Philosophy 251: Handout #16

XXV. For each of the following arguments construct a derivation to prove that the argument is valid.

  1. Universal Quantifierx( Zx Existential QuantifieryKy ) / Universal Quantifierx[ Zx Existential Quantifiery( Ky v Sy )]
  2. Universal Quantifierx( Hx Fx ), Universal Quantifierx[( Fx & Uxx ) Wxx ], Universal QuantifierzUzz / Universal Quantifierx( Hx Wxx )
  3. Universal Quantifierx[( Fx & Gx ) Existential Quantifiery( Axy & Py )], Existential QuantifierxExistential Quantifiery[ Fx & ( Axy & Py )] / Existential Quantifierx( Fx & Gx )
  4. Universal Quantifierx( Px Qx ) / ( Existential QuantifierxPx & Existential QuantifierxQx ) Existential Quantifierx( Px & Qx )
  5. Universal QuantifierxUniversal Quantifiery[( Ry v Dx ) ~Ky ], Universal QuantifierxExistential Quantifiery( Ax ~Ky ), Existential Quantifierx( Ax v Rx ) / Existential Quantifierx~Kx
  6. Universal Quantifiery( My Ay ), Existential QuantifierxExistential Quantifiery[( Bx & Mx) & (Ry & Syx )], Existential QuantifierxAx Universal QuantifieryUniversal Quantifierz( Syz Ay )
    /
    Existential Quantifierx( Rx & Ax )
  7. ~Universal Quantifierx( Fx & Aix ) ~Universal QuantifierxKx , Universal Quantifiery[ Existential Quantifierx~( Fx & Aix ) & Ryy ] / ~Universal QuantifierxKx
  8. Existential Quantifierx( Jxa & Ck ), Existential Quantifierx( Sx & Hxx ), Universal Quantifierx[( Ck & Sx ) ~Ax ] / Existential Quantifierz( ~Az & Hzz )
  9. Existential Quantifierx[ Cx v Universal Quantifiery( Wxy Cy )], Universal Quantifierx( Wxa & ~Ca ) / Existential QuantifierxCx
  10. Universal QuantifierxUniversal Quantifiery( Dxy Cxy ), Universal QuantifierxExistential QuantifieryDxy, Universal QuantifierxUniversal Quantifiery( Cyx Dxy ) / Existential QuantifierxExistential Quantifiery( Cxy & Cyx )
  11. / Universal Quantifierx( Ax Bx ) Universal Quantifierx( ~Bx ~Ax )
  12. / ~Existential Quantifierx( Ex v Fx ) Universal Quantifierx~Ex
  13. / ( Existential QuantifierxJx Existential QuantifierxKx ) Existential Quantifierx( Jx Kx )
  14. / Universal QuantifierxExistential Quantifiery( Nx v Oy ) Existential QuantifieryUniversal Quantifierx( Nx v Oy )
  15. Universal Quantifierx{( Fx & ~Kx ) Existential Quantifiery[( Fy & Hyx ) & ~Ky ]},
    Universal Quantifierx[( Fx & Universal Quantifiery[( Fy & Hyx ) Ky ]) Kx ] Mp / Mp

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