Consistency

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consistency

, consistence
degree of viscosity or firmness

Consistency

 

the aggregate of rheological properties of a viscous liquid or of a viscoplastic or viscoelastic body. The concept of consistency does not always have a clearly defined meaning in physics.

Unlike other rheological concepts with a similar connotation (for example, viscosity, fluidity, and plasticity), “consistency” as a technological term denotes the mobility (thickness) of liquid (”semiliquid”) and solid (”semisolid”) products and materials. It is usually used in describing systems whose flow (or deformation) characteristics vary with changes in applied stress.

Consistency is sometimes estimated qualitatively, by comparison of a given system with that of commonly known products, such as “consistency of honey” or “consistency of butter.” Special instruments called consistometers, which express consistency in arbitrary units or in indexes of viscosity and strength, are more frequently used.


Consistency

 

a property of a deductive theory (or system of axioms by means of which the theory is specified) such that it is impossible to derive from it a contradiction, that is, any two sentences A and ┐A, each of which is the negation of the other. Consistency for a broad class of formal theories, including the axiom A & ┐A 3 B (“a contradiction implies any assertion”), is equivalent to the existence in the given theory of at least one unprovable sentence.

Consistency, which is necessary in order that a system be considered a description of an “intuitive situation,” does not by any means guarantee the existence of such a situation. However, abstract models can be indicated in each case for any consistent system of axioms. Consistency therefore serves for representatives of “classical” trends in the foundations of mathematics and logic (and even more so for adherents of model theory) if not as the foundation for the “existence” of sets of abstract objects described by the axioms, then at least as a sufficient reason for an intuitive examination and study of such objects. Insofar as a “situation” that can be described by a theory lies outside the theory itself, the aforementioned concept of consistency, which may be called “internal” (in other words, syntactic, or logical) consistency, is closely related to what is called “external” (semantic) consistency. External consistency consists in the unprovability in a given theory of any sentence contradicting (in the ordinary intuitive sense) the facts of the “reality” described by the theory. In spite of this relation, syntactic and semantic consistency are equivalent only for such “poor” logical theories as, for example, the propositional calculus. However, in general, internal consistency is stronger than external consistency. The role of the “reality” reflected by a given concrete theory can also be played by some other deductive theory, so that the external consistency of the original theory can be understood as relative consistency, and the proof of this consistency is the indication of a system of corresponding semantic rules for translating concepts, expressions, and assertions from the second theory into the first, yielding an interpretation (model) of the original theory.

In classical mathematics, set theory serves in the final analysis as a source for the construction of models for such proofs. However, the discovery of paradoxes (antinomies) in set theory has led to the necessity of searching for new, in a sense, “absolute,” methods of proving consistency that are fundamentally different from the method of interpretations. The need for new methods of proving consistency also arises from the noncoincidence of the concepts of internal and external consistency.

One may choose an intermediate method, requiring an absolute proof of consistency only for the axiomatic theory of sets (to which it would be possible to reduce the consistency problem of specific mathematical theories by using purely model-theoretic means) or even for such a relatively simple part of it as the formalized arithmetic of natural numbers, since a set-theoretic universe of discourse of the principal divisions of classical mathematics can be constructed by means of this formalized arithmetic. Such a path was taken by D. Hubert, who proposed a broad program, in the carrying out of which the theories for which a foundation was to be provided would first undergo formalization and the resulting formal systems (calculi) would be checked for syntactic consistency by finitary, that is, intuitive, means, but not using questionable set-theoretic abstractions.

Such absolute consistency proofs constituted the fundamental content of the metamathematics (proof theory) developed by Hilbert’s school. But by 1931, K. Godel had demonstrated the fundamental unfeasability of Hilbert’s program and thereby the limitations of the axiomatic method, within the framework of which the requirements of consistency and completeness, for sufficiently rich formal theories, are inconsistent. The requirement of completeness loses its meaning in the case of intuitive, or “nonformal,” deductive (including mathematical) theories, so that their consistency remains, as before, the most important necessary criterion for their having meaning and being applicable in practice.

REFERENCE

Kleene, S. C. Vvedenie v metamatematiku. Moscow, 1957. (Translated from English; contains list of references).

IU. A. GASTEV

consistency

[kən′sis·tən·sē]
(materials)
The degree of solidity or fluidity of a material such as grease, pulp, or slurry.

consistency

1. The degree of firmness, or the relative ability of freshly mixed concrete, grout, or mortar to flow; usually measured by the slump test for concrete, and by the flow test for mortar, plaster, cement paste, or grout. Also see viscosity.
2. The property of a cohesive soil that describes its physical state.
References in classic literature ?
Spenlow, 'having experience of what we see, in the Commons here, every day, of the various unaccountable and negligent proceedings of men, in respect of their testamentary arrangements - of all subjects, the one on which perhaps the strangest revelations of human inconsistency are to be met with - but that mine are made?
As an example of motiveless degradation of character, we have Menelaus in the Orestes: of character indecorous and inappropriate, the lament of Odysseus in the Scylla, and the speech of Melanippe: of inconsistency, the Iphigenia at Aulis,--for Iphigenia the suppliant in no way resembles her later self.
I rapidly ran over these thoughts, and then I laughed a low and involuntary laugh, moved thereto by a slight inward revelation of the inconsistency of man.
Said Peter 'Though I cannot sound The depths of such a man as you, Yet in your character I've found An inconsistency or two.
No man can like being driven into the appearance of such inconsistency.
Nioche meditated: there was an inconsistency in his position; it made him chronically uncomfortable.
Yet such is human inconsistency that one of the interests of the new place to her was the accidental virtues of its lying near her forefathers' country (for they were not Blakemore men, though her mother was Blakemore to the bone).
To Catherine's simple feelings, this odd sort of reserve seemed neither kindly meant, nor consistently supported; and its unkindness she would hardly have forborne pointing out, had its inconsistency been less their friend; but Anne and Maria soon set her heart at ease by the sagacity of their "I know what"; and the evening was spent in a sort of war of wit, a display of family ingenuity, on one side in the mystery of an affected secret, on the other of undefined discovery, all equally acute.
With this resolve firmly impressed upon her mind, she awoke next day; but during the morning, after her early lunch, she fell into a condition of remarkable inconsistency.
With dim lights and tangled circumstance they tried to shape their thought and deed in noble agreement; but after all, to common eyes their struggles seemed mere inconsistency and formlessness; for these later-born Theresas were helped by no coherent social faith and order which could perform the function of knowledge for the ardently willing soul.
He had attributed this weakness to the fact that she was a woman, although he had been rather troubled by the apparent inconsistency of his hatred for her and his repeated protection of her when danger threatened.
Then he turned once more to face her enemies with the strange inconsistency which had ever marked his methods.