Solve Exercises in Qute and Model Construction:

Part A. Construct models showing that the following sequents are incorrect.

a. xyz ((~Rxy&~Ryz) ~Rxz) |= xRxx

c. xyz((Rxy&Ryz) Rxz) |= xyz ((~Rxy & ~Ryz) ~Rxz)

e. xFx, x(Fxy(Rxy & Fy)), x(Fxy(Rxy & ~Fy)) |= xy(Rxy (Fx v Fy)) v x(FxyRyx)

Part B. Determine whether the following sets of sentences (or constraints) are consistent or inconsistent. If they are consistent, demonstrate whether they are finitely consistent by providing a model with a finite domain which satisfies those constraints. [As a challenge, try to find a finite model with the minimum number of members in its domain.]

a. x~Fxx, xyFxy

c. xy(Fxy ~Fyx), xyFxy

e. xyz ((Fxy & Fyz) ~Fxz), xyFxy

g. xyz ((Fxy & Fyz) Fxz), xyz ((Fxy & Fyz) ~Fyz)

i. xyz ((Fxy & Fyz) Fxz), xy(Fxy ~Fyx), xyFxy

k. xy (Fxy & ~Fyx), xyz ((Fxy & Fyz) Fxz)

m. xy (Fxy & ~Fyx), xy ~(Fxy v Fyx), xyz ((Fxy & Fyz) Fxz)