# FOL Translation

I'm learning first order logic right now and I'm stuck on a translation.

Key:

P(x) : x is a planet

M(x) : x is a moon

S(x) : x is a star

O(x,y) : x orbits around y

1. Only stars have planets orbiting around them.

My attempt:

Ax (x is a star and y orbits x -> y is a planet)

AxAy(S(x) /\ O(y,x)) -> P(y))

1. Every moon orbits around some planet.

Ax (x is a moon -> x orbits some planet y)

Ax (M(x) -> Ey (P(y) /\ O(x,y)))

Are these correct? Note A is "for all" and E is "there exists". Thanks

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1. Only stars have planets orbiting around them.
AxAy(S(x) /\ O(y,x)) -> P(y))
1. "Only A B" means "if B then A", not "if A then B": "Only stars have planets which orbit them" translates as `if there is a planet orbiting x, then x must be a star`, which by the truth table of `->` is false exactly when something has a planet orbiting it but is not a star, but can still be true for objects which are stars but don't have orbiting planets (because "only" means that at most stars have such planets, not necessarily all of them). But `if x is a star, then x has an orbiting planet` makes these predictions precisely the wrong way round. So antecedent and succedent in the implication have to be switched.
2. You got the connectives the wrong way round: The statement should be `if y is a planet that orbits x, then x must be a star` (which means that planets orbit only starts) , not `if x is a star and y orbits x, then y must be a planet` (which would mean that stars have only planets as orbiting objects).

This gets you

`all x all y ((P(y) ^ O(x,y)) -> S(x))`

1. This is correct, but a little unintuitive: Instead of moving the existence of a planet out of the implication to a universal quantification (which makes for a logically equivalent statement), keep the quantifier where the English sentence has it:

`all x (exists y (P(y) ^ Q(x,y)) -> S(x))`
("For all objects, if there exists a planet which orbits it, then it must be a star.")

1. Every moon orbits around some planet.
Ax (M(x) -> Ey (P(y) /\ O(x,y)))

Correct.

• Thank you for the insightful reply. – Ford Wendy Nov 8 '20 at 23:59