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Let W={w,u,v} and let the relation R on W be euclidean and symmetric;

Suppose R(w,u) and R(u,v).

-by symmetry we get R(u,v) -by euclidean we get R(v,w) and R(w,v).

Similarly, by euclidean we get R(u,u) & R(v,v) & R(w,w). In other words, every world has access to itself.

This looks identical to S5 which has a universal frame.

My three questions:

1: Does every world access itself like i think it does given the Set W and relation R on W?

2: if it does, are the two systems: KB5 and S5 identical or is S5 a sub logic of KB5?

3: or, is this just a coincidence that in this case the frame <W, R, v> is identical to a fully universal frame.

This is what I think:

This apparent reflexivity is just a coincidence given W and R; the two systems KB5 and S5 are not identical neither is S5 a sublogic of KB5.

S5 is definitely not a sublogic of KB5 (i.e. there are formulas provable in S5 that are not provable in KB5). Naturally, the two are not equivalent either.

Am i correct?

The apparent universality of this frame is throwing me off. Any ideas...

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To put briefly, notice that S5 is equivalent to KTB5. A comparison of the descriptive names KTB5 and KB5 hints at reflexivity is not a general property of KB5.

While wandering through the systems of modal logic, it may be helpful to keep the following cube at hand (figure credit: C. Benzmüller; 'M' is the alternative symbol for the axiom T):

The Modal Logic Cube - credit for the figure Christoph Benzmüller

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  • Thank you. by the way I just posted another question feel free to have a look and answer if you can. – ryan Aug 8 '20 at 8:11

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