1. (20 points) Determine whether each of the following strings is a sentence of
LFOL. If it is not, indicate why not. If it is, indicate whether it is an
open or a closed sentence, and what the connective of largest scope is
(include quantifiers as connectives):
(a) (x)(Fx Gx)
(b) (x)(Fx → Gx Hx)
(c) (x)(y)Rxy → Ryx
(d) Gb → (x)(x = b → Gb)
(e) (x)(y)(Rxy → Ryx)
(f) (F)Fx → Rba
(g) (x)Fx Fa
(h) (x)((y)Rxy → Ryx)
(i) (x)((y)(Rxy → Ryx))
(j) (x,y)(Ryx Rxx)
2. (20 points) Determine whether each of the following sentences is true
relative to the following interpretation I:
• D(I) = {1,2,3,4,5}
• I(a) = 2
• I(b) = 1
• I(c) = 1
• I(F) = {3,4,5}
• I(G) = {1,2,4,5}
• I(R) = {, , , , , , }
(a) (x)Gx → (x)Fx
(b) b = c → (x)(Fx → (Gx Rax))
(c) (x)(Fx Gx) (x)(Gx Fx)
(d) (x)(Rxx & (Gx Fx))
(e) (x)(y)(Rxy & Rxa & Ray)
(OVER) Problem Set 8 2
3. (48 points) For each of the following implication claims, construct an
interpretation which shows that the implication claim is false by making
the premises true and the conclusion false:
(a) (x)(Fx Gx) |= (x)(Fx Gx)
(b) (x)Fx → (x)Gx |= (x)(Fx → Gx)
(c) (x)(Fx Gx), (x)(Fx → Hx), (x)Hx |= (x)Gx
(d) (x)Fx (x)Gx |= (x)(Fx Gx)
(e) (x)Fx (x)Gx |= (x)(Fx Gx)
(f) (x)(y)(Fx → Gy) |= (y)(x)(Fx → Gy)
4. (12 points) Provide a formal proof justifying the following claim:
Q → P, R → S, Q → S, P → (Q R) |-NK Q
02_sh_b_ps_08.pdf Problem Set 8 1 PHIL 114 Problem Set #8 1. (20 points) Determine whether each of the following strings is a sentence of LFOL. If it is not, indicate why not. If it is, indicate whether it is an open or a closed sentence, and what the connective of largest scope is (include quantifiers as connectives): (a) (x)(Fx Gx) (b) (x)(Fx → Gx Hx) (c) (x)(y)Rxy → Ryx (d) Gb → (x)(x = b → Gb) (e) (x)(y)(Rxy → Ryx) (f) (F)Fx → Rba (g) (x)Fx Fa (h) (x)((y)Rxy → Ryx) (i) (x)((y)(Rxy → Ryx)) (j) (x,y)(Ryx Rxx) 2. (20 points) Determine whether each of the following sentences is true relative to the following interpretation I: • D(I) = {1,2,3,4,5} • I(a) = 2 • I(b) = 1 • I(c) = 1 • I(F) = {3,4,5} • I(G) = {1,2,4,5} • I(R) = {<1,2>, <2,5>, <3,1>, <3,2>, <3,3>, <3,4>, <5,5>} (a) (x)Gx → (x)Fx (b) b = c → (x)(Fx → (Gx Rax)) (c) (x)(Fx Gx) (x)(Gx Fx) (d) (x)(Rxx & (Gx Fx)) (e) (x)(y)(Rxy & Rxa & Ray) (OVER) Problem Set 8 2 3. (48 points) For each of the following implication claims, construct an interpretation which shows that the implication claim is false by making the premises true and the conclusion false: (a) (x)(Fx Gx) |= (x)(Fx Gx) (b) (x)Fx → (x)Gx |= (x)(Fx → Gx) (c) (x)(Fx Gx), (x)(Fx → Hx), (x)Hx |= (x)Gx (d) (x)Fx (x)Gx |= (x)(Fx Gx) (e) (x)Fx (x)Gx |= (x)(Fx Gx) (f) (x)(y)(Fx → Gy) |= (y)(x)(Fx → Gy) 4. (12 points) Provide a formal proof justifying the following claim: Q → P, R → S, Q → S, P → (Q R) |-NK Q ***