T5.6 4 of 6
OK, then, our derivation can be done quite simply if we can only do the right pattern matching:
If you think about HS as a way to prove your ultimate goal 'F>H'. This makes you think about a preliminary goal (here 'G>H') that will suggest IM. Seeing the problem in this way, doing the goal analysis to the point of having preliminary goals based on seeing the right patterns, takes time. But with a little practice, it saves lots of time.
However, if you don't see the patterns fairly quickly, just assume the opposite of your ultimate goal and go
through the longer proof by RD:
This may seem like too much to do at first; but a little practice makes a huge difference.
Either way, keep our four point strategy in mind:
Let's see how this applies to showing an SL sentence is logically true in SD.
Logical Truth in SD
What is it for a sentence P of SL to be logically true?
P couldn't be false.
P couldn't be true.
None of the above.
Right: A logical truth could not possibly be false. It is true no matter what. For example, it's true no matter what that it's raining or it's not. So, 'Rv~R' is a logical truth. We don't need any evidence to prove it.
This suggests that to show a sentence P logically true we won't need any premises.
Another way to look at this, is that if we have a logical truth P that must be true, it's negation must be false: a self-contradiction. So, if P is logically true, it will lead to contradiction in a derivation.
We can use both these ideas together
to get our test for logical truth.
To show a sentence like 'Rv~R' logically true, we assume it -- without taking any premises and derive a contradiction. If we can get to the contradiction , this will mean that we've shown 'Rv~R' must be true no matter what...no premises/evidence required: we will cite RD.
Let's see...
So, from the assumption of the negation of our goal, 'Rv~R' we can indeed dervie a contradiction....
We have shown that '~(Rv~R)' could not possibly be true (it's a self-contradiction) and we can apply RD to finish:
General Test:
A derivation shows P is logically true if the derivation has no premises but assumes ~P , subderives a contradiction, and so derives P by RD.