Consequence operator

Consequence operator theory is a mathematical model that employs basic abstract set-theory to model the most significant aspects of logical discourse. There are two types of consequence operators, the general, and the finite or finitary. In 1930, Tarski (1954) introduced the more significant of these two, the finite consequence operator and, today, consequence operator theory is a subcategory of universal logic. From the abstract algebraic viewpoint, a consequence "operator" is an example of a closure operator. Since being introduced, it has carried various names. Originally, due to its application to a simple form of logical discourse, a consequence operator was termed as a "consequence operation." As an operation, it is a "unary" operation and as such generates an equivalent mathematical function. Within the published literature through 1978, it has also been called a "consequence," or "closure operation" or simply "closure operator."

In 1978, Robert A. Herrmann, initiated the nonstandard investigation of these objects using nonstandard analysis. For this reason, among others, the name was changed to "consequence operator." Usually, nonstandard consequence operators are members of a nonstandard extension of a nonempty set of finite consequence operators. Certain highly technical aspects of the original operators are not necessary preserved exactly by the nonstandard consequence operators. The nonstandard consequence operators do retain "similar" aspects of the standard operators and often display many new properties. In particular, each of the nonstandard consequence operators is, at the last, a closure operator and has a type of finitary property that behaves like the "finite" concept but may be viewed, in a special way, as having non-finite qualities as well.

Consequence operator axioms
A consequence operator is a mathematical function (operator, map, mapping) defined on the set of all subsets of a nonempty set L. Due to their application to logical discourse, the set L is often called a language. As an example, consider the language {a,b,c}. Then the set of all subsets of L is P(L) = {empty set,{a},{b},{c},{a,b},{a,c}, {b,c}, {a,b,c}}. A consequence operator associates each member of P(L), with a single member of P(L) so that the association satisfies a set of axioms. This association can depend upon the language-forms used (Tarski, 1954, p. 71) as well as additional properties imposed on L. A function C defined on all the subsets of a nonempty language L is a consequence operator if and only if the following axioms are satisfied.

(1) For each X, a subset of L, there is a single set C(X) (the image, value, second coordinate) associated with X, such that X is a subset of C(X), C(X) is a subset of L and only such an association defines the operator C.

(2) For each X and Y subsets of L, the set C(X) is a subset of C(Y).

(3) For each X, a subset of L, C(X) = Y is a subset of L and, as such, C applies to Y and C(Y) = Y.

The above three axioms satisfy the axioms for a closure operator where the operator is defined on the set of all subsets of L and the mathematical ordering is the "subset ordering." The more significant finite consequence operator must satisfy the following additional axiom.

(4) For each X, a subset of L, and each x in C(X), there exists a finite F subset of X such that x is a member of set C(F).

(Note: Axiom (4) can be expressed in a different manner so that it logically implies axiom (2))

A finite consequence operator models various aspects of ordinary logical discourse where the informal deductive processes are fixed. Call a subset X of L a "set of hypotheses." Then the "set of deductions C(X)" represents all of the results obtained, by logical deduction, using the set of hypotheses X. The deductive processes used are not altered in the sense that the same processes are applied to each set of hypotheses. The finite requirement of axiom (4) stems from the fact that when a single result x in C(X) is deduced only a finite set of the set of hypotheses X is used. Further, only a finite number of steps is necessary to deduce x. The requirement that the logical processes used be fixed and these finite properties lead to axiom (3). Axiom (1) states that all of the deduced statements should remain members of the language L and, technically, when deductions are actually made, the hypotheses can be trivially deduced. Axiom (2) follows from the experience that, when fixed fixed logical processes are used, adjoining hypotheses to the original set of hypotheses X does not eliminate any of the previously deduced members of C(X).

Consequence operators are used for various purposes within universal logic. They lead to simple characteristics for the consistency or completeness of a set of hypotheses X. Collections of consequence operators defined on the same language have significant abstract algebraic properties. In particular, such collections form what is called a lattice.

General logic-systems
Logical discourse is not usually viewed via the consequence operator. Logical discourse is most often considered as a mental process that applies fixed implicit or explicit general rules of inference to sets of hypotheses expressed in various language-forms. Deductive processes use implicit or explicit general rules of inference coupled with an informal algorithm. Set-theory can model the general rules of inference. General rules of inference and the algorithm taken together form a general logic-system. It has been shown that a general logic-system determines a finite consequence operator C that yields the same results, C(X), as the set of all deduced results obtained from a set of hypotheses X by application of the algorithm and general rules of inference,.

Conversely, given any finite consequence operator C, then C generates a set of general rules of inference in such a manner that for each set of hypotheses X the set of all deductions obtain from X using the defined general rules of inference and the algorithm is the same as C(X). Using general logic-systems, finite consequence operators with distinctive features are easily obtained. It is the strict association of finite consequence operators with general logic-systems that indicates exactly how finite consequence operators model aspects of logical discourse.

Physical modeling via finite consequence operators
Descriptions for accepted physical laws and scientific theories can be expressed as general logic-systems. The General Grand Unification Model uses the notion of the event sequence to produce a developing physical-system. The model uses standard and nonstandard finite consequence operators to model the production of and alterations in the behavior of physical-systems. The application of standard physical laws and standard physical theories to real and specific standard physical-systems within a universe via event sequences requires that each corresponding general logic-system be specified, at the least, relative to primitive or observer time (Herrmann, 2002, p. 71). Such a specification need comprise but one statement within the physical axioms. Further, the statement can be generalized for the standard physical laws and standard physical theories that are all assumed to be universal in "time" through the use of a single symbol t, where the deduction C(X), for completeness, contains a symbol t' not assumed the same value as t. Each general logic-system that describes a standard physical law or standard physical theory includes such a specification statement prior to application to a specific standard physical-system. Technically, this yields slightly different expressions for the general logic-systems that represent standard physical laws or standard physical theories for each moment in, at least, primitive time.

Ultralogics
An ultralogic is a nonstandard consequence operator that has a special purpose. Such operators are employed extensively in the General Grand Unification Model. Previously, ultralogics were termed as superdeductions, a term no longer employed. Depending upon their application, ultralogics are also termed as IUN-processes (intrinsic ultranatural processes) as well as force-like processes and representations for a higher intelligence. As examples, each standard finite consequence operator has an nonstandard extension. If a set of standard finite consequence operators represent physical-laws or scientific theories, then their nonstandard extensions are ultralogics. Other ultralogics are the nonstandard consequence operators that yield the probabilistic results predicted by scientific theories and *S, the basic universe generating operator.