# Quantificational Logic Question (Determining Truth-Values)

I was working through these questions and I just wanted to verify my answers since I don't have access to any solutions and I wanted to make sure I was on the right track:

Let the following be the non-logical symbols of CQL (classical quantificational logic): IS = {a, b, c}, VR (variables) = {x,y,z}, RS1 (one-place predicates) = {F,G,H} and RS2 (two-place predicates) = {R,S,T}.

1. Determine the truth-value of the formulae listed below, in light of the following model: M = ⟨U, i⟩, where U (universe of discourse) = {1, 2} and i: a􏰀→1; b 􏰀→ 2; S 􏰀→ ∅; R 􏰀→ {⟨1, 1⟩, ⟨2, 2⟩}

(a) ∀x∀y(Sxy → Rxy); Proposed Answer: True since nothing satisfies S so the antecedent is always False.

(b) ∀x∀y((Rxy ∧ Sxy) → Syx); Proposed Answer: True (same reasoning as a)

(c) ∀x∀y((Rxy ∧ ¬Sxy) → Ryx); Proposed Answer: True

(d) ∀x∀y((Rxy ∧ Ryx) → ∃z(Rxz ∨ Syz)); Proposed Answer: True

1. State the truth-value of the formulae listed below, in light of the following model: M = ⟨U, i⟩, where U = Z+10 (that is, the natural numbers 1 through 10) and R and S are interpreted as < and ≤ respectively (that is, Rab is interpreted as a < b and Sab is interpreted as a ≤ b).

(a) ∀xSxx; Answer: True, since every number is equal to itself.

(b) ∀x∀y(Rxy → ¬Ryx); Answer: True

(c) ∀x∀y∀z((Rxy ∧ Ryz) → Rzx); Answer: False

(d) ∃x∀ySyx; Answer: True (the natural number 10 satisfies this formula)