Chapter Five, Tutorial Seven
SD+: Short-Cut Rules

SD derivations are a little like real, natural language deduction. But only a little: for one thing they are much too cumbersome. So, to make our derivations a more natural account of reasoning, we will provide means of short-cutting. The basic idea is that we will give new rules that allow one to do what amount to a number of steps in one.

For example, we will have a new rule called Modus Tolens:

MT
(Modus
Tolens)
input 1:
input 2:

output:
P >Q
~Q

~P


First notice that this rule makes sense: If P implies Q but Q is false, then P can't very well be true!

But also, notice that MT isn't really necessary. One can make the inference in SD without MT:

Premise 1. A>B  
Premise 2. ~B  
Assumption 3. what if ..... A
1,3 >E 4. then ........ B
2 R 5. then ........ ~B
3-5 ~I 6. ~A  

But this derivation is unwieldy for such a simple inference. So, in SD+ we can make the inference in one step beyond the premises.

Premise 1. A>B
Premise 2. ~B
1,2 MT 3. ~A

This may not seem like a huge difference. But as the premises get longer or more numerous and as we get more shortcut rules, the time saved and the clarity gained will be substantial.

Now, let's produce another "short-cut" rule.

input 1:
input 2:

output:
P vQ
~P

???

How should we fill in for the question marks? Fill in so that this is a helpful valid inference:

  1. Q
  2. ~Q
  3. P
  4. ~P