# Necessary and sufficient condition

Necessary condition and sufficient condition are terms from the theory of scientific explanations that divide conditions into two different types. The different relationships between the conditioned and the conditioned are also dealt with in logic, especially propositional logic .

Understood causally , both terms concern the question of whether certain events are irreplaceable as causes of other events, and whether the other events would inevitably occur if the particular events were to occur ( see also counterfacticity ).

## Necessary condition

Ground beans are a necessary condition for making coffee - you cannot do without them. But they are not sufficient.

From a propositional point of view, a necessary condition for a statement is a statement that must necessarily be true (fulfilled) if is true. So it does not happen that is fulfilled without that being fulfilled. ${\ displaystyle B}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle B}$

The connection is expressed by the symbolic notation , i.e. “K implies B” or “from K follows B”. The arrow that symbolizes the context stands for the possible conclusion . When it is certain that it is fulfilled, one can be sure that it is also fulfilled; so it can be closed from on . It is immaterial whether temporally before or after taking place. Often the point is to draw conclusions about the previous conditions from the existence of a conclusion. Are there several necessary conditions i. H. holds , all must be fulfilled at the same time, if is fulfilled ( logical conjunction ):${\ displaystyle K \ Rightarrow B}$${\ displaystyle K}$${\ displaystyle B}$${\ displaystyle K}$${\ displaystyle B}$${\ displaystyle K}$ ${\ displaystyle B}$${\ displaystyle K}$${\ displaystyle B_ {1}, B_ {2}, \ dotsc}$${\ displaystyle K \ Rightarrow B_ {1}, K \ Rightarrow B_ {2}, \ dotsc}$${\ displaystyle K}$${\ displaystyle K \ Rightarrow B_ {1} \ land B_ {2} \ land \ dotsb}$

Are there different from each other logically independent, necessary conditions so that for all pairs of conditions to apply, so no one can by itself sufficiently to be, as this would contradict the that the others are necessary. A necessary condition is therefore irreplaceable for an event to occur. But if it is not sufficient at the same time, it alone is not enough for the event to occur. In other words: Without it it is not possible (hence the Latin term conditio sine qua non , see also conditio-sine-qua-non formula ), but something else may be necessary for the occurrence of . ${\ displaystyle \ lnot (B_ {j} \ Rightarrow B_ {k})}$${\ displaystyle j \ neq k}$ ${\ displaystyle K}$

## Sufficient condition

Hay to eat, satisfy hunger hamster sufficiently, but the hay is known for its saturation unnecessary. It can also be achieved in other ways (e.g. with carrots).

A sufficient condition inevitably ensures (or at least ceteris paribus ) the occurrence of the conditional event. If the condition is not necessary at the same time, then there are other sufficient conditions that also lead to the occurrence of the event. The sufficient, unnecessary condition is therefore replaceable or circumventable (multiple satisfiability). In other words, if there is a sufficient condition, then the conditional event inevitably occurs. If the event has already occurred, however, conclusions can only be drawn about its necessary conditions, because if a sufficient condition considered is not necessary, there must always be other possible conditions that are just as sufficient. Which of the sufficient conditions exists cannot be decided based on the conditional event.

From a propositional perspective: If a statement has several sufficient conditions , i. H. apply Subordinators so, it is sufficient that at least one is satisfied ( logical disjunction ) so that the following applies: . ${\ displaystyle K}$${\ displaystyle B_ {1}, B_ {2}, \ dotsc}$ ${\ displaystyle B_ {1} \ Rightarrow K, B_ {2} \ Rightarrow K, \ dotsc}$${\ displaystyle K}$${\ displaystyle B_ {1} \ lor B_ {2} \ lor \ dotsb \ Rightarrow K}$

## Equivalent condition

A condition that is both necessary and sufficient is called an equivalent condition. Is logical statements for the abbreviation iff - English. if and only if usual; German-language equivalents are g. d. w. , abbreviated for if and then and only then , formula symbols . ${\ displaystyle \ Leftrightarrow}$

For every conditioned there can be only one single simultaneously necessary-and-sufficient condition. If there were alternative sufficient conditions, it would not be necessary; if there were additional necessary conditions, it would not be sufficient. Condition and conditioned are thus in the logical relation of the biconditional : they are equivalent. ${\ displaystyle A \ Leftrightarrow B}$

## Propositional context

Necessary and sufficient condition are closely related. In propositional logic, (pronounced “K implies B”) means both ${\ displaystyle K \ Rightarrow B}$

• " Is a sufficient condition for " as well${\ displaystyle K}$${\ displaystyle B}$
• " Is a necessary condition for ".${\ displaystyle B}$${\ displaystyle K}$

In propositional logic, necessary and sufficient conditions alone do not allow any further conclusions to be drawn about the type of relationship between condition and conditional. This requires further considerations and often empirical studies; see also Paradoxes of Material Implication .

## INUS condition

The INUS condition of the Australian philosopher John Leslie Mackie represents a nested concept: What is meant is an insufficient but necessary part of an unnecessary but sufficient condition. This concept is intended in particular to do justice to the knowledge that it is seldom possible to identify equivalent conditions for empirical events, even under ceteris paribus clauses.