Intuitively, why is a Necessary Condition, weaker than and opposite of, a Sufficient Condition?

Question

Abbreviate Modus Tollens to MT, Necessary Condition to NC, and Sufficient Condition to SC.
_{I pursue only intuition; please do not answer with formal proofs or Truth Tables.}

1. It is the negation that does the trick. Think of a "condition" as a restriction on the class of things that satisfy it, the stronger the restriction the narrower the class. Normally NC are weaker than SC, but negation always reverses the order of strength:
¬{a weaker condition} is always stronger than ¬{a stronger condition}.

2. (3rd last ¶) The opposite of a SC is a NC, and vice versus.

Please aid me in diagnosing my problem.

Answer 1

In general, we have

SufficientConditionForX => X
X => NecessaryConditionForX

so we can see that they behave in opposite ways in regards to how statements imply one another.

Given the question, I think you want to apply this to how negation reverses implication. Recall that the following are equivalent:

• X => Y
• ¬Y => ¬X

So we have

¬X => ¬SufficientConditionForX
¬NecessaryConditionForX => ¬X

or otherwise

• ¬SufficientConditionForX is a necessary condition for ¬X
• ¬NecessaryConditionForX is a sufficient condition for ¬X

Answer 2

Necessary and sufficient are not opposites: at least not in the sense of logic.

If 'A' is a necessary condition for 'B', then whenever 'B' is true, so is 'A'. If 'C' is a sufficient condition for 'B', then whenever 'C' is true, then so is 'B'. But necessary conditions can be too broad and sufficient conditions can be too restrictive.

I will provide examples from the perspective of Euclidean geometry. Consider the following three definitions

• (R1) a rectangle is a quadrilateral all of whose angles are right angles.
• (P1) a parallelogram is a quadrilateral whose opposite sides are parallel.
• (S1) a square is a regular quadrilateral (regular means that all sides are congruent and all angles are congruent).

Then P1 is a necessary condition for R1. S1 is a sufficient reason for R1.

But, not all parallelogram are rectangles (hence the necessary condition is too broad). And there are rectangles that are not squares (hence the sufficient condition is too restrictive). However, it is possible for a condition to be simulanteously necessary and sufficient. For example

• (E1) All the angles of the quadrilateral are congruent.

If you have 'E1', then you have a rectangle (thus E1 is sufficient). If you have a rectangle, then 'E1' is true (thus E1 is necessary).

I hope this clarifies the differences between these two concepts.

Answer 3

(I am not sure this answers the question 'Why?', but hopefully it normalizes the feeling and gives it some context.)

This happens, grammatically, because necessity and sufficiency are a complementary pair of modes.

Modal verbs have natural complements. Should, and May (in the most proper sense, not the sense of ambiguity.) are complementary: What one is not obligated not to do is something one is allowed to do. In a short form dodging the risk implicit in English negations: not(should(not X)) = may X.

So are Can and Will (again in the most proper sense): What one is not capable of not doing is something one is predictably always going to do. Parallel to the above, not(can(not X)) == will X.

In this case, not(must(not X)) = might X. Sufficient conditions can always be put in the form "whenever X happens Y must happen". And necessary conditions can always be put in the form "only when X happens might Y happen." So from the sufficient condition "whenever X does not happen, Y must not happen" we can deduce the necessary condition "only when X happens, might Y happen".

These are complementary (or dual), not opposite; because in this case the two negations do not get you back to the original statement. You end up, instead, in the complementary stance. The traditional notation for this is ¬[]¬p = <>p, (as distinct from ¬[]p = <>p, which is false). Here the box represents the more restrictive of two complementary modes and the diamond the more constructive one.

Unfortunately, sticking with this basis in the language is difficult because our use of modality slides around all the time. In particular, the phrasing of necessary conditions varies because "can" and "may" are often used to express "might", and vice versus. We simply are not consistent enough to be specific. (The Germans lent us nice things to play with, and look what we have done with them.)

Answer 4

Sufficient conditions are not the opposite of necessary conditions. A necessary condition is one which cannot be lacking for something to happen. However, it being present doesn't entail that it does happen. A sufficient condition is one which could be absent and the event still happen, but it being present is enough to make the event happen (in the presence of all necessary conditions).

Answer 5

If X is true, then a necessary condition N must be true.
If N is false, then X must be false.

If a sufficient condition S is true, then X is true.
If X is false, then S must be false.

The possible combinations for N, X and S are: 1. (false, false, false), 2. (true, false, false), 3. (true, true, false), 4. (true, true, true).

So 1 and 4 are the two cases where N is not the opposite of S.