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.

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 without distribution. I

1996 ◽  
Vol 61 (2) ◽  
pp. 353-378 ◽  
Author(s):  
Ross T. Brady
Keyword(s):  
Left Handed ◽  
Analytic Tableaux ◽  
Relevant Logics ◽  
History Of ◽  
Single Formula ◽  
The Right

The history of the Gentzenization of relevant logics goes back to Kripke [17], who in 1959 Gentzenized R→ and went on to prove its decidability. Formulae were separated by commas on the left side of the turnstile, the commas just representing nested implications. Kripke employed just a singleton formula to the right of the turnstile. He also considered adding negation, as well as other connectives, but it was not until 1961 that Belnap and Wallace, in [5], Gentzenized and proved its decidability, though their Gentzenization employed commas on both sides of the turnstile. Subsequently, in 1966, the logic R without distribution, now called LR (for lattice R), was Gentzenized in a similar style by Meyer in [20]. He also went on to show decidability for LR by extending Kripke's argument. Later, in 1969, Dunn Gentzenized R+ (published in [1], pp. 381–391) using two structural connectives (commas and semicolons) to the left of the turnstile, and with a single formula to the right. Here, the commas represent conjunction and the semicolons represent an intensional conjunction, called “fusion”. This is all nicely set out in McRobbie [19], where he also introduces left-handed Gentzenizations and analytic tableaux for a number of fragments of relevant logics. In 1979, further work on distributionless logic was done by Grishin, in a series of papers, including [16], in which he produced a Gentzenization of quantified RW without distribution (which we will call LRWQ), and used it to prove the decidability of this quantified logic.

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 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.

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+.

A via-free left-handed Transmission Line with Radial Stubs

2009 ◽  
Author(s):  
G. Naga Satish ◽  
K. V. Srivastava ◽  
A. Biswas ◽  
D. Kettle
Keyword(s):  
Transmission Line ◽  
Left Handed ◽  
Free Left

Single molecule optical diffraction: A tool for understanding the structure of triple stranded left-hand helical cellulose

1989 ◽  
Vol 47 ◽  
pp. 1024-1025
Author(s):  
George C. Ruben ◽  
William Krakow
Keyword(s):  
Single Molecule ◽  
Helical Structure ◽  
Optical Diffraction ◽  
Maximum Diameter ◽  
Primary Cell Wall ◽  
Cellulose Synthesis ◽  
Left Hand ◽  
Left Handed ◽  
Freeze Dried ◽  
Diffraction Patterns

Tobacco primary cell wall and normal bacterial Acetobacter xylinum cellulose formation produced a 36.8±3Å triple-stranded left-hand helical microfibril in freeze-dried Pt-C replicas and in negatively stained preparations for TEM. As three submicrofibril strands exit the wall of Axylinum , they twist together to form a left-hand helical microfibril. This process is driven by the left-hand helical structure of the submicrofibril and by cellulose synthesis. That is, as the submicrofibril is elongating at the wall, it is also being left-hand twisted and twisted together with two other submicrofibrils. The submicrofibril appears to have the dimensions of a nine (l-4)-ß-D-glucan parallel chain crystalline unit whose long, 23Å, and short, 19Å, diagonals form major and minor left-handed axial surface ridges every 36Å.The computer generated optical diffraction of this model and its corresponding image have been compared. The submicrofibril model was used to construct a microfibril model. This model and corresponding microfibril images have also been optically diffracted and comparedIn this paper we compare two less complex microfibril models. The first model (Fig. 1a) is constructed with cylindrical submicrofibrils. The second model (Fig. 2a) is also constructed with three submicrofibrils but with a single 23 Å diagonal, projecting from a rounded cross section and left-hand helically twisted, with a 36Å repeat, similar to the original model (45°±10° crossover angle). The submicrofibrils cross the microfibril axis at roughly a 45°±10° angle, the same crossover angle observed in microflbril TEM images. These models were constructed so that the maximum diameter of the submicrofibrils was 23Å and the overall microfibril diameters were similar to Pt-C coated image diameters of ∼50Å and not the actual diameter of 36.5Å. The methods for computing optical diffraction patterns have been published before.

Left handed GDPs and students

BDJ ◽  
1995 ◽  
Vol 178 (12) ◽  
pp. 448-448 ◽  
Author(s):  
J M Brown
Keyword(s):  
Left Handed

Composite right/left-handed transmission line with array of thermocouples for generating terahertz radiation

2020 ◽  
Vol 92 (2) ◽  
pp. 20502
Author(s):  
Behrokh Beiranvand ◽  
Alexander S. Sobolev ◽  
Anton V. Kudryashov
Keyword(s):  
Transmission Line ◽  
Terahertz Radiation ◽  
Transmission Lines ◽  
Software Package ◽  
Optical Pulses ◽  
Capacitively Coupled ◽  
Voltage Difference ◽  
Left Handed ◽  
Microwave Transmission ◽  
Commercial Software Package

We present a new concept of the thermoelectric structure that generates microwave and terahertz signals when illuminated by femtosecond optical pulses. The structure consists of a series array of capacitively coupled thermocouples. The array acts as a hybrid type microwave transmission line with anomalous dispersion and phase velocity higher than the velocity of light. This allows for adding up the responces from all the thermocouples in phase. The array is easily integrable with microstrip transmission lines. Dispersion curves obtained from both the lumped network scheme and numerical simulations are presented. The connection of the thermocouples is a composite right/left-handed transmission line, which can receive terahertz radiation from the transmission line ports. The radiation of the photon to the surface of the thermocouple structure causes a voltage difference with the bandwidth of terahertz. We examined a lossy composite right/left-handed transmission line to extract the circuit elements. The calculated properties of the design are extracted by employing commercial software package CST STUDIO SUITE.

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