What does "only if" mean exactly?

Question

I learned the followings; 'if', 'only if', and 'if and only if'.

I understood like the followings;

'$A$ if $B$' means that 'That $B$ holds implies that $A$ holds'

'$A$ only if $B$' means that 'That $A$ holds implies that $B$ holds'

'$A$ if and only if $B$' means that 'That $B$ holds implies that $A$ holds. At the same time, that $A$ holds implies that $B$ holds'

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However, in the textbook states 'DP can be applied numerically only if the dimensions of the spaces are relatively small.'

I do not understand why the author uses only if instead of if.

Is there any exception of these notations with respect to 'if' that I do not know?

Answer 1

"A if B". When B is true then A must be true (sufficient).

"A only if B". When B is false then A must be false (necessary).

"A if and only if B". When B is true A must be true, When B is false A must be false. (necessary and sufficient)

"if" allows A to be any truth value when B is false. "only if" allows A to be any truth value when B is true.