15. Conditional Exchange and Informal Proofs

 

What enhances the formal proof is the ability we have to transform information from less usable to more usable forms. The transformation principles with which we are concerned are DeMorgan's conjunctive and disjunctive laws (a method of losing parentheses), contraposition (a DC based inversion of antecedent and consequent), biconditional exchange (exchanging the triple bar for a conjunction of two sufficient conditions), and tautology (when two disjuncts or two conjuncts are the same constant, then by tautology we infer that constant).

Although we have reached a point of some flexibility in processing information in our proofs, we have been limited by the fact that the rules for conjunctions/disjunctions operate on a separate "turf" from the conditional rules. However, we also have a rule by which we can easily cross over from the one turf to the other. This cross over rule is called conditional exchange (CE).

1A. The cross over rule

Conditional exchange means that any conditional can be transformed into a disjunction by negating the first part. It also works in reverse so that any disjunction can be transformed into a conditional by negating the first part.

The basis for conditional exchange can be "thought through" by means of the "can't have" principle of the sufficient conditional. If p is sufficient for q, then we can't have p and not q. What we can't have unfolds like this:

By means of DeMorgan's conjunctive law, ~(p.~q) becomes the disjunction: ~p v q and we have therefore the equivalence of the conditional with the disjunction by negating the first part. Note that the following are equivalent:

It can be argued that these are not precisely equivalent. Still they do materially imply one another; hence, the relation here is called material implication. Thus, you cannot have p without q is the same as saying either you do not have p or you have q: ~p v q, because if you do have p then you have q. Recall the dating example of Joe who says, "No gal I've ever dated was both beautiful and wealthy." It translates ~(B.W) which by DeMorgan's becomes ~B v ~W and in turn implies that all those dated were either not beautiful or not wealthy. Now note the intuitively performed conditional exchange that occurs when we say: "Poor Joe, if she was beautiful she was not wealthy; if she was wealthy she was not beautiful."

Try the tulip example. If my planting the bulbs is sufficient to make them ready for Spring then it is true that either I have not planted them or they are ready for Spring (On one hand, if I have not planted them then they are or are not ready for Spring depending on my wife's gardening, or I have planted them and they are ready).

Now let's put this into an example argument. Consider the following data that I am given to reach a policy conclusion: From ~F > ~D and F > T can we conclude ~D v T ? I note that the premises are both conditionals but the conclusion is a disjunction:

I also note that the simple statements D and T of the conclusion are in the premises, D is in the first premise and T is in the second. What I want to do first is get D and T onto the same line. But how can I do this? They are conditionals but there is no linkage for a chain. Perhaps linkage can be established if I invert one of the statements by contraposition.

So, which do I contrapose? Not premise two because that will put T in the first position of the statement but I need it in the second position to reach the conclusion. So I will do contraposition of line 1 on line 3:

1. ~F > ~D

2. F > T / .. ~D v T

3. D > F by contra from 1

Now I can get D and T on the same line by the chain inference rule from line 3 to line 2 to get D > T. Then I can make the final move to the conclusion by conditional exchange:

1. ~F > ~D

2. F > T / .. ~D v T

3. D > F by contra from 1

4. D > T by chain 3, 2

5. ~D v T by cond exch from 4

 

2A. Reductio ad absurdum/Informal Proof

The reductio, or indirect proof, is an argument form in which you prove a conclusion to be valid by showing that its opposite is impossible. The conclusion is sought indirectly. The reductio is also called an informal proof because it uses the structure of a formal proof but it does so in an atypical or informal way (as we shall see, it works toward a different goal as the last line of the proof).

1B. The basis

The basis of the reductio is the principle that a deductively valid argument cannot have true premises and a false conclusion. Thus, in a DV argument a denied conclusion will be inconsistent with the premises.

2B. The procedure

The reductio begins with the base set up of a formal proof but it does not proceed along formal lines (hence the title: informal proof). The steps are as follows: 1) negate the conclusion, 2) add the negated conclusion to the list of premises, 3) process the information of the premises as would be done in a formal proof. Include the negated conclusion as a premise in the process, and 4) try to expose a contradiction as the key goal for the last line of the proof. If a contradiction can be exposed through this process, then the original argument is DV.  Consider the following argument as a reductio:

 

1. O > B

2. ~ B

3. O v K /.. K

4. ~K by reductio (negation of the conclusion to begin the proof)

5. O by DSD 4, 3

6. B by AA 5, 1

7. B . ~ B adj 6, 2

Because the contradiction of  B and ~B surfaced, the argument must be DV.

 

 

 

 

 

 

 

 

Informal Proofs ws1

For practice how would the following be done both formally and informally?

A. 1. ~Z v ~P                                     B. 1. ~A > ~D

     2. P v W /.. ~W > ~Z                         2. B > C

                                                                3. ~(~D . C) /.. ~A > ~B

 

 

 

 

 

 

 Informal Proofs ws2

Note a good practice example demanding many steps informally is Churchill, A, 5, p. 289.

1. Translate into symbolic language using the following constants: A, D, O, K, E, B (example: A = God is all- powerful).

If God is all-powerful, then he can do anything. If he is omniscient, then he knows that evil exists. If he knows that evil exists and can do anything, then he would eradicate evil if he was also beneficent. But evil continues unabated in the world. Therefore, either God is not omniscient, or if he can do anything, he is not beneficent. 1. 2. 3. 4. /..

 

 

 

 

 

 

 

 

 

 

2. Prove that the following are DV both formally and informally:

A. 1.   K > T                                       1.   K > T

     2. ~K > G / .. T v G                       2. ~K > G / .. T v G

 

 

 

 

 

 

 

 

B. 1. ~(Q > R)                                             1. ~(Q > R)

2. Q > S                                                      2. Q > S

3. T > R / .. ~(S > T)                                   3. T > R / .. ~(S > T)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. Prove DV either formally or informally:

1. 1. (A > ~B) > Y                                 2.  1. A . B

    2. C > Y                                                 2. ~(A . C)

    3. Y > ~A                                               3. (~C v F) > E /.. E

    4. A / .. B

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.  1. ~P > Q                                             4. 1. ~(B.~C)

     2. ~(O . ~P)                                             2. B . T

     3. ~Q > ~( ~O > ~Q)                              3. (~C > H) > ~G

     4. O > ~L                                                 4. (G > K) > V / ..V

     5. ~P v [ ~Q v (L v R) ]

     6. ~Q v O / .. R

A good place to begin # 3 is to derive p by tautology, then DSD can be done on line 5. This tautology emerges from first doing a dilemma.