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Consider the following argument:

Turtles see everything. Seeing is asymmetric (for the sake of argument). Therefore, something is not a turtle.

I have problems symbolizing these statements.

My attempt:

Predicates:

Tx: x is a turtle

Sxy: x sees y

Premises:

1) (∃xTx) Λ ∀x∀y(Tx → Sxy)

2) ∀x∀y (Sxy → ¬Syx)

Using this set-up I cannot deduce the conclusion, which is ∃x¬Tx.

I feel like my set-up is wrong. I think I should be able to complete the proof after correctly symbolizing the statements. Please let me know how to proceed.

First, your premises are inconsistent: your second premise implies that turtles do not see other turtles, or themselves, yet, according to the first premise, they see everything. So, taking y=x, we can deduce both ∀xSxx and ∀x¬Sxx. After that you can deduce whatever you want directly using the law of explosion, contradiction implies anything.

If you fix this, e.g. by specifying that turtles see everything but turtles, your desired conclusion will not be deducible. This is because in the standard predicate calculus the universal quantifier does not have what is called existential import. In other words, ∀yP(y) → ∃yP(y) is generally false. So ∀x∀y(Tx → Sxy) may hold even if there is no y for x to see. It is interpreted conditionally: if there was a non-turtle y then any turtle x would see it. Hence, your modified premises will hold even in a world of turtles only. Just as you specified that your world is not empty, with ∃xTx, you'll have to specify explicitly that it is not devoid of non-turtles, with ∃x¬Tx. Or, you can add existential import as a separate axiom, and apply it to P(y)=∀x¬Syx. Then ∃y∀x¬Syx, together with the first premise, will give you ∃y¬Ty.