Chapter 5, Tutorial 3
Rules of Replacement: DM, DN, IM, EQ
All the rules described in this chapter are old friends. Well, at least we've seen them all before though in different guise: as principles of logically equivalent symbolization. If you've not done it recently, now would be a good time to go back to exercise 4.5e to review these principles.
One way to symbolize "neither P nor Q" is ~(PvQ). But "neither are true" can just as well be symbolized as ~P&~Q. We have two equivalent ways of saying the same thing.
"Not both P and Q" is ~(P&Q). But it's logically equivalent to say at least one is not true: ~Pv~Q.
Tables prove these are logically equivalent:
P  Q  ~  (P  &  Q)  ,  [~  P  v  ~  Q]  
T  T  
T  F  
F  T  
F  F 
We called this De Morgan's principle back in 4.5e. Now we'll use it as a derivation rule.
But we'll state this a little differently:




The double arrow means that the two flanking expressions are logically equivalent forms.
We'll call these doublearrow rules "rules of replacement". We'll see that they are more powerful than our 5 older rules of inference.
You may use either of these two forms. This is really two rules in one. Looking at the first form, the first doublearrow: whenever you have something of the form ~Pv~Q on a line of a derivation, you can just replace this by its equivalent: ~(P&Q).
For example:
Premise  1  ~Av~B  
1 DM  2  ~(A&B) 
Look at the citation: 1 DM. This means that line 2 comes from line 1 by replacement in accordance with the rule DM.
Now, how could this be useful?
Suppose you have two premises and you need to see what follows from them:
J>(L&S)
~Lv~S
???
Then you can derive a conclusion by replacing premise two with it's equivalent form by DM. This helps because
then MT applies:
Premise  1  J>(L&S)  
Premise  2  ~Lv~S  
2 DM  3  ~(L&S)  
1,3 MT  4  ~J 
It's hard to see how to do this in advance. But that's why we'll practice!
Now, let's see how you can use this rule again but in the other direction. If you're derivation starts out like so...
Premise  1  ~(K&L)  
???  2  ??? 
...then what can you do next? What will replace the question marks?