Quantified temporal alethic boulesic doxastic logic
The paper presents a very rich version of the first-order modal language. The language includes four different groups of modalities which can be described as follows: \begin{itemize} \item Alethic modalities: necessity and possibility plus universal modality and its associated dual, the universal possibility. \item Temporal modalities: sometimes (resp. always) in the future (resp. in the past). Plus the absolute modalities of sometimes and always. The latter are not definable in terms of past and future since it is not generally assumed that any two given moments are comparable in terms of the temporal order. \item Doxastic modalities: believing and conceiving. \item So-called boulesic modalities: wanting, accepting, rejecting, and being indifferent. \end{itemize} The language also includes the equality predicate plus the special designated unary predicates \(R\) for being rational and \(E\) for existing. The language is supplied with a sort of Kripke semantics where the domain of (possible) objects is assumed to be constant. Beside the object domains, the models include the set of worlds and the set of time moments, and the formulas get evaluated at world-moment pairs. The models include a set of modal accessibility relations corresponding to the different groups of modalities allowed in the language. Whereas the temporal relation is given in the usual format, the alethic modal accessibility relation is ternary with the tuple of arguments of the form \((\mathrm{World}, \mathrm{World}, \mathrm{Moment})\), so that one world can be alethically accessible from another one at a particular time moment. In case of boulesic and doxastic accessibility relations, the argument tuple is even quaternary and has the form \((\mathrm{Object}, \mathrm{World}, \mathrm{World}, \mathrm{Moment})\). The object argument plays the role of the agent index since in the present setting the agents live as objects in the domain assigned to a given world. The author does not offer much in terms of justification for this peculiar setting, and proceeds to a consideration of different constraints one might impose on the general class of the Kripke models loosely described in the previous paragraph of this review. Naturally, given the lavish set of modalities present in the language, the number of possible constraining principles is astronomical; the author considers more than a hundred of them, organized in 16 tables, and the whole discussion sounds rather monotonic. The author displays yet another group of principles and writes down, for each principle, a formula schema that becomes valid in the subclass of models satisfying the principle. No attempt is made to argue in the reverse direction and show that some of the formulas also define their corresponding frame classes. It is worth noting that the author clearly prefers very special type of constraining principles. All of them are first-order definable, moreover, it seems like the author does not attempt to use the additional flexibility inherent in the new argument places for modal accessibility relations as they are invariably taken ``out of the game'' by means of a universal closure. The author then turns to proof theory and provides a basic system of labeled analytic tableaux which is not completely cut-free in that it seems to require a weakened version of the cut rule. However, the system can be extended with additional rules accommodating all -- or almost all -- of the constraints considered in the previous section. It is hard to tell whether some of the 100-plus constraints end up not having their counterpart, since the presentation of the rules takes up the space of 26 tables. This is followed by a list of examples of provable theorem schemas classified into 21 different groups. A standard completeness proof is then supplied by the method of saturating a non-closed table branch and reading off a counter-model. The paper ends with an attempt to motivate the applicability of the system; more precisely, the author looks into two examples of realistic reasoning that seem to require the full set of resources present in the language of the system under consideration. Whereas it is intuitively clear that many reasoning patterns used in the real world freely combine several different kinds of modalities, the two particular examples chosen by the author seem to be somewhat inconspicuous in the sense that they haven't attracted much interest in the previously existing literature (the author does not seem to give any relevant references, and a quick Internet search finds nothing).
scientific article; zbMATH DE number 7344022
scientific article