scholarly journals Ternary Relational Semantics for the Variants of BN4 and E4 which Contain Routley and Meyer's Logic B

2021 ◽  
Author(s):  
Sandra M. López
Keyword(s):  
Relational Semantics ◽  
Basic Logic ◽  
Relevant Logics ◽  
The Family ◽  
Soundness And Completeness ◽  
Completeness Theorems

Six hopefully interesting variants of the logics BN4 and E4 – which can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively – were previously developed in the literature. All these systems are related to the family of relevant logics and contain Routley and Meyer's basic logic B, which is well-known to be specifically associated with the ternary relational semantics. The aim of this paper is to develop reduced general Routley-Meyer semantics for them. Strong soundness and completeness theorems are proved for each one of the logics.

Relational semantics and a relational proof system for full Lambek calculus

1998 ◽  
Vol 63 (2) ◽  
pp. 623-637 ◽  
Author(s):  
Wendy MacCaull
Keyword(s):  
Proof Theory ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Complete Proof ◽  
Proof Search ◽  
Proof Tree ◽  
Relational Logic ◽  
Completeness Theorems

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

scholarly journals A Logic for Vagueness

2010 ◽  
Vol 8 ◽  
Author(s):  
John Slaney
Keyword(s):  
Natural Deduction ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Relational Semantics ◽  
Deduction System ◽  
First Order ◽  
Natural Deduction System ◽  
Soundness And Completeness ◽  
Propositional System

This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree.

scholarly journals Complete decision procedure for the theory of bounded pointer arithmetic based on quantifier instantiation and SMT

2021 ◽  
Vol 33 (4) ◽  
pp. 177-194
Author(s):  
Rafael Faritovich Sadykov ◽  
Mikhail Usamovich Mandrykin
Keyword(s):  
Program Verification ◽  
Source Code ◽  
Decision Procedures ◽  
C Programs ◽  
Smt Solver ◽  
Smt Solvers ◽  
Relevant Logics ◽  
Soundness And Completeness ◽  
C Program ◽  
Quantifier Instantiation

The process of developing C programs is quite often prone to errors related to the uses of pointer arithmetic and operations on memory addresses. This promotes a need in developing various tools for automated program verification. One of the techniques frequently employed by those tools is invocation of appropriate decision procedures implemented within existing SMT-solvers. But at the same time both the SMT standard and most existing SMT-solvers lack the relevant logics (combinations of logical theories) for directly and precisely modelling the semantics of pointer operations in C. One of the possible ways to support these logics is to implement them in an SMT solver, but this approach can be time-consuming (as requires modifying the solver’s source code), inflexible (introducing any changes to the theory’s signature or semantics can be unreasonably hard) and limited (every solver has to be supported separately). Another way is to design and implement custom quantifier instantiation strategies. These strategies can be then used to translate formulas in the desired theory combinations to formulas in well-supported decidable logics such as QF_UFLIA. In this paper, we present an instantiation procedure for translating formulas in the theory of bounded pointer arithmetic into the QF_UFLIA logic. We formally proved soundness and completeness of our instantiation procedure in Isabelle/HOL. The paper presents an informal description of this proof of the proposed procedure. The theory of bounded pointer arithmetic itself was formulated based on known errors regarding the correct use of pointer arithmetic operations in industrial code as well as the semantics of these operations specified in the C standard. Similar procedure can also be defined for a practically relevant fragment of the theory of bit vectors (monotone propositional combinations of equalities between bitwise expressions). Our approach is sufficient to obtain efficient decision procedures implemented as Isabelle/HOL proof methods for several decidable logical theories used in C program verification by relying on the existing capabilities of well-known SMT solvers, such as Z3 and proof reconstruction capabilities of the Isabelle/HOL proof assistant.

scholarly journals S5-Style Non-Standard Modalities in a Hypersequent Framework

2021 ◽  
pp. 1-30
Author(s):  
Yaroslav Petrukhin
Keyword(s):  
Cut Elimination ◽  
Hypersequent Calculi ◽  
Soundness And Completeness ◽  
Completeness Theorems

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

scholarly journals Multisequent Gentzen Deduction Systems For B2 2-Valued First-Order Logic

2018 ◽  
Vol 7 (1) ◽  
pp. 53
Author(s):  
Wei Li ◽  
Yuefei Sui
Keyword(s):  
Boolean Algebra ◽  
Order Logic ◽  
First Order Logic ◽  
Deduction System ◽  
Truth Values ◽  
First Order ◽  
Soundness And Completeness ◽  
Completeness Theorems

For the four-element Boolean algebra B22, a multisequent Г|Δ|∑|∏ is a generalization of sequent Г→Δ in traditional B22 valued first-order logic. By defining the truth-values of quantified formulas, a Gentzen deduction system G22 for B22-valued first-order logic will be built and its soundness and completeness theorems will be proved.

TRANSITIVE PRIMAL INFON LOGIC

2013 ◽  
Vol 6 (2) ◽  
pp. 281-304 ◽  
Author(s):  
CARLOS COTRINI ◽  
YURI GUREVICH
Keyword(s):  
Access Control ◽  
Linear Time ◽  
Kripke Models ◽  
Control Applications ◽  
Traditional Logic ◽  
Soundness And Completeness ◽  
Image Position ◽  
Small Models ◽  
Completeness Theorems ◽  
Subformula Property

AbstractPrimal infon logic was introduced in 2009 in connection with access control. In addition to traditional logic constructs, it contains unary connectives p said indispensable in the intended access control applications. Propositional primal infon logic is decidable in linear time, yet suffices for many common access control scenarios. The most obvious limitation on its expressivity is the failure of the transitivity law for implication: $x \to y$ and $y \to z$ do not necessarily yield $x \to z$. Here we introduce and investigate equiexpressive “transitive” extensions TPIL and TPIL* of propositional primal infon logic as well as their quote-free fragments TPIL0 and TPIL0* respectively. We prove the subformula property for TPIL0* and a similar property for TPIL*; we define Kripke models for the four logics and prove the corresponding soundness-and-completeness theorems; we show that, in all these logics, satisfiable formulas have small models; but our main result is a quadratic-time derivation algorithm for TPIL*.

scholarly journals Relational semantics for the 4-valued relevant logics BN4 and E4

2016 ◽  
Author(s):  
Gemma Robles ◽  
José M. Blanco ◽  
Sandra M. López ◽  
Jesús R. Paradela ◽  
Marcos M. Recio
Keyword(s):  
Relational Semantics ◽  
Relevant Logics

Semantics for relevant logics

1972 ◽  
Vol 37 (1) ◽  
pp. 159-169 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Modal Logic ◽  
Relational Structure ◽  
Material Implication ◽  
First Order ◽  
Relevant Logics ◽  
Order Language ◽  
Meta Level ◽  
Semantic Treatment ◽  
Completeness Theorems

In what follows there is presented a unified semantic treatment of certain “paradox-free” systems of entailment, including Church's weak theory of implication (Church [7]) and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6]). We shall refer to these systems generally as relevant logics.The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice—interpreted informally as the semilattice of possible pieces of information. Completeness theorems can be given relative to these semantics for the implicational fragments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems—in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the “paradoxes of material implication”.The systems discussed are formulated in fragments of a first-order language with → (entailment), &, ⋁, ¬,(x) and (∃x) primitive, omitting identity but including a denumerable list of propositional variables (p, q, r, p1,…etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, A1,… are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The semantics of the systems are given in informal terms; it is an easy matter to turn the informal descriptions into formal set-theoretical definitions.

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