Towards a many‐valued logic of quantified belief: The information lattice

In a previous article we introduced extended logical operators, based on the Dubois family of T‐norms and their dual T‐conorms, to induce a semantics for a language involving and, or , and negation. Thus, given these logical operators and an arbitrary set‐up S (a mapping from atomic formulas into a set of truth‐values), we extended S to a mapping of all formulas into a set of truth‐values defined as belief/disbelief pairs. Then using a particular partial order between belief/disbelief pairs to define entailment we were able to derive a many‐valued variant of the so‐called relevance logic. Here we introduce the notion of the so‐called information lattice built upon another type of partial order between belief/disbelief pairs. Furthermore, we introduce specific meet and join operations and use them to provide answers to three fundamental questions: How does the reasoning machine represent belief and/or disbelief in the validity of the constituents of a complex formula when it is supplied with belief and/or disbelief in the validity of this complex formula as a whole; how does it determine the amount of belief and/or disbelief to be assigned to complex formulas in an epistemic state , that is, a collection of set‐ups ; and finally, how does it change its present belief and/or disbelief in the validity of formulas already in its data base, when provided with an input bringing in new belief and/or disbelief in the validity of these formulas.