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CS245  Winter 2010, Lecture 11
Shai BenDavid
Formal proofs are part of the syntax aspect of propositional logic. Formal
provability is deﬁned by a strict rules concerning the characters in a proposition
without addressing their meaning (the truth values).
There is yet another important syntactic notion  consistency.
Deﬁnition
A set of propositions, Σ is
consistent
if, there exists no proposition
α
for which both Σ
‘
α
and
σ
‘
(
¬
α
). A set is
inconsistent
if it is not consistent.
Intuitively speaking, a set of propositions is consistent if it domes no con
tradiction can be derived from it.
Examples:
The following sets are inconsistent.
1. Σ
1
=
{
p,
(
¬
p
)
}
.
2. Σ
2
=
{
a,
(
a
→
b
)
,
(
¬
b
)
}
.
I leave it to you to prove that these sets are inconsistent by showing that
Σ
1
‘
p
, Σ
1
‘
(
¬
p
)
}
, Σ
2
‘
b
and Σ

2
‘
(
¬
b
).
Can we come up with examples of consistent sets? This seems to be a bit
more diﬃcult, since, showing that a set Σ is consistent amounts to showing
that, for
every
proposition
α
if Σ
‘
α
then no sequence of propositions consists
a formal proof of (
¬
α
) from Σ. Recalling the monotonicity of propositional logic
(Namely, that Σ
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 Spring '10
 Mr.Lushman

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