Translate the following sentences into predicate logic. Use

*S**x*for “*x*is a student”,*P**x*for “*x*is a professor”,*R**x**y*for “*x*respects*y*”, and use the equality symbol “=” where relevant. You may assume that quantifiers range over people, so you don’t need any additional symbol for “*x*is a person.” [2 points each]Any professor whom some students respect is respected by some professor.

There is someone who, although not him or herself a student, respects any student who respects no professor.

There is exactly one professor who respects only those students who respect him.

Prove the following sequents. You may use cut and/or replacement with any results, so long as you refer to a fully rigorous proof (e.g. in textbook or a previous pset). [4 points each]

(∀

*x**F**x*→∀*x**G**x*) → ∀*x**F**x*⊢ ∀*x**F**x*Solution: This is just an instance of Peirce’s law, and its validity depends only on its propositional structure.

∀

*x*∀*y*∀*z*(*R**x**y*→¬*R**y**z*) ⊢ ∃*y*∀*x*¬*R**x**y*Solution: As usual, there are several different ways that this result can be proven. For example: use excluded middle to infer ∃y∃xRxy∨¬∃y∃xRxy. The second disjunct implies ∀y∀x¬Rxy and hence ∃y∀x¬Rxy. The first disjunct says that there are two things, say a and b, such that Rab. The first premise then implies that, for arbitrary c, Rca→¬Rab. Therefore ¬Rca and it follows that ∃y∀x¬Rxy.

(I’m fairly confident that this result cannot be proven in intuitionistic logic, and so requires the use of EM or another result derived from the full power of DNE.)

Provide

*two*interpretations to show that the following sentence is a contingency. [4 points]∀

*x*(*F**x*→*G**x*) ∨ ∀*x*(*G**x*→*F**x*)Can there be a correctly written proof with the following line fragments? Justify your answer by reference to the existence (or non-existence) of relevant interpretations, and by citing soundness or completeness. [4 points]

`1 (1) ∀y∃xRxy A 1 (n) ∃x∃y(Rxy∧Ryx)`