Logical opposites - Questions

[Blueballoon5%]

1.) How do we convert this sentence to its logical opposite: "Without any apples in our basket, we will not be able to make our pie." Which negative do we convert (the "without" or the "not")?

2.) "Unless we protect our rights, we will lose them." In this sentence, the book explains that the logical opposite is: "Even if we protect our rights, we will lose them." However, could we also say: "Unless we protect our rights, we will not lose them"?

3.) "Experts believe that an increase in pollution will lead to an increase in the sea level." The book put: "Experts do not believe..." However, could we also say, "Experts believe that an increase in pollution will not lead to an increase in the sea level."

4.) "Many of the cafe's patrons are unaware that the cafe does not bake the doughnuts it sells." The book put: "Many of the cafe's patrons are aware..." However, could we also say, "Many of the cafe's patrons are unaware that the cafe does bake the doughnuts it sells."

[David Boyle]

1.) How do we convert this sentence to its logical opposite: "Without any apples in our basket, we will not be able to make our pie." Which negative do we convert (the "without" or the "not")?

2.) "Unless we protect our rights, we will lose them." In this sentence, the book explains that the logical opposite is: "Even if we protect our rights, we will lose them." However, could we also say: "Unless we protect our rights, we will not lose them"?

3.) "Experts believe that an increase in pollution will lead to an increase in the sea level." The book put: "Experts do not believe..." However, could we also say, "Experts believe that an increase in pollution will not lead to an increase in the sea level."

4.) "Many of the cafe's patrons are unaware that the cafe does not bake the doughnuts it sells." The book put: "Many of the cafe's patrons are aware..." However, could we also say, "Many of the cafe's patrons are unaware that the cafe does bake the doughnuts it sells."

Hello Blueballoon5%:

Four questions at once; that's a lot!!!! --Here we go:

1. This is sort of like an "unless", and basically says, "No pie unless apples in basket." Convertible to
pie apples in basket.
One could say the logical opposite, then, is simply
pie slash apples in basket.
People must be careful, though. If one reads the latter diagram too literally, it sounds like, "To have a pie, there MUST BE NO apples in the basket", which sounds sort of crazy. Some people even slash the arrow instead of what's after it, to show that what's after the arrow is just not really needed--as opposed to saying that you cannot have that thing at all.
But if "A slash B" works for you as the logical opposite of "A B", and you know exactly what it means in a given circumstance, fine.

2. This looks interesting. "Not lose rights protect" makes sense as a diagram for the first part. So if we made it "not lose rights not protect", for the logical opposite, then contrapositive it to "protect lose rights"...it may sound strange, but "Even if we protect our rights, we will lose them." may be correct, then.
(Again, from what I said in problem 1, maybe "Not losing your rights does not need you to protect your rights" might also be an acceptable rendition of a logical opposite. Some may differ with that, though.)
"Unless we protect our rights, we will not lose them"? sounds a little strange, and is diagrammable as
"lose rights protect rights", which may be a mistaken reversal of what the book says, "protect lose rights".

3. "Experts believe that an increase in pollution will not lead to an increase in the sea level." is not the same as
"Experts do not believe..." In the latter, the experts may not have formed an opinion yet. In the former, you're saying that they definitely have formed an opinion.

4. "Many of the cafe's patrons are unaware that the cafe does bake the doughnuts it sells." is wrong, since it changes the whole meaning (e.g., it asserts that the cafe does bake the doughnuts, rather than focusing on awareness).

Hope this helps,
David