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In ordinary English (also natural language) "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being a man is a necessary condition for being a brother, but it is not sufficient—while being a man sibling is a necessary and sufficient condition for being a brother.
The absence these conditions guarantees the outcome cannot occur, and no other condition can overcome the lack of this condition. Further, necessary conditions are not always sufficient. For example, AIDS necessitates HIV, but HIV does not always cause AIDS. In such instances, the condition demonstrates its necessity but lacks sufficiency.
However, only the occurrence of the necessary condition x may not always result in y also occurring. [2] In other words, when some factor is necessary to cause an effect, it is impossible to have the effect without the cause. [3] X would instead be a sufficient cause of y when the occurrence of x implies that y must then occur.
3. P is not necessary but it is sufficient: If (P OR Q) then S. We don't NEED P. We can get S from Q. But on the other hand if we DO have P, P is enough in itself to create S. No need for anything else. 4. P is not necessary AND not sufficient: If (P OR Q) AND R then S. P is not necessary. We can get S from Q even though we do not have P.
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
Example 1. One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example: If someone lives in San Diego, then they live in California. Joe lives in California. Therefore, Joe lives in San Diego. There are many places to live in California other than San Diego.
An example of a necessary condition that is used for finding weak extrema is the Euler ... The Euler–Lagrange equation is a necessary, but not sufficient, ...
In philosophical terminology, "cause" can refer to necessary, sufficient, or contributing causes. In examining correlation, "cause" is most often used to mean "one contributing cause" (but not necessarily the only contributing cause). Dinosaur illiteracy and extinction may be correlated, but that would not mean the variables had a causal ...