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“Not necessarily” versus “necessarily not”: can someone explain the difference with an everyday example?

Thamarai Raman Fellow
· 4 replies
Modal & non-classical logicAnalyticIndianJainism#modal-operators#necessity#possibility#scope
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My older child, who is nine, has started catching me out on words.

Yesterday I said "it won't necessarily rain at the picnic". She said, "So it necessarily won't rain?" I said no, that's different. She said, "How?" and I realised I couldn't explain it cleanly. I tried to look it up and found symbols, boxes and diamonds, and got more confused.

Could someone explain:

  1. The difference between "not necessarily P" and "necessarily not P", in plain words.
  2. What the box and diamond symbols mean.
  3. Whether "not necessarily" is the same as "possibly not".

Examples a nine-year-old would accept are especially welcome. She is a tough audience.

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4 answers

Accepted answer

  1. Tomasz Szewczak

    ✓ Accepted answer Fellow

    Your daughter has found the most common confusion in the whole subject, so she's in good company.

    Logicians write □P for "necessarily P" and ◇P for "possibly P". Then:

    • "Not necessarily rain" is ¬□(rain). It denies that rain is guaranteed. It leaves open that it rains.
    • "Necessarily not rain" is □¬(rain). It says rain is ruled out entirely.

    The position of "not" is everything. In the first, "not" applies to the necessity; in the second, it applies to the rain.

    For a nine-year-old: "You won't necessarily get dessert" means dessert isn't guaranteed; you might, you might not. "You necessarily won't get dessert" means there is no way you are getting dessert. Children grasp this difference very quickly when dessert is involved.

    And yes, to question 3: "not necessarily P" is equivalent to "possibly not P". If rain isn't guaranteed, then it's possible it won't rain. Likewise "not possibly P" is the same as "necessarily not P". The box and diamond are each other's mirror images, with a "not" on each side.

    Helpful · 1
    1. Thamarai Raman

      Fellow

      The dessert example worked immediately, thank you, Tomasz. She then asked whether "I will possibly not tidy my room" is a legitimate answer. I said that was a different branch of logic.

      Helpful · 3

Other answers

  1. Hsieh Wei-ting

    Fellow

    Game designer here: we use this daily without the symbols. "This level is not necessarily winnable" (some players will get stuck) versus "this level is necessarily unwinnable" (a bug, file a ticket). Mixing them up has caused real arguments in our studio.

    Helpful · 2
  2. Hetal Shah

    Fellow

    A Jain footnote, since it's close to my heart. Jain logic qualifies every claim with syāt, often translated "perhaps" or "in some respect": in some respect it is, in some respect it is not, and so on. It is tempting to read that as a modal logic of possibility. Many scholars think "from a certain standpoint" is the better reading, which makes it about perspectives rather than possibilities. Either way, my grandmother would have told your daughter that "it won't necessarily rain" is the wise answer about any picnic.