*Question {8.1}*

1. Anthea is a geologist.

2. Brian is not a hairdresser.

3. Either Anthea is a hairdresser or Brian is a geologist.

4. Both Anthea and Brian are hairdressers.

5. Brian is both a hairdresser and a geologist.

6. Brian is not a geologist.

7. If Brian is a geologist, he’s a hairdresser.

8. Anthea is larger than Brian.

9. Brian is not larger than Anthea.

10. All geologists are hairdressers.

11. Not all geologists are hairdressers.

12. All geologists are not hairdressers.

13. Some hairdressers are people.

14. All hairdressers are larger than Brian.

15. For any person there’s a person larger than that person.

16. For any person there’s a geologist larger than that person.

17. For any person, anything that that person is larger than is not larger than that person. (check errata)

18. Someone’s not a geologist.

19. There’s a person larger than any person.

20. Take any three people you like: If the first is larger than the second and the second is larger than the third, then the first is larger than the third.

*Question {8.2}*

1. *Pb*

2. *Hb* ⊃ ~ *Gb*

3. ~ *Gb* ⊃ ~ *Ga*

4. ~(∃ *x*)(*Px* & *Gx*)

5. (∀ *x*)(*Px* ⊃ *Gx*)

6. (∀ *x*)(*Gx* ⊃ *Px*)

7. (∃ *x*)(*Gx* & *Px*)

8. (∃ *x*)(*Px* & *Gx*)

9. (∃ *x*)(*Gx* & *Hx*)

10. (∃ *x*)(*Hx* & ~ *Px*)

11. (∃ *x*)(~ *Px* & ~ *Gx*)

12. (∃ *x*)(*Hx* & (*Gx* & *Px*))

13. *Ga* ⊃ (∃ *x*)(*Px* & *Lax*)

14. (∀ *x*)(*Gx* ⊃ (∃ *y*)(*Hy* & *Lxy*))

15. (∀ *x*)((*Gx* & *Hx*) ⊃ ~ *Px*)

16. (∃ *x*)(*Gx* & (∀ *y*)(*Hy* ⊃ *Lxy*))

17. (∀ *x*)(*Px* ⊃ ~(*Gx* & *Hx*))

18. Which is it? (∀ *x*)(*Px* ⊃ (∃ y)(*Gy* & *Lxy*)) or (∀ *x*)(*Gx* ⊃ (∀ *y*)(*Py* ⊃ *Lyx*))?

19. (∀ *x*)(*Gx* ⊃ (∀ *y*)(*Hy* ⊃ *Lxy*))

20. (∀ *x*)((*Hx* & (∃ *y*)(*Gy* & *Lxy*)) ⊃ (∃ *z*)(*Hz* & *Lxz*))

21. (∃ *x*)(*Gx* & (∃ *y*)(*Hy* & *Lxy*))

22. (∃ *x*)(*Gx* & (∀ *y*)(*Hy* ⊃ *Lxy*))

23. (∀ *x*)(*Hx* ⊃ (*Lxa* & *Lxb*)

24. (∀ *x*)(*Lxa* ⊃ *Lxb*) ⊃ *Lab*

25. (∀ *x*)(*Lxa* ⊃ *Gx*) ⊃ (∀ *x*)(*Hx* ⊃ *Lax*)

26. (∃ *x*)(*Hx* & *Lax*) ⊃ (∃ *x*)(*Gx* & *Lax*)

27. (∃ *x*)(*Px* & (∀ *y*)(*Hy* ⊃ *Lxy*))

28. (∃ *x*)(*Hx* &~(∃ *y*)(*Gy* & *Lxy*))

29. (∀ *x*)(*Px* ⊃ (*Gx* ⊃ (*Lxb* & ~ *Lxa*)))

30. (∀ *x*)((*Hx* & (∃ *y*)(*Gy* & *Lxy*)) ⊃ (∀ *z*)(*Pz* ⊃ *Lxz*))

*Question {8.3}*

In these answers variables are bound by the quantifier that marked by the same type of *. If a variable has a * above it, then it is bound by the quantifier with a * above it, similarly for variables with *‘s below.

10:

(∀ y) | (G | y | ⊃ | H | y) |

* | * | * |

11:

~ | (∀ x) | (G | x | ⊃ | H | x) |

* | * | * |

12:

(∀ x) | (G | x | ⊃ | ~ | H | x) |

* | * | * |

13:

(∃ x) | (H | x | & | P | x) |

* | * | * |

14:

(∀ x) | (H | x | ⊃ | L | x | b) |

* | * | * |

15:

* | * | * | ||||||||

(∀ x) | (P | x | ⊃ | (∃ y) | (P | y | & | L | y | x)) |

* | * | * |

16:

* | * | * | ||||||||

(∀ x) | (P | x | ⊃ | (∃ y) | (G | y | & | L | y | x)) |

* | * | * |

17:

* | * | * | ||||||||||

(∀ x) | (P | x | ⊃ | (∀ y) | (L | x | y | ⊃ | ~ | L | y | x)) |

* | * | * | * |

18:

(∃ y) | (P | y | & | ~ | G | y) |

* | * | * |

19:

* | * | * | ||||||||

(∃ x) | (P | x | & | (∀ y) | (P | y | ⊃ | L | x | y)) |

* | * | * |

20:

* | * | * | |||||||||||

(∀ x) | (∀ y) | (∀ z) | (L | x | y | ⊃ | (L | y | z | ⊃ | L | x | z)) |

* | * | * | |||||||||||

* | * | * |

*Question {8.4}*

1. *Lb*, *Rb*, therefore *Lb* & *Rb*.

*Lx* = *x* studies linguistics, *Rx* = belongs to the rockclimber’s club, *b* = Brian.

2. (∃ *x*)*Lx*, (∃ *x*)*Rx*, therefore (∃ *x*)(*Lx* & *Rx*)

3. (∀ *x*)(*Sx* ⊃ (∃ *y*)(*Ly* & *Oxy*)) therefore (∃ *y*)(*Ly* & (∀ *x*)(*Sx* ⊃ *Oxy*))

*Sx* = *x* is a solid, *Lx* = *x* is a liquid, *Oxy* = *x* is soluble in *y*.

4. (∀ *x*)(*Ex* ⊃ (*Sx* ∨ *Ax*)), *Ea* therefore *Sa* & *Aa*

*Ex* = *x* is eligible for the clean desk prize. *Sx* = *x* is a secretary. *Ax* = *x* is an administrator. *a* = Ian.

5. (∀ *x*)*Mx*, therefore (~(∃ *x*)*Mx* ∨ ((∃ *x*)(*Mx* & *Tx*) & (∀ *x*)(*Tx* ⊃ *Mx*))) & ~(~(∃ *x*)*Mx* & ((∃ *x*)(*Mx* & *Tx*) & (∀ *x*)(*Tx* ⊃ *Mx*))) {This was a hard one!}

*Mx* = *x* is material. *Tx* = *x* is mental.

6. (∃ *x*)(*Mx* & (∀ *y*)((*My* & ~ *Syy*) ⊃ *Sxy*)) therefore (∃ *x*)(*Mx* & *Sxx*)

*Mx* = *x* is a man in town. *Sxy* = *x* shaves *y*.

7. (∀ *x*)(*Hx* ⊃ *Ax*), therefore (∀ *x*)((∃ *y*)(*Hy* & *Oxy*) ⊃ (∃ *y*)(*Ay* & *Oxy*))

*Hx* = *x* is a horse. *Ax* = *x* is an animal. *Oxy* = *x* is a head of *y*.

8. (∀ *x*)(∀ *y*)((*Rxy* & *Py*) ⊃ *Nx*), ~(∃ *x*)(*Nx* & *Fx*), (∀ *x*)(∀ *y*)((*Rxy* & (*Ny* & ~ *Py*)) ⊃ ~ *Fx*), therefore (∀ *x*)(∀ *y*)((*Rxy* & *Ny*) ⊃ ~ *Fx*)

*Rxy* = *x* is a square root of *y*. *Px* = *x* is a perfect square. *Nx* = *x* is a natural number. *Fx* = *x* is a fraction.

9. ~(∃ *x*)(*Px* & *Cx*) ⊃ (∃ *x*)(*Px* & *Dx*), therefore (∃ *x*)(*Px* & (~ *Cx* ⊃ *Dx*))

*Px* = *x* is a person. *Cx* = *x* contributes to Oxfam. *Dx* = *x* dies of hunger.

10. ~(∃ *x*)(*Ax* & ~ *Kx*), (∀ *x*)(*Gx* ⊃ *Ix*), ~(∃ *x*)(*Ix* & *Kx*) therefore (∀ *x*)(*Gx* ⊃ ~ *Ax*).

{I have treated “no-one” as “nothing” here.}

*Ax* = *x* really appreciates Beethoven. *Kx* = *x* keeps silent while the Moonlight Sonata is played. *Gx* = *x* is a guinea pig. *Ix* = *x* is hopelessly ignorant of music.

*Questions {8.5}* and *{8.6}*:

Discuss these with your tutor.