1. Victoria King Logic Summaries Dr. Lovern
1
7.5: Conditional Proof, p. 391
A conditional proof is a method for obtaining a line in a proof sequence (either the conclusion or some
intermediate line).
- Why it’s useful:
o Shorter and simpler than direct method
o Some arguments have conclusions that do not work with direct method, so
conditional proof must be used on them.
 Lines 3-7 are indented to show they are hypothetical; that is, they depend upon the assumption
given in Line 3 via ACP (Assumption for Conditional Proof).
o The conditional sequence ends at line 8 upon reaching the conclusion. At that point,
because the conditional sequence has been discharged, Line 8 is not indented.
Rules of Constructing Conditional Proofs:
1. Decide what should be initially assumed. (The antecedent is assumed first; tag it ACP.)
2. Obtain consequent of desired conditional statement at the end of the conditional sequence.
3. Discharge conditional sequence in a conditional statement (labeled CP).
Conditional proof can also be used to obtain a line other
than the conclusion of an argument:
* (Extra rule): After a conditional proof sequence has
been discharged, no line in the sequence may be used
as a justification for a subsequent line in the proof.
Here’s why:
- Because no mere assumption can provide any
genuine support for anything, neither can any line that
depends on such an assumption. When a conditional
statement is discharged, the assumption on which it
rests is expressed as the antecedent of a conditional
statement. The conditional statement can be used to
support following lines since its value is assumed true.

2. Victoria King Logic Summaries Dr. Lovern
2
However…
One conditional sequence can be used within
the scope of another to obtain a desired
statement.
Right:
 No line in 5-8 can be used to support
any line subsequent to 9.
 No line in 3-10 can be used to support
any line subsequent to line 11.
 Lines 3 or 4 could be used to support
an line in sequence 5-8.
7.6: Indirect Proof (p. 397)
Indirect proof is similar to conditional proof and can be used on any argument to derive either the
conclusion or some intermediate line leading to the conclusion. It consists of assuming the negation of
the statement to be obtained, using this assumption to derive a contradiction, and then concluding that
the original assumption is false. This last step establishes the truth of the statement to be obtained.
Above: Example of method
 First, assume negation of conclusion (A).
 If contradiction is found (D * ~D), then discharge the indirect sequence (line 11) by asserting the
negation of the original assumption.

3. Victoria King Logic Summaries Dr. Lovern
3
Indirect proof can also be used to derive an
intermediate line leading to the conclusion.
Example:
The indirect proof sequence begins with the
assumption of F on line 8, ends with a
contradiction of (J * ~J) on line 14, and is
discharged as ~F on line 15 (by asserting the
negation of the assumption).
As in conditional proofs, no line in the indirect
proof sequence is to be used to justify the
remainder of the argument after it has been
discharged. 8-14 cannot be used again after 15 or
any subsequent lines.
In the argument on the right, lines 6 and 7 could
have been included in the indirect proof
sequence; however, it was more efficient not to
include them in it, because line 7 was used again
in line 16, after the indirect proof sequence had
been discharged. Had they have been used in
the indirect proof sequence, it would have been
necessary to repeat them subsequently.
A conditional sequence may be constructed within
the scope of an indirect sequence, and, conversely,
an indirect sequence may be constructed within the
scope of either a conditional sequence or another
indirect sequence.
The photo to the left provides an example of an
indirect sequence within the scope of a conditional
sequence.

4. Victoria King Logic Summaries Dr. Lovern
4
Indirect proof provides a convenient way for proving the validity of an argument having a tautology for its
conclusion. In fact, the only way in which the conclusion of many such arguments can be derived is
through either conditional or indirect proof.
The example on the left shows an instance of when indirect proof is easier to use.
There’s example on the right shows an instance of when conditional proof is the easier of the two, being
that the conclusion is a conditional statement.
Indirect proof can be viewed as a variety of conditional proof in that it amounts to a modification of the
way in which the indented sequence is discharged, resulting in an overall shortening of the proof for many
arguments. The next two photos give an example of this by providing a comparison:
Above (right): Indirect proof is used here. In this
instance, it is more efficient to use this method.
Above (left): Conditional proof is used here.

5. Victoria King Logic Summaries Dr. Lovern
5
7.7: Proving Logical Truths, p. 402
Both conditional and indirect proof can be used to establish the truth of a logical truth (tautology).
Tautological statements can be treated as if they were the conclusions of arguments having no premises.
This is suggested by the fact that any argument ending in a tautology is treated as valid regardless of its
premises. The proof for such an argument derives the conclusion as the exclusive consequence of either
a conditional or indirect sequence.
Using this strategy for logical truths, we write the statement to be proved as if it were the conclusion of an
argument, and we indent the first line in the proof and tag it as being the beginning of either a conditional
or an indirect sequence. In the end, this sequence is discharged to yield the desired statement form.
Tautologies expressed in conditional form are
most easily proved via a conditional sequence.
This example expresses two such sequences,
one within the scope of the other:
(Line 6 restores the proof to the original margin.
The first line is indented since it introduces the
conditional sequence.)
The same argument is shown on p. 403 using
an indirect proof (Hurley, 10th ed., A Concise
Introduction to Logic). It takes 11 lines to
complete rather than 6. This is why conditional sequences are better used for conditional forms rather
than indirect sequence on conditional forms.
Complex conditional statements are proved by extending the technique used in the first proof. The
following example shows that each conditional sequence begins by asserting the antecedent of the
conditional statement to be obtained:

6. Victoria King Logic Summaries Dr. Lovern
6
Tautologies expressed as equivalences are
usually proved using two conditional
sequences, one after the other. Example (from
p. 404):
(Personal side note: What was done on line 10
is brilliant efficient…{line 6} * {line 9})
Chapter 7 Summary:
 Natural Deduction is used to establish
the validity of arguments symbolized in
terms of the 5 logical operators. The
method consists in applying 1+ rules of
inference to the premises and deriving
the conclusion as the last line in a
sequence of lines.
 The 8 rules of implication are “one-way” rules since the premises of each rule can be used to
derive the conclusion, but the conclusion cannot be used to derive the premises.
 The 10 rules of replacement are “two-way” rules and are expressed as logical equivalences, and
the axiom of replacement states that these rules is used to derive a conclusion, the process can
be reversed, and the rule can be used to derive the premise. This allows us to deconstruct a
conclusion in order to find the best way of deriving it in a proof sequence.
 Conditional proof is used for deriving a line in a proof sequence—either the conclusion or another
line. It consists in assuming a certain statement as the first line in a conditional sequence, using
that line to derive one or more additional lines, and then discharging the conditional sequence in
a conditional statement having the first line of that sequence as antecedent and the last line as
consequent. This procedure expresses the notion that if the antecedent of a true conditional
statement is true, then the consequent is also true.
 Indirect proof is another method used to derive a line in a proof sequence. It consists in assuming
the negation of the line to be obtained as the first line in an indirect sequence, using that line to
derive a contradiction, and then discharging the indirect sequence in a statement that is the
negation of the first line in that sequence. This technique conveys the notion that any assumption
that necessarily leads to a contradiction is false.
 Natural deduction can also be used to prove logical truths (tautologies). The technique employs
conditional proof and indirect proof. Conditional proof is often used for logical truths expressed as
conditionals. The antecedent of the logical truth is assumed as the first line of a conditional
sequence, the consequent is derived, and the sequence is then discharged in statement
expresses the logical truth to be proved. Indirect proof can be used for proving any logical truth.
The negation of the logical truth is assumed as the first line of an indirect sequence, a
contradiction is derived, and the sequence is then discharged in a statement consisting of the
negation of the first line in that sequence.