Leibniz and Whitehead as ontologists of agents and on metaphysicians of perspective agree that when we consider the whole process (or the whole class of monads) there are no contradictions. Yet, finite beings can only coordinate what they perceive within perspectives and are guided by finite lab-like simplifications of the whole. These simplifications cannot be taken apart from the whole, and yet they provide some sort of mathesis localis. Maybe we can think of different logical systems in this way: they capture something but only by failing to be fully coordinated with all the rest.
In our investigation of galaxies (classes of possible worlds associated to each logic), we are now wondering whether there are classes of possible world (that we call constellations) that cannot be galaxies. That is, there is no signature F of formulae that could formulate a logic that would make possible exactly the worlds in these constellation. Take a constellations formed by two or more worlds with nothing in common (intersection of the classes of truth in all worlds of the constellation is empty). We conjecture (and provided a proof for simple cases) that there could be a galaxy in any F. In no F there could be a formulation of a logic for such constellation. It would follow that there are constellations that are not galaxies in any F and therefore that some collections of possible worlds could never be such that there is an underlying logic to them. At least not one that can be formulated.
To be sure, we are assume a classical meta-logic for most of our operations and therefore there is some relativity to these results. Further, one could think of logics as something that require no language - and assume every constellation provide a logic even if it is not something that could be expressed otherwise. However, it seems like there are limits to what is possible when we feel the pressure of something beyond the mathesis localis.
In our investigation of galaxies (classes of possible worlds associated to each logic), we are now wondering whether there are classes of possible world (that we call constellations) that cannot be galaxies. That is, there is no signature F of formulae that could formulate a logic that would make possible exactly the worlds in these constellation. Take a constellations formed by two or more worlds with nothing in common (intersection of the classes of truth in all worlds of the constellation is empty). We conjecture (and provided a proof for simple cases) that there could be a galaxy in any F. In no F there could be a formulation of a logic for such constellation. It would follow that there are constellations that are not galaxies in any F and therefore that some collections of possible worlds could never be such that there is an underlying logic to them. At least not one that can be formulated.
To be sure, we are assume a classical meta-logic for most of our operations and therefore there is some relativity to these results. Further, one could think of logics as something that require no language - and assume every constellation provide a logic even if it is not something that could be expressed otherwise. However, it seems like there are limits to what is possible when we feel the pressure of something beyond the mathesis localis.
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