#### Chapter 8 Exercises

Here is a statement of the problems in chapter 5 of the Logic Café.

 Q8.1 1. From the following premise:       (^x)(Tx&Lx)  derive: Ta   2. From the following premise:       Db&Lac  derive: (%y)Dy   3. From the following premise:       (^x)(Bx&Lxe)  derive: (%y)Lyy 8.1ex I 1. From the following premise:       (^x)Ax  derive: Aa&Ab   2. From the following premises:       Ba&Ca       ~La&Ja  derive: (%x)(Cx&~Lx)   3. From no premises,  derive: (^x)(Tx&~Ux)>(%y)Ty (to show the sentence is a logical truth).   4. From the following premise:       (^x)Ax&(^x)~Bx  derive: (%x)(Ax&~Bx)   5. From the following premises:       (^y)(Ty>Uy)       Ta  derive: (%z)(Uz&Tz) 8.1ex II 1. From the following premise:       (^x)(AxvBx)  derive: (^x)(BxvAx)   2. From the following premises:       (^x)(AxvBx)       ~Aa  derive: (%y)By   3. From no premises,  derive: (%x)(Txv~Tx) (to show the sentence is a logical truth).   4. From the following premises:       Aa&Ba       (%y)Ay>Lj       Lj=K  derive: K   5. From the following premises:       (^y)((TyvKy)>Uy)       Ka  derive: (%z)Uz Q8.2 1. From the following premise:       (^x)(Tx&Lx)  derive: (^x)Tx   2. From the following premise:       (%x)Jx  derive: (%y)Jy   3. From the following premises:       (^x)(Mx=Bxc)       (%y)My  derive: (%y)Byc 8.2ex I 1. From the following premise:       (^x)(Ax&Bx)  derive: (^y)By   2. From the following premises:       (^x)Ax       (^x)Bx  derive: (^x)(Ax&Bx)   3. From no premises,  derive: (^x)(Tx&~Ux)>(^x)Tx (to show the sentence is a logical truth).   4. From the following premise:       Aa>(^x)~Bx  derive: (^x)Ax>(^y)~By   5. From the following premise:       ~(%x)Px  derive: (^x)~Px 8.2ex II 1. From the following premise:       (^x)(^y)Lxy  derive: (^y)(^x)Lxy   2. From the following premise:       (%x)Ax>Ba  derive: (^x)(Ax>Ba)   3. From the following premise:       (^y)Ty&(^z)Sz  derive: (^x)(Tx&Sx)   4. From the following premise:       (^x)(Tx&Sx)  derive: (^y)Ty&(^z)Sz   5. From the following premise:       (^x)((Mx&Lxa)>Nx)  derive: (^x)(Mx>(Lxa>Nx)) 8.2ex III 1. From the following premise:       (%x)(Tx&Lx)  derive: (%y)Ty   2. From the following premise:       (%x)Jx  derive: (%y)Jy   3. From the following premises:       (%x)Mxc       (^x)[(%y)Myx>Tx]  derive: Tc   4. From the following premise:       ~Ja  derive: ~(^x)Jx   5. From the following premise:       (%x)(^y)Lxy  derive: (^y)(%x)Lxy   6. From the following premises:       (%x)(Jx&~Lxc)       (^x)(Jx=Wx)  derive: (%x)Wx 8.2ex IV Derivations with %E 1. From the set of premises:       (%z)Tz    derive the conclusion:       (%z)(TzvLzz) 2. From the set of premises:       (%x)(Bx&Tt)    derive the conclusion:       (%x)Bx&Tt 3. Show that       (%y)By>(%x)Bx    is logically true by    giving a derivation. 4. From the set of premises:       (%x)Txa       Taa>Ja       (^x)(^y)Txy>Ja    derive the conclusion:       Ja 8.3ex I 1. From the following premises:       (^x)(Tx>Jx)       (%y)(TyvTy)  derive: (%z)Jz   2. From no premises,  derive: (^x)(Txa>Txa) (to show the sentence is a logical truth).   3. From the following premise:       (%x)(Ax&Bx)  derive: (%x)Ax&(%y)By   4. From no premises,  derive: (^x)Ax>(^x)(AxvBx) (to show the sentence is a logical truth).   5. From the following premise:       (%x)Bxx>Baa  derive: (%x)Bxx=Baa 8.3ex II 1. From the following premise:       (%x)~Px  derive: ~(^x)Px   2. From the following premise:       ~(^x)Px  derive: (%x)~Px   3. From the following premises:       (%x)(%y)Gxy       (^x)(^y)(Gxy>~Gyx)  derive: (%x)(%y)~Gxy   4. From the following premises:       ~Ja       ~(^x)(Gxa>(%y)~Jy)v(^x)Gbx  derive: Gba   5. From the following premise:       (%x)Lxx>J  derive: (^y)(Lyy>J) 8.3ex III Logical Truth 1. Show that       (^x)[Mx>(Jx>Mx)]    is logically true by    giving a derivation. 2. Show that       (^x)(^y)(Gxy>Gxy)    is logically true by    giving a derivation. 3. Show that       (^z)(Nz>Tz)>[(^z)Nz>(^z)Tz]    is logically true by    giving a derivation. 4. Show that       (^x)(Gxv~Gx)    is logically true by    giving a derivation. 8.3ex IV Logical Equivalence in PD 1. Show that       (^w)(Bw&Cw)    is logically equivalent to       (^w)Bw&(^y)Cy    by providing two appropriate derivations. 2. Show that       Na>(^x)Tx    is logically equivalent to       (^x)(Na>Tx)    by providing two appropriate derivations. 3. Show that       (^x)(^y)Lxy    is logically equivalent to       (^y)(^x)Lyx    by providing two appropriate derivations. 4. Show that       (%y)My>Ma    is logically equivalent to       (^y)(My>Ma)    by providing two appropriate derivations. 8.3ex V Logical Falsity and Inconsistency 1. Show that       (^x)(Ax&~Ax)    is logically false by    giving a derivation. 2. Show that       (%x)(Ax&~Ax)    is logically false by    giving a derivation. 3. Show the following set:       { (%x)Axv(%x)Bx,       ~(%y)(AyvBy),            }    is inconsistent by giving a derivation. 4. Show the following set:       { (^x)(%y)(Txy>Bx),       (%y)(^x)(Tyx&~By),            }    is inconsistent by giving a derivation. 8.3ex VI Preparation for PD+ These exercises show four entailments and in so doing show that two groups of PL sentences are logically equivalent pairs. 1. From the set of premises:       ~(^x)Ax    derive the conclusion:       (%x)~Ax 2. From the set of premises:       (%x)~Ax    derive the conclusion:       ~(^x)Ax 3. From the set of premises:       ~(%x)(Ax&Bx)    derive the conclusion:       (^x)(Ax>~Bx) 4. From the set of premises:       (^x)(Ax>~Bx)    derive the conclusion:       ~(%x)(Ax&Bx) Q8.4 QN Derivations 1. From the set of premises:       ~(^x)(Ax&Bx)    derive the conclusion:       (%x)~(Ax&Bx) 2. From the set of premises:       Tjv(%y)~Kmy    derive the conclusion:       ~(^y)KmyvTj 3. From the set of premises:       Lst>(^x)~Jx       Lst    derive the conclusion:       ~(%x)Jx 4. From the set of premises:       ~((%x)Txjv(^x)Kx)    derive the conclusion:       (^x)~Txj 8.4ex I 1. From the following premises:       ~(^x)Gx       (%x)~Gx>(^y)Py  derive: (^y)Py   2. From the following premise:       ~(%x)Ax  derive: ~Aj   3. From the following premise:       ~(^x)(%y)Lxy  derive: (%x)(^y)~Lxy   4. From the following premise:       ~(^x)~Jx  derive: (%x)Jx   5. From the following premises:       (%x)~Lxx       (^x)Lxx v (^x)~Lxx  derive: ~Lpp 8.4ex II 1. From the following premise:       (^x)(Bx=Cx)  derive: (^z)(Bz>Cz)   2. From the following premises:       (^x)Bx       (^y)Cy  derive: ~(%z)(~Bzv~Cz)   3. From the following premise:       ~(%x)(^y)(Bxy=Tyx)  derive: (^x)(%y)~(Tyx=Bxy)   4. From no premises,  derive: ~(%x)(Px&Mxa)>(^x)(Mxa>~Px) (to show the sentence is a logical truth).   5. From the following premise:       ~(^x)(%y)~Fxy  derive: (^y)(%x)Fxy 8.4ex III Logical Truth 1. Show that       ~(^y)Gyy>(%y)~~~Gyy    is logically true by    giving a derivation. 2. Show that       (^y)~Py>~(%z)Pz    is logically true by    giving a derivation. 3. Show that       (^x)Axv(%x)~Ax    is logically true by    giving a derivation. 4. Show that       ~(^x)Ax>(%x)(Ax>Bx)    is logically true by    giving a derivation. 8.4ex IV Logical Equivalence 1. Show that       ~(%x)~Jx    is logically equivalent to      (^x)Jx    by providing two appropriate derivations. 2. Show that       ~(^x)(Ax&Bx)    is logically equivalent to      (%x)(~Axv~Bx)    by providing two appropriate derivations. 3. Show that       ~(^x)~Kxa    is logically equivalent to      (%x)Kxa    by providing two appropriate derivations. 4. Show that       ~(%x)(Px&Qx)    is logically equivalent to      ~(%x)(Qx&Px)    by providing two appropriate derivations. 8.4ex V Logical Falsehood and Inconsistency 1. Show the following set:       { ~(^x)(AxvBx),       (^x)Ax,            }    is inconsistent by giving a derivation. 2. Show the following set:       { (^x)(Px>Qx),       (%x)(Px&~Qx),            }    is inconsistent by giving a derivation. 3. Show that       ~(%x)Lxa&Lja    is logically false by    giving a derivation. 4. Show that       (%x)Px=(^x)~Px    is logically false by    giving a derivation.