Nonstandard Consequence Operators

In 1963, Abraham Robinson applied his newly discovered nonstandard analysis to formal firstorder languages and developed a nonstandard logic [9] relative to the “truth” concept and structures. Since that time not a great deal of fundamental research has been attempted in this specific area with one notable exception [2]. However, when results from this discipline are utilized they have yielded some highly significant and important developments such as those obtained by Henson [3]. The major purpose for this present investigation is to institute formally a more general study than previously pursued. In particular, we study nonstandard logics relative to consequence operators [1] [5] [10] [11] defined on a nonstandard language. Since the languages considered are not obtained by the usual constructive methods, then this will necessitate the construction of an entirely new foundation distinctly different from Robinson’s basic embedding techniques. Some very basic results of this research were very briefly announced in a previous report [5]. In 2, we give the basic definitions, notations and certain new standard results are obtained that indicate the unusual behavior of the algebra of all consequence operators defined on a set. In 3, some new standard properties relative to subalgebras and chains in the set of all consequence operators are investigated. Finally, the entire last section is devoted to the foundations of the theory of nonstandard consequence operators defined on a nonstandard language.