Each of a, b, and c gives a structure in one of the two sorts of presentation described in this section—by a diagram or by tables. Present each of them in the other way.
a.
b.
τ
Fτ
0
T
1
T
2
F
τ
Gτ
0
F
1
F
2
T
R
0
1
2
0
T
T
T
1
F
T
F
2
F
T
T
c.
τ
Fτ
0
T
1
T
2
F
τ
Gτ
0
F
1
T
2
T
τ
Hτ
0
T
1
F
2
T
R
0
1
2
0
F
T
F
1
T
F
F
2
F
T
F
2.
Calculate a truth value for each of the following sentences on the structure used as the chief example in this section (see, for example, Figure 6.4.2-7):
a.
(Fa ∨ Gb) → Rab
b.
R(fca)(fac)
c.
fab = fba
3.
Use derivations to check each of the claims below; if a claim of entailment fails, use either tables or a diagram to present a counterexample that lurks in an open gap.
a.
a = a → Fa ⊨ Fa
b.
¬ (Fa ∧ Fb) ⊨ ¬ Fa → ¬ Fb
c.
a = b ∨ b = a ⊨ a = b ∧ b = a
d.
Fa → a = b, ga = b, Ra(ga) → Fa, F(ga) ⊨ Raa → R(ga)(ga)