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Three Model Answers Involving the “Short” Method of Constructing Interpretations
Philosophy 12A
September 23, 2008
1 Example #1 — Page 66 #3
Answer
.
A
→
(C
∨
E),B
→
D
ø
(A
∨
B)
→
(C
→
(D
∨
E))
Explanation
.
1
Assume that ‘
A
→
(C
∨
E)
’ is true, ‘
B
→
D
’ is true, and ‘
(A
∨
B)
→
(C
→
(D
∨
E))
’ is false. In order for
‘
(A
∨
B)
→
(C
→
(D
∨
E))
’ to be false, both ‘
A
∨
B
’ and ‘
C
’ must be true, and both ‘
D
’ and ‘
E
’ must be false. This
guarantees
that the ﬁrst premise is true (since ‘
A
→
(C
∨
E)
’
must
, at this point, have a true consequent). We can also make the second
premise true, simply by making ‘
B
’ false. So, as the following singlerow truthtable shows, we have
succeeded
in ﬁnding an
interpretation on which ‘
A
→
(C
∨
E)
’ and ‘
B
→
D
’ are both true, but ‘
(A
∨
B)
→
(C
→
(D
∨
E))
’ is false.
A
B
C
D
E
A
→
(C
∨
E)
B
→
D
(A
∨
B)
→
(C
→
(D
∨
E))
>
⊥
>
⊥
⊥
>
>
⊥
Therefore, by the deﬁnition of
±
,
A
→
(C
∨
E),B
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 Spring '08
 FITELSON
 Philosophy

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