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Introduction
Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018
Moreover, from a computational and cognitive perspective, Kato and Nishimura’s focus on time-ordering reflects tense logic (which is concerned with temporal notions such as: henceforth, eventually, hitherto, previously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being . . .) while the main thrust of their paper exemplifies geometric logic (which is concerned with transitions from local to global). Nevertheless, as the topos theorist, Robert Goldblatt (2006: 3), has explained, other forms of logic, of relevance to both computation and cognition, include: deontic logic (it is obligatory/forbidden/permitted/unlawful that . . .); epistemic logic (it is known to X that, it is common knowledge that . . .); doxastic logic (it is believed that . . .); dynamic logic (after the program/computation/action finishes, the program enables, throughout the task . . .); and metalogic (it is valid/satisfiable/provable/consistent that . . .).
Doxastic logic: a new approach
Published in Journal of Applied Non-Classical Logics, 2018
There are many other possible solutions to the problem of logical omniscience. One can use classical modal logic and not normal modal logic as a model for doxastic logic, one can treat the belief-operator as a possibility-operator and not as a necessity-operator, one can make a distinction between implicit and explicit beliefs or between awareness and explicit beliefs, one can introduce the notions of local reasoning and opaque knowledge and beliefs, and principles of implicit beliefs, one can use fusion models or impossible world semantics and one can use various non-modal approaches to doxastic logic to try to solve this problem.2 It is beyond the scope of this paper to discuss all solutions that have been suggested in the literature in detail. However, every solution that I am aware of seems to me to be either intuitively too strong or intuitively too weak or simultaneously both too strong and too weak. A system is too strong if we can prove too much in it, that is, if we can prove things in this system that are counterintuitive; and it is too weak, if we can prove too little in it, that is, if we cannot prove everything that we want to be able to prove. For example, classical systems seem to be too strong, since Belief of equivalent formulas (see above) still holds in such systems, and systems that use impossible worlds often appear to be too weak. It seems to be intuitively plausible that a perfectly rational individual will not believe any contradiction. However, in systems based on some kind of impossible world semantics we cannot show this. Of course, our intuitions are not infallible, but if we can construct a system that is consistent with our intuitions such a system appears to be preferable to systems that have counterintuitive consequences. Some of the solutions also postulate various new kinds of entities that might be ontologically problematic, for example impossible objects and impossible worlds. The systems in this paper are both partly weaker and partly stronger than many standard systems. Several intuitively problematic sentences and arguments that are valid in standard systems are not valid in our systems, and several intuitively plausible sentences and arguments that are not valid in standard systems are valid in our systems. Consequently, we can avoid many problems with classical doxastic logic and with many other solutions to the problem of logical omniscience. This is a good reason to be interested in the results in this paper.