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.