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I was trying to understand the meaning or the definition of what a "stronger statement" is formally. I came across the following definition (from Mattuck's Analysis book):

Stronger and weaker. If $A \implies B$ is true, but $B \implies A$ is false, we say: A is a stronger statement than B; B is weaker than A.

and I was wondering, why is the definition that way? Is there a conceptual/intuitive way to explain this? I know this is just "the definition" but I was trying to understand why it is that way.

In fact I create some type of "memory device" (not sure what else to call it) to remember/justify it to myself. I draw the following diagram:

enter image description here

and then notice that whenever x is in A it means it must be is B also. Therefore, being in A implies being in B. Furthermore, if x is in B it doesn't necessarily always be in A, so the converse is not always true automatically. The only issue I have with my memory device is that I obviously just re-define what the statement A means to be a very specific set membership statement. So I assume its a fine memory device but its oversimplifing things "cheating" in some way. i.e. its not a proof nor I expect it to be the "real reason" why the definition holds.

So I was looking to understand a better way to understand what a stronger statement means without "cheating".

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?

Charlie Parker
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    $A$ is a stronger statement than $B$ because you're saying more when you say $A$, you're not only saying everything $B$ says, but you're saying something else as well. – Set Dec 10 '17 at 17:29
  • @Thoth I guess this is where I also get confused, "more" for me would have meant a bigger set but somehow that translated to a smaller set. Why is that? – Charlie Parker Dec 10 '17 at 17:34
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    There is a contravariance between how strong the statement is, and how big the set of things satisfying the statement is. The smaller the set is, the more specific it is, the more statements you can prove about it. – ziggurism Dec 10 '17 at 17:39
  • See Modus ponens: if $A \to B$ is true, from the truth of $A$ we can deduce the truth of $B$. I.e. $A$ is "enough" for asserting $B$. But if $B \to A$ is false, this happens only when $B$ is trueand $A$ is false. I.e. the truth of $B$ is not "enough". When $A \to B$ is true we can say also that "$A$ is a sufficient condition for $B$". – Mauro ALLEGRANZA Dec 10 '17 at 17:40
  • ziggurism has given the correct interpretation, to give a simple example, everything satisfies a vacuous statement (a statement where you say nothing). – Set Dec 10 '17 at 17:45
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    strong = proves a lot of statements = true about a more specific class of models = the set of models is smaller. weak = proves fewer statements = but those statements are true of a larger more general class of models. – ziggurism Dec 10 '17 at 17:55
  • Is there some deeper explanation or description of this contravariance? I don't know. I bet there is. – ziggurism Dec 10 '17 at 18:19
  • " "more" for me would have meant a bigger set " You don't care about "big" you care about precision. The set of human beings is a "big" set. The set of 6'2" males that were born on May 14, 1958 in Topeka Kansas is a "small" set. "Kuvak is a human being" is a weak statement whereas "Kuvak is a 5'2" male that was born on May 14, 1958 in Topeka Kansas" is a strong statement. It is strong because it is a small sets. Large sets mean weak statements because it is easy to be in a big set. It is hard (takes strength) to be in small set. – fleablood Dec 10 '17 at 18:24
  • A slogan I've heard somewhere: syntax and semantics are adjoint functors. Persumably whatever those adjoint functors are, they are contravariant? – ziggurism Dec 10 '17 at 18:29
  • "syntax and semantics are adjoint functors." Kind of. "more" information = "fewer" options. "less" information = "more" options. So "more" = "bigger" is really not the way to think of it. Strong = more information/fewer options/harder to be true. Weak= Less information/more options/easier to be true. – fleablood Dec 10 '17 at 18:42
  • Related question: https://math.stackexchange.com/questions/2543452/what-are-the-strong-and-weak-in-mathematics/2543472#2543472 – Bram28 Dec 10 '17 at 18:49
  • See page 405: "stronger" because $A$ put a "stricter" condition: "if a triangle is equilateral, then it is isosceles", but not all isosceles triangles are equilateral. In "traditional" verbeage: to be equilateral is a sufficient condition to be isosceles, but to be isosceles is not sufficient to be equilateral. – Mauro ALLEGRANZA Dec 11 '17 at 11:01

5 Answers5

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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.

Noah Schweber
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"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.

fleablood
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I think it will be useful to connect to the original example the OP presented.

Let's build on Noah's answer. Let's start by interpreting the set related to a statement as the ways the statement can be true. So a weaker statement is a broader statement (I assume it's considered weaker because so many things are by default obviously true that it's weak at providing useful information to the reader). The good example he has if the proposition P = "the colored object" then everything is essentially colored so this doesn't tell us anything and the set related to P $S_P$ is basically everyting. It's too broad (to weak). But if P = "the colored pencil" then this set is much stronger since it tells us a lot more information about the set (only colored pencils).

Now if the statement is $P_A = x \in A$ and $P_B = x \in B$ and we know $A \subset B$ then we know $P_A$ is stronger than $P_B$ because the set $P_A$ is smaller or, $P_B$ is weaker (broader) than $P_A$ because it has so many things true.

I think it is also interesting to notice the second interpretation where stronger does mean bigger. The interpretation is the set of statements that P consequently makes true. I find it useful to think of implication $P \implies Q$. Let's define the set of consequences $S_P$ as the set of statements $P$ proves (if you're like me, you prefer to think of implication in proof theory rather than in model theory, i.e. assuming $P$ can we conclude using the proof rules of the system to arrive at $Q$, say using natural deduction ND):

$$ S_P = \{ Q : P \implies Q \}$$

notice how if $P = False$ then everything is there! Thus, False is the strongest statement since it implies everything. Similarly True is much smaller since if $P=True$ then $Q$ must be things that are True.

From the comments section:

strong = proves a lot of statements = true about a more specific class of models = the set of models is smaller. weak = proves fewer statements = but those statements are true of a larger more general class of models.

Charlie Parker
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In your diagram $A$ and $B$ are sets ... but in $A \Rightarrow B$, $A$ and $B$ are statements .. these are of course not the same thing. Indeed, what would be the elements $x$ that go into $A$? That is far from clear.

There is a way to relate statements to sets though: we could draw a circle $A$ as containing 'all the situations/worlds/scenarios in which statement $A$ is true'. But as such, since $B$ is true whenever $A$ is true, we have that in all the situations/worlds/scenarios in which statement $A$ is true, statement $B$ is true as well, i.e every element of the circle corresponding to $A$ is an element of the circle corresponding to $B$, and that means that the circle of $A$ indeed ends up being inside the circle of $B$.

So, if you work with this interpretation of what the circles represent, the picture works!

Bram28
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In this answer, $Px$ denotes the propositional function or predicate $P(x).$

enter image description here

More accurately, the above Venn diagram corresponds to the first-order-logic sentence $$\forall x \big(Ax{\implies}Bx\big)\;\text{and}\;\exists x\big(Bx{\kern.6em\not\kern-.6em\implies}Ax\big),$$ where the elements of set $P$ are precisely the values (in the universe of discourse) that satisfy predicate $Px$ (i.e., such that $Px$ is true).

Here, $A$ being a proper subset of $B$ means that every element that satisfies predicate $Ax$ satisfies predicate $Bx$ but the converse need not be true, in other words, $Ax$ is harder to fulfil than $Bx,$ in the sense that the former needs everything the latter needs and more.

A is stronger because it says everything B says and more.

Voila.

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.

Set $A$ being smaller than set $B$ means that it contains fewer elements; in other words, predicate $Ax$ is true for fewer values than predicate $Bx;$ because $Ax$ is a stronger claim than $Bx$ about variable $x.$

ryang
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