Homework 2
PHIL 151
Due Wed. Jan 19th in class at 10 am
1. (2 points) Problem 1b on page 20.
2. (8 points)Prove or find a counterexample to
(a) If Γ

=
ϕ
∧
ψ
then Γ

=
ϕ
and Γ

=
ψ
(b) If Γ

=
ϕ
∨
ψ
then Γ

=
ϕ
or Γ

=
ψ
where Γ is a set of propositional formulas (or in other words a subset of
PROP
) and
ϕ
and
ψ
are individual propositional formulas (elements
of
PROP
).
3. Suppose that
ϕ
1
, ..., ϕ
k
and
ψ
are in
PROP
.
(a) (5 points) Show that [

ϕ
1
∧
...
∧
ϕ
k

]
v
= 1 if and only if [

ϕ
i

]
v
= 1 for
each
i
≤
k
, where
ϕ
1
∧
...
∧
ϕ
k
abbreviates (
..
(
ϕ
1
∧
ϕ
2
)
∧
ϕ
3
)
...
)
∧
ϕ
k
).
(What has to be shown, specifically, is that the bicondtional holds
for
any
natural number
k
.)
(b) (5 points) Using 3(a), prove that
{
ϕ
1
, ..., ϕ
k
} 
=
ψ
if and only if

= (
ϕ
1
∧
...
∧
ϕ
k
)
→
ψ
.
4. Define the
simultaneous substitution
of
ϕ
1
, ..., ϕ
k
for propositional sym
bols
q
1
, ..., q
k
in
ψ
recursively as follows
•
For atomic
ψ
,
ψ
[
ϕ
1
, ..., ϕ
k
/q
1
, ..., q
k
] =
ϕ
i
if
ψ
=
q
i
,
and
ψ
[
ϕ
1
, ..., ϕ
k
/q
1
, ..., q
k
] =
ψ
otherwise.
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 Winter '09
 Proposition, propositional symbols

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