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“M-Tee”
Basic Trees for Modal Logic
1. So in modal logic, we have the two new modal operations symbolized by the box and diamond. We now need to expand our methods of tree development and branch closure in order to test the validity of sequents incorporating formulas with modal operations, including rules for developing the box and diamond on the left and on the right.
2. The great innovation in modal tableau is going to be the notion of a tableau subordinate to a path. This will be represented by placing an entire tableau underneath (or “downstream”) that path, and then by drawing an arrow from the original path to the subordinate tableau. If that path subsequently splits, then duplicate copies of that subordinate tableau are to be placed under each branch and then developed separately. A first lesson to be learned then is that one should develop a path as much as possible before having to create downstream subordinate tableaus.
3. If a tableau subordinate to a path ever closes completely, then that path will close as well.
4. So what is the point of subordinate paths? Intuitively, they correspond to contexts of reasoning or worlds other than the one from “whence they sprang.” The arrow represents the accessibility of the subordinate context or world to the context of the path which led to its creation.
5. The creation of these subordinate contexts correspond to the development of either a diamond formula on the left, or a box formula on the right.
So schematically, here are the two relevant rules:
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In the terms of alethic modality, these rules tell us that when some formula is possible, then there must be a world or context in which that formula is true, and when it is false that some formula is necessary, then there must be a world or context in which that formula is false.
5. So what do we do with the two remaining rules, box on the left and diamond on the right? Basically, they tell us how further to develop any and all subordinate tableaus once they’ve been created.
Schematically:
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These two rules, box on the left and diamond on the right, resemble the “strong” or “continuous” rules for the quantifiers in predicate logic, in that one must keep in mind that they will need to developed in any subordinate context that ever needs to be created. For that reason, it is advisable to apply the “weak” quantifier rules, before developing the “strong” quantifier rules.
6. An example: Determine whether □( P(Q) --||-- ◊P(◊Q.
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7. That’s just about all there is to it! Let’s get cracking with some exercises. Demonstrate whether the following hold:
1. ◊( P(Q) --||-- ◊P(◊Q
2. □P&□Q --||-- □(P&Q)
3. ◊P&◊Q --||-- ◊(P&Q)
4. □Pv□Q --||-- □(PvQ)
5. ◊Pv◊Q --||-- ◊(PvQ)
6. P(□Q --||-- □(P(Q)
7. ◊(P(Q)--||-- P(◊Q
8. ◊(P(Q)--||-- ◊P(Q
9. ◊P&◊Q --||-- ◊(PvQ)
10. □P, □( P(Q)|-- ◊R(□Q
11. □( Q(P)v□(~ P(Q), □(~PvQ) |-- □( ~P(~Q)
12. □Q , ◊Pv◊Q |-- ◊(P&Q)
13. ◊P |-- □ (◊Pv◊Q)
14. |--□(~□P((◊~P&◊Q))
15. ◊~Q(□□~~P |-- □□Pv□Q
8. The basic rules above are adequate for the modal system K, but that is perhaps too weak of a modal system to be of general interest. So we might wonder then about what we need to do in order to tweak Tableau to accommodate other systems, such as D, S4, or S5. Recall that these systems impose progressively stronger constraints on the accessibility relation. The key to extending tableau is to make sure that the accessibility relations represented by the arrows in the tableau diagrams obey the constraints for the specific type of modal system deployed. Once that is done, then one needs to ensure that all the strong modal operators (boxes) on the left and weak modal operators (diamonds) on the right are witnessed in every context accessible to the one in which it appears.
For instance, in D, every context must have an accessible context (not necessarily itself). Unlike K, then, all of the boxes on the left and diamonds on the right are going to require a witness. In all of the other systems, every context is going to be at least self-accessible, so boxes on the left and diamonds on the right are going to have to be witnessed in their own contexts.
Things can get messy, however. In B for example, where the accessibility relation is symmetric, contexts must be mutually accessible. So if a box on the left or diamond on the right appears on a path in a subordinate tableau, then one will have to witness it for the superordinate context as well. However, one does NOT then go about placing that witnessing formula on the path above the subordinate tableau, for the requirement of that witness might be along only one path of the subordinate context. What we must do here, then, is to place an extension of the superordinate path underneath the path in the subordinate context, and label or tag it so that we know the path to which it belongs.
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