Erratum to splitting an operator

Journal of Symbolic Logic, 1991

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered:1. There exist sentences that are neither paradoxical nor grounded.2. There are fixed points.3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n ∣ A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the sets.4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the sets.5. The closure ordinal for Kripke's construction of the minimal fixed point is .In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen:1. There may or may not exist nonparadoxical, ungrounded sentences.2. The number of fixed points may be any positive finite number, ℵ0, or .3. In the minimal fixed point, the sets that are weakly definable may range from a subcla...

Publications of the Research Institute for Mathematical Sciences, 1970

In [10], Kripke gave a definition of the semantics of the intuitionistic logic. Fitting [2] showed that Kripke's models are equivalent to algebraic models (i.e., pseudo-Boolean models) in a certain sense. As a corollary of this result, we can show that any partially ordered set is regarded as a (characteristic) model of a intermediate logic ^ We shall study the relations between intermediate logics and partially ordered sets as models of them, in this paper. We call a partially ordered set, a Kripke model. 2^ At present we don't know whether any intermediate logic 'has a Kripke model. But Kripke models have some interesting properties and are useful when we study the models of intermediate logics. In §2, we shall study general properties of Kripke models. In §3, we shall define the height of a Kripke model and show the close connection between the height and the slice, which is introduced in [7]. In §4, we shall give a model of LP» which is the least element in n-ih slice S n (see [7]). §1. Preliminaries We use the terminologies of [2] on algebraic models, except the use of 1 and 0 instead of V and /\, respectively. But on Kripke models, we give another definition, following Schiitte [13]. 3) Definition 1.1. If M is a non-empty partially ordered set, then

2020

There are several finite axiomatizations of stratified comprehension. The famous two are Hailperin's and Randall Holmes's. However, the system presented here could be the shortest known one written in the first order language of set theory. This axiomatization uses unordered pairs, unlike the prior axiomaizations which used ordered pairs, so this makes a simpler formulation in the first order language of set theory. The proof works under absence of Extensionality, so it can serve to complete an axiomatization of NFU or NF.

Proceedings of the Japan Academy, 1973

2012

For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no non-trivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variable-free fragment with natural number modalities, denoted GLPω. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each ordinals Θ,Λ we build a Kripke model IΛ and show that GLP 0 Λ is sound for this structure. In our notation, Ignatiev’s original model becomes I0 ω .

Logic and Logical Philosophy

This paper extends Fitting's epistemic interpretation of some Kleene logics, to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting's "cut-down" operator is discussed, rendering a "track-down" operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations.

Motivated by model-theoretic properties of the Bernays-Schönfinkel-Ramsey (BSR) class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ. A class in this family is denoted EBS Σ (σ), where σ ⊆ Σ. Formulae in EBS Σ (σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates in Σ \ σ. We study several properties of the family EBS Σ = {EBS Σ (σ) | σ ⊆ Σ}. For example, classes in EBS Σ are spectrally indistinguishable, the smallest class, EBS Σ (Σ), is semantically equivalent to BSR over Σ, and the largest class, EBS Σ (∅), is the set of all FO formulae over Σ with finite or co-finite spectra. Furthermore, (EBS Σ , ⊆) forms a lattice that is isomorphic to the powerset lattice (℘(Σ), ⊆). We also show that if Σ contains at least one predicate of arity ≥ 2, there exist semantic gaps between EBS Σ (σ 1 ) and EBS Σ (σ 2 ) if σ 1 = σ 2 . This gives a natural semantic generalization of BSR as ascending chains in the lattice (EBS Σ , ⊆).

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1991

Annals of Pure and Applied Logic, 1989

The complexity of subclasses of Magical theories (mainly Presburger and Skolem arithmetic) is studied. The subclasses are defined by the structure of the quantifier prefix.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1971