Section 10: Practice Exercises
Practice Exercises
*Part A Provide a justification (rule and line numbers) for each line of proof that requires one.
*Part B Give a proof for each argument in SL.
1. \(K\) & \(L\), .˙. \(K\) ↔ \(L\)
2. \(A\) → (\(B\) → \(C\)), .˙.(\(A\)&\(B\)) → \(C\)
3. \(P\) &(\(Q\)∨\(R\)), \(P\) →¬\(R\), .˙.\(Q\)∨\(E\)
4. (\(C\) & \(D\))∨\(E\), .˙. \(E\)∨\(D\)
5. ¬\(F\) → \(G\), \(F\) → \(H\), .˙. \(G\)∨\(H\)
6. (\(X\) & \(Y\) )∨(\(X\) & \(Z\)), ¬(\(X\) & \(D\)), \(D\)∨\(M\).˙. \(M\)
Part C Give a proof for each argument in SL.
1. \(Q\) → (\(Q\) & ¬\(Q\)), .˙. ¬\(Q\)
2. \(J\) →¬\(J\), .˙. ¬\(J\)
3. \(E\)∨\(F\), \(F\)∨\(G\), ¬\(F\), .˙. \(E\)&\(G\)
4. \(A\) ↔ \(B\), \(B\) ↔ \(C\), .˙. \(A\) ↔ \(C\)
5. \(M\) ∨(\(N\) → \(M\)), .˙. ¬\(M\) →¬\(N\)
6. \(S\) ↔ \(T\), .˙. \(S\) ↔ (\(T\)∨\(S\))
7. (\(M\)∨\(N\))&(\(O\)∨\(P\)), \(N\) → \(P\), ¬\(P\), .˙. \(M\) & \(O\)
8. (\(Z\) & \(K\))∨(\(K\)&\(M\)), \(K\) → \(D\), .˙. \(D\)
Part D Show that each of the following sentences is a theorem in SL.
1. \(O\) → \(O\)
2. \(N\) ∨¬\(N\)
3. ¬(\(P\) &¬\(P\))
4. ¬(\(A\) →¬\(C\)) → (\(A\) → \(C\))
5. \(J\) ↔ [\(J\) ∨(\(L\)&¬\(L\))]
Part E Show that each of the following pairs of sentences are provably equivalent in SL.
1. ¬¬¬¬\(G\), \(G\)
2. \(T\) → \(S\), ¬\(S\) →¬\(T\)
3. \(R\) ↔ \(E\), \(E\) ↔ \(R\)
4. ¬\(G\) ↔ \(H\), ¬(\(G\) ↔ \(H\))
5. \(U\) → \(I\), ¬(\(U\) &¬\(I\))
Part F Provide proofs to show each of the following.
1. \(M\) &(¬\(N\) →¬\(M\)) ⊢ (\(N\) &\(M\))∨¬\(M\)
2. {\(C\) → (\(E\) &\(G\)), ¬\(C\) → \(G\)}⊢ \(G\)
3. {(\(Z\) & \(K\)) ↔ (\(Y\) &\(M\)), \(D\)&(\(D\) → \(M\))}⊢ \(Y\) → \(Z\)
4. {(\(W\) ∨\(X\))∨(\(Y\) ∨\(Z\)), \(X\) → \(Y\), ¬\(Z\)}⊢ \(W\) ∨\(Y\)
Part G For the following, provide proofs using only the basic rules. The proofs will be longer than proofs of the same claims would be using the derived rules.
1. Show that MT is a legitimate derived rule. Using only the basic rules, prove the following: \(\mathcal{A}\)→\(\mathcal{B}\), ¬\(\mathcal{B}\), .˙. ¬\(\mathcal{A}\)
2. Show that Comm is a legitimate rule for the biconditional. Using only the basic rules, prove that \(\mathcal{A}\) ↔\(\mathcal{B}\) and \(\mathcal{B}\) ↔\(\mathcal{A}\) are equivalent.
3. Using only the basic rules, prove the following instance of DeMorgan’s Laws: (¬\(A\)&¬\(B\)), .˙. ¬(\(A\)∨\(B\))
4. Without using the QN rule, prove ¬∃\(x\)¬\(\mathcal{A}\) ⊢∀\(x\)\(\mathcal{A}\)
5. Show that ↔ex is a legitimate derived rule. Using only the basic rules, prove that \(D\) ↔ \(E\) and (\(D\) → \(E\))&(\(E\) → \(D\)) are equivalent.
*Part H
1. Identify which of the following are substitution instances of ∀\(xRcx\): \(Rac\), \(Rca\), \(Raa\), \(Rcb\), \(Rbc\), \(Rcc\), \(Rcd\), \(Rcx\)
2. Identify which of the following are substitution instances of ∃\(x\)∀\(yLxy\): ∀\(yLby\), ∀\(xLbx\), \(Lab\), ∃\(xLxa\)
*Part I Provide a justification (rule and line numbers) for each line of proof that requires one.
*Part J Provide a proof of each claim.
1. ⊢∀\(xFx\)∨¬∀\(xFx\)
2. {∀\(x\)(\(Mx\) ↔ \(Nx\)),\(Ma\)&∃\(xRxa\)}⊢∃\(xNx\)
3. {∀\(x\)(¬\(Mx\)∨\(Ljx\)),∀\(x\)(\(Bx\) → \(Ljx\)),∀\(x\)(\(Mx\)∨\(Bx\))}⊢∀\(xLjx\)
4. ∀\(x\)(\(Cx\)&\(Dt\)) ⊢∀\(xCx\)&\(Dt\)
5. ∃\(x\)(\(Cx\)∨\(Dt\)) ⊢∃\(xCx\)∨\(Dt\)
Part K Provide a proof of the argument about Billy on p. 62.
Part L Look back at Part B on p. 73. Provide proofs to show that each of the argument forms is valid in QL.
Part M Aristotle and his successors identified other syllogistic forms. Symbolize each of the following argument forms in QL and add the additional assumptions ‘There is an \(A\)’ and ‘There is a \(B\).’ Then prove that the supplemented arguments forms are valid in QL.
Darapti:
All \(A\)s are \(B\)s. All \(A\)s are \(C\)s. .˙. Some \(B\) is \(C\).
Felapton:
No \(B\)s are \(C\)s. All \(A\)s are \(B\)s. .˙. Some \(A\) is not \(C\).
Barbari:
All \(B\)s are \(C\)s. All \(A\)s are \(B\)s. .˙. Some \(A\) is \(C\).
Camestros:
All \(C\)s are \(B\)s. No \(A\)s are \(B\)s. .˙. Some \(A\) is not \(C\).
Celaront:
No \(B\)s are \(C\)s. All \(A\)s are \(B\)s. .˙. Some \(A\) is not \(C\).
Cesaro:
No \(C\)s are \(B\)s. All \(A\)s are \(Bs\). .˙. Some \(A\) is not \(C\).
Fapesmo:
All \(B\)s are \(C\)s. No \(A\)s are \(B\)s. .˙. Some \(C\) is not \(A\).
Part N Provide a proof of each claim.
1. ∀\(x\)∀\(yGxy\)⊢∃\(xGxx\)
2. ∀\(x\)∀\(y\)(\(Gxy\) → \(Gyx\))⊢∀\(x\)∀\(y\)(\(Gxy\) ↔ \(Gyx\))
3. {∀\(x\)(\(Ax\) → \(Bx\)),∃\(xAx\)}⊢∃\(xBx\)
4. {\(Na\) →∀\(x\)(\(Mx\) ↔ \(Ma\)),\(Ma\),¬\(Mb\)}⊢¬\(Na\)
5. ⊢∀\(z\)(\(Pz\)∨¬\(Pz\))
6. ⊢∀\(xRxx\) →∃\(x\)∃\(yRxy\)
7. ⊢∀\(y\)∃\(x\)(\(Qy\) → \(Qx\))
Part O Show that each pair of sentences is provably equivalent.
1. ∀\(x\)(\(Ax\) →¬\(Bx\)), ¬∃\(x\)(\(Ax\)&\(Bx\))
2. ∀\(x\)(¬\(Ax\) → \(Bd\)), ∀\(xAx\)∨\(Bd\)
3. ∃\(xPx\) → \(Qc\), ∀\(x\)(\(Px\) → \(Qc\))
Part P Show that each of the following is provably inconsistent.
1. {\(Sa\) → \(Tm\), \(Tm\) → \(Sa\), \(Tm\)&¬\(Sa\)}
2. {¬∃\(xRxa\), ∀\(x\)∀\(yRyx\)}
3. {¬∃\(x\)∃\(yLxy\), \(Laa\)}
4. {∀\(x\)(\(Px\) → \(Qx\)), ∀\(z\)(\(Pz\) → \(Rz\)), ∀\(yPy\), ¬\(Qa\)&¬\(Rb\)}
*Part Q Write a symbolization key for the following argument, translate it, and prove it:
There is someone who likes everyone who likes everyone that he likes. Therefore, there is someone who likes himself.
Part R Provide a proof of each claim.
1. {\(Pa\)∨\(Qb\),\(Qb\) → \(b\) = \(c\),¬\(Pa\)}⊢ \(Qc\)
2. {\(m\) = \(n\)∨\(n\) = \(o\),\(An\)}⊢ \(Am\)∨\(Ao\)
3. {∀\(xx\) = \(m\),\(Rma\)}⊢∃\(xRxx\)
4. ¬∃\(xx\) ≠ \(m\) ⊢∀\(x\)∀\(y\)(\(Px\) → \(Py\))
5. ∀\(x\)∀\(y\)(\(Rxy\) → \(x\) = \(y\)) ⊢ \(Rab\) → \(Rba\)
6. {∃\(xJx\),∃\(x\)¬\(Jx\)}⊢∃\(x\)∃\(y\) \(x\) ≠ \(y\)
7. {∀\(x\)(\(x\) = \(n\) ↔ \(Mx\)),∀\(x\)(\(Ox\)∨¬\(Mx\))}⊢ \(On\)
8. {∃\(xDx\),∀\(x\)(\(x\) = \(p\) ↔ \(Dx\))}⊢ \(Dp\)
9. {∃\(xKx\)&∀\(y\)(\(Ky\) → \(x\) = \(y\))&\(Bx\),\(Kd\)}⊢ \(Bd\)
10. ⊢ \(Pa\) →∀\(x\)(\(Px\)∨\(x\) ≠ \(a)
Part S Look back at Part D on p. 74. For each argument: If it is valid in QL, give a proof. If it is invalid, construct a model to show that it is invalid.
*Part T For each of the following pairs of sentences: If they are logically equivalent in QL, give proofs to show this. If they are not, construct a model to show this.
1. ∀\(xPx\) → \(Qc\), ∀\(x\)(\(Px\) → \(Qc\))
2. ∀\(xPx\)&\(Qc\), ∀\(x\)(\(Px\)&\(Qc\))
3. \(Qc\)∨∃\(xQx\), ∃\(x\)(\(Qc\)∨\(Qx\))
4. ∀\(x\)∀\(y\)∀\(zBxyz\), ∀\(xBxxx\)
5. ∀\(x\)∀\(yDxy\), ∀\(y\)∀\(xDxy\)
6. ∃\(x\)∀\(yDxy\), ∀\(y\)∃\(xDxy\)
*Part U For each of the following arguments: If it is valid in QL, give a proof. If it is invalid, construct a model to show that it is invalid.
1. ∀\(x\)∃\(yRxy\), .˙. ∃\(y\)∀\(xRxy\)
2. ∃\(y\)∀\(xRxy\), .˙. ∀\(x\)∃\(yRxy\)
3. ∃\(x\)(\(Px\)&¬\(Qx\)), .˙. ∀\(x\)(\(Px\) →¬\(Qx\))
4. ∀\(x\)(\(Sx\) → \(Ta\)), \(Sd\), .˙. \(Ta\)
5. ∀\(x\)(\(Ax\) → \(Bx\)), ∀\(x\)(\(Bx\) → \(Cx\)), .˙. ∀\(x\)(\(Ax\) → \(Cx\))
6. ∃\(x\)(\(Dx\)∨\(Ex\)), ∀\(x\)(\(Dx\) → \(Fx\)), .˙. ∃\(x\)(\(Dx\)&\(Fx\))
7. ∀\(x\)∀\(y\)(\(Rxy\)∨\(Ryx\)), .˙. \(Rjj\)
8. ∃\(x\)∃\(y\)(\(Rxy\)∨\(Ryx\)), .˙. \(Rjj\)
9. ∀\(xPx\) →∀\(xQx\), ∃\(x\)¬\(Px\), .˙. ∃\(x\)¬\(Qx\)
10. ∃\(xMx\) →∃\(xNx\), ¬∃\(xNx\), .˙. ∀\(x\)¬\(Mx\)
Part V
1. If you know that \(\mathcal{A}\)⊢\(\mathcal{B}\), what can you say about (\(\mathcal{A}\)&\(\mathcal{C}\))⊢\(\mathcal{B}\)? Explain your answer.
2. If you know that \(\mathcal{A}\)⊢\(\mathcal{B}\), what can you say about (\(\mathcal{A}\)∨\(\mathcal{C}\))⊢\(\mathcal{B}\)? Explain your answer.