# Logical Implication Question

$A: \text{Humans are at most 12 feet tall}$

$B: \text{Humans are at most 9 feet tall}$

Am I correct?

• You might wanna have a look at the square of opposition. In logic terminology, two sentences $\alpha$ and $\beta$ are said to be in a contradiction iff it is the case that $\alpha$ is true when $\beta$ is false and vice-versa, You can see that this is not the case. Particularly, B and A can be both false. – Bruno Bentzen May 11 '15 at 6:59
• But am I right that A does not imply B and B does not imply A? – akuryo May 11 '15 at 7:02
• Yes, the sentences do not imply each other (they are contraries, so they cannot be both true, then, if you assume one the other will be always false. Hence both $A \rightarrow B$ and $B \rightarrow A$ cannot be the case) – Bruno Bentzen May 11 '15 at 7:27
• @akuryo No. "Humans are at most $x$ feet tall" is not the same statement as "the tallest human is exactly $x$ feet tall". – Graham Kemp May 11 '15 at 8:35

In modern logic terminology, back from the Aristotelian doctrine of the square of opposition, two sentences $\alpha$ and $\beta$ are said to be in a contradiction iff it is the case that $\alpha$ is true when $\beta$ is false and vice-versa, that is, $\alpha$ is false when $\beta$ is true.

This relation is found in both diagonals of the square:

Particularly, Two sentences are are

• contradictory iff they cannot both be true and they cannot both be false
• contraries iff they cannot both be true but can both be false.
• subcontraries iff they cannot both be false but can both be true.

Particularly, B and A can be both false simultaneously, what disqualifies their logical relation as a contradiction. Still, B and A cannot be simultaneously true. Hence, B and A are contraries.

Now in regard of your question as to whether

A does not imply B and B does not imply A

Yes it is true. For the former, let humans be at most 12 feet tall. Then it follows humans are not at most 9 feet tall. For the latter, let humans be at most 9 feet tall. Then humans are not at most 12 feet tall.

Most importantly, A and B are contraries, so remember that they cannot be both true. Then, if you assume one the other will be always false. The conclusion of our analysis is that both $A→B$ and $B→A$ cannot be the case.

• Ah, it was a trick question! It asked as if it was one or the other, but it was neither. That's a convenient diagram. Thanks for showing it to me and thank you for helping. – akuryo May 11 '15 at 7:33
• @akuryo No problem! :) – Bruno Bentzen May 11 '15 at 7:35
• A:Humans are at most 12 feet tall
• B:Humans are at most 9 feet tall

We have:

• A:= $\quad\forall h\in \operatorname{Humans}: \operatorname{tallness}(h)\leq 12$
• B:= $\quad\forall h\in \operatorname{Humans}: \operatorname{tallness}(h)\leq 9$

That is that neither statements' truth requires that the tallest possible human is of their given height, only that no human is taller that it.

So $B$ does not contradict $A$.   In fact there is a material implication: $B\to A$.   It is sufficient to know that if humans are at most $9$ feet tall, then we know humans are at most $12$ feet tall.