Order the following propositions from the strongest to the weakest: A, B, A → B?

Question

So I was recently given a number of questions in class but this one in particular, despite scouring my notes and the internet, has escaped me.

"Given formal statements P and Q, P is said to be stronger than Q if P =⇒ Q (In other words, P =⇒ Q is a tautology).

Answer the following: a) Order the following propositions from the strongest to the weakest:

(i) A (ii) B (iii) A → B

(Hint: It is possible that a proposition may not be stronger or weaker than another proposition. In that case, they are at the same level. Truth tables can be used to derive the relationships between any two of the above statements individually.)

b) Prove that A ∧ B is stronger than A.

c) Is A ∧ B stronger or weaker than A ⇒ B? Justify your answer."

I just cannot wrap my head around this. I've done out truth tables which I can show you if it helps explain it to me; but overall I can't understand how to fully explain why one of these is stronger than the other except to say one has more Truths than another. As a whole I'm not sure how to correctly evaluate the strengths. I was directed here via StackOverflow as they believed it was more relevant.

Answer 1

$A\to B$ is not a tautology, and neither is $B\to A$, so $A$ and $B$ are at the same level: neither is stronger than the other. On the other hand, $B\to(A\to B)$ is a tautology, as we can see from the following truth table:

$$\begin{array}{c|c|c} A&B&A\to B&B\to(A\to B)\\ \hline T&T&T&T\\ T&F&F&T\\ F&T&T&T\\ F&F&T&T \end{array}$$

Thus, $B$ is stronger than $A\to B$. That leaves only $A$ and $A\to B$ to compare. Neither $A\to(A\to B)$ nor $(A\to B)\to A$ is a tautology, so these two propositions are at the same level.

To summarize, $B$ is stronger than $A\to B$, and this is the only such relationship amongst the three propositions $A,B$, and $A\to B$.

Answer 2

It is not just a matter of counting the number of truths in a truth-table, but also how the truth-values are related. In general, a statement $\phi$ is stronger than a statement $\psi$ when $\psi$ is true whenever $\phi$ is true. In terms of truth-tables: statement $\phi$ is stronger than a statement $\psi$ if for every row where $\phi$ is True, $\psi$ is also True.

If you do this, you should find that:

i) $B$ is stronger than $A \rightarrow B$ (but otherwise there are no stronger than relations here

b) $A \land B$ is indeed stronger than $A$ (in every row where $A \land B$ is True, $A$ is also True)

c) $A \land B$ is indeed stronger than $A \rightarrow B$ (which you can also see as follows: $A \land B$ is stronger than $B$, and $B$ is stronger than $A \rightarrow B$, so (by transitivity )$A \land B$ is indeed stronger than $A \rightarrow B$