A contractionless semilattice semantics

1987 ◽  
Vol 52 (2) ◽  
pp. 526-529 ◽  
Author(s):  
Steve Giambrone ◽  
Robert K. Meyer ◽  
Alasdair Urquhart
Keyword(s):  
Natural Deduction ◽  
Model Structure ◽  
Commutative Monoid ◽  
Central Idea ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Gentzen System ◽  
The Way

Semilattice semantics for relevant logics were discovered independently by Routley and Urquhart over 10 years ago. A semilattice semantics was first published in [10], where the weak theory of implication of [8] and [3] (i.e., R →, the pure implication fragment of the system R of relevant implication) is shown to be consistent and complete with respect to it. That result was extended in [11], But the semantics is explored in greatest detail in [12]. As reported in [4], Fine outfitted the positive semilattice semantics for R+ with a suitable Hilbert-style axiomatisation. (We refer to the system as ◡R+.) In 1980 Charlwood supplied a subscripted system of natural deduction. (See [1] and [2].) A subscripted Gentzen system was devised in [5] and [6].Obviously, the central idea of the semilattice semantics is to impose relevant-style valuations on a semilattice (with an identity) used as the underlying model structure. However, in [12] the contractionless semantics are obtained (quite reasonably) by dropping the idempotence postulate and thus changing the relatively simple semilattice structure into a commutative monoid. Here we show that the semilattice structure can be regained for positive, contractionless relevant implication. Although we have no proofs as yet, we think that this semantics will pave the way for showing completeness for the corresponding subscripted Gentzen and natural deduction systems, as well as the Hilbert-style axiomatization, ◡RW+.

scholarly journals Arithmetic Formulated Relevantly

2021 ◽  
Vol 18 (5) ◽  
pp. 154-288
Author(s):  
Robert Meyer
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Critical Problem ◽  
First Order ◽  
Order Form ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Natural Way

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that (1) it is trivial that relevant arithmetic is absolutely consistent, but (2) classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under (1), I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under (2), I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment.

TRANSLATIONS BETWEEN LINEAR AND TREE NATURAL DEDUCTION SYSTEMS FOR RELEVANT LOGICS

2021 ◽  
pp. 1-22
Author(s):  
SHAWN STANDEFER
Keyword(s):  
Natural Deduction ◽  
The Other ◽  
Relevant Logics

Abstract Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

scholarly journals El lado de afuera. Idea de la historia en "Historia argentina" de Rodrigo Fresán

2020 ◽  
Vol 17 ◽  
Author(s):  
Christian Snoey
Keyword(s):  
The State ◽  
History Taking ◽  
Politics And Literature ◽  
Central Idea ◽  
Ricardo Piglia ◽  
Narrative Forms ◽  
Argentine History ◽  
The Relationship ◽  
The Way

El objetivo de este trabajo es reflexionar en torno a las categorías de pensamiento mediante las que se trata de aprehender la historia en el volumen, híbrido entre cuento y novela, Historia argentina, de Rodrigo Fresán, a partir de un análisis de las formas narrativas mediante las que se construyen los relatos, puesto que la estructura de la obra, a modo de cuentos que funcionan por resonancia, espejea la manera en que se concibe la historia. Para ello, parto fundamentalmente de las ideas de Ricardo Piglia acerca de la relación entre política y literatura, para quien la ficción reproduce el lenguaje del Estado y crea su reverso; y también acudo a las teorías de Eloy Fernández Porta, ensayadas en Afterpop, para deslindar la manera en que el uso de referencias pop en esta obra responde a una crítica de las formas de la cultura oficial. Por tanto, el punto de llegada de este trabajo consiste en el análisis de la revisión del lenguaje con el que se ha articulado la historia argentina, y la búsqueda de un lenguaje otro para escribir y aprehenderla en Historia argentina, tomando como idea central el concepto de distanciamiento, puesto que cifra la actitud, tanto emocional como intelectual, respecto a la escritura de la historia. The objective of this work is to reflect on the categories of thought through which it is a question of apprehending the history in the book, hybrid between story and novel, Historia argentina, by Rodrigo Fresán, from an analysis of the narrative forms through those that build the stories, since the structure of the work, by way of stories that work by resonance, reflects the way in which history is conceived. To do this, I fundamentally start form the ideas of Ricardo Piglia about the relationship between politics and literature, for whom fiction reproduces the language of the State and creates its reverse. And I also turn to the theories of Eloy Fernández Porta, studied in Afterpop, to demarcate the way in which the use of pop references in this work responds to a critique of the forms of official culture. Therefore, the objective of this work is the analysis of the revision of the language with which Argentine history has been articulated, and the search for another language to write and apprehend in Argentine history, taking as a central idea the concept of distancing, since it figures the attitude, both emotional and intellectual, regarding the writing of history.

E, R AND γ

1969 ◽  
Vol 34 (3) ◽  
pp. 460-474 ◽  
Author(s):  
Robert K. Meyer ◽  
J. Michael Dunn
Keyword(s):  
Relevant Implication ◽  
Relevant Logics

By γ, we mean the rule, “From ├ A and ├ Ā V B, infer ├ B”.1 This rule has played an important and a controversial role in a set of relevant logics free of certain well-known paradoxes of implication, like AĀ-→B and A-→(B-→B). Among these logics we count the pioneering systems of strenge Implikation presented by Ackermann in [1],2 as well as the Anderson-Belnap systems E of entail-ment and R of relevant implication.3

Gentzenizations of relevant logics with distribution

1996 ◽  
Vol 61 (2) ◽  
pp. 402-420 ◽  
Author(s):  
Ross T. Brady
Keyword(s):  
Basic System ◽  
Relevant Implication ◽  
Left Handed ◽  
Relevant Logics ◽  
Free Left ◽  
Contractionless Logic

We establish cut-free left-handed Gentzenizations for a range of major relevant logics from B through to R, all with distribution. B is the basic system of the Routley-Meyer semantics (see [15], pp. 287–300) and R is the logic of relevant implication (see [1], p. 341). Previously, the contractionless logics DW, TW, EW, RW and RWK were Gentzenized in [3], [4] and [5], and also the distributionless logics LBQ, LDWQ, LTWQo, LEWQot, LRWQ, LRWKQ and LRQ in [6] and [7]. This paper provides Gentzenizations for the logics DJ, TJ, T and R, with various levels of contraction, and for the contractionless logic B, which could not be included in [4] using the technique developed there. We also include the Gentzenization of TW in order to compare it with that in [4]. The Gentzenizations that we obtain here for DW and RW are inferior to those already obtained in [4], but they are included for reference when constructing other systems. The logics EW and E present a difficulty for our method and are omitted. For background to the Gentzenization of relevant logics, see [6], and for motivation behind the logics involved, see [6], [1] and [15]. Because of the number of properties that are brought to bear in obtaining these systems, we prefer to consider Gentzenizations for particular logics rather than for arbitrary bunches of axioms.

A Puzzle in the Three Dialogues and Its Platonic Resolution

2018 ◽  
Author(s):  
John Russell Roberts
Keyword(s):  
Image Of God ◽  
Central Idea ◽  
Innate Ideas ◽  
Abstract Ideas ◽  
Cambridge Platonism ◽  
Key Aspects ◽  
The Image Of God ◽  
Shed Light ◽  
The Way

This essay suggests that Berkeley’s Neoplatonism may be profitably viewed as developed under the influence of Cambridge Platonism. A brief account of some key aspects of Cambridge Platonism are reviewed, specifically the central idea of the Image of God Doctrine (IGD) and Cudworth’s Axiarchism. Then possible points of influence of these aspects on Berkeley’s views are explored. In support of its possible usefulness, this approach to Berkeley’s Neoplatonism is used to shed light on his otherwise puzzling embrace of the pure intellect and abstract ideas. If Berkeley is drawing on the Cambridge Platonism tradition in the way suggested, he can have his pure intellect and its innate ideas without dragging along a commitment to a faculty of abstraction and its abstract ideas. Instead, the pure intellect is seen as a reflective faculty directed to the perfectly particular, concrete self.

Relational proof system for relevant logics

1992 ◽  
Vol 57 (4) ◽  
pp. 1425-1440 ◽  
Author(s):  
Ewa Orlowska
Keyword(s):  
Natural Deduction ◽  
Proof System ◽  
Algebraic Semantics ◽  
Proof Systems ◽  
Relevant Logics ◽  
Deduction Rules

AbstractA method is presented for constructing natural deduction-style systems for propositional relevant logics. The method consists in first translating formulas of relevant logics into ternary relations, and then defining deduction rules for a corresponding logic of ternary relations. Proof systems of that form are given for various relevant logics. A class of algebras of ternary relations is introduced that provides a relation-algebraic semantics for relevant logics.

Equivalences between logics and their representing type theories

1995 ◽  
Vol 5 (3) ◽  
pp. 323-349 ◽  
Author(s):  
Philippa Gardner
Keyword(s):  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Logical Framework ◽  
Simple Formulation ◽  
First Order ◽  
Relevant Logics ◽  
First Time ◽  
New Framework

We propose a new framework for representing logics, called LF+, which is based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions that capture how well a logic has been represented. These definitions are possible because we are able to distinguish in a generic way that part of the LF+ entailment corresponding to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction first-order logic can be well-represented in LF+, whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between type-theoretic and categorical approaches to frameworks.

scholarly journals A non-transitive relevant implication corresponding to classical logic consequence

2019 ◽  
Vol 16 (2) ◽  
pp. 10
Author(s):  
Peter Verdée ◽  
Inge De Bal ◽  
Aleksandra Samonek
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Modus Ponens ◽  
Relevant Logic ◽  
Object Language ◽  
Cut Rule ◽  
Consequence Relation ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Traditional Sense

In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair <premises, conclusions> such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid L-consequence.          Subsequently we apply this denition to the classical logic (CL) consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to thesequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word.       By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism (A and ~AvB relevantly imply B) and Adjunction (A and B relevantly imply A&B) are valid, and neither Modus Ponens nor the Cut rule are admissible.

scholarly journals Derivation of some new distributions in statistical mechanics using maximum entropy approach

2014 ◽  
Vol 24 (1) ◽  
pp. 145-155 ◽  
Author(s):  
Amritansu Ray ◽  
S.K. Majumder
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
The Other ◽  
Central Idea ◽  
Entropy Principle ◽  
Bose Einstein ◽  
The Way

The maximum entropy principle has been earlier used to derive the Bose Einstein(B.E.), Fermi Dirac(F.D.) & Intermediate Statistics(I.S.) distribution of statistical mechanics. The central idea of these distributions is to predict the distribution of the microstates, which are the particle of the system, on the basis of the knowledge of some macroscopic data. The latter information is specified in the form of some simple moment constraints. One distribution differs from the other in the way in which the constraints are specified. In the present paper, we have derived some new distributions similar to B.E., F.D. distributions of statistical mechanics by using maximum entropy principle. Some proofs of B.E. & F.D. distributions are shown, and at the end some new results are discussed.

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