Example 111 Here we illustrate “semantic reasoning”, based on the formal semantics of first-order formulae, for proving or disproving first-order logical consequences.. Show the logical
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4 ∃ xP(x)∧ ∃ xQ(x)∃ x(P(x)∧ Q(x )).
and Q(x ) is interpreted as ‘ x is odd’ is a counter-model:
N |= ∃ xP(x)∧ ∃ xQ(x ), but N ∃ x(P(x)∧ Q(x )).
Example 111 Here we illustrate “semantic reasoning”, based on the formal semantics
of first-order formulae, for proving or disproving first-order logical consequences.
1 Show the logical validity of the following argument:
“If Tinkerbell is a Disney fairy and every Disney fairy has blue eyes, then some-one has blue eyes.”
Proof Let us first formalize the argument in first-order logic by introducing predicates:
P(x ) meaning “ x is a Disney fairy”, Q(x ) meaning “ x has blue eyes”, and a
The argument can now be formalized as follows:
1 P(c)∧ ∀ x(P(x)→ Q(x))|= ∃ yQ(y ).
2 S , v |= ( P(c)∧ ∀ x(P(x)→ Q(x )).
We have to show that:
3 S , v |= ∃ yQ(y ).
4 S , v |= P(c ) and
5 S , v |= ∀ x(P(x)→ Q(x )).
Then:
6 S , v |= P(x ).
7 S , v |= P(x)→ Q(x ).
From (6) and (7) it follows that:
8 S , v |= Q(x ).
fol-lows: v y) =v (x) =c S We then have:
9 S , v |= Q(y ).
(3) S , v |= ∃ yQ(y ), so we are done.
2 Prove that the following argument is not logically valid:
“If everything is black or white then everything is black or everything is white.”