Help with Relational Predication Logic

Question

13. Translate into predicate form. (Relational predication is required.)

Every animal lives somewhere. Mars lives nowhere. So, Mars is not animal.

[A = “is a animal”, L = “lives”, P = “is a place”, M = m “Mars” N = nowhere]

∀x(Ax⟶∃y(Py & Lxy), ∃x(Lx & Nmx) ⊢ ∼Am

14. Translate into predicate form. (Relational predication is required.)

Gluons look like nothing. Everything looks like itself. Hence, there are no gluons.

[G = “gluons”, L = “look like”, I = “itself”,

∀x(Gx⟶∀y∼Lxy) or ∀x(Gx⟶∼∃y Lxy), ∀x(Lx⟶Ix) ⊢ ∼∃xGx

I need help because I am not entirely sure if my translation is correct so any help would be great. Here is an example of correct relational predication translation to help anyone who may be confused.

Relational Predication Example

Answer 1

Some comments:

13:

• Your second statement is unclear. What does $\exists x$ introduce - why having an existential introduction in addition when you are already using a constant name for Mars? Why is $L$ suddenly a one-place predicate when it was tro-place in the first statement? What exactly is the relation $N$ expresses between the two arguments it takes? In general, when giving the translation keys, you should always include all of the arguments the predicate symbol takes, to make clear which argument takes over which role in the relation: e.g. "$Ax$ = x is an animal, $Lxy$ = x lives at y, ..."
• If you formalize "x lives somewhere" as $\exists y (Py \land Lxy)$, then it makes sense to formalize "x lives nowhere" as $\neg \exists y (Py \land Lxy)$. With the first argument to the living-relation ("x") being mars ($m$), your second sentence reads $\neg \exists y(Py \land Lmy)$ (= "There is no thing y such that y is a place and Mars lives at it").
• You could actually omit the place predicate altogether (since "place" doesn't occur in the original English sentence either) and just go with "There is a/no thing such that x lives at it": $\forall x(Ax \to \exists y Lxy), \neg \exists y (Lmy)$.
• Also, are you sure you are supposed to use "$\vdash$" B convention, this means derivability in a formal system, but I suppose you rather want to express logical consequence (the right-hand side logically follows from the left-hand size), which is symbolized as "$\vDash$", or a simple "therefore", as used in syllogisms, which is symbolized as "∴".
• With this, 13 will be:
  $Ax$ = "$x$ is an animal", $Lxy$ = "$x$ lives at $y$", $m$ = "Mars"
  $\forall x(Ax \to \exists y Lxy), \neg \exists y Lmy) ∴ \neg Am$.

14:

• Again, include the predicate arguments in your translations: "$Gx$ = x is a gluon, $Lxy$ = x looks like y, ..."
• There are similar issues with your second sentence as in 13: Why is $L$ now a one-place predicate when it used to be 2-place in the first sentence? What exactly does $Ix$ mean - $x$ is itself? Right now, your second sentence reads "For all x it holds that if it looks like, then it is itself". Instead, You want to express that the look-like relation is reflexive, i.e. that for all x, x looks like x: $\forall x(Lxx)$. You don't need a predicate $I$ for that, but instead express the "itself" by referencing the same variable ($x$).
• So 14 will be $Gx$ = "$x$ is a gluon", $Lxy$ = "$x$ looks like $y$"
  $\forall x(Gx \to \neg \exists y Lxy), \forall x(Lxx) ∴ \neg \exists x Gx$.