Philosophy 251: Handout #18

XXXIX. For each of the following arguments construct a countermodel to prove that the argument is invalid.

1. x( Fx Gx ), x( Hx Gx ) / x( Hx & Fx )
2. y( Fy Fa ), Fa / ~Fb
3. x( Bx Cx ), xBx / xCx
4. x( Bx Cx ), xCx / xBx
5. x( Fx Gx ), x( Hx ~Fx ) / x( Hx Gx )
6. xy~Lxy / x~Lxx
7. x( Fx Gx ) xNx, x( Nx Gx ) / x( ~Fx v Gx )
8. ( ~yFy yFy ) v ~Fa / zFz
9. x( Fx & Gx ), x( Fx & Hx ) / x( Gx & Hx )
10. x( Fx Gx ), x( Hx Gx ) / x( Fx v Hx )

XL. For each of the following arguments determine whether the argument is valid or invalid. For those that are valid construct a derivation; for those that are invalid construct a countermodel.

1. xy( Hxy v Jxy ), xy~Hxy / xyJxy
2. z( Lz Hz ), x~( Hx v ~Bx ) / ~Lb
3. z[ Kzz ( Mz & Nz )], z~Nz / x~Kxx
4. Fa v yGya, Fb v y~Gyb / yGya
5. xGx, x( Gx Dxx ) / xy( Gx & Dxy )
6. x( Fx & Gx), xFx, Fa & Gb / x( Fx v Gx )
7. ~x( Ax Bx ) / x( Ax & ~Bx )
8. xy( Mxy Nxy ) / xy[ Mxy ( Nxy & Nyx )]
9. x~Jx, y( Hby v Ryy ) xJx / y~( Hby v Ryy )
10. x( Fx v Hx ) yGy / z[( Fz v Hz ) Gz ]