The starting point is an analog of interpretation in the context of defeasible reasoning, viz. the notion of an extension of a theory, where a theory is regarded as a set of sentences. An extension of a theory can be thought of as an interpretation of the theory as a set of defeasible statements. In an extension, a theory's sentences can not only be justified, but also defeated. This is in contrast with the standard, non-defeasible interpretation of a theory in terms of models, where all sentences of the theory are assigned the same positive value, viz. true. In an extension of a theory, the justified part of the theory must provide an argument against the entire defeated part.
The search for an analog of valid consequence and proof started naïvely, in work on the graphical presentation of dialectical arguments in which statements can be supported by reasons and also attacked by counterarguments. The development of naïve dialectical arguments for the experimental argument assistance system ArguMed resulted in the discovery and investigation of the notion of dialectical justification: an argument is dialectically justifying if and only if the argument attacks all arguments that are incompatible with it.
Dialectical justification is analogous to valid consequence in the following two relevant ways. First, a dialectically justifying argument can be regarded as a set of premises justifying its conclusions, in the context of defeasible reasoning. The premises provide a basis justifying a conclusion, that is as solid as possible in the context of defeasible reasoning. Second, the investigation of the internal structure of a dialectically justifying argument leads to the notion of a justifying dialectical argument, that is a direct generalization of that of a proof, but incorporates counterarguments. A major difference between dialectical justification and valid consequence is of course that dialectical justification is nonmonotonic relative to a theory: when an argument is dialectically justifying with respect to a theory, it need not be dialectically justifying with respect to a larger theory. Another difference is the phenomenon of dialectical ambiguity: it can be the case that a statement is both dialectically justifiable and dialectically defeasible with respect to a theory. Dialectical ambiguity is analogous to inconsistency, but is not trivializing: the existence of a dialectically ambiguous statement with respect to a theory does not imply that any statement is dialectically justifiable.
The notion of dialectical justification plays a central role in an interesting necessary and sufficient condition for the existence of an extension of a theory. A characterization of the number of extensions (which is as usual zero, one or several) is given in terms of the notion of dialectical justification.
The notion of dialectical justification is closely related to the notion of admissibility that is currently regarded as state of the art: an argument is admissible if and only if it attacks all arguments that attack it. It is shown that the notion of dialectical justification is more satisfactory than the notion of admissibility, as a tool in the analysis of extensions. By a meta-analysis it is shown that three properties of dialectical justification are crucial: the union property, the localization property and the separation property. Admissibility lacks the latter, and as a result of that, does not allow a characterization of the existence of extensions analogous to that in terms of dialectical justification.
A useful instrument in the analysis of the dialectical interpretation of theories is the notion of a theory's stages. A stage of a theory is a partial dialectical interpretation of the theory, i.e., a dialectical interpretation of a subset of the theory. The stages of a theory correspond extensionally to the theory's satisfiable subsets (where satisfiability is used in the standard sense of having a model). There is an interesting intensional difference, which is relevant for the maximization of stages. Instead of maximizing the stage's justified part (which corresponds to maximizing a satisfiable subset), it is natural to maximize the stage's scope, i.e., the part of the theory that is interpreted in the stage, whether justified or defeated.
Verheij, Bart (2000). DefLog - a logic of dialectical justification and defeat. Technical report. http://www.ai.rug.nl/~verheij/publications.htm/DefLog15.htm.
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An adapted and abbreviated version of this text appeared as "DefLog: on the Logical Interpretation of Prima Facie Justified Assumptions" in the Journal of Logic and Computation 13 (3), 319-346. See http://www.ai.rug.nl/~verheij/publications/jlc2003.htm.