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What is a necessary vs. a sufficient condition in logic?

itsemmarobynitsemmarobyn Legacy Member
in General 272 karma

I'm just starting the intro to logic section and I'm a bit confused. What is a sufficient condition? What is a necessary condition?

Comments

  • jknaufjknauf Alum Member
    edited June 2017 1741 karma

    Disclaimer: I wrote a lot. Don't let this overwhelm you!

    A sufficient condition is something that implies if we have this thing, a necessary condition must come along with it.

    If it rains, the ground must be wet.

    A necessary condition is something that must be true when something else is true.

    (Sufficient) If it rains, the ground must be wet (Necessary)

    So if it rains, it must be true that the ground is wet! Why? because our sufficient condition guarantees the necessary. But what if the ground is wet? Must it be true that it rained? No! Just because our necessary condition is met, it does not mean the sufficient follows. But, if the sufficient condition is met, our necessary condition must follow.

    By virtue of this logical relationship, our contrapositive must also be true.
    If the ground is NOT wet, It must NOT have rained.

    And to dive a little further than just sufficient and necessary conditions, lets look at sufficient and necessary assumptions.

    A sufficient assumption is a missing assumption which, if provided, will take our argument to complete 100% validity

    Premise 1: All Jedi use the force
    Conclusion: All Jedi eat chicken wings

    Wait! How can we conclude all Jedi eat chicken wings? We aren't given any information about chicken wings..

    Well, we are missing a premise. Or rather, the author of our argument is making an assumption which isn't stated in our argument. If we supply a sufficient assumption, it will make the argument 100% valid. So what can our assumption be? Well, we have a couple of choices. But to keep it simple I'll just use the basic argument form.

    If we are given the premise, All force users eat chicken wings, does that mean all Jedi eat chicken wings? Yes!

    Premise 1: All Jedi use the force.
    Premise 2: All force users eat chicken wings (sufficient assumption)
    Conclusion: All Jedi eat chicken wings

    More abstractly, our argument looks like this.

    Premise 1: A-->B
    Premise 2: B-->C (sufficient assumption)
    Conclusion: A-->C

    or

    Premise: A-->B-->C
    Conclusion: A-->C

    Necessary assumptions are a little more abstract which tends to make them slightly more difficult. Also, necessary assumptions aren't required to get to the level of 100% validity, so keep that in mind.

    So if I make the argument:

    @itsemmarobyn is the fastest, strongest, and most intelligent athlete ever.
    Therefore, @itsemmarobyn is the best basketball player ever. (Crazy because we all know Lebron is, GO CAVS!)

    So what assumptions are necessary to make this argument?

    Well, there is a bunch. I must assume you know how to play basketball. To go a little deeper, I must assume you know how to dribble a basketball, I must assume you know how to shoot a basketball et cetera.

    Even though these assumptions won't move our argument to the level of 100% validity, I need to make these assumptions.

  • SamiSami Alum Member Sage 7Sage Tutor
    10721 karma

    Hey,

    A sufficient condition implies that it is enough for something else to also be true.
    For example in this statement: all girls are smart.
    Being a girl is enough for you to declare that it also has the quality of being smart.

    So if I say Rita is a girl. That information about Rita being a girl is enough for you to conclude Rita is also smart.

    Necessary condition on the other hand is something that has to be true when something is true.
    So in our example above smart is necessary for being a girl. That means if someone is not smart, they are not a girl.

    If you need more information about this check out these lessons on 7sage. They do a great job at explaining.
    https://7sage.com/lesson/sufficiency-necessity-what-does-the-arrow-mean/
    https://7sage.com/lsat-logic-sufficient-and-necessary-conditions/

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