CONDITIONAL PROOFS



 
 
 
 
 
 

1, What is a "conditional proof" or "CP"?

2. What is the "right attitude" to have towards CP's?

3. When do I use the CP tool?

4. What is the relationship between the CP subproof and the whole proof of which it is a part?

5. How do I set up a CP?

6. How do I do the CP?

7. How do I know when to end the CP?

8. How do I get back into the main proof once the CP is over?


 
1, What is a "conditional proof" or "CP"?
 
CP's can be regarded as mini subproofs within the context of a longer proof; they can be thought of as subroutines needed to deduce some statement which is then used by the full proof to derive the required conclusion. CP can also be regarded as a new rule to add to our toolchest to round out our total to an even twenty. The rule that allows you to construct a CP is a tool used to deduce statements needed by the whole proof.


2. What is the "right attitude" to have towards CP's?
 

Think of CP's as your "friends," and NOT as yet another damn thing to memorize. To extend our "tool" analogy, they are like power tools. CP's make it easier to construct proofs; they make life happier. Welcome them.


3. When do I use the CP tool?
 

As said above, CP's are tools used to derive statements needed by the whole proof. What sorts of statements? Answer: Statements which are either implications themselves or can be transformed easily into implications. You can use a CP WHENEVER you need an implication. This resolves into three possible cases:
a) The required conclusion of the whole proof is itself an implication.
b) The required conclusion can be easily transformed into
    i) an implication (for example, a disjunction) or
    ii) a conjunction of implications (for example an equivalence statement or "biconditional")
c) The required conclusion is not a) or b) above, but an implication is needed to derive the required conclusion.
4. What is the relationship between the CP subproof and the whole proof of which it is a part?
 
The CP is doing a very specific job: it is always deriving a statement which is an implication. This cannot be overemphasized: every CP ends up with an implication. In case a) above where the implication is itself the required conclusion of the whole proof, the CP in effect provides the "guts" of the proof, it does all the work, and the whole proof in which it is set is nothing more than setting up and terminating the CP. However, in cases b) and c) above, the CP does only part of the work and then the implication which is produced by the CP must be used by the whole proof to deduce the required conclusion. In the case where the required conclusion is a conjunction of implications (b ii), use two separate CP's each to derive one of the required conjuncts in the conjunction. You may use as many CP's in a whole proof as are needed to deduce as many implications as are needed, as well as CP's inside of other CP's if you need an implication in a CP.

 

5. How do I set up a CP?
 

First you have to know what implication you are trying to derive, as in cases a), b), or c) above. The CP subproof ALWAYS begins with a line in which you "assume" the antecedent of the desired implication. Write the whole antecedent (everything to the left of the (horseshoe)) of the desired implication as your step. Now to the right hand side (where you usually cite the rule and step numbers), write a new " / " (the slash and triple dot conclusion indicator), and then write the whole consequent of the desired implication. This line now puts you "inside" the CP subroutine, which is indicated by drawing an arrow in the margin next to the step number, bending it a right angle downward, and extending that line downward to the last step of the CP subroutine. The effect of this arrow is to define a "box" which indicates you are "inside" the CP. The CP is not finished until that arrow line cuts off or terminates the CP.

6. How do I do the CP?

Proceed just as you would in any other proof; use all 19 rules and any steps above the step where you are at that point in the CP. In effect what you have done is now added to the "original" premises of the argument the new premise which consists of the antecedent of the desired implication assumed at the opening line of the CP. But now your conclusion has changed! You must now derive not the original conclusion of the argument, but the new CP conclusion, which is ALWAYS the consequent of the desired implication, that which is indicated to the right of the conclusion indicator in the opening line of the CP.


7. How do I know when to end the CP?
 

Like any other proof, the CP is over when you have derived the conclusion of the CP, i.e., the consequent of the desired implication. Once you have done this, "close off" the box that defines the CP by extending the arrow line in the margin past this step number, and then bending it as right angles to the right underneath this last line.


8. How do I get back into the main proof once the CP is over?
 

You do so by "discharging" the assumption you made at the opening line of the CP, the line in which you assumed the antecedent of the desired implication and indicated the new conclusion of the consequent of the desired implication. You show this by the very next step after you have derived that consequent and closed off the box. You write simply the desired implication, which will ALWAYS consist of the first line of the CP (where you assumed the antecedent of the desired implication), then the horseshoe sign, then the consequent of the desired implication, which you derived in the last line of the CP. As your justification for this step you cite the rule "CP" preceded by the numbers of the opening steps through ("-") the final step of the CP.