Is it correct to say that "we did not assume that a or b is positive." is equivalent to "we did not assume that a and b are both positive"?

Question

Given a is a real number.

We did not assume that a is positive.

For me, that sentence means either a is positive or a is not positive.

But given a and b are real numbers.

I read a sentence that "we did not assume that a or b is positive." in my textbook.

I wonder what meaning it has.

Are they

we did not assume that a or b is positive.

we did not assume that a and b are both positive

equivalent to each other?

Is it correct to say that

"we did not assume that a or b is positive."

is equivalent to

"we did not assume that a and b are both positive"?

I built a truth table that showed "tautology"

So can we say that?

p:='a is positive', q:='b is positive', so the 1st claim (we did not assume that a or b is positive) can be translated as ¬(p∨q) ⊢ ¬p∧¬q in classic propositional logic. It's not equivalent to 2nd claim (we did not assume that a and b are both positive) which should be translated as ¬(p∧q) ⊢ ¬p∨¬q... — Double Knot, Jul 25 '22 at 05:23

score 3 · Accepted Answer · answered Jul 25 '22 at 06:45

You're probably running into the confusion that there are 2 definitions of "OR" in logic. The common definition of OR in logic allows for a, b and a&b to be true in order to have a true statement. Whereas the colloquial use of OR is often called XOR (exclusive or) and only allows for one of the two to be true in order to have a true statement.

But either way these two statements are not the same, it could just be that they both fulfill the requirements for OR.

Is it correct to say that "we did not assume that a or b is positive." is equivalent to "we did not assume that a and b are both positive"?

1 Answers1