What Is Mathematical Logic? by C. J. Ash, J. N. Crossley

By C. J. Ash, J. N. Crossley

Publish 12 months note: initially released in 1972

This creation to the most rules and result of mathematical good judgment is a significant therapy aimed toward non-logicians. beginning with a old survey of good judgment in precedent days, it strains the 17th-century improvement of calculus and discusses smooth theories, together with set idea, the continuum speculation, and different rules.

From 1972 edition.

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Arithmetic and logical operations on RL-numbers are straightforward and unique extensions of the operations on crisp numbers, verifying the following: • They verify all the usual properties of crisp arithmetic and logical operations. • The imprecision does not necessarily increase through operations, and can even diminish. The maximum imprecision is related to the number of restriction levels employed. 3 Evaluation of Quantified Sentences We shall consider the evaluation of quantified sentences of type II because of lack of space, and since type I sentences are a particular case of type II sentences, under the following assumptions: – Q is a fuzzy quantifier – A,D are imprecise properties defined on a finite, crisp set X by RLrepresentations (ΛA , ρA ) and (ΛD , ρD ), respectively.

G. triangular. This assumption, combined with assumptions 1, 2 and 3 leads to the conclusion that ξt have independent L-R possibility distributions of the same membership function shape with expected value equal zero and a variance σ 2 . Knowing the realisations ut of the fuzzy random component, we can determine the estimators of the mean value and the variance of the model error: respectively T T 2 t=1 ut t=1 ut , T −k−1 . Further, making use of (7) - (8) or (9) - (10) or (12) depending T on the form of the probabilistic distribution of the random component, we can determine the parameters of the fuzzy variable if it is a symmetric triangular fuzzy variable ξ = (mξ , αξ , αξ ).

G. [9], [10], [11]. We are proposing an approach analogous to the classical regression concept, taking as input for the fuzzy econometric model the observations of the variables, both the independent and the dependent ones. We give an example of the application of the proposed method in the energy load forecasting. 2 Classical Regression Model In the classical econometric approach the input data for the regression equation construction are observations (yt , xt1 , . . , xtk ), t = 1, . . , T .

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