Section 6: Practice Exercises

*Part A Determine whether each sentence is true or false in the model given.

UD = {Corwin, Benedict}
extension(\(A\)) = {Corwin, Benedict}
extension(\(B\)) = {Benedict}
extension(\(N\)) = ∅
referent(\(c\)) = Corwin

1. \(Bc\)
2. \(Ac\) ↔¬\(Nc\)
3. \(Nc\) → (\(Ac\)∨\(Bc\))
4. ∀\(xAx\)
5. ∀\(x\)¬\(Bx\)
6. ∃\(x\)(\(Ax\)&\(Bx\))
7. ∃\(x\)(\(Ax\) → \(Nx\))
8. ∀\(x\)(\(Nx\)∨¬\(Nx\))
9. ∃\(xBx\) →∀\(xAx\)

*Part B Determine whether each sentence is true or false in the model given.

UD = {Waylan, Willy, Johnny}
extension(\(H\)) = {Waylan, Willy, Johnny}
extension(\(W\)) = {Waylan, Willy}
extension(\(R\)) = {<Waylan, Willy>,<Willy, Johnny>,<Johnny, Waylan>}
referent(\(m\)) = Johnny

1. ∃\(x\)(\(Rxm\)&\(Rmx\))
2. ∀\(x\)(\(Rxm\)∨\(Rmx\))
3. ∀\(x\)(\(Hx\) ↔ \(Wx\))
4. ∀\(x\)(\(Rxm\) → \(Wx\))
5. ∀\(x\)[\(Wx\) → (\(Hx\)&\(Wx\))]
6. ∃\(xRxx\)
7. ∃\(x\)∃\(yRxy\)
8. ∀\(x\)∀\(yRxy\)
9. ∀\(x\)∀\(y\)(\(Rxy\)∨\(Ryx\))
10. ∀\(x\)∀\(y\)∀z[(\(Rxy\) & \(Ryz\)) → \(Rxz\)]

*Part C Determine whether each sentence is true or false in the model given.

UD = {Lemmy, Courtney, Eddy}
extension(\(G\)) = {Lemmy, Courtney, Eddy}
extension(\(H\)) = {Courtney}
extension(\(M\)) = {Lemmy, Eddy}
referent(\(c\)) = Courtney
referent(\(e\)) = Eddy

1. \(Hc\)
2. \(He\)
3. \(Mc\)∨\(Me\)
4. \(Gc\)∨¬\(Gc\)
5. \(Mc\) → \(Gc\)
6. ∃\(xHx\)
7. ∀\(xHx\)
8. ∃\(x\)¬\(Mx\)
9. ∃\(x\)(\(Hx\)&\(Gx\))
10. ∃\(x\)(\(Mx\)&\(Gx\))
11. ∀\(x\)(\(Hx\)∨\(Mx\))
12. ∃\(xHx\)&∃\(xMx\)
13. ∀\(x\)(\(Hx\) ↔¬\(Mx\))
14. ∃\(xGx\)&∃\(x\)¬\(Gx\)
15. ∀\(x\)∃\(y\)(\(Gx\)&\(Hy\))

*Part D Write out the model that corresponds to the interpretation given.

UD: natural numbers from 10 to 13
\(Ox\): \(x\) is odd.
\(Sx\): \(x\) is less than 7.
\(Tx\): \(x\) is a two-digit number.
\(Ux\): \(x\) is thought to be unlucky.
\(Nxy\): \(x\) is the next number after \(y\).

Part E Show that each of the following is contingent.

*1. \(Da\)&\(Db\)
*2. ∃\(xTxh\)
*3. \(Pm\)&¬∀\(xPx\)
4. ∀\(zJz\) ↔∃\(yJy\)
5. ∀\(x\)(\(Wxmn\)∨∃\(yLxy\))
6. ∃\(x\)(\(Gx\) →∀\(yMy\))

*Part F Show that the following pairs of sentences are not logically equivalent.

1. \(Ja\), \(Ka\)
2. ∃\(xJx\), \(Jm\)
3. ∀\(xRxx\), ∃\(xRxx\)
4. ∃\(xPx\) → \(Qc\), ∃\(x\)(\(Px\) → \(Qc\))
5. ∀\(x\)(\(Px\) →¬\(Qx\)), ∃\(x\)(\(Px\)&¬\(Qx\))
6. ∃\(x\)(\(Px\)&\(Qx\)), ∃\(x\)(\(Px\) → \(Qx\))
7. ∀\(x\)(\(Px\) → \(Qx\)), ∀\(x\)(\(Px\)&\(Qx\))
8. ∀\(x\)∃\(yRxy\), ∃\(x\)∀\(yRxy\)
9. ∀\(x\)∃\(yRxy\), ∀\(x\)∃\(yRyx\)

Part G Show that the following sets of sentences are consistent.

1. { \(Ma\), ¬ \(Na\), \(Pa\), ¬ \(Qa\) }
2. { \(Lee\), \(Lef\), ¬ \(Lfe\), ¬ \(Lff\) }
3. {¬( \(Ma\) &∃ \(xAx\) ), \(Ma\) ∨ \(Fa\), ∀ \(x\) ( \(Fx\) → \(Ax\) )}
4. { \(Ma\) ∨ \(Mb\), \(Ma\) →∀ \(x\) ¬ \(Mx\) }
5. {∀ \(yGy\), ∀ \(x\) ( \(Gx\) → \(Hx\) ), ∃ \(y\) ¬ \(Iy\) }
6. {∃ \(x\) ( \(Bx\) ∨ \(Ax\) ), ∀ \(x\) ¬ \(Cx\), ∀ \(x\) ( \(Ax\) & \(Bx\) ) → \(Cx\) }
7. {∃ \(xXx\), ∃ \(xY\) \(x\), ∀ \(x\) ( \(Xx\) ↔¬ \(Y\) \(x\) )}
8. {∀ \(x\) ( \(Px\) ∨ \(Qx\) ), ∃ \(x\) ¬( \(Qx\) & \(Px\) )}
9. {∃ \(z\) ( \(Nz\) & \(Ozz\) ), ∀ \(x\) ∀ \(y\) ( \(Oxy\) → \(Oyx\) )}
10. {¬∃ \(x\) ∀ \(yRxy\), ∀ \(x\) ∃ \(yRxy\) }

Part H Construct models to show that the following arguments are invalid.

1. ∀ \(x\) ( \(Ax\) → \(Bx\) ), .˙. ∃ \(xBx\)
2. ∀ \(x\) ( \(Rx\) → \(Dx\) ), ∀ \(x\) ( \(Rx\) → \(Fx)\), .˙. ∃ \(x\) ( \(Dx\) & \(Fx\) )
3. ∃ \(x\) ( \(Px\) → \(Qx\) ), .˙.∃ \(xPx\)
4. \(Na\) & \(Nb\)& \(Nc\), .˙. ∀ \(xNx\)
5. \(Rde\), ∃\(xRxd\), .˙. \(Red\)
6. ∃\(x\)(\(Ex\)&\(Fx\)), ∃\(xFx\) →∃\(xGx\), .˙. ∃\(x\)(\(Ex\)&\(Gx\))
7. ∀\(xOxc\), ∀\(xOcx\), .˙. ∀\(xOxx\)
8. ∃\(x\)(\(Jx\)&\(Kx\)), ∃\(x\)¬\(Kx\), ∃\(x\)¬\(Jx\), .˙. ∃\(x\)(¬\(Jx\)&¬\(Kx\))
9. \(Lab\) →∀\(xLxb\), ∃\(xLxb\), .˙. \(Lbb\)

Part I

*1. Show that {¬\(Raa\),∀\(x\)(\(x\) = \(a\)∨\(Rxa\))} is consistent.
*2. Show that {∀\(x\)∀\(y\)∀\(z\)(\(x\) = \(y\)∨\(y\) = \(z\)∨\(x\) = \(z\)),∃\(x\)∃\(y\) \(x\) ≠ \(y\)} is consistent.
*3. Show that {∀\(x\)∀\(y\) \(x\) = \(y\),∃\(x\) \(x\) ≠ \(a\)} is inconsistent.
4. Show that ∃\(x\)(\(x\) = \(h\)&\(x\) = \(i\)) is contingent.
5. Show that {∃\(x\)∃\(y\)(\(Zx\) & \(Zy\) & \(x\) = \(y\)), ¬\(Zd\), \(d\) = \(s\)} is consistent.
6. Show that ‘∀\(x\)(\(Dx\) →∃\(yTyx\)) .˙. ∃\(y\)∃\(z\) \(y\) ≠ \(z\)’ is invalid.

Part J

1. Many logic books define consistency and inconsistency in this way: “ A set {\(\mathcal{A}\)₁,{\(\mathcal{A}\)₂,{\(\mathcal{A}\)₃,···}is inconsistent if and only if{{\(\mathcal{A}\)₁,{\(\mathcal{A}\)₂,{\(\mathcal{A}\)₃,···}|= ({\(\mathcal{B}\) &¬{\(\mathcal{B}\)) for some sentence {\(\mathcal{B}\). A set is consistent if it is not inconsistent.” Does this definition lead to any different sets being consistent than the definition on p. 84? Explain your answer.

*2. Our definition of truth says that a sentence {\(\mathcal{A}\) is true in \(\mathbb{M}\) if and only if some variable assignment satisfies {\(\mathcal{A}\) in \(\mathbb{M}\). Would it make any difference if we said instead that {\(\mathcal{A}\) is true in \(\mathbb{M}\) if and only if every variable assignment satisfies {\(\mathcal{A}\) in \(\mathbb{M}\)? Explain your answer.