Advanced Symbolic Logic
Relational Predicate Translations with Identity
H. Hamner Hill
Department of Political Science, Philosophy, and Religion
Southeast Missouri State University

1. Everyone envies exactly one person.
(x)(y)[Exy (z)(Exz z = y)]

2. Everyone envies exactly one other person.

(∀x)(y)[(Exy x y) z)(Lxz z = y)]

3. Everyone trusts only her/himself.

(x)[Txx (y)(y x -Txy)]

4. At most, everyone trusts only her/himself.

(x)(y)[Txx (y)(Txy x = y)]

5. Everyone lies to someone sometime.

(x)(y)(z)(Tz Lxyz)
{Tx = x is a time; Lxyz = x lies to y at z}

6. Some people lie only to other people.

(x)(y)(Lxy x y)

7. Only Carla has sympathy for Arturo (C does have sympathy for A).

Sca (x)(Sxa x = c)

8. Every planet except the earth is uninhabitable.

Ie ()[(Px x e) -Ix]

9. At least one woman ran for Vice-President of the United States.

(x)(Wx Rx)

10 At most two women have been President of the American Philosphical Assoc.

(x)(y){[(Wx Px) (Wy Py)] (z)[(Wz Pz) (z = x v z = y)]}

11 Everyone admires the most intelligent person in the world.

(x)[(y)(x y Ixy) (z)Azx]
{Ixy = x is more intelligent than y}

12 The most intelligent person in the world is also the most admired.

(x)(y){[(z)(x y Ixz) (u)[(y = u Ayu) x = y]}

13 The most intelligent person in the world admires only intelligent people.

(x)[(y)(x y Ixy) (z)(Axz Iz)]

14 The person most admired by Arturo is also admired by Carla.

(x)[(y)(x y Aaxy) Acx]
{Axyz = x admires y more than z}

15 Carla's father admires her.

(x)[(Fxc Axc)

16 Carla's father admires the most intelligent person in the world.

(x){(y)(x y Ixy) (z)[Fzc (u)(Fuc u = z) Azx]}

17 Every woman admires a well-groomed man.

(x){Wx (y)[(My Gy) Axy]}
{Gx = x is well-groomed}

18 No ill-groomed man admires any ill-groomed man.

(x){(-Wx Mx) [(y)(-Wy My) -Ayx]}

19 Some, but not all, weak-willed husbands are respected by their wives.

(x)(y)[(Wx Hx) (Wyx Rxy)] (x)(y)[(Wx Hx) (Wyx -Rxy)]
{Wx = x is weak-willed; Wxy = x is a wife to y; Rxy = x is respected by y}

20 Every logical positivist thinks all metaphysical propositions are meaningless.

(x)[Lx (y)(My Txy)]
{Txy = x thinks y is meaningless}

21 No two idealists agree on the nature of reality.

(x)(y){[(Ix Iy) x y] -Axy}

22 No three surfers agree on the nature of the perfect wave.

(x)(y)(z)((((Sx Sy) Sz) ((x y x z) y z))) ((-Axy -Axz) -Ayz))

23 No two sets of fingerprints are identical.

(x)(y){[(Fx Fy) x y] x y}
{Fx = x is a set of fingerprints}