Sentential Logic Question: Translating "at most"

Question

B: Barack will make it to the top of the peak

D: Dallas will make it to the top of the peak

A: Anise will make it to the top of the peak

R: Rod will make it to the top of the peak

Between Barack, Dallas, Anise, and Rod, at most two of them will make it to the top of the peak

Answer 1

The way to think about this in sentential logic is that if you have 4 statements, and at most 2 are true, then at least 2 are false. The way to say that at least 2 statements are false is to enumerate the ways that can happen, and say that at least one of those ways is true. In this case, that would be:

(¬B ∧ ¬D) ∨ (¬B ∧ ¬A) ∨ (¬B ∧ ¬R) ∨ (¬D ∧ ¬A) ∨ (¬D ∧ ¬R) ∨ (¬A ∧ ¬R)

If you were doing this in quantificational logic, where you schematize the sentences as Pb, Pd, Pa, and Pr respectively, then the way to think about this is that if 3 people reach the top of the peak, then at least two of them are actually the same person:

(∀x)(∀y)(∀z)((Px ∧ Py ∧ Pz) ⊃ (x=y ∨ x=z ∨ y=z))