It consists of a general theory of definitions, including (but not limited to) circular and interdependent concepts.
The philosophical background of revision theory is developed by Gupta and Belnap.
[3] Gupta and Belnap maintain that circular concepts are meaningful and logically acceptable.
Its meaning, rather, is a rule of revision that determines how to generate a new hypothetical extension given an initial one.
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.
Gupta and Belnap recommend siding with material adequacy in cases in which the two criteria conflict.
Repeated application of a rule of revision generates sequences of hypotheses, which can be used to define logics of circular concepts.
appealing to the meaning of the undefined expressions in the definition, namely blue and to the left of.
-long, since transfinite revision sequences require the additional specification of what to do at limit stages.
As shown in the following table, all hypotheses for the ground model of the previous example are revised to the set {a, b} .
down to a possibly empty initial segment of the natural numbers and subsequent revisions will build it back up.
This rule reflects the fact that formulas from the ground language do not change their interpretation throughout the revision process.
needs infinitely many revisions, unless the initial hypothesis already assigns all the natural numbers as the extension of
Maricarmen Martinez has identified some syntactic features under which the set of finite definitions is closed.
Its definiens and definiendum will not have the same truth value after any revision, so the material biconditional will not be valid.
For example, if the definientia of contain only symbols from the ground language, then the material counterparts will be valid.
[16] This includes three-valued schemes, such as Strong Kleene, with exclusion negation, whose truth table is the following.
Notably, many approaches to truth, such as Saul Kripke’s Strong Kleene theory, cannot be used with exclusion negation in the language.
Relatedly, revision theory does not postulate any restrictions on the syntactic form of definitions.
are all positive, then revision sequences will reach fixed points, as long as the initial hypothesis has the feature that
, if the initial hypothesis assigns the empty extension to all definienda, then the revision sequence will reach the minimal fixed point.
The concept of truth is circular because some Tarski biconditionals use an ineliminable instance of ‘is true’ in their definiens.
This and other examples show that truth, defined by the Tarski biconditionals, is a circular concept.
The extension and anti-extension of the truth predicate in these theories will not exhaust the set of sentences of the language.
FS is a subtheory of the theory of truth for arithmetic, the set of sentences valid in
Benedikt Löwe has shown that there are close connections between computations of infinite-time Turing machines and revision processes.
André Chapuis has argued that the reasoning agents use in rational choice exhibits an interdependence characteristic of circular concepts.
Revision theory has been used by Gupta to explicate the logical contribution of experience to one's beliefs.