Canonical forms of fuzzy truthoods by meta-theory based upon modal logic
In this paper, we redefine, with the meta-theory based on modal logic, the operations between fuzzy sets of verity, i.e., fuzzy sets of truthood. With this new approach, we can unify the different formulas for the operations AND, OR and NOT between the fuzzy sets of verity, i.e., truthood. The operations between the fuzzy sets of verity become sensitive to the logic value true or false that agents, persons, sensors, assign to the worlds (contexts). The operations are also sensitive to the difference of the worlds and time of synchronisation. It should be pointed out that when we use the logic operations as AND, OR and NOT, we generally assume that the worlds are the same and change their truth-value at the same time. But it is known that there are cases where this synchronic situation and identity is not always valid. That is, there exist transformations that change one world for one proposition to another world for another proposition. In conclusion, the linguistic AND, OR, NOT operations become dependent on the particular truth-value of a world, on the synchronisation and on the worlds assigned to the two propositions via transformations. Thus all of these possible changes in the structure of the worlds, in the modal logic, cause the gradation of the linguistic operations, e.g., AND, OR, and NOT. An individual world (person, agent, sensor, ) assigns to an atomic sentence either a true or false value and uses the classical two value logic operations of AND, OR, NOT. That is the crisp true or false responses (assignments) of worlds generate gradation of truthood value. The uncertainty in a fuzzy set is represented with sets of worlds in a conflict situation, i.e., the same proposition may be true in one world and false in another. Consonant or dissonant relations between sets of worlds that depends on the synchronisation and/or transformation of worlds cause the generation of gradation by the linguistic operations AND OR and NOT. That is, the linguistic operations change for different concrete situations (set of worlds) caused by the generation of a representation in each world, where the descriptive set membership assignment D may be two or infinite valued but the verity assignment V is two-valued in a particular world representation. Whereas membership values of the fuzzy set of truth verifications associated with the set of worlds are in [0,1]. Furthermore, the combination of fuzzy membership values generates a Type II fuzzy set that is captured by FDCF and FCCF formulas. In this paper, we show that there exist a deeper connection between the set membership assignment and the verity assignment of truthood by modal logic with suitable extensions.
canonical forms fuzzy truthoods meta-theory formulas synchronous transformation modal-logic disjunctive conjunctive