I'm a novice to modal logic and only have a passing familiarity with classical logic. I started reading 'Modal Logic for Open Minds'. It is very readable, but then on page 16 the author introduces a graph structure that is very opaque to me.

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What do the numbers and arrows mean in the graph? And what does the notation "2,p" or "4, p, q" mean?

The books says "the valuation is written in the diagram in an obvious manner", but I cannot understand what that means. How do we go from the graph to the "facts" listed below it?

Any help in how to read the statements at the top of page 17 would also be greatly appreciated. How do I 'pronounce' these:

"□⊥", "◇□⊥" etc?

  • 2
    Re your "how to read the statements at the top of page 17", we can use modal formula to uniquely identify and thus define these 4 sample worlds in the propositions stripped relational model, and world 4 as a dead end is the most easy one to get, ie, M,s⊨□⊥ only if s=4. World 2 is the most difficult to come up with a clever formula as shown in your book, and the author also mentioned such cleverness is unnecessary, you can also use the conjunction of negated formulas of the other 3 worlds... To interpret and read such a model in your book case you'd better use deontic/nonalethic modality... Jun 24 at 3:45

1 Answer 1


The nodes, identified by numbers, are the worlds W.

The arrows between worlds encode the accessibility relation R; an arrow going from a to b means that aRb holds.

The letters are the propositions true at that world according to the valuation function V; letters not listed near a world are false at it.

For example, a written representation of the first diagram is
M = (W, R, V) where
W = {1, 2, 3, 4}
R = {<1,2>, <1,3>, <1,4>, <2,2>, <2,4>, <3,4>}
V : (p,1) ↦0, (p,2) ↦1, (p,3) ↦ 0, (p,4) ↦ 1, (q,1) ↦ 0, (q,2) ↦ 0, (q,3) ↦ 1, (q,4) ↦ 0

An agnostic pronunciation of the modal operators is simply "box" and "diamond". Later, once you interpret the modality as e.g. alethic or deontic, the operators can correspondingly be read as "necessarily"/"possibly", "it is obligatory"/"it is allowed to" or however one wants to think of them.

  • Thank you so much! So what does it mean for a world to be accessible to itself, as in the case of 2 mapping back to itself? Jun 24 at 0:25
  • Also, why is □(p->q) true in world 2 if q is false there? It's listed as one of the facts. And more generally, how can □p be true in worlds where p is false? Jun 24 at 0:31
  • 1
    What do you mean what does it mean? It just means that the world itself is to be counted among the relevant worlds to check when evaluating □ or ◇.
    – lemontree
    Jun 24 at 1:11
  • 1
    □(p->q) is not true in w2, □(q->p) is. □p can be true in worlds where p is false by the world not accessing itself and therefore not being relevant in the valuation of □.
    – lemontree
    Jun 24 at 1:14
  • Thank you so much for taking the time to explain this! Jun 24 at 20:19

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