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”.

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.

From Compound Houses to Villas: The Incremental Transformation of Dakar’s Urban Landscape

2016 ◽  
Vol 41 (2) ◽  
pp. 15-22
Author(s):  
Emilie Pinard
Keyword(s):  
Urban Landscape ◽  
Housing Policies ◽  
Global Networks ◽  
Urban Transformations ◽  
Architectural Survey ◽  
Housing Typology ◽  
The City ◽  
House Building ◽  
Constructive Logics ◽  
Spatial Logics

This paper examines the transformation of the housing typology in informal neighbourhoods located on the periphery of Dakar, Senegal. More specifically, it documents the spatial logics and factors guiding the construction of new multi-storey houses called “villas”, which are significantly transforming the landscape of the city. Studies have thus far examined villas through the lenses of migrants’ investments and lifestyles, associating these houses with new functions and decorative elements and materials inspired by time spent abroad, with innovative ways of building and dwelling that disrupt more popular housing practices. Based upon an architectural survey of seventeen houses and the detailed stories of their construction, this paper argues that while the Senegalese villa is influenced by global networks and symbols of success, it is also deeply rooted in popular housing forms and building practices. Moreover, because house-building processes are predominantly incremental, the construction of this new house type is not limited to migrants and other privileged dwellers. Although at different speeds, most residents are building and transforming their houses according to spatial and constructive logics characteristic of villas. These results have implications for housing policies and programmes because they contribute to challenging assumptions about residential production, new housing typologies and the pivotal actors of these urban transformations.

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.

scholarly journals Some results on intermediate constructive logics.

1989 ◽  
Vol 30 (4) ◽  
pp. 543-562 ◽  
Author(s):  
Pierangelo Miglioli ◽  
Ugo Moscato ◽  
Mario Ornaghi ◽  
Silvia Quazza ◽  
Gabriele Usberti
Keyword(s):  
Constructive Logics

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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