Biconditionals

Lsatbreakingnews

Hi guys, I'm going through the biconditional part of the curriculum. Im wondering, for the Or, but not both biconditional, why don't we just write it out like this A -> /B and B->/A, that way you can link it up as well if a chain comes up?

Sami

@Euthyphro said:
Hi guys, I'm going through the biconditional part of the curriculum. Im wondering, for the Or, but not both biconditional, why don't we just write it out like this

A -> /B and B->/A,
If you just have the above sentence as it is written it ends up translating as:
if A is in B is out. And if B is in A is out.
This sentence above only accounts for the "not both" part of the sentence. Because as soon as one of them is in the other is out. But it does not mean one of A or B has to be in. So this sentence is missing the "or" part of the sentence.

To get the "Or" you need to have the sentence written as:
/A---->B and /B--->A
This means if A is out out B is in. And if B is out A is in. So one of A or B has to be in.

These four conditional chain link up easily and can be written as:
A<---->/B and B<------>/A

I hope this helps.

nessa.k13.0

Hi @Euthyphro ! The statement you expressed symbolically could lead us to make an error and translate into English half the inverse of the statement A or B but not both like Sami described above. You can't change the value of variables just to link them up--you'll make invalid inferences. In Boolean logic we have two values for truth: what is true and what is untrue---false. Accordingly, there are only 2 outcomes that are logically possible for a variable, a negative one and a positive one. We either have 1, A or 2, not A and we either have 1, B or 2, not B. The "exclusive not both or" means we can only have one of those two variables occur. De Morgan's law gives us a way to validly distribute negation with the word "or" (for the variable that is untrue/false). We then must represent the logical necessity for one of the variables(A or B ) to occur (and be true/positive) rather than representing the necessity of a variable to not occur (being untrue/false). In the simplified symbolic logic statements, we have to express the "exclusive or" with a negative term in the sufficient condition because the absence of one variable necessitates the other variable occurring. In games or situations where the variables come into play multiple times or you have more than 2 groups, it's crucial to be careful with that translation. In a 2 group and variables placed once setup the "exclusive or"/not both variables absent and not both variables present expression, it is logically true that if we have A then we cannot have B; it's a bit of a paradoxical situation for translating symbolic logic to English though.

akistotle

A -->/B means "not both" so A and B could both be out. If A is OUT, sufficient condition fails and B could be OUT or IN.

A<---->/B means "always apart, never together" so they cannot be out together. If A is OUT, B has to be IN.

Lsatbreakingnews

So helpful guys! Thanks