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.

The relative efficiency of propositional proof systems

1979 ◽  
Vol 44 (1) ◽  
pp. 36-50 ◽  
Author(s):  
Stephen A. Cook ◽  
Robert A. Reckhow
Keyword(s):  
Relative Efficiency ◽  
Natural Deduction ◽  
Type Systems ◽  
Linear Increase ◽  
Proof System ◽  
Proof Systems ◽  
System A ◽  
Frege Systems ◽  
Propositional Proof Systems ◽  
Hilbert Type

We are interested in studying the length of the shortest proof of a propositional tautology in various proof systems as a function of the length of the tautology. The smallest upper bound known for this function is exponential, no matter what the proof system. A question we would like to answer (but have not been able to) is whether this function has a polynomial bound for some proof system. (This question is motivated below.) Our results here are relative results.In §§2 and 3 we indicate that all standard Hilbert type systems (or Frege systems, as we call them) and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes. In §4 we introduce extended Frege systems, which allow introduction of abbreviations for formulas. Since these abbreviations can be iterated, they eliminate the need for a possible exponential growth in formula length in a proof, as is illustrated by an example (the pigeonhole principle). In fact, Theorem 4.6 (which is a variation of a theorem of Statman) states that with a penalty of at most a linear increase in the number of lines of a proof in an extended Frege system, no line in the proof need be more than a constant times the length of the formula proved.

The deduction rule and linear and near-linear proof simulations

1993 ◽  
Vol 58 (2) ◽  
pp. 688-709 ◽  
Author(s):  
Maria Luisa Bonet ◽  
Samuel R. Buss
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Upper Bounds ◽  
Proof System ◽  
Proof Systems ◽  
Linear Speedup ◽  
Frege Systems ◽  
Deduction Rule ◽  
Gentzen Sequent Calculus

AbstractWe introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

scholarly journals What is the Meaning of Proofs?

2020 ◽  
Author(s):  
Sara Ayhan
Keyword(s):  
Natural Deduction ◽  
The Other ◽  
Proof System ◽  
Sequent Calculi ◽  
Singular Terms ◽  
Proof Systems ◽  
Logical Connectives

Abstract The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference that govern the use of the connective. However, what if we go a step further and ask about the meaning of a proof as a whole? In this paper we address this question and lay out a framework to distinguish sense and denotation of proofs. Two questions are central here. First of all, if we have two (syntactically) different derivations, does this always lead to a difference, firstly, in sense, and secondly, in denotation? The other question is about the relation between different kinds of proof systems (here: natural deduction vs. sequent calculi) with respect to this distinction. Do the different forms of representing a proof necessarily correspond to a difference in how the inferential steps are given? In our framework it will be possible to identify denotation as well as sense of proofs not only within one proof system but also between different kinds of proof systems. Thus, we give an account to distinguish a mere syntactic divergence from a divergence in meaning and a divergence in meaning from a divergence of proof objects analogous to Frege’s distinction for singular terms and sentences.

Normalized natural deduction systems for some relevant logics I: The logic DW

2006 ◽  
Vol 71 (1) ◽  
pp. 35-66 ◽  
Author(s):  
Ross T. Brady
Keyword(s):  
Classical Logic ◽  
Natural Deduction ◽  
Relevant Logic ◽  
Full Detail ◽  
Relevant Logics ◽  
Sentential Logic ◽  
Wide Range ◽  
Deduction Rules ◽  
Infinite Set ◽  
Subformula Property

Fitch-style natural deduction was first introduced into relevant logic by Anderson in [1960], for the sentential logic E of entailment and its quantincational extension EQ. This was extended by Anderson and Belnap to the sentential relevant logics R and T and some of their fragments in [ENT1], and further extended to a wide range of sentential and quantified relevant logics by Brady in [1984]. This was done by putting conditions on the elimination rules, →E, ~E, ⋁E and ∃E, pertaining to the set of dependent hypotheses for formulae used in the application of the rule. Each of these rules were subjected to the same condition, this condition varying from logic to logic. These conditions, together with the set of natural deduction rules, precisely determine the particular relevant logic. Generally, this is a simpler representation of a relevant logic than the standard Routley-Meyer semantics, with its existential modelling conditions stated in terms of an otherwise arbitrary 3-place relation R, which is defined over a possibly infinite set of worlds. Readers are urged to refer to Brady [1984], if unfamiliar with the above natural deduction systems, but we will introduce in §2 a modified version in full detail.Natural deduction for classical logic was invented by Jaskowski and Gentzen, but it was Prawitz in [1965] who normalized natural deduction, streamlining its rules so as to allow a subformula property to be proved. (This key property ensures that each formula in the proof of a theorem is indeed a subformula of that theorem.)

Suppose and Tell

2020 ◽  
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
Special Kind ◽  
Context Sensitivity ◽  
Modal Operator ◽  
Counterfactual Conditionals ◽  
Deduction Rules ◽  
Psychological Sense

The book argues that our use of conditionals is governed by imperfectly reliable heuristics, in the psychological sense of fast and frugal (or quick and dirty) ways of assessing them. The primary heuristic is this: to assess ‘If A, C’, suppose A and on that basis assess C; whatever attitude you take to C conditionally on A (such as acceptance, rejection, or something in between) take unconditionally to ‘If A, C’. This heuristic yields both the equation of the probability of ‘If A, C’ with the conditional probability of C on A and standard natural deduction rules for the conditional. However, these results can be shown to make the heuristic implicitly inconsistent, and so less than fully reliable. There is also a secondary heuristic: pass conditionals freely from one context to another under normal conditions for acceptance of sentences on the basis of memory and testimony. The effect of the secondary heuristic is to undermine interpretations on which ‘if’ introduces a special kind of context-sensitivity. On the interpretation which makes best sense of the two heuristics, ‘if’ is simply the truth-functional conditional. Apparent counterexamples to truth-functionality are artefacts of reliance on the primary heuristic in cases where it is unreliable. The second half of the book concerns counterfactual conditionals, as expressed with ‘if’ and ‘would’. It argues that ‘would’ is an independently meaningful modal operator for contextually restricted necessity: the meaning of counterfactuals is simply that derived compositionally from the meanings of their constituents, including ‘if’ and ‘would’, making them contextually restricted strict conditionals.

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.

ON THE NOTION OF CANONICAL DERIVATIONS FROM OPEN ASSUMPTIONS AND ITS ROLE IN PROOF-THEORETIC SEMANTICS

2015 ◽  
Vol 8 (2) ◽  
pp. 296-305 ◽  
Author(s):  
NISSIM FRANCEZ
Keyword(s):  
Natural Deduction ◽  
Proof System ◽  
Logical Constants ◽  
Local Completeness ◽  
Definition Of ◽  
The Impact

AbstractThe paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of local-completeness of a natural-deduction proof-system, underlying its harmony.

Consequences of the Suppositional Rule

2020 ◽  
pp. 31-67
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
David Lewis ◽  
Linguistic Practices ◽  
Deduction Rules

This chapter argues that the Suppositional Rule is a fallible heuristic, because it has inconsistent consequences. They arise in several ways: (i) it implies standard natural deduction rules for ‘if’, and analogous but incompatible rules for refutation in place of proof; (ii) it implies the equation of the probability of ‘If A, C’ with the conditional probability of C on A, which is subject to the trivialization results of David Lewis and others; (iii) its application to complex attitudes generates further inconsistencies. The Suppositional Rule is compared to inconsistent principles built into other linguistic practices: disquotation for ‘true’ and ‘false’ generate Liar-like paradoxes; tolerance principles for vague expressions generate sorites paradoxes. Their status as fallible, semantically invalid but mostly reliable heuristics is not immediately available to competent speakers.

scholarly journals A duality between proof systems for cyclic term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 439-484 ◽  
Author(s):  
CLEMENS GRABMAYER
Keyword(s):  
Close Connection ◽  
Proof System ◽  
Consistency Checking ◽  
Duality Result ◽  
First Order ◽  
Proof Systems ◽  
Similar Proof ◽  
Equational Specifications ◽  
Cyclic Term

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

scholarly journals Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption

2001 ◽  
Vol 8 (37) ◽  
Author(s):  
Ronald Cramer ◽  
Victor Shoup
Keyword(s):  
Special Kind ◽  
Public Key ◽  
Proof System ◽  
Reasonable Assumption ◽  
Encryption Scheme ◽  
Public Key Encryption ◽  
Proof Systems ◽  
Encryption Schemes ◽  
Hash Proof System ◽  
Chosen Ciphertext Attack

We present several new and fairly practical public-key encryption schemes and prove them secure against adaptive chosen ciphertext attack. One scheme is based on Paillier's Decision Composite Residuosity (DCR) assumption, while another is based in the classical Quadratic Residuosity (QR) assumption. The analysis is in the standard cryptographic model, i.e., the security of our schemes does not rely on the Random Oracle model.<br /> <br />We also introduce the notion of a universal hash proof system. Essentially, this is a special kind of non-interactive zero-knowledge proof system for an NP language. We do not show that universal hash proof systems exist for all NP languages, but we do show how to construct very efficient universal hash proof systems for a general class of group-theoretic language membership problems.<br /> <br />Given an efficient universal hash proof system for a language with certain natural cryptographic indistinguishability properties, we show how to construct an efficient public-key encryption schemes secure against adaptive chosen ciphertext attack in the standard model. Our construction only uses the universal hash proof system as a primitive: no other primitives are required, although even more efficient encryption schemes can be obtained by using hash functions with appropriate collision-resistance properties. We show how to construct efficient universal hash proof systems for languages related to the DCR and QR assumptions. From these we get corresponding public-key encryption schemes that are secure under these assumptions. We also show that the Cramer-Shoup encryption scheme (which up until now was the only practical encryption scheme that could be proved secure against adaptive chosen ciphertext attack under a reasonable assumption, namely, the Decision Diffie-Hellman assumption) is also a special case of our general theory.

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