Sentence examples for consequence operations from inspiring English sources

Abstract consequence operations are known as closure operators in universal algebra and lattice theory, for instance.

Consequence operations are present not only in logic but in many areas of mathematics.

There is a dual correspondence between abstract consequence operations on a set A and closed-set system on A. Given an abstract consequence operation C on A, the set C(C) of C-closed sets is a closed-set system and given a closed-set system C the operation C C) defined by C C XX) = ∩ {Y ∈ C: X ⊆ Y}, for every X ⊆ A, is an abstract consequence operation.

If in addition C satisfies (3) we say that it is a finitary consequence operation.

In order to simplify the notation we denote this consequence operation by CL.

He also contributed to general metamathematics (the theory of consequence operation) and intuitionistic logic.

Logicians of this school also made precise many important metalogical concepts, including those of the logical matrix, consequence operation, deductive system and model.

In other words, the theories of L are the closed sets of the consequence operation C⊢L on the set of L-formulas.

To achieve this, we employ an abstract consequence operation to "look beneath the water level", generalising the minimal inconsistent set concept and the related free formula notion.

We will say that an abstract consequence operation C on an arbitrary set A is an operation that applied to subsets of A gives subsets of A and for all X, Y ⊆ A satisfies conditions (1), (2) and (4) above.

The characterization of the (classical first-order) consequence relation given above was semantic, in that we described it in terms of the consequence operation Th ○ Mod which invokes the structures for the language and what is true in them.