Question

Construct expansions in a two-individual universe of discourse for the following sentence:

(x) [Fx --> (Gx v Hx)]

Answer 1

For the interpretation:
X = {1, 2, 3, 4, 5}
Ref(m) = 2 Ref(n) = 4
Ext(F x) = {1, 2, 3} Ext(Gx) = {4} Ext(Hx) = {5}
1. (x)(−F x ∨ −Gx): This sentence is true relative to this interpretation,
since the elements 1, 2, 3, and 5 are not in Ext(Gx) and element 4 is
not in Ext(F x) (thus, for all x, x is either not in Ext(F x) or not in
Ext(Gx)).
2. (x)((Gx&Hx) → Hx): This sentence is true relative to this interpre-
tation, since there is no element in the domain that is in both Ext(F x)
and Ext(Gx), making the antecedent of the conditional false. Thus,
the entire conditional is true.
3. −(x)F x → (∃x)(F x&Gx): This sentence is false relative to this inter-
pretation. The sentence −(x)F x is true relative to this interpretation,
since the elements 4 and 5 are not members of Ext(F x), and thus not
every element of the domain is a member of Ext(F x). The consequent
of the conditional is false relative to this interpretation, since there are
no elements that are members of both Ext(F x) and Ext(Gx). Finally,
since the antecedent of the conditional is true and the consequent false,
the conditional is false.
4. −Gm&(∃x)(F x&Gn): This sentence is true relative to this interpreta-
tion. The first conjunct is true relative to this interpretation because
the element Ref(m), 2, is not in Ext(Gx). The second conjunct is also
true, because the element Ref(n), 4, is in Ext(Gx) and the elements 1,
2, and 3 are all in Ext(F x).
5. (∃x)(−F x&−Gx): This sentence is true relative to this interpretation,
since the element 5 is not in Ext(F x) or in Ext(Gx).
6. (∃x)(F x → Gx): This sentence is true relative to this interpretation,
since the elements 4 and 5 are not in Ext(F x), making the antecedent
of the conditional false, and the conditional as a whole true.