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 universe of people, let's make these laws: (i) if someone likes you, you like them back;
(ii) if someone is kind, then cindy likes them. Now prove, if don is kind, then he likes cindy:
∀x∀y(Flikes(x,y) —> Flikes(y,x)), ∀x(Fkind(x) —> Flikes(cindy,x)) |− Fkind(don) —> Flikes(don,cindy)
Use ∀e along with the logic rules for ->.
2.
"Forall" and "and" interact well with each other.
In class we proved,
∀x F(x), ∀x G(x) |− ∀x(F(x) ∧ G(x))
Prove the converse, using both ∀e and ∀i and the rules for ^:
∀x(F(x) ∧ G(x)) |− (∀z F(z)) ∧ (∀z G(z))
3.
"Forall" and "not" interact in just this one direction; prove it:
∀y ~F(y) |− ~(∀x F(x))
Next, think about the following so that we can discuss it on Friday:
Say that F means ``has brown eyes.'' How does the above sequent read in English words?
How does the following sequent read in English words: ~(∀x F(x)) |− ∀y ~F(y) ? Is this sequent true for the universe of all living humans? Should our logic rules prove the sequent?
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: ∀x(F(x) ∨ G(x)) |− (∀x F(x)) ∨ (∀x G(x)).
Here's a counter example:
Say that the universe consists of all humans. F means ``is left-handed'' and
G means ``is right-handed.'' The formula, ∀x(F(x) ∨ G(x)), says
everyone is either left- or right-handed. But does this guarantee that (∀x F(x)) ∨ (∀x G(x))?