T5.7 6 of 9
Rules of Replacement
It is fairly obvious by now that 'LvO' and 'OvL' mean the same thing, i.e., they are logically equivalent. It should be no surprise then, that the following argument is valid:
(LvO)&Y
(OvL)&Y
But showing that this is valid in SD is rather cumbersome as is relies on vE which in turn requires two applications of >I for the case at hand:
| Premise | 1 | (LvO)&Y | |
| 1 &E | 2 | LvO | |
| Assumption | 3 | what if... | L |
| 3 vI | 4 | then...... | OvL |
| 3-4 >I | 5 | L>(OvL) | |
| Assumption | 6 | what if... | O |
| 6 vI | 7 | then...... | OvL |
| 6-7 >I | 8 | O>(OvL) | |
| 2,5,8 vE | 9 | OvL | |
| 1 &E | 10 | Y | |
| 9,10 &I | 11 | (OvL)&Y |
This is a pain to do when the logical equivalence of 'LvO' and 'OvL' makes the point so obvious. So, we will hereafter be able to utilize a new sort of short-cut rule: a rule of replacement which simply allows up to replace 'LvO' with 'OvL' in one step beyond the premise (rather than ten!).
We will call this rule "commutation" or "CM" because it switches (commutes) the order. Then, in SD+, the following will be allowed:
| Premise | 1 | (LvO)&Y |
| 1CM | 2 | (OvL)&Y |
So we are finished with this problem without all the fuss and suddenly our derivations will become more elegant, efficient and understandable.
Commutation of disuncts may be stated as follows.
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The idea is that one can replace any disjunction with a new disjunction having the disjuncts in opposite order.
The double arrow indicates that one may replace in either direction. This will prove a significant change from rules of inference which are one directional, from input to output.
The most important difference between rules of inference and replacement is illustrated by our one application above:
| Premise | 1 | (LvO)&Y |
| 1CM | 2 | (OvL)&Y |
Notice that the wedge is not the main connective of either line. Although, rules of inference work only with sentences that have appropriate main connectives, rules of replacement work on any component having the appropriate connective.
In other words, one is allowed to commute around any wedge. In our example, we use CM to replace one disjunction with another even though wedge is not the main connective.
The point, of course, is that order of disjuncts doesn't matter to meaning. Thus commuting of disjuncts allows a valid inference.
Now, we know that we can validly commute about a wedge. What other connectives allow for valid commutation?