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Contrapositives of Conjunctions and Disjunctions

evanescenceevanescence Core Member
edited August 2023 in General 194 karma

Hello!
I understand that when doing LG, to help chain up the rules, sometimes you need to contrapose conjunctions and disjunctions ("and" turns into "or," and "or" turns into "and"). However, I'm struggling to remember how the flipping and negating goes. Does anyone have any tips on this?

E.g.
Original: G and S --> /I
Contrapositive: I --> /G or /S
Am I doing this correctly?

OR is it:
/G or /S --> I
(Do we flip conjunctions/disjunctions when contraposing, or just negate?)

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Comments

  • rdyoung12rdyoung12 Alum Member
    306 karma

    I think

    I --> /G or /S

    is right.

    Maybe think about like a more real situation. Perhaps,
    "If the sun is down, then I'm tired and hungry"
    SD -> T and H
    "If i'm not tired nor hungry, then the sun is not down"
    /T or /H -> /SD

    So if either one of the necessary conditions is broken, then it's impossible for the sun to be up? Correct me if I'm wrong but it kinda makes sense when you think of it like that. Feel free to critique or correct me.

    ( https://7sage.com/lesson/negating-conjunction-and-disjunction-de-morgans-laws/?ss_completed_lesson=25317 )

  • markymymarkymarkmarkymymarkymark Alum Member
    edited August 2023 50 karma

    Hi, the first example you did was correct.

    Flip and negate first, then switch the "and"/"or".

    1. A -> B or C

    /B and /C -> /A

    In order to fail A, you must fail BOTH B and C.

    1. A -> B and C

    /B or /C -> /A

    In order to fail A, you must fail ONE OF B or C (you can fail both, but at least one must be failed)

    1. A and B -> C

    /C -> /A or /B

    If you fail C, then you cannot have both A AND B (because if you did, then the rule triggers). So like #2, one of A or B must be failed (although failing both A and B is always an option)

    1. A or B -> C

    /C -> /A and /B

    Since we only need one of A or B to trigger C, if we don't have C we CANNOT have A nor B, because if we do, the rule triggers.

    Hope this helps!

  • evanescenceevanescence Core Member
    194 karma

    Thank you @rdyoung12 and @markymymarkymark !!!

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