B iff A is defined as "If B then A and if A then B".
If that doesn't make it clear enough for you, then try writing out the truth table for both statements.
I think I understand now, but what has left me scratching my nose (metaphorically):
Why is it called "B if and only if A", if what it really means is "B only if A and vice versa"? (Am I correct in thinking that's what it means?)
I just don't understand how that translates grammatically. To me, "B if and only if A" sounds the same as "B only if A". I can accept that they mean different things in the context of logic, just like I can assign any meaning to any label, like I could say that "dog" now means "kite" in a certain context. But it seems unintuitive and doesn't really make sense to me. Does that make sense?
"B if and only if A" is a shorthand way to write "B if A and B only if A". It's like how "He is young and thin" means "He is young and he is thin". We could write it the second way without trouble, but the first way is shorter, we agree that they mean the same thing, and we prefer to conserve energy when writing.
The form "if and only if" is merely a convenient shorthand. Shorthand is usually more convenient for the writer than for the reader. 🤷♂️
Imagine these natural language sentences and analyze how they are different:
I'll go outside if it's not snowing
I'll go outside only if it's not snowing
Hint: what do you do in each case when it's snowing?
"If A then B" means if A is true, then B is guaranteed to be true. Note that if B is true and A is false, "if A then B" is still true.
"B only if A" means the only way for B to be true is for A to be true. It's weird, but it has the inverse truth table as "(not A) and B".
No.
B iff A is defined as "If B then A and if A then B".
If that doesn't make it clear enough for you, then try writing out the truth table for both statements.
I think I understand now, but what has left me scratching my nose (metaphorically):
Why is it called "B if and only if A", if what it really means is "B only if A and vice versa"? (Am I correct in thinking that's what it means?)
I just don't understand how that translates grammatically. To me, "B if and only if A" sounds the same as "B only if A". I can accept that they mean different things in the context of logic, just like I can assign any meaning to any label, like I could say that "dog" now means "kite" in a certain context. But it seems unintuitive and doesn't really make sense to me. Does that make sense?
"B if and only if A" is a shorthand way to write "B if A and B only if A". It's like how "He is young and thin" means "He is young and he is thin". We could write it the second way without trouble, but the first way is shorter, we agree that they mean the same thing, and we prefer to conserve energy when writing.
The form "if and only if" is merely a convenient shorthand. Shorthand is usually more convenient for the writer than for the reader. 🤷♂️
Imagine these natural language sentences and analyze how they are different:
Hint: what do you do in each case when it's snowing?
"If A then B" means if A is true, then B is guaranteed to be true. Note that if B is true and A is false, "if A then B" is still true.
"B only if A" means the only way for B to be true is for A to be true. It's weird, but it has the inverse truth table as "(not A) and B".