Chapter Five, Tutorial Three
Another derived rule: Conditional Proof

We can see conditional proof, or >I, as simply a deried rule. But it may be better to revisit the intuitions behind subderivartions anew. So, in order to think about this tutorial's subject matter, recall the informal proofs of chapter one...

If an argument is sound, then what can we deduce about its premises and conclusion? A few things may come to mind. Here's about the simplest thinking:

We know from the definition of soundness that if (1) an argument is sound, then (2) it's valid.
Also, by definition of a valid argument, if (2) it's valid, then (3) it does not have both true premises and false conclusion.
So, it's easy to jump to the conclusion that if (1) an argument is sound, then (3) it doesn't have both true premises and false conclusion.

All this is correct and I hope it seems somewhat natural to you. But how might we justify it to someone? Here's a common way we do it in English, a way relying on "what-if?" thinking.

Suppose an argument is sound. Then by the first definition cited above, that argument is valid. But by the definition of validity, that argument does not have true premises and false conclusion. So, we have just shown that if an argument is sound, then it does not have true premises and a false conclusion.

What-if thinking? Yes, the above passage starts by making an assumption: "suppose an argument is sound". This is a way of asking "what if?". We see what results if an argument, any arbitrary argument, is sound. (This should be familiar from exercises in chapter one. You may want to review this material.)

Anyway, the standard means to prove a conditional is to make the additional "what if" assumption. Now lets try to formalize this thinking within SD. Use the following interpretation...

S: The (arbitrary) argument is sound.
V: The argument is valid.
T: The argument has all true premises.
F: The argument has a false conclusion.

... to mimic the English thinking above:

Premise (from the definition of "sound"): 1. S>V  
Premise (from the definition of "valid"): 2. V>~(T&F)  
Assumption (we ask "what if" the argument is sound): 3. what if..... S
1,3 >E 4. then........ V
2,4 >E 5. then........ ~(T&F)
Conclusion from our assumption 3. and the argument through 5 6. S>~(T&F)  

We move sentences 3-5 (in blue) over to the right in order to emphasize that they are the "what if" thinking.

It is especially important to keep our extra "assumption" apart from the premises. After all, we know the premises here are true. They come immediately from the definitions of "valid" and of "sound". But our additional assumption, 'S', meaning that the argument is sound, is just a hypothesis. It's about an arbitrary argument that might or might not really be valid. We just try to find out what happens if it is valid.

Again, then, we need to separate our premises -- which are taken to be true -- from our assumptions and their consequences -- which are just hypothetical. In SD we do this by pushing the assumptions and their consequences off to the right.

And notice that at line 6, the big step, we move our final conclusion back to the left underneath the premises. This is because line 6 is justified by the little argument in 3 through 5. See if this makes sense.*

 

OK, I'm ready to move on...