Week 5

01 Apr


7.4 EXERCISES Using the predicates listed for each assertion, “translate” each of the following into quantified logical form.


  1. No managers are sympathetic. (Mx, Sx)
  2. Everything is in its right place. (Rx)
  3. Some cell phones have no service here. (Cx, Sx)
  4. Not everything is settled. (Sx)
  5. Radiohead concerts are amazing. (Rx, Ax)
  6. Nothing is everlasting. (Ex)
  7. Not every earthquake is destructive. (Ex, Dx)
  8. Very few people do not like Mac computers. (Px, Mx)
  9. Only registered voters can vote in the next election. (Rx, Vx)
  10. Not everyone disapproves (i.e., does not approve) of the president’s cabinet selections. (Ax)



7.6 EXERCISES Translate each of the following arguments into quantified form and prove that each is valid using natural deduction. The letters that follow each argument give the predicate letters to use in symbolizing the argument.

  1. If all store supervisors are wise, then some employees benefit. If there are some store supervisors who are not wise, then some employees benefit. As you can see, either way, some employees benefit. (S, W, E, B)
  2. If someone studies philosophy, then all students benefit. If someone studies literature, then there are some students. So if someone studies philosophy and literature, then someone benefits. (P, B, L, S)
  3. Everyone is a Democrat or a Republican, but not both. If someone is a Democrat, then she is a liberal or a conservative. All conservatives are Republican. So all Democrats are liberal. (D, R, L, C)
  4. If there are any mavericks, then all politicians are committed to change. If there are any politicians, then anyone who is committed to change is pandering. So, if there are any mavericks, politicians are pandering. (M, P, C, A (for “pandering”))
  5. If everyone is a liberal, then no one is a conservative. There is a governor of Alaska and she is a conservative. So at least someone is not a liberal. (L, C, G)



7.9.1. Prove the following syllogisms valid first using natural deduction and then using the method of tableaux:

First Figure, Moods EAE, EIO

  • Second Figure, Moods AEE, AOO
  • Third Figure, Moods AII, OAO

Fourth Figure, Moods AEE, IAI



7.9.2. Construct formal proofs for all the arguments below. Use equivalence rules, truth functional arguments, and the rules of instantiation and generalization. These may also be proven using the method of tableaux.

  1. ∀x(Cx ⊃ ¬Sx), Sa ∧ Sb ∴ ¬(¬Ca ⊃ Cb)
  2. ∃xCx ⊃ ∃x(Dx ∧ Ex), ∃x(Ex ∨ Fx) ⊃ ∀xCx ∴ ∀x(Cx ⊃ Gx)
  3. ∀x(Fx ⊃ Gx), ∀x[(Fx ∧ Gx) ⊃ Hx] ∴ ∀x(Fx ⊃ Hx)
  4. ∃xLx ⊃ ∀x(Mx ⊃ Nx), ∃xPx ⊃ ∀x ¬Nx ∴ ∀x[(Lx ∧ Px) ⊃ ¬Mx]
  1. ∀x(Fx ≡ Gx), ∀x[(Fx ⊃ (Gx ⊃ Hx)], ∃xFx ∨ ∃xGx ∴ ∃xHx
  2. ∃x(Cx ∨ Dx), ∃xCx ⊃ ∀x(Ex ⊃ Dx), ∃xEx ∴ ∃xDx
  3. ∀x[(¬Dx ⊃ Rx) ∧ ¬(Dx ∧ Rx)], ∀x[Dx ⊃ (¬Lx ⊃ Cx)], ∀x(Cx ⊃ Rx) ∴ ∀x(Dx ⊃ Lx)



7.9.3. Using the method of tableaux, give an assignment of values for the predicates of each argument that shows that each argument is invalid.

  1. ∀x(Ax ⊃ Bx), ∀x(Ax ⊃ Cx) ∴ ∀x(Bx ⊃ Cx)
  2. ∃x(Ax ∧ Bx), ∀x(Cx ⊃ Ax) ∴ ∃x(Cx ∧ Bx)
  3. ∀x[(Cx ∨ Dx) ⊃ Ex], ∀x[(Ex ∧ Fx) ⊃ Gx] ∴ ∀x(Cx ⊃ Gx)
  4. ∃xMx, ∃xNx ∴ ∃x(Mx ∧ Nx)
  5. ∀x[Dx ∨ (Ex ∨ Fx)] ∴ ∀xDx ∨ (∀xEx ∨ ∀xFx)
  6. ∃x(Cx ∧ ¬Dx), ∃x(Dx ∧ ¬Cx) ∴ ∀x(Cx ∨ Dx)


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