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"Sufficient" and "necessary" conditions refer to the antecedent and consequent, respectively, of a conditional statement. In other words, when you have an "if-then" sentence, phrase, etc., what you're saying is that that "if" particle, when true, is by itself sufficient to guarantee the truth of the "then" particle. Let's use an example:
"If you were born in the 1950s, then you're a baby boomer."
So, we have our sufficient condition, "born in the 50s," and we're asserting that that claim, if true, means that anyone who meets that criterion is a baby boomer. Let's illustrate this with a syllogism:
Anyone born in the 1950s is a baby boomer. (A--->B)
Susan was born in 1958. (A)
Therefore, Susan is a baby boomer. (B)
Because being born in the 50s is sufficient to establish that a person is a baby boomer, we can say with certainty that Susan is a baby boomer. Now, let's see what happens when we deny the sufficient condition:
Anyone born in the 1950s is a baby boomer. (A--->B)
Susan was born in 1961. (~A)
Therefore, Susan is not a baby boomer. (~B)
Is this a valid argument? No, because people born in the early 60s are still considered baby boomers. I.e., being born in the 50s is sufficient to establish that someone is a boomer, but it's not necessary. I know this was a pretty long explanation, so I hope it's helpful.
LSAC official says this isn't true: https://www.reddit.com/r/LSAT/comments/1avud0y/addressing_the_august_2024_and_beyond_lsat_rumors/