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17.3 Difficult Translations

In this section we look at  a couple of difficult but important expressions of conditionality.

Necessary and sufficient conditions are conditional claims.

In science, math and philosophy you will often encounter claims about necessary or sufficient conditions:

  • Being a horse is sufficient for being a mammal.
  • Being a mammal is sufficient for being an animal.
  • The ability to reproduce is necessary for the survival of a species.
  • Being divisible by 2 is necessary and sufficient for being even.

All of these expressions can be translated with conditionals.

X sufficient for Y: X->Y.

If being a horse is sufficient for being a mammal, then anything that is a horse is also a mammal.

Or, to put it in conditional terms: if Ed is a horse, then Ed is a mammal. So sufficiency is just our conditional arrow.

Now think carefully about the next one before answering:

Here’s why “necessary” and “sufficient” can be tricky to translate into PROP: they are both conditionals, but they go in opposite directions.

P is sufficient for Q is P->Q

P is necessary for Q is Q->P

That means that “P is sufficient for Q” and “Q is necessary for P” are really two ways of expressing the same conditional. It might sound like those expressions in English mean different things, but the way they are used in science and mathematics, they are equivalent, so we will translate them the same way.