Logical proof for simple statement,

Question

This seems like a simple question... but I am struggling to find a conclusive proof.

(a) If Robinson's squash is double the strength of Johnson's squash, then you need to use double the volume of Johnson's squash as Robinson's squash for equivalent amount of drink.

(b) If you need to use double the volume of Johnson's squash as Robinson's squash for equivalent amount of drink, then Robinson's squash is double the strength of Johnson's squash.

I am almost certain that both statements (a) and (b) are valid, however I am struggling to prove it in mathematical terms.

How would you approach it?

Answer 1

A priori we need to define the relationship between "strength", "amount", and "volume".   Let $s(x), a(x), v(x)$ be the functions for these on some entity $x$ in the domain of discussion ($\mathcal S$, squashes).

Now, it seems reasonable that we should define strength as amount per volume.   Then our unstated axiom is: $\forall x{\in}\mathcal S~\big(s(x)=a(x)/v(x)\big)$

If Robinson's squash is double the strength of Johnson's squash, then you need to use double the volume of Johnson's squash as Robinson's squash for equivalent amount of drink.

Well, let $j,r$ be the entities "Johnson's squash" and "Robinson's squash".

• $s(r)=2s(j)$ says "Robinson's squash is double the stength of Johnson's squash."
• $v(j)=2v(r)$ says "Johnson's squash is double the volumn of Robinson's squash."
• $a(j)=a(r)$ says "Johnson's squash is the same amount(of drink) as Robinson's squash."

Your task is now to use these predicates, and logical connectives, to express the claim as a nested conditional statement, then apply the definition to prove it.