• Will a direct proof work for any of these?
• Can the proof be performed more efficiently by using different equivalence rules?

1. There are rights that cannot be waived. But alienable rights can be waived. It follows that some rights are inalienable. (Rx: x is a right; Wx: x can be waived; Ax: x is an alienable right)
2. All contingent beings are causally dependent. No necessary beings are causally dependent. Every physical entity is contingent. All atoms are physical entities. We may conclude that no atom is a necessary being. (Cx: x is a contingent being; Dx: x is causally dependent; Nx x is a necessary being; Px: x is a physical entity; Ax: X is an atom)
3. There is an entity that is more powerful than all entities. Therefore, at least one entity is more powerful than itself. (Mxy: x is more powerful than y)
4. All brain processes are physical processes. No mental processes are tangible. Therefore, every brain process that is a mental process is also an intangible process. (Bx: x is a brain process; Px: x is a physical process; Mx: x is a mental process; Tx: x is tangible.)