Skip to main content
Humanities LibreTexts

Section 6: Practice Exercises

  • Page ID
    1062
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    *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.


    This page titled Section 6: Practice Exercises is shared under a CC BY-SA license and was authored, remixed, and/or curated by P.D. Magnus (Fecundity) .

    • Was this article helpful?