Elementary Classes in Basic Modal Logic

Dealing with topics of definability, this paper provides some interesting insights into the expressive power of basic modal logic. After some preliminary work it presents an abstract algebraic characterization of the elementary classes of basic modal logic, that is, of the classes of models that are definable by means of (sets of) basic modal formulas. Taking that for a start, the paper further contains characterization results for modal universal classes and modal positive classes.     

Some Embedding Theorems for Conditional Logic

We prove some embedding theorems for classical conditional logic, covering ‘finitely cumulative’ logics, ‘preferential’ logics and what we call ‘semi-monotonic’ logics. Technical tools called ‘partial frames’ and ‘frame morphisms’ in the context of neighborhood semantics are used in the proof.     

Identical Twins,Deduction Theorems,and Pattern Functions: Exploring the Implicative BCSK Fragment of S5

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives.     

Expressivity of Second Order Propositional Modal Logic

We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML.     

Constructing Finite Least Kripke Models for Positive Logic Programs in Serial Regular Grammar Logics

A serial context-free grammar logic is a normal multimodal logicL characterized by the seriality axioms and a set of inclusionaxioms of the form _ts1..._sk. Such an inclusion axiom correspondsto the grammar rule t s₁... s_k. Thus the inclusion axioms ofL capture a context-free grammar . If for every modal index t, the set of words derivable fromt using is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positivemultimodal logic program P and a set of finite automata specifyinga serial regular grammar logic L, constructs a finite leastL-model of P. (A model M is less than or equal to model M' iffor every positive formula , if M then M' .) A least L-modelM of P has the property that for every positive formula , P iff M . The algorithm runs in exponential time and returnsa model with size 2^O(n³). We give examples of P and L, for bothof the case when L is fixed or P is fixed, such that every finiteleast L-model of P must have size 2⁽ⁿ⁾. We also prove that ifG is a context-free grammar and L is the serial grammar logiccorresponding to G then there exists a finite least L-modelof _s p iff the set of words derivable from s using G is a regularlanguage.     

ORGANON The Web Tutor for Basic Logic Courses

We would like to introduce the web tutor ORGANON, which aimsto support basic logic courses at the University of West Bohemiain Pilsen (Czech Republic). The application was designed tofulfill two requirements. Firstly, it should help students duringtheir study to practice exercises on their own (providing permanentcontrol during students’ practicing exercises as wellas answering students’ questions immediately as they arise).Secondly, it should reduce teachers’ burden (diminishingthe amount of consultations as well as administrating students’homework including correcting and grading). The tutor is availableat http://kfi.zcu.cz/lide/dostalova/organon.     

A Two Dimensional Tense-modal Sortal Logic

We consider a formal language whose logical syntax involves both modal and tense propositional operators, as well as sortal quantifiers, sortal identities and (second order) quantifiers over sortals. We construct an intensional semantics for the language and characterize a formal logical system which we prove to be sound and complete with respect to the semantics. Conceptualism is the philosophical background of the semantic system.     

Critical thinking in North America: A new theory of knowledge,learning, and literacy

The pace of change in the world is accelerating, yet educational institutions have not kept pace. Indeed, schools have historically been the most static of social institutions, uncritically passing down from generation to generation outmoded didactic, lecture-and-drill-based, models of instruction. Predictable results follow. Students, on the whole, do not learn how to work by, or think for, themselves. They do not learn how to gather, analyze, synthesize and assess information. They do not learn how to analyze the diverse logic of the questions and problems they face and hence how to adjust their thinking to those problems. They do not learn how to enter sympathetically into the thinking of others, nor how to deal rationally with conflicting points of view. They do not learn to become critical readers, writers, speakers and listeners. They do not learn how to use their native languages clearly, precisely, or persuasively. They do not, therefore, become literate, in the proper sense of the word. Neither do they gain much in the way of genuine knowledge since, for the most part, they could not explain the basis for what they believe. They would be hard pressed to explain, for example, which of their beliefs were based on rational assent and which on simple conformity to what they have been told. They have little sense as to how they might critically analyze their own experience, or identify national or group bias in their own thinking. They are much more apt to learn on the basis of irrational than rational modes of thought. They lack the traits of mind of a genuinely educated person: intellectual humility, courage, integrity, perseverance, and faith in reason.Happily, there is a movement in education today striving to address these problems in a global way, with strategies and materials for the modification of instruction at all levels of education. At its foundation is an emerging new theory of knowledge, learning, and literacy, one which recognizes the centrality of independent critical thinking to all substantial learning, one which recognizes that higher-order, multilogical thinking is as important to childhood as to adult learning, and as important to foundational learning in monological as in multilogical disciplines. This educational reform movement is not proposing an educational miracle cure, for its leading proponents recognize that many social and historical forces must come together before the ideals of the critical thinking movement will become a full academic reality. Schools do not exist in a social vacuum. To the extent that the broader society is uncritical so, on the whole, will be society's schools. Nevertheless, the social conditions necessary for fundamental changes in schooling are increasingly apparent. The pressure for fundamental change is growing. Whether and to what extent these needed basic changes will be delayed or side-tracked, thus requiring new periodic resurgences of this movement, with new, more elaborate articulations of its ideals, goals, and methods — only time will tell.     

NARROW HEDONISM

Narrow hedonism is defined and defended, as a view according to which pleasurable states are individuated as concrete and total experiential situations of a sentient being at a time. Typical of such situations is that, when we are in them, we are at a certain hedonic level. They feel in a certain way for the creature in them. On this understanding of narrow hedonism, which is the only one making good sense of the theory and which was probably also intended by classical hedonists such as Bentham and Edgeworth, standard objections to hedonism, based on the claim that different pleasures have nothing in common, can be set to one side as misplaced and irrelevant. It is also hard to see how this kind of hedonism can be refined, or revised, in the direction indicated by J.S. Mill, when he wants to distinguish “higher” pleasures from “lower” ones. On this understanding of hedonism, we must claim that, those who want to follow Mill will have to rely on non-hedonistic intuitions and thus desert the hedonist camp altogether.     

Completeness and Decidability Results for Some Propositional Modal Logics Containing “Actually” Operators

The addition of actually operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing actually operators has concentrated entirely upon extensions of KT5 and has employed a particular model-theoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing actually operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.