Consequence operator

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.

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.

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.

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).

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.