\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\TextOrMath }[2]{#2}\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand {\LWRabsorboption }[1][]{}\) \(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\) \(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\) \(\def \mathcode #1={\mathchar }\) \(\let \delcode \mathcode \) \(\let \delimiter \mathchar \) \(\def \oe {\unicode {x0153}}\) \(\def \OE {\unicode {x0152}}\) \(\def \ae {\unicode {x00E6}}\) \(\def \AE {\unicode {x00C6}}\) \(\def \aa {\unicode {x00E5}}\) \(\def \AA {\unicode {x00C5}}\) \(\def \o {\unicode {x00F8}}\) \(\def \O {\unicode {x00D8}}\) \(\def \l {\unicode {x0142}}\) \(\def \L {\unicode {x0141}}\) \(\def \ss {\unicode {x00DF}}\) \(\def \SS {\unicode {x1E9E}}\) \(\def \dag {\unicode {x2020}}\) \(\def \ddag {\unicode {x2021}}\) \(\def \P {\unicode {x00B6}}\) \(\def \copyright {\unicode {x00A9}}\) \(\def \pounds {\unicode {x00A3}}\) \(\let \LWRref \ref \) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \( \newcommand {\multicolumn }[3]{#3}\) \(\require {textcomp}\) 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\\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) \(\let \Dot \dot \) \(\let \Ddot \ddot \) \(\let \Breve \breve \) \(\let \Bar \bar \) \(\let \Vec \vec \) \(\require {mathtools}\) \(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\) \(\newcommand {\approxcolon }{\approx \vcentcolon }\) \(\newcommand {\Approxcolon }{\approx \dblcolon }\) \(\newcommand {\simcolon }{\sim \vcentcolon }\) \(\newcommand {\Simcolon }{\sim \dblcolon }\) \(\newcommand {\dashcolon }{\mathrel {-}\vcentcolon }\) \(\newcommand {\Dashcolon }{\mathrel {-}\dblcolon }\) \(\newcommand {\colondash }{\vcentcolon \mathrel {-}}\) \(\newcommand {\Colondash }{\dblcolon \mathrel {-}}\) \(\newenvironment {crampedsubarray}[1]{}{}\) \(\newcommand {\smashoperator }[2][]{#2\limits }\) \(\newcommand {\SwapAboveDisplaySkip }{}\) \(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\) \(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\) \(\Newextarrow 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#1\LWRnicearraywithdelimtwo }{\end {array}\LWRnicearrayrightdelim }\) \(\newenvironment {pNiceArray} {\begin {NiceArrayWithDelims}{(}{)}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {bNiceArray} {\begin {NiceArrayWithDelims}{[}{]}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {BNiceArray} {\begin {NiceArrayWithDelims}{\{}{\}}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {vNiceArray} {\begin {NiceArrayWithDelims}{\vert }{\vert }} {\end {NiceArrayWithDelims}} \) \(\newenvironment {VNiceArray} {\begin {NiceArrayWithDelims}{\Vert }{\Vert }} {\end {NiceArrayWithDelims}} \) \(\newenvironment {NiceMatrix}[1][]{\begin {matrix}}{\end {matrix}}\) \(\newenvironment {pNiceMatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\) \(\newenvironment {bNiceMatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\) \(\newenvironment {BNiceMatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\) \(\newenvironment {vNiceMatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\) \(\newenvironment {VNiceMatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\) \(\newcommand {\LWRnicematrixBlock }[1]{#1}\) \(\def \LWRnicematrixBlockopt <#1>#2{#2}\) \(\newcommand {\Block }[2][]{\ifnextchar <\LWRnicematrixBlockopt \LWRnicematrixBlock }\) \(\newcommand {\diagbox }[2]{\begin {array}{l}\hfill \quad #2\\\hline #1\quad \hfill \end {array}}\) \(\let \hdottedline \hdashline \) \(\newcommand {\Hline }[1][]{\hline }\) \(\newcommand {\CodeBefore }{}\) \(\newcommand {\Body }{}\) \(\newcommand {\CodeAfter }{}\) \(\newcommand {\line }[3][]{}\) \(\newcommand {\RowStyle }[2][]{}\) \(\newcommand {\LWRSubMatrix }[1][]{}\) \(\newcommand {\SubMatrix }[4]{\LWRSubMatrix }\) \(\newcommand {\OverBrace }[4][]{}\) \(\newcommand {\UnderBrace }[4][]{}\) \(\newcommand {\HBrace }[3][]{}\) \(\newcommand {\VBrace }[3][]{}\) \(\newcommand {\ShowCellNames }{}\) \(\newcommand {\tabularnote }[2][]{}\) \(\newcommand {\cellcolor }[3][]{}\) \(\newcommand {\rowcolor }[3][]{}\) \(\newcommand {\LWRrowcolors }[1][]{}\) \(\newcommand {\rowcolors 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{\mathunder }[1]{#1}\) \(\newcommand {\mathaccent }[1]{#1}\) \(\newcommand {\mathbotaccent }[1]{#1}\) \(\newcommand {\mathalpha }[1]{\mathord {#1}}\) \(\def\upAlpha{\unicode{x0391}}\) \(\def\upBeta{\unicode{x0392}}\) \(\def\upGamma{\unicode{x0393}}\) \(\def\upDigamma{\unicode{x03DC}}\) \(\def\upDelta{\unicode{x0394}}\) \(\def\upEpsilon{\unicode{x0395}}\) \(\def\upZeta{\unicode{x0396}}\) \(\def\upEta{\unicode{x0397}}\) \(\def\upTheta{\unicode{x0398}}\) \(\def\upVartheta{\unicode{x03F4}}\) \(\def\upIota{\unicode{x0399}}\) \(\def\upKappa{\unicode{x039A}}\) \(\def\upLambda{\unicode{x039B}}\) \(\def\upMu{\unicode{x039C}}\) \(\def\upNu{\unicode{x039D}}\) \(\def\upXi{\unicode{x039E}}\) \(\def\upOmicron{\unicode{x039F}}\) \(\def\upPi{\unicode{x03A0}}\) \(\def\upVarpi{\unicode{x03D6}}\) \(\def\upRho{\unicode{x03A1}}\) \(\def\upSigma{\unicode{x03A3}}\) \(\def\upTau{\unicode{x03A4}}\) \(\def\upUpsilon{\unicode{x03A5}}\) \(\def\upPhi{\unicode{x03A6}}\) \(\def\upChi{\unicode{x03A7}}\) \(\def\upPsi{\unicode{x03A8}}\) \(\def\upOmega{\unicode{x03A9}}\) \(\def\itAlpha{\unicode{x1D6E2}}\) \(\def\itBeta{\unicode{x1D6E3}}\) \(\def\itGamma{\unicode{x1D6E4}}\) \(\def\itDigamma{\mathit{\unicode{x03DC}}}\) \(\def\itDelta{\unicode{x1D6E5}}\) \(\def\itEpsilon{\unicode{x1D6E6}}\) \(\def\itZeta{\unicode{x1D6E7}}\) \(\def\itEta{\unicode{x1D6E8}}\) \(\def\itTheta{\unicode{x1D6E9}}\) \(\def\itVartheta{\unicode{x1D6F3}}\) \(\def\itIota{\unicode{x1D6EA}}\) \(\def\itKappa{\unicode{x1D6EB}}\) \(\def\itLambda{\unicode{x1D6EC}}\) \(\def\itMu{\unicode{x1D6ED}}\) \(\def\itNu{\unicode{x1D6EE}}\) \(\def\itXi{\unicode{x1D6EF}}\) \(\def\itOmicron{\unicode{x1D6F0}}\) \(\def\itPi{\unicode{x1D6F1}}\) \(\def\itRho{\unicode{x1D6F2}}\) \(\def\itSigma{\unicode{x1D6F4}}\) \(\def\itTau{\unicode{x1D6F5}}\) \(\def\itUpsilon{\unicode{x1D6F6}}\) \(\def\itPhi{\unicode{x1D6F7}}\) \(\def\itChi{\unicode{x1D6F8}}\) \(\def\itPsi{\unicode{x1D6F9}}\) \(\def\itOmega{\unicode{x1D6FA}}\) \(\def\upalpha{\unicode{x03B1}}\) \(\def\upbeta{\unicode{x03B2}}\) \(\def\upvarbeta{\unicode{x03D0}}\) \(\def\upgamma{\unicode{x03B3}}\) \(\def\updigamma{\unicode{x03DD}}\) \(\def\updelta{\unicode{x03B4}}\) \(\def\upepsilon{\unicode{x03F5}}\) \(\def\upvarepsilon{\unicode{x03B5}}\) \(\def\upzeta{\unicode{x03B6}}\) \(\def\upeta{\unicode{x03B7}}\) \(\def\uptheta{\unicode{x03B8}}\) \(\def\upvartheta{\unicode{x03D1}}\) \(\def\upiota{\unicode{x03B9}}\) \(\def\upkappa{\unicode{x03BA}}\) \(\def\upvarkappa{\unicode{x03F0}}\) \(\def\uplambda{\unicode{x03BB}}\) \(\def\upmu{\unicode{x03BC}}\) \(\def\upnu{\unicode{x03BD}}\) \(\def\upxi{\unicode{x03BE}}\) \(\def\upomicron{\unicode{x03BF}}\) \(\def\uppi{\unicode{x03C0}}\) \(\def\upvarpi{\unicode{x03D6}}\) \(\def\uprho{\unicode{x03C1}}\) \(\def\upvarrho{\unicode{x03F1}}\) \(\def\upsigma{\unicode{x03C3}}\) \(\def\upvarsigma{\unicode{x03C2}}\) \(\def\uptau{\unicode{x03C4}}\) \(\def\upupsilon{\unicode{x03C5}}\) \(\def\upphi{\unicode{x03D5}}\) \(\def\upvarphi{\unicode{x03C6}}\) \(\def\upchi{\unicode{x03C7}}\) \(\def\uppsi{\unicode{x03C8}}\) \(\def\upomega{\unicode{x03C9}}\) \(\def\italpha{\unicode{x1D6FC}}\) \(\def\itbeta{\unicode{x1D6FD}}\) \(\def\itvarbeta{\unicode{x03D0}}\) \(\def\itgamma{\unicode{x1D6FE}}\) \(\def\itdigamma{\mathit{\unicode{x03DD}}}\) \(\def\itdelta{\unicode{x1D6FF}}\) \(\def\itepsilon{\unicode{x1D716}}\) \(\def\itvarepsilon{\unicode{x1D700}}\) \(\def\itzeta{\unicode{x1D701}}\) \(\def\iteta{\unicode{x1D702}}\) \(\def\ittheta{\unicode{x1D703}}\) \(\def\itvartheta{\unicode{x1D717}}\) \(\def\itiota{\unicode{x1D704}}\) \(\def\itkappa{\unicode{x1D705}}\) \(\def\itvarkappa{\unicode{x1D718}}\) \(\def\itlambda{\unicode{x1D706}}\) \(\def\itmu{\unicode{x1D707}}\) \(\def\itnu{\unicode{x1D708}}\) \(\def\itxi{\unicode{x1D709}}\) \(\def\itomicron{\unicode{x1D70A}}\) \(\def\itpi{\unicode{x1D70B}}\) \(\def\itvarpi{\unicode{x1D71B}}\) \(\def\itrho{\unicode{x1D70C}}\) \(\def\itvarrho{\unicode{x1D71A}}\) 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{x2A08}}\limits }\) \(\newcommand {\bigtimes }{\mathop {\unicode {x2A09}}\limits }\) \(\newcommand {\modtwosum }{\mathop {\unicode {x2A0A}}\limits }\) \(\newcommand {\sumint }{\mathop {\unicode {x2A0B}}\limits }\) \(\newcommand {\intbar }{\mathop {\unicode {x2A0D}}\limits }\) \(\newcommand {\intBar }{\mathop {\unicode {x2A0E}}\limits }\) \(\newcommand {\fint }{\mathop {\unicode {x2A0F}}\limits }\) \(\newcommand {\cirfnint }{\mathop {\unicode {x2A10}}\limits }\) \(\newcommand {\awint }{\mathop {\unicode {x2A11}}\limits }\) \(\newcommand {\rppolint }{\mathop {\unicode {x2A12}}\limits }\) \(\newcommand {\scpolint }{\mathop {\unicode {x2A13}}\limits }\) \(\newcommand {\npolint }{\mathop {\unicode {x2A14}}\limits }\) \(\newcommand {\pointint }{\mathop {\unicode {x2A15}}\limits }\) \(\newcommand {\sqint }{\mathop {\unicode {x2A16}}\limits }\) \(\newcommand {\intlarhk }{\mathop {\unicode {x2A17}}\limits }\) \(\newcommand {\intx }{\mathop {\unicode {x2A18}}\limits }\) \(\newcommand {\intcap 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Exam version: exam4sample

David W. Agler

August 20, 2026

This exam has 47 questions, for a total of 100 points and X bonus points. Place proofs on the blank space on the answersheet. Good luck!

Q1. A model \(M\) is a two-part structure consisting of what? Select two choices.

  • 1. A valuation function \(v\)

  • 2. A set of second-order predicates that can be quantified over, e.g., \((\forall P)Pa\)

  • 3. *A domain \(\mathcal {D}\)

  • 4. *An interpretation function \(\mathscr {I}\)

Q2. An interpretation of QL is a function that does what (indicate all that apply):

  • 1. specifies what objects are in the domain.

  • 2. assigns truth values to n-place predicate terms followed by n terms.

  • 3. *for each name in QL it assigns that name one and only one item in \(\mathcal {D}\)

  • 4. *for each \(n\)-place predicate term in QL assigns, it assigns that predicate term a set of \(n\)-tuples composed of elements from \(\mathcal {D}\)

Q3. What is a derivation / proof of \(\psi \) from \(\Gamma \) using \(\mathbf {QD}\)?

  • 1. a finite string of wffs starting with some premises \(\Gamma \) and ending with \(\psi \).

  • 2. a finite string of wffs starting with some premises \(\Gamma \) or assumptions and ending with \(\psi \).

  • 3. *a finite string of formulas from a set \(\Gamma \) of QL wffs where (i) the last formula in the string is \(\psi \) and (ii) each formula is either a premise, an assumption, or is the result of the preceding formulas and the deductive apparatus.

Are the following wffs open or closed?

KEY: (A) = Open. (B) = Closed.

Q6. \(\neg Fab\) --- Answer: closed

Q7. \(\neg Qy\) --- Answer: open

Q8. \((\exists x)(\forall y)\neg Lxy\) --- Answer: closed

Determine whether the following wffs are T or F by using the following model: \(\mathcal {D}= \{1,2,3\}\), \(\mathscr {I}(a)=1\), \(\mathscr {I}(b)=2\), \(\mathscr {I}(c)=3\), \(\mathscr {I}(N)=\{1,2,3\}\), \(\mathscr {I}(G)=\{\langle 2,1 \rangle , \langle 3,2 \rangle , \langle 3,1\rangle \}\), \(\mathscr {I}(I)=\varnothing \), \(\mathscr {I}(E)=\{2\}\), \(\mathscr {I}(O)=\{1,3\}\)

KEY: (A) = T. (B) = F.

Q9. \((\exists x)\neg Ix\) --- Answer: T

Q10. \(Gab\) --- Answer: F

Q11. \(Gba\) --- Answer: T

Q12. \((\exists y)Ey\) --- Answer: T

Q13. \((\exists y)\neg Ey\) --- Answer: T

Q14. \(\neg (\exists y)Ey\) --- Answer: F

Q15. \((\forall x)Ox\) --- Answer: F

Q16. \((\exists x)(Nx\wedge Gxc)\) --- Answer: F

Q17. \((\exists x)(Ex\wedge Nx)\) --- Answer: T

Q18. \((\forall y)(Ey\rightarrow Nx)\) --- Answer: T

Select the correct translation of the QL wffs below using the following: \(Ax\): x is an athlete. \(Sx\): x is a student.

KEY: (A) = Some athletes are students. (B) = Some athletes are not students. (C) = All students are athletes. (D) = All athletes are students. (E) = No athletes are students.

Q20. \((\exists x)(Ax\land Sx)\) --- Answer: A

Q21. \((\forall x)(Ax\to \lnot Sx)\) --- Answer: E

Q22. \((\exists x)(Ax\land \lnot Sx)\) --- Answer: B

Q23. \((\forall x)(Ax\to Sx)\) --- Answer: D

Q24. \((\forall x)(Sx\to Ax)\) --- Answer: C

Q25. \(\lnot (\exists x)(Ax\land Sx)\) --- Answer: E

Select the correct translation of the QL wffs below using the following: \(a\): Ava; \(Lxy\): x loves y.

KEY: (A) = Someone loves Ava. (B) = Ava loves someone. (C) = Someone loves someone. (D) = Someone loves themselves.

Q26. \((\exists x)Lxa\) --- Answer: Someone loves Ava.

Q27. \((\exists x)Lax\) --- Answer: Ava loves someone.

Q28. \((\exists x)(\exists y)Lxy\) --- Answer: Someone loves someone.

Q29. \((\exists x)Lxx\) --- Answer: Someone loves themselves.

Select the derivation rule described in the following:

KEY: (A) = \(\exists I\), (B) = \(\exists E\), (C) = \(\forall I\), (D) = \(\forall I\), (E) = \(QN\)

Q39. \(\neg Qab\wedge Pa \vdash (\exists z)(\neg Qzb\wedge Pz)\) --- Answer: \(\exists I\)

Q40. \((\forall z)(Fz\wedge \neg Fz)\vdash Fd\rightarrow \neg Fd\) --- Answer: \(\forall E\)

Q41. From \(Qc\wedge Fc\) to \((\forall y)(Qy\wedge Fy)\) provided (1) \(c\) is not in a premise or in an assumption of an active subproof and (2) \(c\) is not in \((\forall y)(Qy\wedge Fy)\)? --- Answer: \(\forall I\)

Q42. \(\neg (\forall z)Fz\vdash (\exists z)\neg Fz\) --- Answer: \(QN\)

Provide proofs for the following:

Q43. \((\exists x)Fx\rightarrow (\forall y) By, Fa\vdash Ba\)

--- Answer: \((\exists x)Fx\rightarrow (\forall y) By, Fa\vdash Ba\)

(A natural deduction proof with five lines. Line 1: there exists an x such that Fx implies for all y By, premise. Line 2: Fa, premise. Line 3: there exists an x such that Fx, from line 2 by existential introduction. Line 4: for

all y By, from lines 1 and 3 by conditional elimination. Line 5: Ba, from line 4 by universal elimination.)

Q44. \(Faa, (\forall x)(\forall y)Lxy\)
\(\vdash (\exists z)(\exists x)Lzx\)

--- Answer: \(Faa, (\forall x)(\forall y)Lxy\vdash (\exists z)(\exists x)Lzx\)

(A natural deduction proof with six lines. Line 1: Faa, premise. Line 2: for all x for all y Lxy, premise. Line 3: for all y Lay, from line 2 by universal elimination. Line 4: Laa, from line 3 by universal elimination. Line 5:

there exists an x such that Lax, from line 4 by existential introduction. Line 6: there exists a z there exists an x such that Lzx, from line 5 by existential introduction.)

Q45. \((\exists x)\neg Mx\vdash (\exists y)(My\lor \neg Py)\)

--- Answer: \((\exists x)\neg Mx\vdash (\exists y)(My\lor \neg Py)\)

(A natural deduction proof with five lines. Line 1: there exists an x such that not Mx, premise. Line 2: not Ma, assumption for existential elimination. Line 3: not Ma or not Pa, from line 2 by disjunction introduction. Line 4:

there exists a y such that My or not Py, from line 3 by existential introduction. Line 5: there exists a y such that My or not Py, from line 1 and lines 2 through 4 by existential elimination.)

Q46. \(\vdash (\forall x)(Lxx\rightarrow (Gx\rightarrow Lxx))\)

--- Answer: \(\vdash (\forall x)(Lxx\rightarrow (Gx\rightarrow Lxx))\)

(A natural deduction proof with six lines. Line 1: Laa, assumption for conditional introduction. Line 2: Ga, assumption for conditional introduction. Line 3: Laa, from line 1 by reiteration. Line 4: Ga implies Laa, from lines 2

through 3 by conditional introduction. Line 5: Laa implies Ga implies Laa, from lines 1 through 4 by conditional introduction. Line 6: for all x Lxx implies Gx implies Lxx, from line 5 by universal introduction.)