1. The Finite Universe Method
Bottom Line Meanings
Where H__ = is happy . . .
(x) Hx = Everything is happy.
(x) Hx = Something is happy.
In a Universe containing only Tom . . .
Everything is happy = Tom is happy
(x) Hx = Ht

2. Something is happy = Tom is happy
(x) Hx = Ht
In a Universe containing only Tom and Dick . . .
Everything is happy = Tom is happy, and Dick is happy.
(x) Hx = (Ht · Hd)
Something is happy = Tom is happy, or Dick is happy.
(x) Hx = (Ht v Hd)

3. In a Universe containing only Tom, Dick, and Harry . . .
Everything is happy = Tom is happy, and Dick is happy, and Harry is happy.
(x) Hx = [(Ht · Hd) · Hh]
Something is happy = Tom is happy, or Dick is happy, or Harry is happy.
(x) Hx = [(Ht v Hd) v Hh]

4. In a Universe of two.
(x) (Lx  ~Qx) =
(La  ~Qa) · (Lb  ~Qb)
(x) (~Nx · Bx) =
(~Na · Ba) v (~Nb · Bb)
(x) (~Gx  Kx) v (x) (Wx · ~Ax) =
[(~Ga  Ka) · (~Gb  Kb)] v [ (Wa · ~Aa) v (Wb · ~Ab)]
(x) ~(Fx · ~Qx) ≡ (x) (~Sx  Tx) =
[~(Fa · ~Qa) v ~(Fb · ~Qb)] ≡ [(~Sa  Ta) · (~Sb  Tb)]

5. In a Universe of Three
(x) ~(~Hx  Gx) =
[~(~Ha  Ga) · ~(~Hb  Gb)] · ~(~Hc  Gc)
(x) (Ox · ~Vx) =
[(Oa · ~Va) v (Ob · ~Vb)] v (Oc · ~Vc)
(x) ~(Fx · ~Gx)  (x) (Cx  ~Zx) =
{[~(Fa · ~Ga) v ~(Fb · ~Gb)] v ~(Fc · ~Gc)}  {[(Ca  ~Za) · (Cb  ~Zb)] · (Cc  ~Zc)}

6. The number of items needed in the universe is always equal to the number of combinations of truth values needed for the Predicates (Capital Letters) to yield all true premises and a false conclusion.