Examples of Computing Truth-Values
in Kripke Models
1
Rules + Helpful Shortcut
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Then we define
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✤
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Rule for □ operator: □p is true at a world w if p is
true at all the worlds
accessible from w.
A helpful shortcut: If a world w
can access exactly one world w’,
(which might be w itself) then:
✤
Rule for ⬦ operator: ⬦p is true at a world w if p is
true at some world accessible
from w.
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✤
2
□p is true at w if and only if p is true at w’.
⬦p is true at w if and only if p is true at w’.
So in this situation we compute
modal statements □p, ⬦p at w
by copying the value of p at w’.
Example 1. Compute the values of □p, □q,
⬦p, ⬦q at each world
w1
w3
p
q
T
T
p
q
F
T
□p □q ⬦p ⬦q
w2
□p □q ⬦p ⬦q
w4
3
p
q
T
F
p
q
F
F
□p □q ⬦p ⬦q
□p □q ⬦p ⬦q
Example 2. Compute the values of □p, □q,
⬦p, ⬦q at each world
w1
w3
p
q
T
T
p
q
F
T
□p □q ⬦p ⬦q
w2
□p □q ⬦p ⬦q
w4
4
p
q
T
F
p
q
F
F
□p □q ⬦p ⬦q
□p □q ⬦p ⬦q
Example 3
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✤
w1
It’s not important for the
computation of the truthvalues that we have all four of
the worlds on the diagram.
We could omit some of them
and the computations work
just the same.
w2
✤
Here’s one where there’s just
two worlds, and each sees itself
and each sees the other.
5
p
q
T
T
p
q
T
F
□p □q ⬦p ⬦q
□p □q ⬦p ⬦q
Example 4
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The computation of more
complex statements now works
just like in truth-tables.
w1
To compute the truth-value of
□p∧ ⬦q at each world, we
simply first use the arrows and
the rules for the operators □, ⬦
w2
to compute □p and ⬦q
individually, and then we use
the truth-table for conjunction
to compute □p∧ ⬦q.
6
p
q
T
T
p
q
T
F
□p
⬦q
□p∧⬦q
□p
⬦q
□p∧⬦q
Example 5
w1
✤
Of course, some of the
computations of truth-values
we might do can involve just
one of the propositional letters.
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For instance, here we first
compute □q in each world, and
then we use the truth-table for w2
the ‘implies’ symbol → to
compute the value for (□q)→q.
7
p
q
T
T
p
q
T
F
□q
(□q)→ q
□q
(□q)→ q