Help in understanding logical statements

[Ryan Dade]

In lieu of taking an extra course I signed up for a class without fulfilling a prerequisite. Now I'm trying to teach myself the stuff I should have learned and am having a hard time with two problems. If anyone can help explain how to get to the solution I would appreciate it.

1.) Show that the following holds true. Derive (\forallx)(If ~Mx thenMx) with the assumption (\forallx)(Mx)

2.) Show that the following is valid.
(\existsx) (If Cx then Ch)
(\existsx) (Cx if and only if Ch)

 

[CompuChip]

Can you clarify on notation? For if I take the ~ symbol to mean "not" then the first question seems to be of the form A \rightarrow \neg A which is a contradiction (if it's raining, it's not raining). Or perhaps you meant: \forall x(\neg\neg Mx \rightarrow Mx) ?

For the second, note that "Cx if and only if Ch" is shorthand for "(If Cx then Ch) and (if Ch then Cx)" and that one of them is already given. The other side is almost trivial: what is it you still need to prove?

 

[HallsofIvy]

CompuChip, notice the "with the assumption (\forallx)(Mx). The first one is really "if, for all x, Mx is true and, for all x, Mx is false, then Mx is true". Since the hypothesis is false, the statement is trivially true.

 

[Ryan Dade]

I'm trying to figure out how to show it. For example

Derive (\forallz)Kzz

1. (\forallx)Kxx Assumption
2. Kcc 1\forallE
3. (\forallz)Kzz 2\forallI

But I do not have a strong background in Predicate Logic so some problems are more difficult for me to get a full understanding of