Discussion
When I teach my 9th grade math students about the converse I tell them that it's one of the most important concepts they will learn in my class. I'll share this lesson with you, for free!
A statement of the form "If ---- then ----." is called an implication. If you swap it around? *That's* the converse.
Statement: "If it rained, then the sidewalk will be wet."
Converse statement: "If the sidewalk is wet, then it rained."
But why does it matter?
@futurebird I see. Excellent teaching point.
A true statement may have a false converse, but many people do not recognize this. It's a common logical error.
If A, then B.
is not equivalent to
If B, then A.
Just because the sidewalk is wet, you can't assume it rained. Maybe someone dumped a bucket of water out there.
In statistics this comes up in the distinction between correlation and causation, which is a related nest of logical errors.
So, now you can say "you can't assume the converse" in an argument when you see it happen.
@futurebird where I live, the most frequent waterer of sidewalks are strip malls, businesses that think they need a little lawn out front, and apartment complexes.
@futurebird I had a longish write-up in my currently offline website about this, the importance of not confusing necessary and sufficient conditions, and how to properly invert them (e.g. the fact that the proper way to invert «if A then B» is «if not B then not A»), and I was surprised by how many people ended up visiting it (this was in the auld days of lore when much of the traffic was organic)
@futurebird I think this a lot but didn't have a succinct way to express it, so thanks!
@futurebird this is maybe a tad too high-level for your stundents, but i also love the "types as propositions" interpretation of this fact.
let's imagine "types" as, uh, named sets, each statement we wanna talk about gets its own named set of "witnesses" for its truth (i.e. false statements cannot have witnesses, so they must be empty). every "if A, then B" corresponds to the set of functions A → B, and "not A" corresponds to a function A → Ø (forcing A to be empty, if there is such a function).
then we get an elegant proof that A → B implies ¬B → ¬A, because if we have a function h : A → B, and a function g : B → Ø (aka ¬B), then we can construct a function q : A → Ø (aka ¬A) by simply applying first h, then g (i.e. q is the composition g ∘ h).
similarly, because we can always trivially construct a function from an empty set to every other set (no work needs to be done since the empty set doesn't contain anything), we can easily show that if every implication implied its converse (i.e. the, uh, class of functions, (A : Set, B : Set, A → 😎 → (B → A)), then we could prove every set is empty (which is a contradiction, since we know there are true statements).
the more "real-world" version of this is, how a statement like "if there is a green unicorn in this room, then i have a blue pencil" being vacuously true (since there (presumably) are no green unicorns) doesn't mean having a blue pencil would magically conjure up a green unicorn into the room (fullfilling the converse).
but yeah, teach them kiddos about basic logic and reasoning, they absolutely will need it, regardless of the more abstract nonsense version i'm rambling about here.
@futurebird I do remember looking at this in terms of computer logic.
A = B does mean that B = A
But A => B (A Implies 😎 is a different statement, that, as you say, does not mean that B => A.
This is a very nerdy way of expressing it, but it was my first development from the usual logic operators, and something I have taken with me ever since.
@futurebird You may have met this before, I got it from Altemeyer's The Authoritarians, but apparently it's a common error in authoritarian followers to say "I know B is true, I know A is true, therefore B must imply A".
@hypolite @futurebird I see your point, and certainly one can characterise various events as loyalty declarations (are you prepared to humiliate yourself and cut yourself off from normal people to demonstrate your loyalty to the leader); but in this case I meant error in the sense of a technical error of logic (B does not imply A).
@futurebird I remember this as "reverse modus ponens" from my Philosophy of Logic class
The whole "modus ponens" set of vocabulary is fun but I think "converse" is about as simple as I've seen it.
I'm even happy with "reversed if, then" ... because the vocabulary words are fun and fancy ... however it's the concepts that matter.
Heck I'll take "A → B does not mean B → A"
@futurebird For sure, the latin terms are not necessary to understand the concept at all and will make a lot of people's eyes glaze over. I did find that working through the symbolic representations of some of the more complex syllogisms was really eye opening for understanding why certain logic was valid or invalid. I really enjoyed that class
@futurebird @leeglidewell i always remember if your main statement is a ->b then the only other definite thing you can derive is the the contrapositive, !b->!a the converse is bollocks.
@futurebird damn too bad most people wont pay any attention to this when its them doing it.
I do think recognizing when you are stating an implication can help, maybe.
@futurebird They don't need to know why the sidewalk is wet, as long as being mad about it is considered correct to their peers.
@futurebird in my experience when you point out any kind of logical fallacy many people will simply turn into it harder and then look for their in-group for support. Kinda like here. I have lost a lot of faith in humanity to perform any higher level functions beyond respond to the cybernetic echo with more echo. Like you are assuming people want to accept the risk of critical thinking. As I now live in a major metro, Down Town San Diego, this never happens.
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@futurebird I remember a similar logic example in my college math and logic course: All poodles are dogs but not all dogs are poodles. This is for set theory of course but may also help explain the converse in a discussion of logical fallacies