Towards a many-valued logic of quantified belief

Abstract: We consider a logic which "truth-values" are represented as quantified belief/disbelief pairs, thus integrating reports on how strongly the truth of a proposition is believed, and how strongly it is disbelieved. In this context a major motive for the logic proposed is, that it should not lead (as in classical logic) to irrelevant conclusions when contradictory beliefs are encountered. The logical machinery is built around the notion of the so-called logical klattice: a particular partial order on belief/disbelief pairs and fuzzy set-theoretic operators representing meet and join. A set of principles (semantically valid and complete) to be used in making inferences is proposed, and it is shown that they are a many-valued variant of the tautological entailments of relevance logic.To treat non truth-functional aspects of knowledge we introduce also the notion of the information lattice together with particular meet and join operators. These are used to provide answers to three fundamental questions: how to represent knowledge about belief/disbelief in the constituents of a formula when supplied with belief/disbelief about the formula as a whole; how to determine the amount of belief/disbelief to be assigned to formulas in an epistemic state (or a state of knowledge), that is, a collection of partial interpretations, and finally, how to change the present belief/disbelief in the truth of formulas, when provided with an input bringing in new belief/disbelief in the truth of these formulas. The answer to all these questions is given by defining a formula as a mapping from one epistemic state to a new state. Such a mapping is constructed as the minimum mutilation of the given epistemic state which makes a formula to be believed true (or false) in the new one. The entailment between formulas is also given the meaning of an input and its properties are studied.We study also if- then inference rules that are not pure tautological entailments, but rather express the causal relationship between the beliefs held with respect to the truth and falsity of the antecedent and the conclusion. Detachment operators are proposed to be used in cases when: (i) it is firmly believed that belief/disbelief in the validity of the conclusion follows from belief and/or disbelief in the validity of the antecedent, and (ii) it is believed, but only to a degree, that belief/disbelief in the validity of the conclusion follows from belief/disbelief in the validity of the antecedent. It is shown that the following four modes of inference are legitimated within the setting of these rules: modus ponens, modus tollens, denial, and confirmation.We consider also inference rules augmented with the so-called exeption condition: if IA/ then /BI unless IC/. The if- then part of the rule expresses the major relationship between A and B, i.e., it is believed (up to a degree) that belief and/or disbelief in Bfollows from belief and/or disbelief in A. Then the unless part acts as a switch that transforms the belief/disbelief pair of B from one expressing belief in its validity to oneindicating disbelief in the validity of B, whenever there is a meaningful enough belief in the exception condition C.We also give a meaning to the inference rules proposed as mappings from epistemic states to epistemic states, thus using them as a tool for changing already existing beliefs as well as for deriving new ones.