Extra PL Derivations

 

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1. Universal Quantifierx( ~Fx v ~Gx ), Fa / ~Universal Quantifierx( ~Fx v Gx ).


2. Universal Quantifierx( Fx Gx ), Fm v Fn / Existential QuantifierxGx


3. Universal Quantifierx( Gx Existential Quantifiery~Kxy ), Existential Quantifierx( Fx &Universal QuantifieryKxy ) / Existential Quantifierx( Fx & ~Gx )


4. Universal Quantifierx[( Fx Hx ) Gx ] /Universal Quantifierx( ~Fx & ~Hx ) Universal QuantifierxGx


5. Universal Quantifierx[ Ax Universal Quantifiery( By Cxy ) ], Am & Bn / Cmn


6. Universal Quantifiery( My Ay ), Existential Quantifiery( Cy & My ) / Existential Quantifiery( Cy & Ay )


7. Universal Quantifierx( Ax v Bx ),Universal Quantifierx( Bx Ax ) /Universal QuantifierxAx


8. /Universal QuantifierxUniversal Quantifiery[( Fx Fy ) ( Fy v ~Fx )]


9. Universal Quantifiery( Jy ( Ky & Ly )), Existential Quantifierx~Kx / Existential Quantifierz~Jz


10. Existential QuantifierxFx Existential Quantifierx( Bx & Cx ), Existential Quantifierx( Cx v Dx ) Universal QuantifierxAx /Universal Quantifierx( Fx Ax )


11. Universal QuantifierxFx Existential QuantifierxGx,Universal Quantifierx~Gx / Existential Quantifierx~Fx


12. Universal Quantifierx( Ax & ~Bx ) Existential QuantifierxCx, ~Existential Quantifierx( Cx v Bx ) / ~Universal QuantifierxAx


13. Existential QuantifierxExistential QuantifieryRxy, Universal QuantifierxUniversal Quantifiery( Rxy Universal QuantifierzRxz), Universal Quantifierx(Universal QuantifierzRxz Universal QuantifieryRyx) /Universal QuantifierxUniversal QuantifieryRxy


14. Universal QuantifierxAx Existential Quantifierx( Bx & Cx ), Universal Quantifierx( Cx Bx ) / Universal QuantifierxAx Existential QuantifierxCx


15. Existential Quantifierx~Ax v Existential Quantifierx~Bx,Universal QuantifierxBx / ~Universal QuantifierxAx


16. /Universal Quantifierx( Ax & Existential Quantifiery~Bxy ) ~Existential Quantifierx[ ~Ax vUniversal Quantifiery( Bxy & Bxy ) ]


17. Universal Quantifierx[ ( Ax & Bx ) Cx ], ~Universal Quantifierx( Ax Cx ) / ~Universal QuantifierxBx


18. Universal QuantifierxExistential QuantifieryAxy Universal QuantifierxExistential QuantifieryBxy, Existential QuantifierxUniversal Quantifiery~Bxy / Existential QuantifierxUniversal Quantifiery~Axy


19. Existential Quantifierx[Ax &Universal Quantifiery( By Cxy )] /Universal Quantifierx[ Bx Existential Quantifiery( Ay & Cyx )]


20. Existential Quantifierx[ Ax & Universal Quantifiery( By Cxy ) ], Existential QuantifierxAx Bj / Existential QuantifierxCxj


21. Universal Quantifierx[ Existential QuantifieryRxy Existential Quantifierz~Wz ], Existential QuantifieryExistential Quantifierz( Ryz & Hz), Universal Quantifierx(~Hx Wx ) / Existential Quantifierz( ~Wz & Hz )


22. / [ Existential QuantifierxAx ( Existential QuantifieryBy Universal QuantifierzCz )] Universal QuantifierxUniversal QuantifieryUniversal Quantifierz (( Ax & By ) Cz )


23. Existential QuantifierzQz Universal Quantifierw(Lww ~Hw), Existential QuantifierxBx Universal Quantifiery(Ay Hy) / Existential Quantifierw(Qw & Bw) Universal Quantifiery(Lyy ~Ay)


24. Universal QuantifierxUniversal Quantifiery(( Ax & By ) Cxy ), Existential Quantifiery( Ey &Universal Quantifierw( Hw Cyw)),
Universal Quantifier xUniversal QuantifieryUniversal Quantifierz[( Cxy & Cyz ) Cxz ], vw( Ew Bw ) / Universal QuantifierzUniversal Quantifierw[( Az & Hw ) Czw ]

 

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