Use the Natural-deduction-proof-checker tool to prove your answers to the following questions, and submit your answers to K-State Online in a zipped folder.
1. In a domain of happy people, we have these laws: (i) if someone likes you, you like them back;
(ii) if someone is kind, cindy likes them. We can prove in this domain that when chris is kind, chris also likes cindy.
Show this by a proof of this sequent:
∀y∀z (Flikes(y,z) —> Flikes(z,y)), ∀x(Fkind(x) —> Flikes(cindy,x)) |− Fkind(chris) —> Flikes(chris,cindy)
Use ∀e along with —>e.
2.
"Forall" and "and" interact nicely with each other. Prove this:
∀z F(z), ∀x G(x) |− ∀y(F(y) ∧ G(y))
Use∀i and ∀e. (By the way, it is easy to prove the converse result: ∀y(F(y) ∧ G(y)) |− (∀z F(z)) ∧ (∀x G(x)), but you don't have to just now.)
3.
"Forall" and "not" interact in just one direction (prove it):
∀y ~F(y) |− ~(∀x F(x))
Say that F means ``has brown eyes.'' How does the above sequent read in English words? Can we use our logic rules to prove the converse claim: ~(∀x F(x)) |− ∀y ~F(y) ? Why not? Think about this (say that F means ``has brown eyes''); you don't need to submit an answer to grade.
4.
"Forall" and "or" interact in just this one direction (prove it):
(∀x F(x)) ∨ (∀x G(x)) |− ∀x(F(x) ∨ G(x))
You cannot prove the converse. Here's why: Say that F means ``is left-handed'' and G means ``is right-handed.'' The formula, ∀x(F(x) ∨ G(x)) merely says everyone is either left- or right-handed. But does this guarantee that (∀x F(x)) ∨ (∀x G(x))? Why not? Think about this; you don't need to submit an answer to grade.