I am having trouble translating this sentence into Predicate Logic: "Anything a knave claims is false" (or "If x is a knave and x claims y, y is untrue.") I have pasted two attempts, along with a key, below:
Kx: x is a knave, Uy: y is untrue, Cxy: x claims y
∀x(Kx⟶∀y(Cxy⟶Uy))
∀x∀y((Kx & Cxy)⟶Uy)
Is any one of them correct? If not, how would you correctly translate the sentence?
Thank you in advance for your time, help, and consideration. I really appreciate it.
It is unusual to use a predicate such as U to indicate untrue. In classical logic a proposition is untrue if and only if its negation is true. So, unless you are trying to do something cute with the semantics of truth and falsehood, it would be better to replace Uy with ¬y. That said, both your formulations are correct.
However, there is another issue. Your variable y is quantifying over propositions rather than things, so your expressions are not first order logic, but second order. If you want to translate your sentence into first order predicate logic, you could express it as an extended conjunction:
(∀x)(Kx → ((CxP → ¬P) ⋀ (CxQ → ¬Q) ⋀ (CxR → ¬R) ...))
This, of course, is unwieldy because there is no obvious limit to the extent of the conjunction. If you choose to live with the second order version you could run into paradoxes such as a version of the liar paradox, but then I suspect that is the intention.
