The purpose of a derivation is to show that a conclusion follows deductively from a set of premises. One can do this successfully, if, and only if, the conclusion does in fact follow from the premises. In other words, if the argument forms used are invalid, then one cannot get from the premises to the conclusion.
Derivations, for one who knows what s/he is doing, can be exciting challenges. But, for the beginner, they can be exasperating enough without the frustrations posed by invalid arguments. So, for now, we will work only with valid argument forms.
Another feature of derivations that often perplexes the would-be deriver is the freedom one has in getting from premises to conclusions. Just as there is usually more than one way to get where you are going in Charlotte, there is more than one way to get from a set of premises to the desired conclusion. Sometimes either route is just as good as the other. Of course, where possible, we prefer the shorter route. But in derivations what may appear quicker may not actually be, for we have a tendency to make some of the more obvious moves in our heads and not on paper. I will insist, for now, that each step in a derivation be explicitly shown. There can be no missed steps. So use commutation and double negation in order to get an expression into a form that matches exactly the rule being used.
Since one cannot proceed mechanically from what is known (the premises) to what is alleged to be true (the conclusion), one must discover, often through trial and error, how to get from the premises to the conclusion. Here are some hints:
1. Work backward from the conclusion. In your head, or even better, on a worksheet, ask what one needs in order to get to this conclusion. Ask yourself what inference rule would be required to produce the conclusion. Then examine the premises to see if some of what is needed is present in them. For instance, if the conclusion is a ~A and you have a A É B premise, then all you would need is a ~B to get a ~A by Modus Tollens. Of course, matters will usually not be this simple. More than likely you will have something such as this:
1. A É B
2. B É C
3. D Ú ~C
4. ~D /\ ~A
Working back from the conclusion (~A) you could look for a Modus Tollens. In this case, you could get it by noticing that lines 1 and 2 would give you, by a Hypothetical Syllogism, the proposition, A É C. But how could you get a ~C? This suggests a second tactic:
2. Pick out some recognizable argument form, such as Disjunctive Syllogism. In the example above you could use lines 3 and 4 and DS to get a ~C. The ~C could then be used with the A É C to deduce a ~A by Modus Tollens. Voila! Here is what the complete derivation would look like:
1. A É B
2. B É C
3. D Ú ~C
4. ~D /\ ~A
5. A É C 1,2 HS
6. ~C 3,4 DS
7. ~A 5,6 MT
3. Look for interchanges. If the premises do not seem very promising, perform an interchange to get something useful. Consider the following argument:
1. K º L
2. ~K
3. L Ú M
4. M É N /\ N
Biconditionals are good candidates for interchanges. Performing one on line 1 gives you two conditionals, one of which can be detached by means of Commutation and Simplification, yielding L É K. Then, by means of Modus Tollens, one could get a ~L. This ~L could be used with line 3 and Disjunctive Syllogism to get an M. Then, using the M and line 4, by means of Modus Ponens, one could get the desired conclusion. Here is the displayed derivation:
1. K º L
2. ~K
3. L Ú M
4. M É N /\ N
5. (K É L) & (L É K) 1 Bicond
6. (L É K) & (K É L) 5 Comm
7. L É K 6 Simpl
8. ~L 7,2 MT
9. M 3,8 DS
10. N 4,9 MP
Step 6, which involved Commutation, is preferable in order to put the conjunction in the right form for the Simplification move in step 7. Notice also that we list the lines in the order in which they would appear in a displayed argument. Thus step 8 (7,2 MT) is actually telling us that the following argument is being made:
Modus Tollens
7. L É K
2. ~K
8. ~LAnyone looking at your derivation should be able to make explicit each of the steps. In this way, s/he can follow and evaluate your reasoning process.
Exercise 13 provides practice opportunities.
Copyright © 1999, Michael Eldridge