scholarly journals Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics

2017 ◽  
Vol 46 (1/2) ◽  
Author(s):  
Nobu-Yuki Suzuki
Keyword(s):  
Disjunction Property ◽  
Existence Property ◽  
Intermediate Predicate Logics

We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.

The disjunction and related properties for constructive Zermelo-Fraenkel set theory

2005 ◽  
Vol 70 (4) ◽  
pp. 1233-1254 ◽  
Author(s):  
Michael Rathjen
Keyword(s):  
Set Theory ◽  
Proof Technique ◽  
Disjunction Property ◽  
Regular Extension ◽  
Existence Property ◽  
Extension Axiom

AbstractThis paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.

The equivalence of the disjunction and existence properties for modal arithmetic

1989 ◽  
Vol 54 (4) ◽  
pp. 1456-1459 ◽  
Author(s):  
Harvey Friedman ◽  
Michael Sheard
Keyword(s):  
Natural Number ◽  
Modal System ◽  
Disjunction Property ◽  
Existence Property

AbstractIn a modal system of arithmetic, a theory S has the modal disjunction property if whenever S ⊢ □φ ∨ □ψ, either S ⊢ □φ or S ⊢ □ψ. S has the modal numerical existence property if whenever S ⊢ ∃x □φ(x), there is some natural number n such that S ⊢ □φ(n). Under certain broadly applicable assumptions, these two properties are equivalent.

Counting the maximal intermediate constructive logics

1993 ◽  
Vol 58 (4) ◽  
pp. 1365-1401 ◽  
Author(s):  
Mauro Ferrari ◽  
Pierangelo Miglioli
Keyword(s):  
Disjunction Property ◽  
Intermediate Predicate Logics ◽  
Constructive Logics

AbstractA proof is given that the set of maximal intermediate propositional logics with the disjunction property and the set of maximal intermediate predicate logics with the disjunction property and the explicit definability property have the power of continuum. To prove our results, we introduce various notions which might be interesting by themselves. In particular, we illustrate a method to generate wide sets of pairwise “constructively incompatible constructive logics”. We use a notion of “semiconstructive” logic and define wide sets of “constructive” logics by representing the “constructive” logics as “limits” of decreasing sequences of “semiconstructive” logics. Also, we introduce some generalizations of the usual filtration techniques for propositional logics. For instance, “fitrations over rank formulas” are used to show that any two different logics belonging to a suitable uncountable set of “constructive” logics are “constructively incompatible”.

On the derivability of instantiation properties

1977 ◽  
Vol 42 (4) ◽  
pp. 506-514 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Second Order ◽  
Disjunction Property ◽  
Second Order Arithmetic ◽  
Recursive Enumerability ◽  
Recursively Enumerable ◽  
Existence Property ◽  
Order Arithmetic

AbstractEvery recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.

scholarly journals CZF does not have the existence property

2014 ◽  
Vol 165 (5) ◽  
pp. 1115-1147 ◽  
Author(s):  
Andrew W. Swan
Keyword(s):  
Existence Property

Some properties of epistemic set theory with collection

1986 ◽  
Vol 51 (3) ◽  
pp. 748-754 ◽  
Author(s):  
Andre Scedrov
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Existential Quantifier ◽  
Modal Interpretation ◽  
Conservative Extension ◽  
Disjunction Property ◽  
First Order ◽  
Slight Extension ◽  
Order Arithmetic ◽  
Epistemic Systems

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.

Intermediate predicate logics determined by ordinals

1990 ◽  
Vol 55 (3) ◽  
pp. 1099-1124 ◽  
Author(s):  
Pierluigi Minari ◽  
Mitio Takano ◽  
Hiroakira Ono
Keyword(s):  
Predicate Logic ◽  
The Other ◽  
Constant Domain ◽  
Kripke Frames ◽  
Other Hand ◽  
Countable Ordinal ◽  
Intermediate Predicate Logics ◽  
Regular Ordinal

AbstractFor each ordinal α > 0, L(α) is the intermediate predicate logic characterized by the class of all Kripke frames with the poset α and with constant domain. This paper will be devoted to a study of logics of the form L(α). It will be shown that for each uncountable ordinal of the form α + η with a finite or a countable η(> 0), there exists a countable ordinal of the form β + η such that L(α + η) = L(β + η). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(α) if α is an uncountable regular ordinal. Moreover, it will be proved that the mapping L is injective if it is restricted to ordinals less than ωω, i.e. α ≠ β implies L(α) ≠ L(β) for each ordinal α, β ≤ ωω.

The decidability of the Kreisel-Putnam system

1970 ◽  
Vol 35 (3) ◽  
pp. 431-437 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Finite Model ◽  
Model Property ◽  
Intuitionistic Propositional Logic ◽  
Disjunction Property ◽  
Finite Model Property ◽  
Image Position

The intuitionistic propositional logic I has the following disjunction property This property does not characterize intuitionistic logic. For example Kreisel and Putnam [5] showed that the extension of I with the axiomhas the disjunction property. Another known system with this propery is due to Scott [5], and is obtained by adding to I the following axiom:In the present paper we shall prove, using methods originally introduced by Segerberg [10], that the Kreisel-Putnam logic is decidable. In fact we shall show that it has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]. The decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.

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