scholarly journals Possibility Semantics for Intuitionistic Logic

2004 ◽  
Vol 2 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Intuitionistic Logic ◽  
Possible Worlds ◽  
Predicate Logic ◽  
Order Logic ◽  
Semantic Interpretation ◽  
First Order Logic ◽  
First Order ◽  
Partial Indices ◽  
Closure Conditions ◽  
First Order Predicate Logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

scholarly journals The completeness of Heyting first-order logic

2003 ◽  
Vol 68 (3) ◽  
pp. 751-763 ◽  
Author(s):  
W. W. Tait
Keyword(s):  
Type Theory ◽  
Disjoint Union ◽  
Predicate Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
First Order Predicate Logic

AbstractRestricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x: A,F(x) is understood as disjoint union, are the projections, and these do not preserve first-orderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting's system.

GLIVENKO AND KURODA FOR SIMPLE TYPE THEORY

2014 ◽  
Vol 79 (2) ◽  
pp. 485-495 ◽  
Author(s):  
CHAD E. BROWN ◽  
CHRISTINE RIZKALLAH
Keyword(s):  
Type Theory ◽  
Predicate Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Simple Type ◽  
Universal Quantifier ◽  
First Order ◽  
Simple Type Theory ◽  
Glivenko’S Theorem ◽  
First Order Predicate Logic

AbstractGlivenko’s theorem states that an arbitrary propositional formula is classically provable if and only if its double negation is intuitionistically provable. The result does not extend to full first-order predicate logic, but does extend to first-order predicate logic without the universal quantifier. A recent paper by Zdanowski shows that Glivenko’s theorem also holds for second-order propositional logic without the universal quantifier. We prove that Glivenko’s theorem extends to some versions of simple type theory without the universal quantifier. Moreover, we prove that Kuroda’s negative translation, which is known to embed classical first-order logic into intuitionistic first-order logic, extends to the same versions of simple type theory. We also prove that the Glivenko property fails for simple type theory once a weak form of functional extensionality is included.

Logical Tools

2021 ◽  
pp. 14-52
Author(s):  
Cian Dorr ◽  
John Hawthorne ◽  
Juhani Yli-Vakkuri
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
System P

This chapter presents the system of classical higher-order modal logic which will be employed throughout this book. Nothing more than a passing familiarity with classical first-order logic and standard systems of modal logic is presupposed. We offer some general remarks about the kind of commitment involved in endorsing this logic, and motivate some of its more non-standard features. We also discuss how talk about possible worlds can be represented within the system.

Relational and partial variable sets and basic predicate logic

1996 ◽  
Vol 61 (3) ◽  
pp. 843-872 ◽  
Author(s):  
Silvio Ghilardi ◽  
Giancarlo Meloni
Keyword(s):  
Predicate Logic ◽  
Order Logic ◽  
Important Case ◽  
First Order Logic ◽  
Partial Functions ◽  
First Order ◽  
Transition Functions ◽  
Independence Results

AbstractIn this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are obtained for different kinds of Beck and Frobenius conditions.

scholarly journals Formalization of Linear Space Theory in the Higher-Order Logic Proving System

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Jie Zhang ◽  
Danwen Mao ◽  
Yong Guan
Keyword(s):  
Linear Space ◽  
Theorem Proving ◽  
Predicate Logic ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Space Theory ◽  
Higher Order Logic ◽  
First Order ◽  
Important Approach

Theorem proving is an important approach in formal verification. Higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and stronger semantics. Higher-order logic is more expressive. This paper presents the formalization of the linear space theory in HOL4. A set of properties is characterized in HOL4. This result is used to build the underpinnings for the application of higher-order logic in a wider spectrum of engineering applications.

scholarly journals Subformula Linking for Intuitionistic Logic with Application to Type Theory

2021 ◽  
pp. 200-216
Author(s):  
Kaustuv Chaudhuri
Keyword(s):  
Theorem Proving ◽  
Intuitionistic Logic ◽  
Type Theory ◽  
Linear Logic ◽  
Interactive Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
Direct Manipulation ◽  
First Order

AbstractSubformula linking is an interactive theorem proving technique that was initially proposed for (classical) linear logic. It is based on truth and context preserving rewrites of a conjecture that are triggered by a user indicating links between subformulas, which can be done by direct manipulation, without the need of tactics or proof languages. The system guarantees that a true conjecture can always be rewritten to a known, usually trivial, theorem. In this work, we extend subformula linking to intuitionistic first-order logic with simply typed lambda-terms as the term language of this logic. We then use a well known embedding of intuitionistic type theory into this logic to demonstrate one way to extend linking to type theory.

scholarly journals Encoding hybridized institutions into first-order logic

2014 ◽  
Vol 26 (5) ◽  
pp. 745-788 ◽  
Author(s):  
RĂZVAN DIACONESCU ◽  
ALEXANDRE MADEIRA
Keyword(s):  
Possible Worlds ◽  
Sufficient Conditions ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Institution Theory ◽  
Wide Range ◽  
Characteristic Features ◽  
Verification Process ◽  
Base Logic

A ‘hybridization’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridized institutions’ we mean the result of this process when logics are treated abstractly as institutions (in the sense of the institution theory of Goguen and Burstall). This work develops encodings of hybridized institutions into (many-sorted) first-order logic (abbreviated $\mathcal{FOL}$) as a ‘hybridization’ process of abstract encodings of institutions into $\mathcal{FOL}$, which may be seen as an abstraction of the well-known standard translation of modal logic into $\mathcal{FOL}$. The concept of encoding employed by our work is that of comorphism from institution theory, which is a rather comprehensive concept of encoding as it features encodings both of the syntax and of the semantics of logics/institutions. Moreover, we consider the so-called theoroidal version of comorphisms that encode signatures to theories, a feature that accommodates a wide range of concrete applications. Our theory is also general enough to accommodate various constraints on the possible worlds semantics as well a wide variety of quantifications. We also provide pragmatic sufficient conditions for the conservativity of the encodings to be preserved through the hybridization process, which provides the possibility to shift a formal verification process from the hybridized institution to $\mathcal{FOL}$.

scholarly journals Conservative Intensional Extension of Tarski's Semantics

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Zoran Majkić
Keyword(s):  
Free Algebra ◽  
Possible Worlds ◽  
Order Logic ◽  
Point Of View ◽  
Relational Algebra ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
Possible World ◽  
Logic Formula ◽  
First Order

We considered an extension of the first-order logic (FOL) by Bealer's intensional abstraction operator. Contemporary use of the term “intension” derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the “extension” of an idea consists of the subjects to which the idea applies, and the “intension” consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of standard FOL, we obtain a commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of the FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from Cylindric algebras). Then we show that this composition corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

Nonmonotonic consequence based on intuitionistic logic

1992 ◽  
Vol 57 (4) ◽  
pp. 1176-1197 ◽  
Author(s):  
Gisèle Fischer Servi
Keyword(s):  
Intuitionistic Logic ◽  
Numerical Approach ◽  
Order Logic ◽  
First Order Logic ◽  
Material Implication ◽  
First Order ◽  
Alternative Approach ◽  
Logical Notion

Research in AI has recently begun to address the problems of nondeductive reasoning, i.e., the problems that arise when, on the basis of approximate or incomplete evidence, we form well-reasoned but possibly false judgments. Attempts to stimulate such reasoning fall into two main categories: the numerical approach is based on probabilities and the nonnumerical one tries to reconstruct nondeductive reasoning as a special type of deductive process. In this paper, we are concerned with the latter usually known as nonmonotonic deduction, because the set of theorems does not increase monotonically with the set of axioms.It is generally acknowledged that nonmonotonic (n.m.) formalisms (e.g., [C], [MC1], [MC2] [MD], [MD-D], [Rl], [R2], [S]) are plagued by a number of difficulties. A key issue concerns the fact that most systems do not produce an axiomatizable set of validities. Thus, the chief objective of this paper is to develop an alternative approach in which the set of n.m. inferences, which somehow qualify as being deductively sound, is r.e.The basic idea here is to reproduce the situation in First Order Logic where the metalogical concept of deduction translates into the logical notion of material implication. Since n.m. deductions are no longer truth preserving, our way to deal with a change in the metaconcept is to extend the standard logic apparatus so that it can reflect the new metaconcept. In other words, the intent is to study a concept of nonmonotonic implication that goes hand in hand with a notion of n.m. deduction. And in our case, it is convenient that the former be characterized within the more tractable context of monotonic logic.

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