Advanced problems

Provide L4 symbolisations for each of the following sentences.


S18. Every cat sees Fido

S19. Some dog loves Aristotle

S20. Some dog likes every cat

S21. A dog chased a cat

S22. Some giraffe likes every baboon that likes no hyena.


Construct annotated derivations showing that the following arguments are valid.


4:1 ∀x∀yRxy ∴ Raa

4:2 ∀x∃yFxy. Ga ∴ ∃yFay

4:3 ∀x∀y∀z((Rxy ∧ Ryz) → Rxz). (Rbc ∧ Rab) ∴ Rac

4:4 ∀x∀y((Fx ∧ Gy) → (Rxy ↔ ¬Ryx)) ∴ ∀x∀y((Fx ∧ Gy) → (Rxy ∨ Ryx))

4:5 ∀x((Gx ∧ Txh) → ∀y(Fy → Txy)). ∃x(Gx ∧ ∃y(Fy ∧ ¬T(xy))) ∴ ∃x(Gx ∧ ¬Txh)

4:6 ∃x∀yFxy ∴ ∀y∃xFxy

4:7 ∃x(Fx ∧ ∀y(By → Sxy)). ∀x∀y(Bx ↔ Cy) ∴ ∀x(Cx → ∃ySyx)

4:8 ∴ ∀x∀yF(xy) ↔ ∀y∀xF(xy)

4:9 ∴ ∀x∃y(Fxy ∧Gy) → ∃x∃y(Fxy ∧Gx)

4:10 ∴ ∃y(∃xFxy↔Gy) ↔ (∃y∀x∃z((Fxy→Gy) ∧ (Gy→Fzy)))


Construct countermodels demonstrating that the following arguments are invalid.


C26. ∀y∃xFxy ∴ ∃x∀yFxy

C27. ∀x(Fx ↔ ∃yRxy). (Fa ∧ Fb) ∴ (Rab ∨ Rba)

C28. ∀x(Fxa ↔ Gxb) ∴ ∃x∃y(Fxy ∧ Gxy)

C29. ∀x∀y∃zRxyz ∴ ∀x∃z∀yRxyz

C30. ∃x∃yFxy ↔ ¬∃y∃xFxy