1

In Modal Logic 1.2 -- truth trees for K about 1:30 into the video, the presenter said:

If you have possibly A then you need to draw an arrow to a new world and derive A in that world because w0 accesses some world in which A is the case.

Later he also said that the rule is different for necessity:

The rule for necessity says that if you have an arrow to another world then you can use necessity to derive A in that world.

Basically, given possibility one can draw a world using a truth tree, but not with necessity. I think this has something to do with the accessibility of w0 to this new world, w1, but I don't understand what that is or whether this rule is peculiar to system K.

Hence my question: Why can one not draw a world in modal logic given necessity?

Here is the diagram completed as of about 3:30:

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For the first and fourth diagram the black circle about w1 means that the world w1 has to be there prior to deriving anything from necessity in that world. For the second and third diagram the red circle about w1 means this world can be drawn and one can derive the possibility in that world.


Reference

Modal Logic 1.2 -- truth trees for system K, Kane B channel https://www.youtube.com/watch?v=fkS7cE8PA5I

2

In Krippke's accessible worlds model:

If a statement is possible then there exists an accessible world where it happens to be true. So if a thing is possible we can definitely find at least one accessible world where it happens.

If a statement is necessary then for all accessible worlds it happens to be so — which does not guarantee that there are in fact worlds accessible. So if a thing is necessary then we may not be able to find any accessible worlds at all but it will definitely be true in any accessible world we might find.

Refer to the notion of vacuous truth. If no worlds are accessible then everything is necessary.

  • I was beginning to suspect that was the reason. Thank you for confirming it. I assume the additional axioms added to the various modal logics above K give one more accessible worlds. +1 – Frank Hubeny Nov 17 '18 at 1:01
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    @FrankHubeny Indeed. Frames with the axioms of Reflexivity (T: □p→p) or Seriality (D: □p→◊p) will guarantee an accessible world for any neccessary proposition. – Graham Kemp Nov 17 '18 at 2:47

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