# 15.5 How to Find Shortcuts

In the previous section we saw a shortcut: when we proved Pv~P, we didn’t have to do the 5-step plan a second time.

You might have figured that out by noticing the recurring pattern: to do the 5-step plan a second time, we would be assuming the same sentence as what we just proved.

Shortcuts involve noticing a recurring pattern.

Most shortcuts work like that: they involve noticing some recurring pattern that lets you find a fast and elegant way to do a proof.

For example, let’s prove this tautology: PvQv(~P&~Q). Don’t read the hints unless you need them!

Once you did the 5-step plan twice, you had the information ~P and ~Q. But if you were to do the 5-step plan a third time, you’d be assuming ~P&~Q.

Instead of assuming that, let’s just use &Intro on ~P and ~Q.

Similar to the first shortcut, we have to notice the common sentences here, and realize that if we are creative we can reproduce the ~P&~Q pattern with the materials we already have.

In the second half of the course, the idea for finding shortcuts will be the same: look for common patterns and be creative!

In the second half of the course, when our logical systems get more complex, there will be much more room for finding proof shortcuts. But the idea will still be the same: look for common patterns and be creative!