Revision theory is a subfield of philosophical logic. It consists of a general theory of definitions, including (but not limited to) circular and interdependent concepts. A circular definition is one in which the concept being defined occurs in the statement defining it—for example, defining a G as being blue and to the left of a G. Revision theory provides formal semantics for defined expressions, and formal proof systems study the logic of circular expressions.
Definitions are important in philosophy and logic. Although circular definitions have been regarded as logically incorrect or incoherent, revision theory demonstrates that they are meaningful and can be studied with mathematical and philosophical logic. It has been used to provide circular analyses of philosophical and logical concepts.
History
Revision theory is a generalization of the revision theories of truth developed by Anil Gupta, Hans Herzberger, and Nuel Belnap. In the revision theories of Gupta and Herzberger, revision is supposed to reflect intuitive evaluations of sentences that use the truth predicate. Some sentences are stable in their evaluations, such as the truth-teller sentence, The truth-teller is true. Assuming the truth-teller is true, it is true, and assuming that it is false, it is false. Neither status will change. On the other hand, some sentences oscillate, such as the liar, The liar sentence is not true. On the assumption that the liar is true, one can show that it is false, and on the assumption that it is false, one can show that it is true. This instability is reflected in revision sequences for the liar.
The generalization to circular definitions was developed by Gupta, in collaboration with Belnap. Their book, The Revision Theory of Truth, presents an in-depth development of the theory of circular definitions, as well as an overview and critical discussion of philosophical views on truth and the relation between truth and definition.
Philosophical background
The philosophical background of revision theory is developed by Gupta and Belnap. Other philosophers, such as Aladdin Yaqūb, have developed philosophical interpretations of revision theory in the context of theories of truth, but not in the general context of circular definitions.
Gupta and Belnap maintain that circular concepts are meaningful and logically acceptable. Circular definitions are formally tractable, as demonstrated by the formal semantics of revision theory. As Gupta and Belnap put it, "the moral we draw from the paradoxes is that the domain of the meaningful is more extensive than it appears to be, that certain seemingly meaningless concepts are in fact meaningful."
The meaning of a circular predicate is not an extension, as is often assigned to non-circular predicates. Its meaning, rather, is a rule of revision that determines how to generate a new hypothetical extension given an initial one. These new extensions are at least as good as the originals, in the sense that, given one extension, the new extension contains exactly the things that satisfy the definiens for a particular circular predicate. In general, there is no unique extension on which revision will settle.
Revision theory offers an alternative to the standard theory of definitions. The standard theory maintains that good definitions have two features. First, defined symbols can always be eliminated, replaced by what defines them. Second, definitions should be conservative in the sense that adding a definition should not result in new consequences in the original language. Revision theory rejects the first but maintains the second, as demonstrated for both of the strong senses of validity presented below.
The logician Alfred Tarski presented two criteria for evaluating definitions as analyses of concepts: formal correctness and material adequacy. The criterion of formal correctness states that in a definition, the definiendum must not occur in the definiens. The criterion of material adequacy says that the definition must be faithful to the concept being analyzed. Gupta and Belnap recommend siding with material adequacy in cases in which the two criteria conflict. To determine whether a circular definition provides a good analysis of a concept requires evaluating the material adequacy of the definition. Some circular definitions will be good analyses, while some will not. Either way, formal correctness, in Tarski's sense, will be violated.