Consequence operator

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In mathematics, consequence operators are entities defined using basic set theory notions. Alfred Tarski introduced the finite consequence operator in the 1930s (Tarski 1956). They are an example of a closure operator and are used to model a variety of notions associated with logic, mathematical logic, physical theory unifications and physical behavior.

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[edit] Definition

Let L denote any nonempty set and P(L) denote the set of all subsets of L. For any X \subset L, let F(X) denote the set of all finite subsets of X. An operator C\colon P(L) \to P(L) is a (general) consequence operator if and only if it satisfies the following two axioms. For each X, \ Y \subset L

(1) X \subset C(X) = C(C(X)) \subset L.
(2) If X \subset Y, then C(X) \subset C(Y).

If C also satisfies for each X \subset L

(3) X = \bigcup\{C(Y)\mid Y \in F(X)\},

then C is a finite (finitary, algebraic) consequence operator.

The above three axioms are not independent. Axioms (1) and (3) imply (2). Consequence operators form a major category within the subject universal logic. For a given X \subset L, C(X) is often considered as the set of all objects deduced from X. It has been shown that under this interpretation consequence operators are equivalent to the sets of objects deduced from X by general logic systems. Hence, consequence operators model the basic aspects of reasoning from hypotheses when L represents a formal language, images, or a special form of information. Such reasoning includes formal and informal deductive, inductive and basic dialectic forms.

[edit] Algebraic properties

The set of general consequence operators defined on nonempty L forms a complete lattice and the subset of all finite consequence operators forms a join-complete lattice. This last fact has been used to show explicitly how to construct the "best possible unification for any collection of physical theories" (Herrmann 2004) without altering any of the theories.

[edit] Nonstandard consequence operators

Besides using consequence operators to investigate specific deductive processes and general logical systems, consequence operators can be viewed from a nonstandard model (Herrmann 1987). Such a model is usually, at the least, an enlargement. Enlargements are nonstandard models that characterize collections of sets that satisfy the finite intersection property. Technically, within model theory, a nonstandard model for a set of formal sentences S is usually a model for S that is not isomorphic to a declared standard model for S. The idea of embedding a language into an enlargement and investigating nonstandard logics was originated, in 1963, by Abraham Robinson. Nonstandard consequence operators, where some are termed as ultralogics, need not have the exact same properties as those of the defined general or finite (standard) consequence operators. Their properties are, however, considered as similar in structure to standard consequence operators. Nonstandard consequence operators form the fundamental mathematical objects used within the testable and falsifiable general grand unification model (i.e. the GGU-model) (Herrmann 1988).

[edit] Criticisms of nonstandard models

There has been some criticisms of the use of nonstandard models in that depending upon the machinery used in their investigations the axiom of choice is often employed. However, the axiom of choice is not necessary in order to do nonstandard analysis if the formal mathematical language is but restricted slightly (Stroyan & Luxemburg 1976). The ultrafilter lemma, which is strictly weaker than the axiom of choice, is the only additional axiom that needs to be adjoined to the first eight ZF axioms for set theory in order to do nonstandard analysis. W. A. J. Luxemburg originated most aspects of this ultrafilter approach to nonstandard analysis. The book listed first under the external links heading is an example of the ultrafilter approach for a simple language.

[edit] References

  • Herrmann, Robert A., "General logic-systems and finite consequence operators," Logica Universalis, 1(2006), 201-208.
  • Herrmann, Robert A., "The best possible unification for any collection of physical theories," Internat. J. Math. and Math. Sci., 17(2004), 861-721.
  • Herrmann, Robert A., "Hyperfinite and standard unifications for physical theories," Internat. J. Math. and Math. Sci., 28(2001), 93-102.
  • Herrmann, Robert A., "Physics is legislated by a cosmogony," Speculations in Science and Technology, 11(1)(1988), 17-24.
  • Herrmann, Robert A., "Nonstandard consequence opreators," Kobe J. Math. 4(1)(1987), 1-14.
  • Loeb, Peter A. and Manfred Wolff (eds), Nonstandard Analysis for Working Mathematicians, Kluwer Academic Publishers, Dordrecht, Netherlands, 2000.
  • Luxemburg, W. A. J., "What is Nonstandard Analysis?" In J. C. Abbott (ed.) Papers in the Foundations of Mathematics, Number 13 of the Herbert Ellsworth Slaught Memorial Papers, Supplement to the American Mathematical Monthly, Vol. 80(6), June-July 1973.
  • Luxemburg, W. A. J., Non-standard Analysis: Lectures in A. Robinson's Theory of Infinitesimals and Infinitely Large Numbers, Mathematics Department, California Institute of Technology, Pasadena (1962, 1973).
  • Robinson, A., "On languages which are based on non-standard arithmetic," Nagoya Math. J., 22(1963), 83-118.
  • Stroyan, K. D. and W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, NY, 1976.
  • Tarski, Alfred, Fundamental concepts of the methodology of deductive sciences. In Logic, Semantics, Metamathematics, Oxford University Press, Oxford, 1956, (Hackett, 1981).

[edit] External links

    • Each of these external links connects to the arxiv.org free e-Print archives in mathematics and physics. Each article or monograph may be retrieved, stored or printed without restrictions.


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