A reduction rule for Peirce formula

A reduction rule is introduced as a transformation of proof figures in implicational classical logic. Proof figures are represented as typed terms in a -calculus with a new constant P ⁽⁽⁾⁾. It is shown that all terms with the same type are equivalent with respect to -reduction augmented by this P-reduction rule. Hence all the proofs of the same implicational formula are equivalent. It is also shown that strong normalization fails for P-reduction. Weak normalization is shown for P-reduction with another reduction rule which simplifies of (( ) ) into an atomic type.This work was partially supported by a Grant-in-Aid for General Scientific Research No. 05680276 of the Ministry of Education, Science and Culture, Japan and by Japan Society for the Promotion of Science. Hiroakira Ono     

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).     

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.     

The Classification of Propositional Calculi

We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.     

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.     

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     

Converse Ackermann Croperty and semiclassical negation

A prepositional logic S has the Converse Ackermann Property (CAP) if (AB)C is unprovable in S when C does not contain . In A Routley-Meyer semantics for Converse Ackermann Property (Journal of Philosophical Logic, 16 (1987), pp. 65–76) I showed how to derive positive logical systems with the CAP. There I conjectured that each of these positive systems were compatible with a so-called semiclassical negation. In the present paper I prove that this conjecture was right. Relational Routley-Meyer type semantics are provided for each one of the resulting systems (the positive systems plus the semiclassical negation).     

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.     

Predicate logics without the structure rules

In our previous paper [5], we have studied Kripke-type semantics for propositional logics without the contraction rule. In this paper, we will extend our argument to predicate logics without the structure rules. Similarly to the propositional case, we can not carry out Henkin's construction in the predicate case. Besides, there exists a difficulty that the rules of inference () and () are not always valid in our semantics. So, we have to introduce a notion of normal models.Dedicated to the memory of the late Professor Hidetosi Takahasi     

Gentzen-type formulation of the prepositional logic LQ

We give a Gentzen-type formulation GQ for the intermediate logic LQ and prove the cut-elimination theorem on it, where LQ is the propositional logic obtained from the intuitionistic propositional logic LI by adding the axioms of the form AV A.     

On two problems of Harvey Friedman

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).The second part includes a proof of the theorem ensuring existence of a greatest structurally complete intermediate logic having the disjunction property. This theorem confirms H. Friedman's conjecture 41 (cf. [2], problem 41).In the third part the reader will find a criterion which allows us to obtain sets satisfying the conditions of Friedman's problem 42, on the basis of intermediate logics satisfying the conditions of problem 41.Finally, the fourth part contains a proof of a criterion which allows us to obtain modal logics endowed with Hallden's property on the basis of structurally complete intermediate logics having the disjunction property.Dedicated to Professor Roman SuszkoThe author would like to thank professors J. Perzanowski and A. Wroski for valuable suggestions.     

A modification of kendall's tau for measuring association in contingency tables

A coefficient of association is described for a contingency table containing data classified into two sets of ordered categories. Within each of the two sets the number of categories or the number of cases in each category need not be the same.=+1 for perfect positive association and has an expectation of 0 for chance association. In many cases also has –1 as a lower limit. The limitations of Kendall's _a and _b and Stuart's _c are discussed, as is the identity of these coefficients to' under certain conditions. Computational procedure for is given.     

A metacompleteness theorem for contraction-free relevant logics

I note that the logics of the relevant group most closely tied to the research programme in paraconsistency are those without the contraction postulate(A.AB).AB and its close relatives. As a move towards gaining control of the contraction-free systems I show that they are prime (that wheneverA B is a theorem so is eitherA orB). The proof is an extension of the metavaluational techniques standardly used for analogous results about intuitionist logic or the relevant positive logics.The main results of this paper were presented at the Paraconsistent Logic conference in Canberra in 1980. The author wishes to thank the participants in that conference for comments and suggestions made at the time.     

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.     

Interpolation and amalgamation properties in varieties of equivalential algebras

Important positive as well as negative results on interpolation property in fragments of the intuitionistic propositional logic (INT) were obtained by J. I. Zucker in [6]. He proved that the interpolation theorem holds in purely implicational fragment of INT. He also gave an example of a fragment of INT for which interpolation fails. This fragment is determined by the constant falsum (), well known connectives: implication () and conjunction (), and by a ternary connective defined as follows: (p, q, r)= ^df (pq)(pr).Extending this result of J. I. Zucker, G. R. Renardel de Lavalette proved in [5] that there are continuously many fragments of INT without the interpolation property.This paper is meant to continue the research mentioned above. To be more precise, its aim is to answer questions concerning interpolation and amalgamation properties in varieties of equivalential algebras, particularly in the variety determined by the purely equivalential fragment of INT.     

The logical structure of linguistic commitment II: Systems of relevant commitment entailment

In The Logical Structure of Linguistic Commitment I (The Journal of Philosophical Logic 23 (1994), 369–400), we sketch a linguistic theory (inspired by Brandom's Making it Explicit) which includes an expressivist account of the implication connective, : the role of is to make explicit the inferential proprieties among possible commitments which proprieties determine, in part, the significances of sentences. This motivates reading (A B) as commitment to A is, in part, commitment to B. Our project is to study the logic of . LSLC I approximates (A B) as anyone committed to A is committed to B, ignoring issues of whether A is relevant to B. The present paper includes considerations of relevance, motivating systems of relevant commitment entailment related to the systems of commitment entailment of LSLC I. We also consider the relevance logics that result from a commitment reading of Fine's semantics for relevance logics, a reading that Fine suggests.     

On defining necessity in terms of entailment

In their book Entailment, Anderson and Belnap investigate the consequences of defining Lp (it is necessary that p) in system E as (pp)p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions { _E Lpp, _EL(pq)(LpLq), _E pLp} if and only if its has the form C ₁.C₂ .... C_np, where each C _iis equivalent in E to either pp or ((pp)p)p.     

Foundations of a new system of probability theory

The aim of my book is to explain the content of the different notions of probability.Based on a concept of logical probability, which is modified as compared with Carnap, we succeed by means of the mathematical results of de Finetti in defining the concept of statistical probability.The starting point is the fundamental concept that certain phenomena are of the same kind, that certain occurrences can be repeated, that certain experiments are identical. We introduce for this idea the notion: concept K of similarity. From concept K of similarity we derive logically some probability-theoretic conclusions:If the events E() are similar —of the same kind - on the basis of such a concept K, it holds good that intersections of n of these events are equiprobable on the basis of K; in formulae: E(₁)...E( _{n K} E('₁)...E(' _n , _{i j} ,' _j ' _j for ij On the basis of some further axioms a partial comparative probability structure results from K, which forms the starting point of our further investigations and which we call logical probability on the basis of K.We investigate a metrisation of this partial comparative structure, i.e. normed -additive functions m _K, which are compatible with this structure; we call these functions m _K measure-functions in relation to K.The measure-functions may be interpreted as subjective probabilities of individuals, who accept the concept K.Now it holds good: For each measure-function there exists with measure one the limit of relative frequencies in a sequence of the E().In such an event, where all measure-functions coincide, we speak of a quantitative logical probability, which is the common measure of this event. In formulae we have: l _K (h _n lim h _n )=1 in words: There is the quantitative logical probability one that the limit of the relative frequencies exists. Another way of saying this is that the event ^* (h_n lim h _n) is a maximal element in the comparative structure resulting from K.Therefore we are entitled to introduce this limit and call it statistical probability P.With the aid of the measure-functions it is possible to calculate the velocity of this convergence. The analog of the Bernoulli inequation holds true: m _K (¦h _n –P¦)1–1/4n².It is further possible in the work to obtain relationships for the concept of statistical independence which are expressed in terms of the comparative probability.The theory has a special significance for quantum mechanics: The similarity of the phenomena in the domain of quantum mechanics explains the statistical behaviour of the phenomena.The usual mathematical statistics are explained in my book. But it seems more expedient on the basis of this new theory to use besides the notion of statistical probability also the notion of logical probability; the notion of subjective probability has only a heuristic function in my system.The following dualism is to be noted: The statistical behaviour of similar phenomena may be described on the one hand according to the model of the classical probability theory by means of a figure called statistical probability, on the other hand we may express all formulae by means of a function, called statistical probability function. This function is defined as the limit of the relative frequencies depending on the respective state of the universe. The statistical probability function is the primary notion, the notion of statistical probability is derived from it; it is defined as the value of the statistical probability function for the true unknown state of the universe.As far as the Hume problem, the problem of inductive inference, is concerned, the book seems to give an example of how to solve it.The developed notions such as concept, measure-function, logical probability, etc. seem to be important beyond the concept of similarity.The present work represents a summary of my book Grundzüge zu einem neuen Aufbau der Wahrscheinlich-keitstheorie [5], For this reason, I have frequently dispensed with providing proof and in this connection refer the interested reader to my book.     

Commentary on Boharts “The Client is the Most Important Common Factor”

The cliché Treatment operates on patient to produce effects is reversed by Bohart into Client operates on treatments and procedures to produce effects. Although this formula has the advantage of underscoring the patient's responsibility and competence, it may also overemphasize his or her role. A more balanced formula could be Process operates on both patient and client to produce effects, as it means that neither the therapist nor the client, but the process is the operator. There seems to be not much to earn, if the old hero (the therapist) is replaced by the new one (the client). A more promising perspective opens if both give up their pretence to be the operator, or the one who knows what is to be done, and listen and submit to the logic of the process that goes beyond both.     

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.