Show that we can obtain complete expressiveness for a language using only the implication connective if we add to our language the “propositional constant” ⊥ which always has the truth value 0 (false).
I found this problem in Introduction to Logic by Paul Herrick. I'm really stumped on how to prove this.
Any help would be great, thanks in advance!
Negation and disjunction are known to be logically complete, so if we can express them in terms of implication and falseness, we are all set.
$$\neg P=P\to\bot$$ $$P\vee Q=(P\to\bot)\to Q$$ and, just to see how to keep building these: $$P\wedge Q=(P\to(Q\to\bot))\to\bot$$ $$P\equiv Q=((P\to Q)\to((Q\to P)\to\bot))\to\bot$$