [Segerberg, 1994a], but only with van Linder, van der Hoek and Meyer [1995] did we get a faithful interpretation of AGM in a DDL language (that is, a model theoretical analysis of Gärdenfors’s characterization in [1988) of the three main doxastic operations: expansion, contraction, and revision of belief sets). However, in neither paper was the question of a complete axiomatization of the set of valid formulæ addressed; here, it is.

In this paper, we consider the logic ${\textsf{ITL}}^{e}$, a variant of intuitionistic linear temporal logic that is interpreted over the class of dynamic Kripke frames. These are bi-relational structures of the form $ \langle{W, \preccurlyeq , f}\rangle $ where $\preccurlyeq $ is a partial order on $W$ and $f: W \to W$ is a $\preccurlyeq $-monotone function. Our main result answers a question recently raised by Boudou et al. (2017, A decidable intuitionistic temporal logic. In Computer Science Logic 2017, pp. 14:1–14:17. Vol. 82 of LIPIcs) about axiomatizing this logic. We provide an axiomatization of ${\textsf{ITL}}^{e}$ and prove its strong completeness with respect to the class of all dynamic Kripke frames. The proposed axiomatization is infinitary; it has two derivation rules with countably many premises and one conclusion. It should be mentioned that ${\textsf{ITL}}^{e}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete.