Harmonious logic: Craig's interpolation theorem and its descendants (original) (raw)
A Note on the Interpolation Theorem in First Order Logic
Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1982
1. The purpose of this paper is to present a model theoretic argument for the interpolation theorem. Model theoretic arguments in the literature are of three basic types : BUCHI-CRAIG, ROBINSON and HENKIN. BUCHI-CRAIG arguments formulate the interpolation theorem in terms of pseudoelementary classes ([3]). Here there are both algebraic and topological arguments. Algebraic arguments include those using ultrapowers ([7]) and ultralimits ([2]), and topological arguments include those which put a topology on the space of the models and use the normality of this topology to establish the interpolation theorem ([14]). ROBIXSON arguments prove the interpolation theorem from ROBINSON'S joint consistency lemma. Arguments here are distinguished one from another by the techniques used to prove ROBINSON'S lemma: elementary chains ([l], [ 5 ] ) ; saturated and special models ([4]. [ 5 ] ) ; and recursively saturated models ([l], [9]). HENKIN arguments are based on refinements of the HENKIN style proof of the compactness theorem. These include those using separable pairs of theories ([6], [5]), and thoye using consistency properties ([12], [ll], [8], [El).
Failure of interpolation in relevant logics
Journal of Philosophical Logic, Vol. 22, 449-479, 1993
Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.
An interpolation theorem in equational logic
1988
The Centre for Mathematics and Computer Science is a research institute of the Stichting Mathematisch Centrum, which was founded on February 11 , 1946, as a nonprofit institution aiming at the promotion of mathematics, computer science, and their applications. It is sponsored by the Dutch Government through the Netherlands Organization for the Advancement of Pure Research (Z.W.0.).
Logique Et Analyse, 2017
We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order logic, showing that if [G is satisfiable] and [D is ot logically true] and G|- D then there is an interpolant c, constructed using only non-logical vocabulary common to both members of G and members of D, such that (i) G entails c in the first-order version of Kleene’s strong three-valued logic (K3), and (ii) c entails D in the first-order version of Priest’s Logic of Paradox (LP). The proof proceeds via a careful analysis of derivations in a cut-free sequent calculus for first-order classical logic. Lyndon’s strengthening falls out of an observation regarding such derivations and the steps involved in the construction of interpolants. The proof is then extended to cover the {&, v, ~, A, E}-fragment of classical first-order logic with identity. Keywords: Craig–Lyndon Interpolation Theorem (for classical first-order logic); Kleene’s strong 3-valued logic;Priest’s Logic of Paradox; Belnap’s f...
L O ] 1 0 A ug 2 01 8 Interpolation in extensions of first-order logic
2018
We provide a constructive proof of the interpolation theorem for extensions of classical first order logic with a special type of geometric axioms, called singular geometric axioms. As a corollary, we obtain a direct proof of interpolation for first-order logic with identity. Interpolation is a central result in first-order logic. It asserts that for any theorem A → B there exists a formula C, called interpolant, such that A → C and C → B are also theorems and C only contains non-logical symbols that are contained in both A and B. The aim of this paper is to extend interpolation beyond first-order logic. In particular, we show how to prove interpolation in Gentzen’s sequent calculi with singular geometric rules, a special case of geometric rules investigated in [5]. Interestingly, singular geometric rules include the rules for first-order logic with identity (as well as those for the theory of strict partial orders and other order theories). Thus, from interpolation for singular geo...
Craig Interpolation in the Presence of Unreliable Connectives
Logica Universalis, 2014
Arrow and turnstile interpolations are investigated in UCL (introduced in [32]), a logic that is a complete extension of classical propositional logic for reasoning about connectives that only behave as expected with a given probability. Arrow interpolation is shown to hold in general and turnstile interpolation is established under some provisos.
New aspects of interpolative reasoning
1996
Using the concept of vague environment described by scaling functions [2] instead of the linguistic terms of the fuzzy partition gives a simple way for fuzzy approximate reasoning. In the next steps I would like to introduce a way of fuzzy interpolative reasoning, using the method of interpolation in vague environment.
A non-classical refinement of the interpolation property for classical propositional logic
2015
We refine the interpolation property of the {∧,∨,¬}-fragment of classical propositional logic, showing that if 2 ¬φ, 2 ψ and φ ψ then there is an interpolant χ, constructed using at most atomic formulas occurring in both φ and ψ and negation, conjunction and disjunction, such that (i) φ entails χ in Kleene’s strong three-valued logic and (ii) χ entails ψ in Priest’s Logic of Paradox.
Craig's interpolation theorem for the intuitionistic logic and its extensions—A semantical approach
Studia Logica, 1986
A semantical proof of Craig's interpolation theorem for the intuitionistie predicate logic and some intermediate propositional Iogies will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.
Uniform interpolation in modal logics
2016
Interpolation has been studied in a variety of settings since William Craig proved that classical predicate logic has interpolation in 1957. Interpolation is considered by many to be a “good ” property because it indicates a certain well-behavedness of the logic, vaguely reminiscent to analycity. In 1992 it was proved by Andrew Pitts that intuitionistic propositional logic IPC, which has interpola-tion, also satisfies the stronger property of uniform interpolation: given a formula ϕ and an atom p, there exist uniform interpolants ∀pϕ and ∃pϕ which are for-mulas (in the language of IPC) that do not contain p and such that for all ψ not containing p: ` ϕ → ψ ⇔ ` ∃pϕ → ψ ` ψ → ϕ ⇔ ` ψ → ∀pϕ. This is a strengething of interpolation in which the interpolant only depends on the premiss (in the case of ∃) or the conclusion (in the case of ∀) of the given implication. As the notation suggests, the fact that the uniform interpolants are definable in IPC also shows that the propositional quan...