6/22/12
Reading
Rod Girle [2000] Modal Logics and Philosophy Chapter 3
K has the rules
{Non-modal propositional rules + Modal Negation + ◊R + □R}
These are described in Review of K Rules.
Basically what you have here are the restricted rules and little or nothing in the way of Access. Use of the possibility elimination rule will provide you with access to the generated world, and, with certain arguments, there may be explicit access given by the premises. But that is it. In particular, even though a formula may hold in a world, that by itself does not give access from that world to that world, so the inference.
a) ∴ □P ⊃P (invalid in K)
will be invalid.
Try to derive it as Tree1 below [switch the Rule Set to K, and do not use any of the Access Rules]. You should not be able to close the tree. You can also try using S5, and you should be able to succeed there.
There are a couple more for you to try
b) ∴◊~P⊃~□P
c) ∴□P⊃□□P (invalid in K)
Roll your own trees with K
Girle's Chapter 3 has a number of exercises. You can do them here (be sure to use the following logical symbols)
∼ & ∨ ⊃ ≡ ∀ ∃ ∴ □ ◊
[Switch the Rule Set to K, and do not use any of the Access Rules].