# Should these two explanations of Conditional Proof be identical to each other?

Brevity motivates abbreviation Conditional Proof as CP, Conditional Statements as CS, and Indirect Proof as IS.

Source: A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

[ p 437, Chapter 7.5 'Conditional Proof' ]
Conditional proof may thus be seen as completing the rules of inference. The method consists of assuming the antecedent of the required conditional statement on one line, deriving the consequent on a subsequent line, and then “discharging” this sequence of lines in a conditional statement that exactly replicates the one that was to be obtained.

[ p 446, Chapter 7.6 'Indirect Proof' ]
This example illustrates how a conditional proof can be used to derive the conclusion of any argument, whether or not the conclusion is a conditional statement. Simply begin by assuming the negation of the conclusion, derive contradictory statements on separate lines, and use these lines to set up a disjunctive syllogism yielding the negation of the assumption as the last line of the conditional sequence. Then, discharge the sequence and use tautology to derive the negation of the assumption outside the sequence.

Because both paragraphs above discuss the same CP, should they cohere with each other 100%? If not, does each logically imply the other? I am confused if the differences cannot be conciliated.

To me, the two quoted paragraphs appear different because
(from p 437) assuming the antecedent of the required conditional statement
differs from (p 446) assuming the negation of the conclusion
(this last bolded phrase says nothing about CS). To wit, an IP's conclusion may NOT be a CS.. In fact, the example referenced in p 446's conclusion is ~A, which is NOT a CS.

I'm sure you already understand the first part of what I am going to say, but for the sake of clarity I shall say it anyway.

In classical logic, a conditional statement, AB, is only false if it is possible for A to be true and B to be false.

The first method of proof described (p.437) works by assuming that A is true and showing that this means that B must be true. We can then conclude that AB.

The second method of proof described (p.446), usually called the contrapositive, works by assuming that B is false, then showing this would mean that A cannot be true. We can then conclude AB.

These two methods of proof are logically/formally equivalent.

Having said all of that, your confusion is the result of the use of the name "conditional statement" in the description of the first method of proof (p.437), while in the description of the second method of proof (p.446) the author draws attention to
♦ the fact that B can be any well-formed statement, conditional or otherwise. ♦
The same (ie: the noun phrase above surrounded by ♦) is true in the first method of proof, although the author does not make this point in his description.

So I would say that you need to keep in mind that in either case, A and B can be any well-formed statements, conditional or otherwise. With this in mind, the logical equivalence of the two methods should be clear.

For example, if A is the conditional statement (CD) and B is the conditional statement (EF), then both methods of proof apply and are equivalent.

• I modified your post marginally and am sorry for any offense. Please feel free to refine. 1. Thank you for locating the source of my confusion so precisely! I fortified the 3rd last paragraph to confirm your correct prediction. 2. I fear that 'the same' may be ambiguous; so I added formatting to clarify its antecedent in the previous sentence. – NNOX Apps Jan 3 '16 at 21:45