Notes

1. "[...] one philosopher's modus ponens is another philosopher's modus tollens." (H. Putnam, "Realism Without Absolutes," International Journal for Philosophical Studies, Vol 1(2), p. 180).

2. "A" ranges over formulas of first-order predicate logic with identity. "D(A)" means "A is formally derivable." "Val(A)" means "A is intuitively valid"--that is, valid in every conceivable circumstance, in every state of affairs. "V(A)" means "A is valid in all set-theoretic structures."

3. Kreisel ([1971)], p. 254).