sufficient condition vs necessary condition

Question

First sentence:

If you live in Los Angeles, then you live in California

Second sentence:

If you don't live in California, then you don't live in Los Angeles.

I am not sure, but I would like to say that the first sentence is NOT a sufficient condition because it is necessary you must live in Los Angeles to live in California.

Is it correct that the first sentence is NOT a sufficient condition, and the second sentence IS a necessary condition?

Answer 1

Being a sufficient or necessary condition is not an absolute property but a relative property. A proposition is a sufficient or necessary condition for another proposition, not in and of itself. So, saying "the first sentence is (or is not) a necessary condition" is meaningless, without saying for what it is (or it not) a necessary condition. The same for sufficient condition.

In general, in a statement of the form $A \to B$ ("if $A$ then $B$", where $A$ and $B$ are propositions), we say that $A$ is a sufficient condition for $B$, and that $B$ is a necessary condition for $A$. Note that, in particular, saying that $A$ is a sufficient condition for $B$ is equivalent to say that $B$ is a necessary condition for $A$.

In your first sentence, the structure is $A \to B$, where

• $A$ is "living in Los Angeles",
• $B$ is "living in California".

So, according to the first sentence, "living in Los Angeles" is a sufficient condition for "living in California", and "living in California" is a necessary condition for "living in Los Angeles".

The structure of your second sentence is $\lnot B \to \lnot A$, which is logically equivalent to $A \to B$, therefore you can conclude the same thing as in the first sentence.

Answer 2

The whole sentence

If you live in Los Angeles, then you live in California

is not a sufficient condition. Rather, you can say that this sentence expresses that living in Los Angeles is a sufficient condition for living in California. Or, in terms of sentences (or claims or propositions) , you can say that the sentence/claim/proposition

you live in Los Angeles

is a sufficient condition for the sentence/claim/proposition

you live in California

Likewise, the whole sentence:

If you don't live in California, then you don't live in Los Angeles.

is not a sufficient or necessary condition all by itself ... Rather, this sentence expresses that living in California is a necessary condition for living in Los Angeles. Or again, in terms of sentences, the sentence/claim/proposition

you live in California

is a necessary condition for the sentence/claim/proposition

you live in Los Angeles

In sum, it's not that the first sentence is a sufficient condition, but rather that it expresses a sufficient condition relationship: that one thing is a sufficient condition for something else. Likewise, it's not that the second sentence is a necessary condition, but rather that it expresses a necessary condition relationship: that one thing is a necessary condition for something else.

And finally, a technical note: while any 'if $P$ then $Q$' sentence is indeed most naturally interpreted as expressing a sufficient condition relationship, namely that $P$ is a sufficient condition for $Q$, it can also be seen as expressing a necessary condition relationship: that $Q$ is necessary for $P$. For example, suppose I say: "If you have taken Calculus II, then you have taken Calculus I". Why do you think I am able to say that? It's because I know that taking Calculus I is a necessary condition for Calculus II.

Likewise, while any 'if not $Q$ then not $P$' sentence is indeed most naturally interpreted as expressing a necessary condition relationship, namely that $Q$ is a sufficient condition for $P$, it can also be seen as expressing a sufficient condition relationship: that $P$ is sufficient for $Q$.

Answer 3

A simple method :

(1) first , rephrase the sentence you are dealing with using the $\rightarrow$ notation.

Note :

• " if A then B " gives : $(A\rightarrow B)$

• " B, if A " also gives $(A\rightarrow B)$

• " A, only if B " also gives $(A\rightarrow B)$

• " not B , if not A " gives : ( $\neg A \rightarrow \neg B$), which is equivalent, by contraposition , to ( $B \rightarrow A$).

(2) everything that comes before the arrow is a sufficient condition of the whole thing that comes after the arrow ( provided the whole conditional sentence is a true one, unless the " sufficient condition" is only an " alledged sufficient condition").

So ,

• in " ( you live in LA $\rightarrow$ you live in California) " the sentence " you live in LA" is a sufficient condition of " you live in California"

• in " ( you do not live in LA $\rightarrow$ you do not live in California) " the sentence " you do not live in LA" is a sufficient condition of " you do not live in California"

(3) everything that comes after the arrow is a necessary condition of the whole thing that comes before the arrow ( provided the whole conditional sentence is a true one, unless the " necessary condition" is only an " alledged necessary condition").

So ,

• in " ( you live in LA $\rightarrow$ you live in California) " the sentence " you live in California" is a necessary condition of " you live in LA"

• in " ( you do not live in LA $\rightarrow$ you do not live in California) " the sentence " you do not live in California" is a necessary condition of " you do not live in LA"

Note : a good thing to do is to get acquainted with the contrapositon rule, in order to perform transformations on conditional sentences; a conditional can be much clearer once transformed in a proper way