A New Representation Theorem for Contranegative Deontic Logic

The logic of an ought operator O is contranegative with respect to an underlying preference relation if it satisfies the property Op & (¬p)(¬q) Oq. Here the condition that is interpolative ((p (pq) q) (q (pq) p)) is shown to be necessary and sufficient for all -contranegative preference relations to satisfy the plausible deontic postulates agglomeration (Op & OqO(p&q)) and disjunctive division (O(p&q) Op Oq).     

The logic of linear tolerance

A nonempty sequence T₁,...,T_n of theories is tolerant, if there are consistent theories T ₁ ⁺ ,..., T _n ⁺ such that for each 1 i n, T _i ⁺ is an extension of T_i in the same language and, if i n, T _i ⁺ interprets T _i+1 ⁺ . We consider a propositional language with the modality , the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOL. It is shown that TOL (resp. TOL) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if (A₁,..., A_n) is understood as (a formalization of) PA+A₁, ..., PA+A_n is tolerant.     

O pojęciu zdania analitycznego

Rezultaty przedstawione w pracy niniejszej pokrywaj si czciowo z wynikami osignitymi przezR. Wójcickiego w pracy:Analityczne komponenty definicji arbitralnych. Studia Logica, t. XIV. Dotyczy to gównie rezultatów zawartych w czci pierwszej. Chciabym podkreli, i wyniki R. Wójcickiego uzyskane zostay cakowicie niezalenie od rezultatów przedstawionych w pracy obecnej.Allatum est die 16 Aprilis 1962     

Axiomatization of the De Morgan type rules

In Section 1 we show that the De Morgan type rules (= sequential rules in L(, ) which remain correct if and are interchanged) are finitely based. Section 2 contains a similar result for L(). These results are essentially based on special properties of some equational theories.     

Intuitionistic uniformity principles for propositions and some applications

This note deals with the prepositional uniformity principlep-UP: p x N A (p, x) x N p A (p, x) ( species of all propositions) in intuitionistic mathematics.p-UP is implied by WC and KS. But there are interestingp-UP-cases which require weak KS resp. WC only. UP for number species follows fromp-UP by extended bar-induction (ranging over propositions) and suitable weak continuity. As corollaries we have the disjunction property and the existential definability w.r.t. concrete objects. Other consequences are: there is no non-trivial countable partition of;id is the only injective function from to; there are no many-place injective prepositional functions; card () is incomparable with the cardinality of all metric spaces containing at least three elements.     

A calculus for the common rules of ∧ and ∨

We provide a finite axiomatization of the consequence , i.e. of the set of common sequential rules for and . Moreover, we show that has no proper non-trivial strengthenings other than and . A similar result is true for , but not, e.g., for ⁺.To the memory of Jerzy Supecki     

Equivalents of Mingle and Positive Paradox

Relevant logic is a proper subset of classical logic. It does not include among its theorems any ofpositive paradox A (B A)mingle A (A A)linear order (A B) (B A)unrelated extremes (A ) (B B¯)This article shows that those four formulas have different effects when added to relevant logic, and then lists many formulas that have the same effect as positive paradox or mingle.     

Is “some-other-time” sometimes better than “sometime” for proving partial correctness of programs?

The main result of this paper belongs to the field of the comparative study of program verification methods as well as to the field called nonstandard logics of programs. We compare the program verifying powers of various well-known temporal logics of programs, one of which is the Intermittent Assertions Method, denoted as Bur. Bur is based on one of the simplest modal logics called S5 or sometime-logic. We will see that the minor change in this background modal logic increases the program verifying power of Bur. The change can be described either technically as replacing the reflexive version of S5 with an irreflexive version, or intuitively as using the modality some-other-time instead of sometime. Some insights into the nature of computational induction and its variants are also obtained.This project was supported by the Hungarian National Foundation for Scientific Research, Grant No. 1810.     

Some modifications of Scott's theorem on injective spaces

D. Scott in his paper [5] on the mathematical models for the Church-Curry -calculus proved the following theorem.A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of Scott's open sets in a continuous lattice.In this paper we prove a generalization of this theorem for the category of , -closure spaces. The main theorem says that, for some cardinal numbers , , absolute extensors for the category of , -closure spaces are exactly , -closure spaces of , -filters in , >-semidistributive lattices (Theorem 3.5).If = and = we obtain Scott's Theorem (Corollary 2.1). If = 0 and = we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If = 0 and = we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary 3.3).     

Modal logic with names

We investigate an enrichment of the propositional modal language with a universal modality having semanticsx iff y(y ), and a countable set of names — a special kind of propositional variables ranging over singleton sets of worlds. The obtained language _c proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment () of, where is an additional modality with the semanticsx iff y(y x y ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in _c. Strong completeness of the normal _c-logics is proved with respect to models in which all worlds are named. Every _c-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from to _c are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.     

Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G ⁰(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G ⁰are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G ⁰is not compact (and therefore not canonical), and we proof with the tableau-method that G ⁰is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G ⁰is decidable and also characterised by the class of all frames for G ⁰.Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.Presented by W. Rautenberg     

Three prepositional calculi of probability

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic ₀ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.The second calculus MP is a usual modal propositional calculus. It has the modal rules x x, x y x y, x x, x y (y x), (y x), in addition to the rules of classical propositional logic. One may read x as x is probable. Imbeddings of NP and of ₀ into MP are given.The third calculus P is a modal extension of ₀. It may be obtained by adding the rule ((xy)y) xy to the modal logic of quantum mechanics Q [5]. One may read x in P as x is observed. An imbedding of NP into P is given.     

On finite approximability of ψ-intermediate logics

The aim of this note is to show (Theorem 1.6) that in each of the cases: = {, }, or {, , }, or {, , } there are uncountably many -intermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each : either all -intermediate logics are finitely approximate or there are uncountably many of them which lack the property.     

Poetry in Existential Psychotherapy with Couples and Families

The use of poetry during the process of existential psychotherapy with couples and families is described and illustrated. In this approach, poems can be utilized to help the couple and/or family notice meaning potentials in the future, actualize and make use of such meaning potentials in the here and now, and re-collect and honor meanings previously actualized and deposited in the past.     

⊃E is Admissible in “true” relevant arithmetic

The system R## of true relevant arithmetic is got by adding the -rule Infer xAx from A0, A1, A2, .... to the system R# of relevant Peano arithmetic. The rule E (or gamma) is admissible for R##. This contrasts with the counterexample to E for R# (Friedman & Meyer, Whither Relevant Arithmetic). There is a Way Up part of the proof, which selects an arbitrary non-theorem C of R## and which builds by generalizing Henkin and Belnap arguments a prime theory T which still lacks C. (The key to the Way Up is a Witness Protection Program, using the -rule.) But T may be TOO BIG, whence there is a Way Down argument that produces a better theory TR, such that R## TR T. (The key to the Way Down is a Metavaluation, on which membership in T is combined with ordinary truth-functional conditions to determine TR.) The result is a theory that is Just Right, whence it never happens that A C and A are theorems of R## but C is a non-theorem.     

On Some Kripke Complete and Kripke Incomplete Intermediate Predicate Logics

The Kripke-completeness and incompleteness of some intermediate predicate logics is established. In particular, we obtain a Kripke-incomplete logic (H* +A+D+K) where H* is the intuitionistic predicate calculus, A is a disjunction-free propositional formula, D = x(P(x) V Q) xP(x) V Q, K = ¬¬x(P(x) V ¬P(x)) (the negative answer to a question of T. Shimura).     

A gentzen system for conditional logic

Conditional logic is the deductive system , where is the set of propositional connectives {, ,} and is the structural finitary consequence relation on the absolutely free algebra that preserves degrees of truth over the structure of truth values C, . HereC is the non-commutative regular extension of the 2-element Boolean algebra to 3 truth values {t, u, f}, andf<u<t. In this paper we give a Gentzen type axiomatization for conditional logic.Presented byJan Zygmunt     

Oral production of linguistically complex sentences with meaning relationships of time

The purpose of this investigation was to determine the abilities of children to use the adjoining mechanism in combining two constituent sentences with the temporal adjoiners: after, before, until, when, and while. To elicit responses, a sentence repetition task was devised that included these five temporal adjoiners in four different syntactic environments: transitive sentences with the adjoiner and the subordinate clause following the main clause, transitive sentences with the adjoiner and the subordinate clause preceding the main clause, intransitive sentences with the adjoiner and the subordinate clause following the main clause, and intransitive sentences with the adjoiner and the subordinate clause preceding the main clause. The 30 were between the ages of 4O and 66 years. They were average children who were free from any known emotional disturbance, who were acquiring Standard American English as a native language, who had normal speech and hearing, and whose parents had neither very high nor very low socioeconomic status. To the extent that the children in this study were representative of normal-speaking children of their ages, certain general conclusions were drawn. Children begin to use the temporal adjoining mechanism early, but they do not master it by the age of 66 years. The ability to use the adjoiners, nor is it equal for different syntactic structures nor for all degrees of semantic complexity. After, before, and when appear earlier than while and until. A rapid period of growth in learning to use the temporal adjoining mechanism occurs between the ages of 4 and 5 years. However, a plateau of learning appears to be reached between the ages of 5 and 6 years. In general, children first learn to use the temporal adjoining mechanism in intransitive sentences with the adjoining link in the middle or at the beginning of the utterance. Next, they learn to use it in transitive sentences with the adjoining link at the beginning of the utterance. Finally, they learn to use it in transitive sentences with the adjoining link in the middle of the utterance. In transitive sentences, children appear to learn the rule for placing the subordinate clause at the beginning of the utterance when temporally adjoining two constituent sentences before they learn the base structure rule. In intransitive sentences, they appear to learn the rule for placing the subordinate clause at the beginning of the utterance when temporally adjoining two constituent sentences at the same time that they learn the base structure rule. The underlying semantic relationships that are expressed by specific temporal adjoiners are important determinants of children's abilities to use these adjoiners. In linguistic evaluations, one should consider the syntactic environment in which the temporal adjoiner occurs and assume that after, before, and when are developmentally earlier than while and until.     

Treating connectionism properly: Reflections on Smolensky

Summary In this essay, I undertake to examine the principal theses of Paul Smolensky's 1988Behavioral and Brain Sciences target article, On the proper treatment of connectionism, from the point of view of the methodology and epistemology of science, that is, the philosophical theory of theories in general. After exploring the instrumentalist and realist views of the relationships between micro- and macrotheories on their home ground in the natural sciences, the procedures by which a phenomenally described cognitive task is prepared for symbolic or subsymbolic modeling, and the contrast between the deliberate conscious reasoning processes of a novice and intuitive behavior of an expert in solving a given family of cognitive problems, I argue that although Smolensky is right about what it would take for connectionist subsymbolic models to relate to symbolic models as micro- to macrotheories, he is wrong in concluding that they do. On the contrary, it would be something of a miracle if the idealized nomological structure of the behavior of stochastic patterns of activity over large numbers of subsymbolic units in a connectionist machine corresponded even approximately to the nomological structure of the conceptual level behavior of a Von Neumann computer running off a program whose syntax had been explicitly designed to structurally operationalize a determinate fragment of intentional semantics — unless, of course, the connectionist machine had been deliberately constructed to implement the symbol processor in the first place.I conclude that the proper treatment of connectionism is to be found among the blandly ecumenical proposals for irenic cooperation and division of labor that Smolensky considers and rejects.     

The pastor as enabler

Out of a personal struggle to understand and help a chronically ill friend the writer evolves a view of the pastor as enabler. The definition of enable, the role of the enabler, and the problems which prevent many from being enabled, are discussed. These definitions are illuminated by the Biblical perspective of Elihu in the book ofJob. The pastor who sees himself as enabler is able to bring perspective, clarity, empathy, compassion, and concrete help to the person in need.Chaplain Rusnak is a graduate of The Luthern School of Theology at Chicago.