A Theory of Truth that Prefers Falsehood

Abstract

We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to have many nice features. It is noted that a similar class of fixed points, preferring truth, can also be studied. The notion of intrinsic is shown to relativize to these two subclasses. The mathematical ideas presented here originated in investigations of so-called stable models in the semantics of logic programming.

Default Reasoning with Imperfect Information in Multivalued Logics

“…Most conventional approaches in defining the program semantics are based on pessimistic/negative default assumptions [6,14] where the value f alse is the default value to be assigned to underivable atoms and less commonly on skeptical default assumptions [2] where a default logical value corresponding to "unknown" is preferred. This is also the case with the most of extended semantics expressing uncertainty [13,8,4,5]. In our work [9] any of the logical value true, f alse, ⊥ and are used to complete the missing information in defining the semantics of an extended program based on multivalued α-fixed models.…”

“…Fitting [4,5] defined the (multivalued) stable models for extended programs in bilattices. This concept extends that of stable model in the conventional bivalued logic [6].…”

“…The connection between the default semantics and mstable k (P ) matches the characteristics of the two interpretations. As Fitting shows in [4], in the definition of the multivalued stable models there exists a preference for falsehood, in the sense that the truth value false is assigned to atoms whenever possible (similarly to using a default reasoning). On the other hand, minimality in the degree of information (i.e.…”

“…the knowledge order) is a characteristic of both interpretations. In [4] it is suggested the mechanism used for defining the multivalued stable models can be extended to build a theory that prefers truthhood, which intuitively speaking, would correspond, in our terms, to using a default interpretation assigning the value true to all ground atoms.…”

Logic programming revisited

“…Moreover, in some complex applications, partial models simply cannot be avoided. An illustration is the theory of truth presented in Fitting [1997]. Fitting uses the well-founded semantics to define the truth predicate and obtains one in which the liar paradox ("I am a liar") is undefined (⊥) but the truth sayer ("I am true") is false.…”