Why the Impossible Sucks In this paper, I will argue that the relevant logician is required to accept philosophically and metaphysically intractable positions in order to preserve the Law of Excluded Middle (LEM). I will do this by first explaining the interpretations of a relevant logic in terms of the ternary relation R, the worlds W, the Routley Star, and content inclusion. I will consider two possible understandings of worlds, one being in the sense used by classical modal logic, and another in terms of information states from authors like Restall. I will explain what the semantics of R in a relevant logic mean with respect to each of these concepts of worlds, and what the Routley Star and its characterization of star worlds offers to
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Content inclusion is a reflexive and transitive binary relation on worlds in W and is a rather intuitive relation. We say that if w ⊑ w’, then everything that is true at w is true at w’. As a result of this introduction, we have a new set of tableaux rules and ways of evaluating validity in our language. The relation of content inclusion allows us to make the crucial inference we needed before to demonstrate LEM’s validity. For instance, we can introduce the following constraint: if a is a normal world, then whatever is true in a’s star world is also true in a itself. …show more content…
Indeed, it might make a lot of intuitive sense that a world contains the truths of the worlds that are included in it. If we consider possible worlds, for instance, it seems obvious that we would be able to organize these worlds in terms of proximity to the actual world, where the nearest of possible worlds are the most like the actual world, such that for almost every fact p both the actual world and the possible worlds agree. It makes even more intuitive sense in the case where our worlds are simply states of information or ways the world can be. Per Restall, “there is a relationship of involvement x ≤ y between states. To say that x ≤ y is just to say that being y includes being x, or that y involves x”. Another way of phrasing this is that if we have a certain amount of information in one state, we do not lose that information in a later one. From this position we may now see where content inclusion and impossible worlds fit together. Indeed, Edwin Mares says that “we must distinguish between the assertion of ¬A and the denial of A… If ¬A obtains in [a situation] s, then it obtains in all ⊑–extensions of s”. What this all effectively means, then, is that what is true in our logically impossible worlds has some kind of bearing on what is true in the actual world. But this seems wildly inaccurate. The constraint on content inclusion we instituted above seems terribly incorrect. What would it mean for the truths
Content inclusion is a reflexive and transitive binary relation on worlds in W and is a rather intuitive relation. We say that if w ⊑ w’, then everything that is true at w is true at w’. As a result of this introduction, we have a new set of tableaux rules and ways of evaluating validity in our language. The relation of content inclusion allows us to make the crucial inference we needed before to demonstrate LEM’s validity. For instance, we can introduce the following constraint: if a is a normal world, then whatever is true in a’s star world is also true in a itself. …show more content…
Indeed, it might make a lot of intuitive sense that a world contains the truths of the worlds that are included in it. If we consider possible worlds, for instance, it seems obvious that we would be able to organize these worlds in terms of proximity to the actual world, where the nearest of possible worlds are the most like the actual world, such that for almost every fact p both the actual world and the possible worlds agree. It makes even more intuitive sense in the case where our worlds are simply states of information or ways the world can be. Per Restall, “there is a relationship of involvement x ≤ y between states. To say that x ≤ y is just to say that being y includes being x, or that y involves x”. Another way of phrasing this is that if we have a certain amount of information in one state, we do not lose that information in a later one. From this position we may now see where content inclusion and impossible worlds fit together. Indeed, Edwin Mares says that “we must distinguish between the assertion of ¬A and the denial of A… If ¬A obtains in [a situation] s, then it obtains in all ⊑–extensions of s”. What this all effectively means, then, is that what is true in our logically impossible worlds has some kind of bearing on what is true in the actual world. But this seems wildly inaccurate. The constraint on content inclusion we instituted above seems terribly incorrect. What would it mean for the truths