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.     

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.     

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.     

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).     

How to bereally contraction free

A logic is said to becontraction free if the rule fromA (A B) toA B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there isanother contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to berobustly contraction free if there is no such operator expressible in its language. We show that a large class of finitely valued logics are each not robustly contraction free, and demonstrate that some other contraction free logics fail to be robustly contraction free. Finally, the sublogics of (with the standard connectives) are shown to be robustly contraction free.     

Parallel action: Concurrent dynamic logic with independent modalities

Regular dynamic logic is extended by the program construct, meaning and executed in parallel. In a semantics due to Peleg, each command is interpreted as a set of pairs (s,T), withT being the set of states reachable froms by a single execution of, possibly involving several processes acting in parallel. The modalities << and [] are given the interpretations<>A is true ats iff there existsT withsRT andA true throughoutT, and[]A is true ats iff for allT, ifsRT thenA is true throughoutT, which make <> and [] no longer interdefinable via negation, as they are in the regular case.We prove that the logic defined by this modelling is finitely axiomatisable and has the finite model property, hence is decidable. This requires the development a new theory of canonical models and filtrations for reachability relations.     

An almost general splitting theorem for modal logic

Given a normal (multi-)modal logic a characterization is given of the finitely presentable algebras A whose logics L A split the lattice of normal extensions of . This is a substantial generalization of Rautenberg [10] and [11] in which is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok [2] that for all cycle-free and finite A L A splits the lattice of normal extensions of K. Although we firmly believe it to be true, we have not been able to prove that if a logic splits the lattice of extensions of then is the logic of an algebra finitely presentable over ; in this respect our result remains partial.     

“The story says that” operator in story semantics

In [2] a semantics for implication is offered that makes use of stories — sets of sentences assembled under various constraints. Sentences are evaluated at an actual world and in each member of a set of stories. A sentence B is true in a story s just when B s. A implies B iff for all stories and the actual world, whenever A is true, B is true. In this article the first-order language of [2] is extended by the addition of the operator the story ... says that ..., as in The story Flashman among the Redskins says that Flashman met Sitting Bull. The resulting language is shown to be sound and complete.     

Maximization of sums of quotients of quadratic forms and some generalizations

Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matricesX with columns x₁, ..., x _r . These functions are h₁(x)= _k (xA _kx)(xC _kx)^–1, H₁(X)= _k tr (XA _k X)(XC _k X)^–1, h₁(X)= _{k l} (x _l A _kx _l ) (x _l C _kx _l )^–1 withX constrained to be columnwise orthonormal, h₂(x)= _k (xA _kx)²(xC _kx)^–1 subject to xx=1, H₂(X)= _k tr(XA _kX)(XA_kX)(XC_kX)^–1 subject toXX=I, and h₂(X)= _{k l} (x _l A _kx _l )² (x _l C _kX _l )^–1 subject toXX=I. In these functions the matricesC _k are assumed to be positive definite. The matricesA _k can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge for stimulating this research and for helpful comments on an earlier version of this paper.     

Omega-consistency and the diamond

G is the result of adjoining the schema (qAA)qA to K; the axioms of G* are the theorems of G and the instances of the schema qAA and the sole rule of G* is modus ponens. A sentence is -provable if it is provable in P(eano) A(rithmetic) by one application of the -rule; equivalently, if its negation is -inconsistent in PA. Let -Bew(x) be the natural formalization of the notion of -provability. For any modal sentence A and function mapping sentence letters to sentences of PA, inductively define A by: p = (p) (p a sentence letter); = ; (AB)su}= (A B); and (qA)= -Bew(A )(S) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A, _G A iff for all , _PA A ; and for every modal sentence A, _G* A iff for all , A is true.I should like to thank David Auerbach and Rohit Parikh.     

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.     

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).     

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 consistent theory of attributes in a logic without contraction

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional.     

Perceptual scission of surface-lightness and illumination: An examination of the gelb effect

Summary An attempt was made to examine how the photometric equation: luminance (L)=albedo (A)×illuminance (I) could be solved perceptually when a test field (TF) was not seen as figure, but as ground. A gray disk with two black or white patches was used as the TF. Illuminance of the TF was changed over 2.3 log units and TF albedo was varied from 2.5 to 8.0 in Munsell value. Albedos of the black- and white-appearing patches were 1.5 and 9.5 in Munsell values, respectively. Two types of category judgments for apparent TF lightness (A) and apparent overall illumination (I) were made on the total of 40 TFs (5 illuminances×4 TF-albedos×2 patch-albedos). The results indicated that when the black patches were added to the TF, A was indistinguishable from I and when the white patches were placed on the TF, A and I could be distinguished from each other. The Gelb effect was interpreted as a manifestation of such A–I scission. It was concluded, therefore, that as far as the Gelb effect was observed, the perceptual system could solve the equation, L=A×I, in the sense that for a fixed L, the product of A and I would be constant.     

Finitude and Hume’s Principle

The paper formulates and proves a strengthening of Freges Theorem, which states that axioms for second-order arithmetic are derivable in second-order logic from Humes Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. Finite Humes Principle also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Freges definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.     

On a certain class ofm-address machines

Conclusion It follows from the proved theorems that ifM =Q, (whereQ={0,q ₁,q ₂,...,q }) is a machine of the classM _F then there exist machinesM _i such thatM _i(1,c)=M (q _i,c) andQ _i={0, 1, 2, ..., +1} (i=1, 2, ..., ).And thus, if the way in which to an initial function of content of memorycC a machine assigns a final onecC is regarded as the only essential property of the machine then we can deal with the machines of the formM ={0, 1, 2, ..., }, and processes (t) (wheret=1,c,cC) only.Such approach can simplify the problem of defining particular machines of the classM _F , composing and simplifying them.Allatum est die 19 Januarii 1970     

Interpretations of the first-order theory of diagonalizable algebras in peano arithmetic

For every sequence |p _n } _n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula ⁰ A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, ⁰ A is a theorem ofPA.     

A note on syntactical and semantical functions

We say that a semantical function is correlated with a syntactical function F iff for any structure A and any sentence we have A F A .It is proved that for a syntactical function F there is a semantical function correlated with F iff F preserves propositional connectives up to logical equivalence. For a semantical function there is a syntactical function F correlated with iff for any finitely axiomatizable class X the class ^–1X is also finitely axiomatizable (i.e. iff is continuous in model class topology).