﻿ [SL18] Limitations

Consider this argument :

All hackers are nerds.
Mitch is a hacker.
So Mitch is a nerd.

This argument is obviously valid, but its validity cannot be demonstrated in SL. This is because the best we can do in SL is to translate the premises and the conclusion using distinct sentence letters, as in :

P, Q R.

Such a sequent is of course not valid in SL. To demonstrate the validity of the first argument, we need to analyse the internal structure of the premises and the conclusion in more details. A more powerful formal system, predicate logic, allows us to do that. Like SL, PL is also a formal system of logic. It is at least as powerful as SL in that PL includes all the WFFs of SL, and any logical truth of SL can also be proved in PL. But in addition, PL can be used to express certain logical connections between statements that SL cannot, and carry out more complicated proofs.

Determine whether these valid arguments can be shown to be valid in SL.

• All birds can fly. Anything that can fly has a moving part. So every bird has a moving part.
• If everyone is happy then Jane is happy. Everyone is happy. So Jane is happy.
• Some philosophers are clever. All philosophers are old people. So some old people are clever.
• Sammy is a girl. Every boy loves Sammy. So there is a girl that every boy loves.
• A horse is an animal. So the head of a horse is the head of an animal.