OSCAR: a system for weak-consistency replication

A large number of formal systems based on combinatory logic or λ-calculus have been extended to include first order predicate calculus. Several of these however have been shown to be inconsistent, all, as far as the author knows, in the strong sense that all well formed formulas (which here include all strings of symbols) are provable. We will call the corresponding consistency notion—an arbitrary wff ⊥ is provable—weak consistency. We will say that a system is strongly consistent if no formula and its negation are provable.Now for some systems, such as that of Kuzichev [11], the strong and weak consistency notions are equivalent, but in the systems of [5] and [6], which we will be considering, they are not. Each of these systems is strong enough to have all of ZF set theory, except Grounding and Choice, interpretable in it, and the system of [5] can also encompass first order arithmetic (see [7]). It therefore seems unlikely that a strong consistency result could be proved for these systems using elementary methods. In this paper however, we prove the weak consistency of both these systems by means that could be formulated, at least within the theory of [5]. The method also applies to the typed systems of Curry, Hindley and Seldin [10] and to Seldin's generalised types [12].

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REGRESSION METHODS FOR HESITANT FUZZY PREFERENCE RELATIONS

Technological and Economic Development of Economy ◽

10.3846/20294913.2014.881430 ◽

2014 ◽

Vol 19 (Supplement_1) ◽

pp. S214-S227 ◽

Cited By ~ 48

Author(s):

Bin Zhu ◽

Zeshui Xu

Keyword(s):

Regression Method ◽

Weak Consistency ◽

Preference Relations ◽

Complete Consistency ◽

Regression Methods ◽

Straightforward Method ◽

Fuzzy Preference Relations ◽

Fuzzy Preference

In this paper, we develop two regression methods that transform hesitant fuzzy preference relations (HFPRs) into fuzzy preference relations (FPRs). On the basis of the complete consistency, reduced FPRs with the highest consistency levels can be derived from HFPRs. Compared with a straightforward method, this regression method is more efficient in the Matlab environment. Based on the weak consistency, another regression method is developed to transform HFPRs into reduced FPRs which satisfy the weak consistency. Two algorithms are proposed for the two regression methods, and some examples are provided to verify the practicality and superiority of the proposed methods.

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