39
XII. CHAPTER TWELVE: PREDICATE LOGIC TRUTH TREES
Use the truth-tree method to determine the validity/invalidity of the following arguments.
If invalid, use the constants in an open path to expand the premises and conclusion and
assign truth-values to show the invalidity.
(1) 1. (x) ~ (Ax ~ Bx)
2. ( x) (~ Bx Cx)
/ ( x) ~ Ax
(2) 1. (x) ~ Lx
2. ( x)Lx ( x)Mx
/ ( x)Mx
(3) 1. ~ ( x) (Ax Bx)
2. (x)[(Cx Dx) Ax]
~ ( x)Cx
(4) 1. ( y) (x)Lyx
/ (x) ( y)Lyx
(5) 1. (x) [Fx ( y)Gxy]
/ (x) ( y) [Fx Gxy]
(6) 1. (x)[Lx (Ax Bx)]
2. (x)[(Mx Lx) ~ Ax]
/ (x)(Mx Lx)
(x)(Mx Bx)
(7) 1. ( x) (Px Qx)
2. ( x) (Px · Rx)
/ ( x) (Qx Rx)
(8) 1. ( x) ( y)Fxy ( x) ( y)Gxy
2. ( x)Fxx
/ ( x)Gxx
40
XII. ANSWERS
(1) 1. (x) ~ (Ax ~Bx) p
(2) 1. (x) ~ Lx p
(3) 1. ~ ( x) (Ax Bx) p
41
(4) 1. ( y) (x)Lyx p/ (x) ( y)Lyx
(6) 1. (x)[Lx (Ax Bx)] p
43
(8) 1. ( x) ( y)Fxy ( x) ( y)Gxy p