scholarly journals On Natural Deduction for Herbrand Constructive Logics III: The Strange Case of the Intuitionistic Logic of Constant Domains

2018 ◽  
Vol 281 ◽  
pp. 1-9
Author(s):  
Federico Aschieri
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Constant Domains ◽  
Constructive Logics

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

scholarly journals Normalization as a consequence of cut elimination

2009 ◽  
Vol 86 (100) ◽  
pp. 27-34
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Cut Elimination ◽  
Deduction System ◽  
Normalization Theorem ◽  
Natural Deduction System

Pairs of systems, which consist of a system of sequents and a natural deduction system for some part of intuitionistic logic, are considered. For each of these pairs of systems the property that the normalization theorem is a consequence of the cut-elimination theorem is presented.

scholarly journals Co-constructive Logics for Proofs and Refutations

Studia Humana ◽  
2015 ◽  
Vol 3 (4) ◽  
pp. 22-40 ◽  
Author(s):  
James Trafford
Keyword(s):  
Intuitionistic Logic ◽  
Proof Theory ◽  
Direct Proof ◽  
Negative Information ◽  
Constructive Logic ◽  
Logic Of Proofs ◽  
Constructive Logics

Abstract This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov's logic of problems.

A new version of Beth semantics for intuitionistic logic

1977 ◽  
Vol 42 (2) ◽  
pp. 306-308 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Tree Structure ◽  
Truth Conditions ◽  
Truth Definition ◽  
Image Position ◽  
Constant Domains ◽  
Condition B

We use the notation of Kripke's paper [1]. Let M = (G, K, R) be a tree structure and D a domain and η a Beth model on M. The truth conditions of the Beth semantics for ∨ and ∃ are (see [1]):(a) η (A ∨ B, H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B, either η(A, H′) = T or η(B, H′) = T.(b) η(∃xA(x), H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B there exists an a ∈ D such that η(A (a), H′) = T.Suppose we change the truth definition η to η* by replacing condition (b) by the condition (b*) (well known from the Kripke interpretation):Call this type of interpretation the new version of Beth semantics. We proveTheorem 1. Intuitionistic predicate logic is complete for the new version of the Beth semantics.Since Beth structures are of constant domains, and in the new version of Beth semantics the truth conditions for ∧, →, ∃, ∀, ¬ are the same as for the Kripke interpretation, we get:Corollary 2. The fragment without disjunction of the logic CD of constant domains (i.e. with the additional schema ∀x(A ∨ B(x))→ A ∨ ∀xB(x), x not free in A) equals the fragment without disjunction of intuitionistice logic.

scholarly journals A Formalisation of Weak Normalisation (with Respect to Permutations) of Sequent Calculus Proofs

2000 ◽  
Vol 3 ◽  
pp. 1-26
Author(s):  
A.A. Adams
Keyword(s):  
Fixed Points ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
The State ◽  
Original Presentation

AbstractDyckhoff and Pinto present a weakly normalising system of reductions on derivations are characterised as the fixed points of the composition of the Prawitz translations into natural deduction and back. This paper presents a formalisation of the system, including a proof of the Weak normalisation property for the formalisation. More details can be found in earlier work by the author. The formalisation has been kept as closes as possible to the original presentation to allow an evaluation of the state of proof assistance for such methods, and to ensure similarity of methods, and not merely similarly of results. The formalisation is restricted to the implicational fragment of intuitionistic logic.

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.

scholarly journals Natural deduction for bi-intuitionistic logic

2017 ◽  
Vol 25 ◽  
pp. S72-S96 ◽  
Author(s):  
Luca Tranchini
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction

Proofs, Snakes and Ladders

Dialogue ◽  
1974 ◽  
Vol 13 (4) ◽  
pp. 723-731 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Tree Structure ◽  
Deduction System ◽  
Syntactical Structure ◽  
Form Theorem ◽  
Structural Identity ◽  
Natural Deduction System ◽  
Normal Form Theorem

Anyone who has worked at proving theorems of intuitionistic logic in a natural deduction system must have been struck by the way in which many logical theorems “prove themselves.” That is, proofs of many formulas can be read off from the syntactical structure of the formulas themselves. This observation suggests that perhaps a strong structural identity may underly this relation between formulas and their proofs. A formula can be considered as a tree structure composed of its subformulas (Frege 1879) and by the normal form theorem (Gentzen 1934) every formula has a normalized proof consisting of its subformulas. Might we not identify an intuitionistic theorem with (one of) its proof(s) in normal form?

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