2
⇒
α
1
,
Γ
3
α
2
,
Γ
1
,
Γ
2
⇒
Γ
3
Γ
1
, (
α
1
→
α
2
),
Γ
2
⇒
Γ
3
Left-
→
α
1
,
Γ
1
⇒
α
2
,
Γ
2
,
Γ
3
Γ
1
⇒
Γ
2
, (
α
1
→
α
2
),
Γ
3
Right-
→
Γ
1
,
Γ
2
⇒
α
,
Γ
3
Γ
1
,
¬α
,
Γ
2
⇒
Γ
3
Left-
¬
α
,
Γ
1
⇒
Γ
2
,
Γ
3
Γ
1
⇒
Γ
2
,
¬α
,
Γ
3
Right-
¬
Γ
1
, (
α
1
→
α
2
), (
α
2
→
α
1
),
Γ
2
⇒
Γ
3
Γ
1
, (
α
1
↔
α
2
),
Γ
2
⇒
Γ
3
Left-
↔
Γ
1
⇒
Γ
2
, (
α
1
→
α
2
),
Γ
3
Γ
1
⇒
Γ
2
, (
α
2
→
α
1
),
Γ
3
Γ
1
⇒
Γ
2
, (
α
1
↔
α
2
),
Γ
3
Right-
↔

Propseq1.doc:1998/04/14:page 9 of 31
At this point, it might appear that we are no better
off with G
′
than we are with the Hilbert system.
We
have an infinite number of axioms, and so a infinite
number of proofs.
Q:
To begin a proof, must we “divine” the correct
set of starting axioms?
A:
No!
With G
′
, we will construct the proof
backwards.
That is, we will start with the statement
which we wish to prove, and work backwards
towards the axioms.
There is an algorithm to
construct such proofs.
Before developing formally notions of soundness
and completeness, we will work out a variety of
examples.

Propseq1.doc:1998/04/14:page 10 of 31
Examples:
First of all, we will provide a proof, within G
′
, that
the wff
((A
1
→
A
2
)
→
(
¬
A
2
→
¬
A
1
))
is a tautology.
As with resolution and semantic tableaux, the proof
is best represented as a directed graph.
A solution
is shown below on the next slide, in both notations.
Notice the following:
•
In constructing this proof, we work backward from
the conclusion, to reach valid atomic sequents.
•
The formula
ϕ
to be validated is represented as
the sequent
⇒
ϕ
.
•
As we work backwards, each step up the tree
produces a “simpler” formula.
(This means that
the tree cannot grow forever.)
•
A path ends whenever an atomic sequent is
encountered.
Such a sequent cannot be
reduced further.
•
The original formula is valid iff each path ends
with a valid atomic sequent.

Propseq1.doc:1998/04/14:page 11 of 31
»…
»
((A
1
→
A
2
)
→
(
¬
A
2
→
¬
A
1
))
…
»…
»
A
1
, (
¬
A
2
→
¬
A
1
)
…
»
¬
A
2
…
»
¬
A
1
, A
1
…
»
(A
1
→
A
2
)
…
»
(
¬
A
2
→
¬
A
1
)
…
»
A
2
…
»
(
¬
A
2
→
¬
A
1
)
…
»
¬
A
2
, A
2
…
»
¬
A
1
…
Right-
→
Right-
→
Left-
→
Right-
→
Valid
Valid
»
A
2
…
⇒
»
A
2
,
¬
A
1
…
»
A
1
,
¬
A
2
…
»
A
1
…
»
A
1
…
»
A
1
,
¬
A
2
…
Right-
¬
Left-
¬
Left-
¬
»
A
1
, A
2
…
⇒
»
A
2
…
⇒
((A
1
→
A
2
)
→
(
¬
A
2
→
¬
A
1
))
⇒
A
1
, (
¬
A
2
→
¬
A
1
)
¬
A
2
⇒
¬
A
1
, A
1
(A
1
→
A
2
)
⇒
(
¬
A
2
→
¬
A
1
)
A
2
⇒
(
¬
A
2
→
¬
A
1
)
¬
A
2
, A
2
⇒
¬
A
1
Right-
→
Right-
→
Left-
→
Right-
→
Valid
Valid
A
2
⇒
A
2
,
¬
A
1
Right-
¬
A
1
,
¬
A
2
⇒
A
1
A
1
, A
2
⇒
A
2
Left-
¬
Right-
¬
A
1
⇒
A
2
, A
1

Propseq1.doc:1998/04/14:page 12 of 31
•
In establishing validity it is not necessary to
expand a node until it becomes an atomic
sequent.
For validity, it suffices that there be a
common proposition name (as a wff; not as a
component of one) in each sequence.
Thus, the
expansions in the previous example could
actually have been halted without expanding the
boxes which are outlined with dashes.
•
It is in fact possible to halt the expansion of a
node when there is a common
wff
in each
sequence.
However, testing for such common
formulas at each step involves substantial
overhead, and so the decision to do this should
be weighed carefully.
In these notes, only
checking for matching proposition names will be
performed.

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