Why is the definition of stronger statement the way it is in logic?

Solution 1:

As pointed out by the comment, A is stronger because it says everything B says and more. I guess for me intuitively that would have meant that A is a bigger set, but in my memory device that translated to a smaller set, which makes it confusing to me. Anyone know why?

When you interpret a statement as a set, there are two basic ways you can do this:

• You can view a statement as the set of its consequences. In this case, a stronger statement corresponds to a bigger set (which matches the intuition "stronger = bigger"), because they tell you more.

• Or, you can view a statement as the set of ways it can be true. Here, a weaker statement yields a bigger set: the set of ways I can be unhappy is much bigger than the set of ways I can have just had a piano dropped on my head, so "I am unhappy" corresponds to a bigger set in this sense than "I just had a piano dropped on my head." In this context, it might be less confusing to replace "weaker" with "broader."

The second approach, by the way, matches how Venn diagrams work. The set of blue objects is bigger than the set of blue dogs, because the property "is a blue object" is broader (weaker) than the property "is a blue dog." I think this is where confusion tends to creep in: we're so used to Venn diagrams - and the general idea that "bigger = stronger" - that the notion of "stronger statement" seems unintuitive.

Solution 2:

"more" is probably a bad choice of words.

A statement that says "more" gives more precission which means there are fewer ways it can be true. Because there are fewer true statements with more information.

Perhaps an easier way to see this is

Weak statement 1: Kuvak is a human being.

Strong statement 2: Kuvak is a 6' 2" male who was born on May 14, 1958 in Topeka Kansas and he has three 7s in his social security number.

Strong Statement 2: Has more information and because it has more information belongs to a smaller set.

Weak Statement 1: Has less information and because it has less information belongs to a bigger set.

When it comes to information: more = strong.

When it comes to possibilities: more = weak = LESS information.

The LESS you say, the easier it is to be right and the weaker you have to be to be right.

Statement 2 $\implies$ Statement 1. So Statement 2 is a stronger statement because it is harder for statement 2 to be true. To be true not only does statement 1 also have to be true, but a bunch of other stuff has to be true as well.

Statement 1 $\not \implies$ Statement 2. So Statement 1 is a weaker statement because it is easier for statement 1 to be true. It's true every single time statement 2 is true. And it's true a lot of other times as well. The number of possibilities that statement 1 is true is a lot BIGGER then the number of possibilities that statement 2 is true. So it is WEAKER because it is easier to be true.