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 superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

A semantical investigation into Leśniewski's axiom of his ontology

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure).     

Model theoretic results for infinitely deep languages

We define a subhierarchy of the infinitely deep languagesN described by Jaakko Hintikka and Veikko Rantala. We shall show that some model theoretic results well-known in the model theory of the ordinary infinitary languages can be generalized for these new languages. Among these are the downward Löwenheim-Skolem and o's theorems as well as some compactness properties.     

Sprachverstehende maschinen

In this paper the author tries to disentangle some of the problems tied up in John Searle's famous Chinese-room-argument. In a first step to answer the question what it would be for a system to have not only syntax, but also semantics the author gives a brief account of the functioning of the language understanding systems (LUS) so far developed in the framework of AI research thereby making clear that systems like Winograd's SHRDLU are indeed doing little more than mere number crunching. But things would be entirely different, the author argues, if the database of a LUS were built up by the system itself via some perceptual component-at least, if this perceptual component had the capacity to distinguish objects having a certain property F from objects which do not. For in this case the system could store an internal representation of the fact that the object has the property F in its database if and only if the object in fact has that property. And this would be a good basis for calling such a system a genuine LUS. But Searle has objected to a very similar account of J. Fodor that nothing could be further from true language understanding. The reason for this complaint seems to be that Searle holds the view that a true LUS must e.g., know that the word hamburgers refers to hamburgers and that he moreover claims that this knowledge must be explicit or that the system must be aware of the reference of hamburgers to hamburgers. The author argues that this is asking too much. For it seems plausible to say that a system is able to understand e.g., the word hamburger even if it has only implicit knowledge of the fact that hamburger refers to hamburgers in the sense that it has the capacity to tell hamburgers from non hamburgers and the capacity to bring the word hamburger together just with objects of the former kind.     

Intensional Completeness in an Extension of Gödel/Dummett Logic

We enrich intuitionistic logic with a lax modal operator and define a corresponding intensional enrichment of Kripke models M = (W, , V) by a function T giving an effort measure T(w, u) {} for each -related pair (w, u). We show that embodies the abstraction involved in passing from true up to bounded effort to true outright. We then introduce a refined notion of intensional validity M |= p : and present a corresponding intensional calculus iLC-h which gives a natural extension by lax modality of the well-known G: odel/Dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : | L |= p : } characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of and completely recovered by a suitable semantic interpretation of proofs.     

Action incompleteness

The author has previously introduced an operator into dynamic logic which takes formulae to terms; the suggested reading of A was the bringing about of A or the seeing to it that A. After criticism from S. K. Thomason and T. J. Surendonk the author now presents an improved version of his theory. The crucial feature is the introduction of an operatorOK taking terms to formulae; the suggested reading of OK is always terminates.     

A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.     

O liczbach kardynalnych

Tre: Wstp. I. Niektóre pojcia i prawa teorii klas stosunków. II. O klasach abstrakcji. III. O równoskadnikowoci. IV. O kardynalnych liczbach skadników. V. O. stosowanej teorii równoskadnikowoci. VI. O relacji równej liczby elementów. VII. Stosunek obu poj liczb kardynalnych. Zakoczenie.Allatum est die 30 Augusti 1961     

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     

Utility theory with inexact preferences and degrees of preference

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.     

On a new property of partially balanced association schemes useful in psychometric structural analysis

In an earlier paper [Psychometrika,31, 1966, p. 147], Srivastava obtained a test for the HypothesisH ₀ : = ₀₀ + ... + _ll, where i are known matrices,i are unknown constants and is the unknown (p ×p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p ×p) covariance matrix of a random variablex (withp components) the condition that ₀, ₁, ..., l form a commutative linear associative algebra and a certain vector, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where ₀, ₁, ..., l are the association matrices of a partially balanced association scheme.This research was partially supported by the U. S. Air Force under Grant No. AF33(615)-3231, monitored by the Aero Space Research Labs.Now at Colorado State University.     

Supererogation in Deontic Logic: Metatheory for DWE and Some Close Neighbours

In "Doing Well Enough: Toward a Logic for Common Sense Morality", Paul McNamara sets out a semantics for a deontic logic which contains the operator It is supererogatory that. As well as having a binary accessibility relation on worlds, that semantics contains a relative ordering relation, . For worlds u, v and w, we say that u w v when v is at least as good as u according to the standards of w. In this paper we axiomatize logics complete over three versions of the semantics. We call the strongest of these logics DWE for Doing Well Enough.     

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.     

Variations in use of the term “or”

Two studies showed that adults' responses to questions involving the term or varied markedly depending upon the type of question presented. When presented with various objects (A's and B's) and asked to circle all things which are A or B subjects tended to circle A's as well as B's, whereas when asked to circle all the A's or B's subjects showed a relatively stronger tendency to circle one or the other. Moreover the nature of the sets of objects (As and Bs) influenced behavior as well. There was also evidence that the effects due to question wording or set type transferred.     

Mathematics of Totalities: an alternative to mathematics of sets

I dare say, a set is contranatural if some pair of its elements has a nonempty intersection. So, we consider only collections of disjoint nonempty elements and call them totalities. We propose the propositional logicTT, where a proposition letters some totality. The proposition is true if it letters the greatest totality. There are five connectives inTT: , , , , # and the last is called plexus. The truth of # means that any element of the totality has a nonempty intersection with any element of the totality . An imbeddingG of the classical predicate logicCPL inTT is defined. A formulaf ofCPL is a classical tautology if and only ifG(f) is always true inTT. So, mathematics may be expounded inTT, without quantifiers.     

Rules in relevant logic — II: Formula representation

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A ₁ .A ₂ . ... .A _n B, the conjunctive form,A ₁&A ₂ & ...A _n B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A ₁&A ₂& ...A _n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C.I acknowledge help from anonymous referees for guidance in preparing Part II, and especially for the suggestion that Theorem 9 could be expanded to fully contraction-less logics.     

Paradoxical consequences of Balzer''s and G?hde''s criteria of theoreticity. Results of an application to ten scientific theories

It is shown that the criteria of T-theoreticity proposed by Balzer and Gähde lead to strongly counterintuitive and in this sense paradoxical results: most of the obviously empirical or at least nontheoretical terms come out as theoretical. This is demonstrated for a lot of theories in different areas. On the way, some improved and some new structuralist theory-reconstructions are given. The conclusion is drawn that the T-theoreticity of a term cannot possibly be proved on the basis of the mathematical structure of theory T alone (as Gähde and Balzer suggest). Rather, an independent notion of pre-T-theoreticity and-more importantly-of empiricity is needed; i.e., not empirical and not pre-T-theoretical are independent, necessary but not sufficient conditions for T-theoretical (this is also a necessary complement of Sneed's original criterion). Finally it is asked whether the structuralist criterion of T-theoreticity complemented by such independent conditions would be a satisfactory answer to Putnam's challenge, and the answer again is negative: the criterion is not able to distinguish between empirically contentful and completely contentless (superfluous) theoretical terms.