I have had to flesh out the logic of this myself. (after a late night of studying necessary and sufficient conditions for LSAT purposes, no less. I was a philosophy major and I couldn't just let it be a surface understanding). First, to all you legitimate technicians on this forum (whether by credential or study) if I have made some errors here please elucidate. (By necessity, this information is in the simplest terms possible)
It comes down to the obvious difference between the normal (read "logically unnecessary") way we express conditional statements in everyday arguments, and the INVERSE RELATIONSHIP between sufficiency and necessity.
We NORMALLY say "If _____, then ______ ." This has its analogue in propositional logic (which you should think of as necessarily different than the way we speak) as material implication and the rule of inference that makes the implication valid is Modus Ponens: P → Q (If P, then Q). Modus Ponens explicitly shows that P is sufficient for Q.
Ok, I know you understand that, but just bear with me.
NECESSITY AS A RULE OF INFERENCE FROM SUFFICIENT CONDITIONS:
What Modus Ponens does not explicitly show is that with P being sufficient with Q, we can infer that Q is necessary for P. We must use transposition and the second rule of inference, Modus Tollens, to explicitly state necessity. Now, we are taught that P implying Q is written or read as "If P, then Q", but we may miss the step of inference--which is also a rule of inference--that allows up to see that P implying Q can also be written or read so as to indicate the necessary relationship of Q to P (Q is necessary for P) as 'Only if Q, P' and 'P, only if Q'. When we write sufficient statements as P, if only Q' in order to show necessity we should understand that we are taking what modus tollens proves for granted. Moreover, seeing Modus Tollens as a rule of inference that explicitly states a backward inference to necessity will, I hope, clarify this. (for both of us!)
HERE'S THE TRICK
Classically, in propositional logic, it seems we are taught necessity by seeing Modus Tollens as a valid deduction from the logic of Modus Ponens, and we are taught the rules of transposition, and 'the directions of inference' by seeing how sufficiency is the negation (read 'inverse') of necessity.
The 'first' step becomes Modus Ponens, it is a forward inference to obtain a conclusion; and, it is an explicit relationship of sufficiency that has a necessary relationship in terms of sufficiency, and this necessary relationship requires a deduction, or inference backwards to make necessity explicit: it requires another rule of inference, a deductive rule, to make the necessary relationship explicit.
You have to come to the explicit relationship of necessary statements through transposition and Modus Tollens (MT).
Definition of transposition: a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" to the truth of "Not-B implies not-A", and conversely.
If P, then Q. (premise -- material implication)
If not Q , then not P. (derived by transposition)
Not Q . (premise)
Therefore, not P. (derived by modus ponens)
[Note that even with Modus Tollens, it is never stated that P is necessary for Q (P, only if Q); rather, necessity is seen as a implication using transposition (inversion). I wonder about this, but I think it is because of the the nature of the subject: conditional statements]
Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition.
Now we can see that Modus Tollens is the explicit statement of the necessary relationship implied by material implication and modus ponens.
Modus Tollens can be understood as the proof--and logical path--to these statements:
'Q is necessary for P'
'P only if Q'
'only if Q, P'
'the absence of Q implies the absence of P'
Notice that these are all equal statements: 'If P, then Q' = 'P, only if Q' = 'only if Q, P'. (you should see here that you need to be familiar with the rules more so than the order of the sufficient or necessary condition)
And we can also see from the conditionals AND the deductive reasoning from Modus Ponens to Modus Tollens how transposition works correctly and incorrectly:
Though Modus Tollens is an inference of transposition and a logical deduction from the rule of Modus Ponens that allows us to explicitly state necessity FROM sufficiency, it does not allow us to change the relationship of the statements, in their non-negated forms. To do so is to commit logical fallacies (denying the antecedent and affirming the consequent).
We can transpose necessity onto sufficiency (denying the antecedent), only if we also transpose sufficiency onto necessity (affirming the consequent), which is to say that:
'If P, then Q; AND if Q, then P' = 'P, only if Q; AND Q, only if P' = 'Only if Q, P; AND only if P, Q'.
Importantly, 'if and only if' (the wording that contains the logic behind the transpostioning that leads to biconditionality) can be seen as a logical leap over the two fallacies.
Q = Madison will eat the fruit
P = if it is an apple
1) "Madison will eat the fruit, if it is an apple." (equivalent to "Only if Madison will eat the fruit, is it an apple;" or "Madison will eat the fruit ← fruit is an apple")
2) "Madison will eat the fruit, only if it is an apple." (equivalent to "If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → fruit is an apple")
3) "Madison will eat the fruit if and only if it is an apple" (equivalent to "Madison will eat the fruit ↔ fruit is an apple")
Note that If and only if statements can be viewed as the joining of non-negated, transposed statements that taken separately contradict each the other's necessary and sufficient conditions: in 2, by transposing P for Q (non-negated forms), we make both logical fallacies in one 'fell' swoop because this new information makes 1 a logical fallacy, too. From 2, the sufficient condition (if it is an apple) becomes the necessary condition (only if it is an apple); AND, from 1, vice versa.
So we see that (2), above, can be restated in the form of if...then as "If Madison will eat the fruit in question, then it is an apple"; taking this in conjunction with (1), we find that (3) can be stated as "If the fruit in question is an apple, then Madison will eat it; AND if Madison will eat the fruit, then it is an apple".
This exercise makes me remember the difference and similarities between mathematical logic and putting ideas into words. Both seek to be as efficient as possible: mathematical logic with truth, and the logic of our ideas with fallacies. Well, I guess it depends on the books we read.