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Taken from the Powerscore LR book, I can understand the applicatino of double arrows for something like:

X if and only if Y
All W's are Z's, and all Z's are W's.

Those are obvious double arrows, however I don't understand why a double arrow is applicable when:

"At least one heron has blue feathers" H <-s-> BF
or
"Not all of the Smallville roads are safe." SmallvilleRoads<-s-> NotSafe

The way I read the heron example, is some herons have blue feathers, but going the opposite direction, blue feather is an indicator of a heron, but could be another bird, like a blue bird.

I hope what I said makes sense, I suppose I don't understand why those relationships go both ways. Thanks.

The difference is that the latter are 'some' statements.

"At least one heron has blue feathers" is another way of saying "There exists at least one heron that has blue feathers" You could reverse that to say "There exists at least one blue feather that is on a Heron". That's why some statements are reversible.

goatness wrote:Taken from the Powerscore LR book, I can understand the applicatino of double arrows for something like:

X if and only if Y
All W's are Z's, and all Z's are W's.

Those are obvious double arrows, however I don't understand why a double arrow is applicable when:

"At least one heron has blue feathers" H <-s-> BF
or
"Not all of the Smallville roads are safe." SmallvilleRoads<-s-> NotSafe

The way I read the heron example, is some herons have blue feathers, but going the opposite direction, blue feather is an indicator of a heron, but could be another bird, like a blue bird.

I hope what I said makes sense, I suppose I don't understand why those relationships go both ways. Thanks.

The "some" relationship goes both ways, because, to use your example, if at least one heron has blue feathers, than at least one thing with blue feathers is a heron. And if at least one Smallville road is not safe, then at least one road that is not safe is a Smallville road.

The double arrow does not work automatically with "most" or "all". It has to be specified this those cases. Hence, <-s->, m->, and a-> (or just simply ->).

goatness wrote:Taken from the Powerscore LR book, I can understand the applicatino of double arrows for something like:

X if and only if Y
All W's are Z's, and all Z's are W's.

Those are obvious double arrows, however I don't understand why a double arrow is applicable when:

"At least one heron has blue feathers" H <-s-> BF
or
"Not all of the Smallville roads are safe." SmallvilleRoads<-s-> NotSafe

The way I read the heron example, is some herons have blue feathers, but going the opposite direction, blue feather is an indicator of a heron, but could be another bird, like a blue bird.

I hope what I said makes sense, I suppose I don't understand why those relationships go both ways. Thanks.

The double arrow you are referring to isn't a true bi-conditional because it uses the quantifier "some" which means at least one. This one uses the some quantifier for both directions not just one.

"At least one heron has blue feathers" H <-s-> BF
This means "some herons have blue feathers" going one direction ("H s-> BF" OR "SOME H -> BF")
"Some blue feathered birds are herons" going the other direction ("BF s-> H" OR "SOME BF -> H")
Emphasis is that the some goes both directions.

Likewise for the Smallville roads.

Best to think of it visually, as with a Venn diagram. This will explain the double arrow easier for "some," and why it doesn't exist for "most" or "all." "Some" indicates any overlap whatsoever, which extends to both/all entities the same; whereas "most" and "all" entail a specific kind of overlap that doesn't necessarily exist for both/all entities.

Ah, very helpful. I also just realized if I kept reading another page it explains the 'reversibility' of the double arrows, haha. I just wanted to stop before I got too confused though, thanks for the responses everyone.