T6: 12 of 12

Part IV: Rules of Replacement

More powerful than rules of inference are "rules of replacement". These rules involve pairs of logically equivalent sentence forms.

For example, 'P&Q' and 'Q&P' form such a pair. We will write this as...

P&Q  Q&P    

(This rule is called "CM" for "commutation".)

Our rules of replacement allow us to replace one member of the pair with the other as one step in any derivation. For example...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

That rule again...

... if line 1 (say) of a derivation is '(A>B)&(C&~D)' then line 2 can switch first and second disjunct: '(C&~D)&(A>B)'.

Justification:       Sentence:

1.
2.

We cite the line number of the sentence from which the replacement is made. (For rules of replacement, it is always one line number to be cited.)

OK...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What was that rule once again?

 

 

Now, rules of replacement are even more powerful than the example above suggests...

  1. as the double arrow indicates, replacement can go in either direction. Also,
  2. replacement can work on the sentential components of any line of a derivation. So, line 2 from the above example could be '(A>B)&(~D&C)' where commutation takes place only within the second conjunct:
    Justification:       Sentence:

    1.
    2.
    here the switch has been make around the second ampersand — one that is not the main connective.

Now, look to the other rules of replacement...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here are the rest of the standard "rules of replacement. Notice that the sentences flanking the arrow are logically equivalent.

SD+ Rules of Replacement
CM
(Commutation)
P&Q  Q&P
or
PvQ  QvP
or
P=Q  Q=P

DN
(Double Negation)
  P ~~P

IM
(Implication)
P>Q  ~PvQ

TR
(Transposition)
P>Q  ~Q>~P

ID
(Idempotence)
P&PP
or
PvPP

DM
(De Morgan's)
~(P&Q)~Pv~Q
or
~(PvQ)~P&~Q

AS
(Association)
P&(Q&R)(P&Q)&R
or
Pv(QvR)(PvQ)vR

DI
(Distribution)
P&(QvR)(P&Q)v(P&R)
or
Pv(Q&R)(PvQ)&(PvR)

EQ
(Equivalence)
P=Q(P&Q)v(~P&~Q)
or
P=Q(P>Q)&(Q>P)

EX
(Exportation)
  P>(Q>R)(P&Q)>R

Enough for now? Try some exercises on these rules of replacement! Or look at all the exercises for this tutorial.