T5.6 4 of 9

So, we finish the subderivation by citing the entire subderivation, 4-6.

Premise 1. [A=(B=L)]&[C=(X&(B>L))]  
Premise 2.  A  
1 &E 3.  A=(B=L)  
Assumption 4. what if ........................  B
2,3 =E 5. then ...........................  B=L
4,5 =E 6. then ...........................  L
4-6 >I 7. B>L  

We are not allowed to cite merely 4 and/or 6 because that would be citing from within a terminated subderivation.


Strategy (in life or derivation) is reasoning toward a goal. When doing complex derivations, haphazard application of the rules may provide no useful results. Instead, one needs to not only apply rules with an eye toward how a derivations should end. We have just seen how this works in the subderivation dissected at length in the last few pages.

Now, think about a new example. From the premises given below, you are asked to derive '~A'.

Premise 1. A>(B&C)
Premise 2. ~B&~X

This example is really pretty simple. But it still illustrates the stratagems you will apply to harder problems.

So, what should we do at line 3 and beyond? Just as in the first example above, we need to think about what we are asked to derive. That's our (ultimate) goal – in this problem it's '~A'. And as with our previous example (the one at the top of this page), we should think about the main connective of this goal sentence to see how it should be derived. Because the main connective is tilde, we may well finish the derivation by an application of ~I.

This sort of stategizing is a way of working backwards. We think about how we will attain our goal in the end and then see how to proceed in that direction.

So, you may begin thinking about this derivation by seeing how it will end! For instance, you may want to write or type the following in:

Premise 1. A>(B&C)  
Premise 2. ~B&~X  
Assumption 3. what if ...................  A
??? 4. then ......................  
??? 5. then ......................  
??? 6. then ...................... New Goal???
3-6 ~I??? 7. ~A???  

We've just guessed that this will be 7 lines long. (When you are doing your own problems, just guess on fairly large number of lines so you leave plenty of space. Extra, blank lines will not be a problem.)

So, what have we done at line 7? The idea here is that we would like to finish the derivation by using rule ~I. So, we write this all down and add the question marks so that we remember that this is just our goal — we have not yet derived it! Instead, we are just setting up the format for the derivation we will now start.

Negation Introduction, the rule we will apply when we do get down to line 7 requires a subderivation assuming the opposite of our goal sentence. That's line 3:'A' .

Question: What sentences do we need at line 6? what's our new goal?

  1. 'B&C' because that is the consequent we need to derive.
  2. '~X' because we are doing negation introduction.
  3. Any contradiction of a previous sentence in the subderivation. This will show our assumption of 'A' could not be true.