Consider the logical statement P -> (Q -> R) Since this involves two implication
ID: 2942292 • Letter: C
Question
Consider the logical statement
P -> (Q -> R)
Since this involves two implications it can be viewed in some sense as two “nested" theorems. Write this in the form of a single implication (theorem) of the form glob -> R, where “glob" involves P, Q, and R (not necessarily all of them). Discuss (with reasoning and examples) why you think your single implication is equivalent to the original theorem, and how you might prove that.
TRY NOT TO INVOKE TRUTH TABLES!
Explanation / Answer
The single implication you are looking for is: (p and q) -> r I don't want to give too much of it away (save some work for you! :) ) but here is a simple justification. Think about (p and q). If p is false, then (p and q) is false, making (p and q)->r true and p ->(q -> r) true as well. If q is false, then (p and q) is false and (for reasons similar to the first case) (p and q)->r and p->(q->r) are both true. If both are false then it is an obvious statement. If both are true, however, then it makes a difference whether r is truthful or not. If p and q are both true then (p and q) is a true statement. If r is false then both statements will be false. If r is true then both statements will be true. Then we have that the truthfulness of p,q, and r affect both logical statements the same. How would we prove this? I would probably try to prove it by method of contrapositive. In terms of examples, replace "p", "q", and "r" with any statement you want and you can show by examples that the logic holds. Hope this helps!