Logic Summaries

Victoria King Logic Summaries Dr. Lovern

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 methodo 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

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antecedent of a conditional statement. The conditional statement can be used to support following lines since its value is assumed true.

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.

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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.

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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:

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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.

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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:

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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

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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.

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