Hey, I have a question about whether a pair of syllogisms is equivalent to another.
The first two are:
-if not A, then not X
-if not B, then not X
Is:
if not A or not B, then not X
equivalent to the first two syllogisms?
Why or why not?
Yes, in most systems of logic, “(A implies C) and (B implies C)” is equivalent to “(A or B) implies C”.
Why? Because both of these have the exact same meaning: that either an A or a B can be exchanged for a C, so to speak.
They’re equivalent.
It’s not IF A and B, then not X.
It’s if A or B, then not X.
The two statements in the first set are not dependent on each other, so you only need A or B to result in ~X
Entering the expression
{(~A > ~X) & (~B > ~X)} = {(~A + ~B) > ~X}
into the truth table generator here
http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/
verifies that they are equivalent.
I agree with the other posters with one proviso. You should be using the standard non-exclusive or. If your “or” means either A or B but not both, then they aren’t the same.
Cool, thanks.
Is there a specific name for the general class of syllogisms that this particular one is an example of? (i.e. De Morgan’s law, commutative law)
Also another logical question:
Say that A and B are both necessary conditions for C, and that there are no additional necessary conditions for C other than A and B. Does this mean that A and B together are sufficient conditions for C?
Depends on exactly what you mean. It might be the case that A is necessary and B is necessary and either X or Y is necessary. And A and B and either X or Y is sufficient. It’s true that A and B together are not sufficient, but it’s not clear if you could say there are no other necessary conditions. It is true that neither X nor Y is necessary, but it is also true that X or Y is necessary, so it depends on how you interpret “there are no additional necessary conditions.”
No. You have C -> A and C -> B, but that doesn’t allow you to conclude that any combination of A and B will imply C.
What does it mean to say “there are no additional necessary conditions for C other than A and B”? One way of reading that is as saying “A and B together are equivalent to C”, in which case, the answer to your question is trivially yes. But it’s not clear that that’s what you meant, since that’s so trivial.
What do you think necessary and sufficient mean? If you are just taking them to accord with (material) implication, then among the necessary conditions will be C -> C; and so among the necessary conditions will be a sufficient condition.
Saying how the later is a reading of the former might not be so trivial, though.
Well, I would say it trivially. I might interpret “There are no additional necessary conditions for Y other than X” to directly mean “X is both necessary and sufficient for X”. What else does “There are no additional necessary conditions” mean?, if not “This is everything; this is all it takes; nothing else is required; this is enough. Aka, this is sufficient.”