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.

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 From Gentzen to Jaskowski and Back: Algorithmic Translation of Derivations Between the Two Main Systems of Natural Deduction

2017 ◽  
Vol 46 (1/2) ◽  
Author(s):  
Jan Von Plato
Keyword(s):  
Natural Deduction ◽  
The Other ◽  
Partially Ordered ◽  
The Way

The way from linearly written derivations in natural deduction, introduced by Jaskowski and often used in textbooks, is a straightforward root-first translation. The other direction, instead, is tricky, because of the partially ordered assumption formulas in a tree that can get closed by the end of a derivation. An algorithm is defined that operates alternatively from the leaves and root of a derivation and solves the problem.

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+.

scholarly journals Lambda terms for natural deduction, sequent calculus and cut elimination

2000 ◽  
Vol 10 (1) ◽  
pp. 121-134 ◽  
Author(s):  
HENK BARENDREGT ◽  
SILVIA GHILEZAN
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
The Other ◽  
Cut Elimination ◽  
Simple Consequence ◽  
Type Assignment ◽  
Two Systems ◽  
The Many

It is well known that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover, normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a one-one map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (λN) and the other to sequent calculus (λL). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the many-one correspondence mentioned above. Moreover, the second type assignment system has a ‘cut-free’ fragment (λLcf). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms possess a normal form.

On natural deduction

1950 ◽  
Vol 15 (2) ◽  
pp. 93-102 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Natural Deduction ◽  
The Other ◽  
Quantification Theory ◽  
Other Hand ◽  
Derived Rules

For Gentzen's natural deduction, a formalized method of deduction in quantification theory dating from 1934, these important advantages may be claimed: it corresponds more closely than other methods of formalized quantification theory to habitual unformalized modes of reasoning, and it consequently tends to minimize the false moves involved in seeking to construct proofs. The object of this paper is to present and justify a simplification of Gentzen's method, to the end of enhancing the advantages just claimed. No acquaintance with Gentzen's work will be presupposed.A further advantage of Gentzen's method, also somewhat enhanced in my revision of the method, is relative brevity of proofs. In the more usual systematizations of quantification theory, theorems are derived from axiom schemata by proofs which, if rendered in full, would quickly run to unwieldy lengths. Consequently an abbreviative expedient is usually adopted which consists in preserving and numbering theorems for reference in proofs of subsequent theorems. Further brevity is commonly gained by establishing metatheorems, or derived rules, for reference in proving subsequent theorems. In natural deduction, on the other hand, proofs tend to be so short that the abbreviative expedients just now mentioned may conveniently be dispensed with—at least until theorems of extraordinary complexity are embarked upon. In natural deduction accordingly it is customary to start each argument from scratch, without benefit of accumulated theorems or derived rules.

scholarly journals Cut-free Sequent Calculus and Natural Deduction for the Tetravalent Modal Logic

Studia Logica ◽  
2021 ◽  
Author(s):  
Martín Figallo
Keyword(s):  
Modal Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
The Other ◽  
Cut Elimination ◽  
Modal Algebras ◽  
General Method

AbstractThe tetravalent modal logic ($${\mathcal {TML}}$$ TML ) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic$${{\mathcal {TML}}}^N$$ TML N ) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so good as in $${{\mathcal {TML}}}^N$$ TML N , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al. (Log Univ 1:41–69, 2006), we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic.

scholarly journals Relevant Logics Obeying Component Homogeneity

2018 ◽  
Vol 15 (2) ◽  
pp. 301 ◽  
Author(s):  
Roberto Ciuni ◽  
Damian Szmuc ◽  
Thomas Macaulay Ferguson
Keyword(s):  
Sufficient Conditions ◽  
Relevant Logic ◽  
The Other ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Characterization Result ◽  
General Characterization ◽  
Relevant Logics ◽  
Necessary And Sufficient ◽  
Open Issues

This paper discusses three relevant logics (S*fde , dS*fde , crossS*fde) that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde , dS*fde , crossS*fde. Second, the paper establishes complete sequent calculi for S*fde , dS*fde , crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define (a given family of) containment logics, we explore the single-premise/single-conclusion fragment of S*fde , dS*fde , crossS*fde and the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde  as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues.

Simplified foundations for mathematical logic

1955 ◽  
Vol 20 (2) ◽  
pp. 123-139 ◽  
Author(s):  
Robert L. Stanley
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Natural Deduction ◽  
Small Body ◽  
The Other ◽  
Consistency Proof ◽  
Basic Notion ◽  
Relative Consistency ◽  
And Mathematics

A system SF, closely related to NF, is outlined here. SF has several novel points of simplicity and interest, (a) It uses only one basic notion, from which all the other concepts of logic and mathematics may be built definitionally. Three-notion systems are common, but Quine's two-notion IA has for some time represented the extreme in conceptual economy, (b) The theorems of SF are generated under just three rules of analysis, which unify into a single postulational principle, (c) SF is built solely in terms of what is commonly, known as the “natural deduction” method, under which each theorem is attacked primarily as it stands, by means of a very small body of rules, rather than less directly, through a very large, potentially infinite backlog of theorems. Although natural deduction is by no means new as a method, its exclusive applications have previously been relatively limited, not even reaching principles of identity, much less set theory, relations, or mathematics proper, (d) SF is at least as strong as NF, yielding all of its theorems, which are expressed here in forms analogous to those of the metatheorems in ML. If NF is consistent, so is SF. The main points in the relative consistency proof are set forth below in section seven.

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