\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\TextOrMath }[2]{#2}\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand {\LWRabsorboption }[1][]{}\) \(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\) \(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\) \(\def \mathcode #1={\mathchar }\) \(\let \delcode \mathcode \) \(\let \delimiter \mathchar \) \(\def \oe {\unicode {x0153}}\) \(\def \OE {\unicode {x0152}}\) \(\def \ae {\unicode {x00E6}}\) \(\def \AE {\unicode {x00C6}}\) \(\def \aa {\unicode {x00E5}}\) \(\def \AA {\unicode {x00C5}}\) \(\def \o {\unicode {x00F8}}\) \(\def \O {\unicode {x00D8}}\) \(\def \l {\unicode {x0142}}\) \(\def \L {\unicode {x0141}}\) \(\def \ss {\unicode {x00DF}}\) \(\def \SS {\unicode {x1E9E}}\) \(\def \dag {\unicode {x2020}}\) \(\def \ddag {\unicode {x2021}}\) \(\def \P {\unicode {x00B6}}\) \(\def \copyright {\unicode {x00A9}}\) \(\def \pounds {\unicode {x00A3}}\) \(\let \LWRref \ref \) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \( \newcommand {\multicolumn }[3]{#3}\) \(\require {textcomp}\) 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\\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) \(\let \Dot \dot \) \(\let \Ddot \ddot \) \(\let \Breve \breve \) \(\let \Bar \bar \) \(\let \Vec \vec \) \(\require {mathtools}\) \(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\) \(\newcommand {\approxcolon }{\approx \vcentcolon }\) \(\newcommand {\Approxcolon }{\approx \dblcolon }\) \(\newcommand {\simcolon }{\sim \vcentcolon }\) \(\newcommand {\Simcolon }{\sim \dblcolon }\) \(\newcommand {\dashcolon }{\mathrel {-}\vcentcolon }\) \(\newcommand {\Dashcolon }{\mathrel {-}\dblcolon }\) \(\newcommand {\colondash }{\vcentcolon \mathrel {-}}\) \(\newcommand {\Colondash }{\dblcolon \mathrel {-}}\) \(\newenvironment {crampedsubarray}[1]{}{}\) \(\newcommand {\smashoperator }[2][]{#2\limits }\) \(\newcommand {\SwapAboveDisplaySkip }{}\) \(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\) \(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\) \(\Newextarrow \xLongleftarrow {10,10}{0x21D0}\) \(\Newextarrow \xLongrightarrow {10,10}{0x21D2}\) \(\let \xlongleftarrow \xleftarrow \) \(\let \xlongrightarrow \xrightarrow \) \(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\) \(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\) \(\let \LWRorigshoveleft \shoveleft \) \(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\) \(\let \LWRorigshoveright \shoveright \) \(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\) \(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\) \(\newcommand {\LWRnicearrayarray }[1]{\begin {array}{#1}}\) \(\def \LWRnicearrayarrayopt #1[#2] {\begin {array}{#1}}\) \(\newenvironment {NiceArray}[2][]{\ifnextchar [{\LWRnicearrayarrayopt {#2}}{\LWRnicearrayarray {#2}}}{\end {array}}\) \(\newcommand {\LWRnicearraywithdelimtwo }[2][]{\ifnextchar [{\LWRnicearrayarrayopt {#2}}{\LWRnicearrayarray {#2}}}\) \(\newenvironment {NiceArrayWithDelims}[2]{\def \LWRnicearrayrightdelim {\right #2}\left #1\LWRnicearraywithdelimtwo }{\end {array}\LWRnicearrayrightdelim }\) \(\newenvironment {pNiceArray} {\begin {NiceArrayWithDelims}{(}{)}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {bNiceArray} {\begin {NiceArrayWithDelims}{[}{]}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {BNiceArray} {\begin {NiceArrayWithDelims}{\{}{\}}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {vNiceArray} {\begin {NiceArrayWithDelims}{\vert }{\vert }} {\end {NiceArrayWithDelims}} \) \(\newenvironment {VNiceArray} {\begin {NiceArrayWithDelims}{\Vert }{\Vert }} {\end {NiceArrayWithDelims}} \) \(\newenvironment {NiceMatrix}[1][]{\begin {matrix}}{\end {matrix}}\) \(\newenvironment {pNiceMatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\) \(\newenvironment {bNiceMatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\) \(\newenvironment {BNiceMatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\) \(\newenvironment {vNiceMatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\) \(\newenvironment {VNiceMatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\) \(\newcommand {\LWRnicematrixBlock }[1]{#1}\) \(\def \LWRnicematrixBlockopt <#1>#2{#2}\) \(\newcommand {\Block }[2][]{\ifnextchar <\LWRnicematrixBlockopt \LWRnicematrixBlock }\) \(\newcommand {\diagbox }[2]{\begin {array}{l}\hfill \quad #2\\\hline #1\quad \hfill \end {array}}\) \(\let \hdottedline \hdashline \) \(\newcommand {\Hline }[1][]{\hline }\) \(\newcommand {\CodeBefore }{}\) \(\newcommand {\Body }{}\) \(\newcommand {\CodeAfter }{}\) \(\newcommand {\line }[3][]{}\) \(\newcommand {\RowStyle }[2][]{}\) \(\newcommand {\LWRSubMatrix }[1][]{}\) \(\newcommand {\SubMatrix }[4]{\LWRSubMatrix }\) \(\newcommand {\OverBrace }[4][]{}\) \(\newcommand {\UnderBrace }[4][]{}\) \(\newcommand {\HBrace }[3][]{}\) \(\newcommand {\VBrace }[3][]{}\) \(\newcommand {\ShowCellNames }{}\) \(\newcommand {\tabularnote }[2][]{}\) \(\newcommand {\cellcolor }[3][]{}\) \(\newcommand {\rowcolor }[3][]{}\) \(\newcommand {\LWRrowcolors }[1][]{}\) \(\newcommand {\rowcolors 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{#2}}\) \(\newcommand {\tcblower }{}\) \(\newcommand {\tcbline }{}\) \(\newcommand {\tcbtitle }{}\) \(\newcommand {\tcbsubtitle [2][]{\mathrm {#2}}}\) \(\newcommand {\tcboxmath }[2][]{\boxed {#2}}\) \(\newcommand {\tcbhighmath }[2][]{\boxed {#2}}\) \(\let \symnormal \mathit \) \(\let \symliteral \mathrm \) \(\let \symbb \mathbb \) \(\let \symbbit \mathbb \) \(\let \symcal \mathcal \) \(\let \symscr \mathscr \) \(\let \symfrak \mathfrak \) \(\let \symsfup \mathsf \) \(\let \symsfit \mathit \) \(\let \symbfsf \mathbf \) \(\let \symbfup \mathbf \) \(\newcommand {\symbfit }[1]{\boldsymbol {#1}}\) \(\let \symbfcal \mathcal \) \(\let \symbfscr \mathscr \) \(\let \symbffrak \mathfrak \) \(\let \symbfsfup \mathbf \) \(\newcommand {\symbfsfit }[1]{\boldsymbol {#1}}\) \(\let \symup \mathrm \) \(\let \symbf \mathbf \) \(\let \symit \mathit \) \(\let \symtt \mathtt \) \(\let \symbffrac \mathbffrac \) \(\newcommand {\mathfence }[1]{\mathord {#1}}\) \(\newcommand {\mathover }[1]{#1}\) \(\newcommand {\mathunder }[1]{#1}\) \(\newcommand {\mathaccent }[1]{#1}\) \(\newcommand {\mathbotaccent }[1]{#1}\) \(\newcommand {\mathalpha }[1]{\mathord {#1}}\) \(\def\upAlpha{\unicode{x0391}}\) \(\def\upBeta{\unicode{x0392}}\) \(\def\upGamma{\unicode{x0393}}\) \(\def\upDigamma{\unicode{x03DC}}\) \(\def\upDelta{\unicode{x0394}}\) \(\def\upEpsilon{\unicode{x0395}}\) \(\def\upZeta{\unicode{x0396}}\) \(\def\upEta{\unicode{x0397}}\) \(\def\upTheta{\unicode{x0398}}\) \(\def\upVartheta{\unicode{x03F4}}\) \(\def\upIota{\unicode{x0399}}\) \(\def\upKappa{\unicode{x039A}}\) \(\def\upLambda{\unicode{x039B}}\) \(\def\upMu{\unicode{x039C}}\) \(\def\upNu{\unicode{x039D}}\) \(\def\upXi{\unicode{x039E}}\) \(\def\upOmicron{\unicode{x039F}}\) \(\def\upPi{\unicode{x03A0}}\) \(\def\upVarpi{\unicode{x03D6}}\) \(\def\upRho{\unicode{x03A1}}\) \(\def\upSigma{\unicode{x03A3}}\) \(\def\upTau{\unicode{x03A4}}\) \(\def\upUpsilon{\unicode{x03A5}}\) \(\def\upPhi{\unicode{x03A6}}\) \(\def\upChi{\unicode{x03A7}}\) \(\def\upPsi{\unicode{x03A8}}\) \(\def\upOmega{\unicode{x03A9}}\) \(\def\itAlpha{\unicode{x1D6E2}}\) \(\def\itBeta{\unicode{x1D6E3}}\) \(\def\itGamma{\unicode{x1D6E4}}\) \(\def\itDigamma{\mathit{\unicode{x03DC}}}\) \(\def\itDelta{\unicode{x1D6E5}}\) \(\def\itEpsilon{\unicode{x1D6E6}}\) \(\def\itZeta{\unicode{x1D6E7}}\) \(\def\itEta{\unicode{x1D6E8}}\) \(\def\itTheta{\unicode{x1D6E9}}\) \(\def\itVartheta{\unicode{x1D6F3}}\) \(\def\itIota{\unicode{x1D6EA}}\) \(\def\itKappa{\unicode{x1D6EB}}\) \(\def\itLambda{\unicode{x1D6EC}}\) \(\def\itMu{\unicode{x1D6ED}}\) \(\def\itNu{\unicode{x1D6EE}}\) \(\def\itXi{\unicode{x1D6EF}}\) \(\def\itOmicron{\unicode{x1D6F0}}\) \(\def\itPi{\unicode{x1D6F1}}\) \(\def\itRho{\unicode{x1D6F2}}\) \(\def\itSigma{\unicode{x1D6F4}}\) \(\def\itTau{\unicode{x1D6F5}}\) \(\def\itUpsilon{\unicode{x1D6F6}}\) \(\def\itPhi{\unicode{x1D6F7}}\) \(\def\itChi{\unicode{x1D6F8}}\) \(\def\itPsi{\unicode{x1D6F9}}\) \(\def\itOmega{\unicode{x1D6FA}}\) \(\def\upalpha{\unicode{x03B1}}\) \(\def\upbeta{\unicode{x03B2}}\) \(\def\upvarbeta{\unicode{x03D0}}\) \(\def\upgamma{\unicode{x03B3}}\) \(\def\updigamma{\unicode{x03DD}}\) \(\def\updelta{\unicode{x03B4}}\) \(\def\upepsilon{\unicode{x03F5}}\) \(\def\upvarepsilon{\unicode{x03B5}}\) \(\def\upzeta{\unicode{x03B6}}\) \(\def\upeta{\unicode{x03B7}}\) \(\def\uptheta{\unicode{x03B8}}\) \(\def\upvartheta{\unicode{x03D1}}\) \(\def\upiota{\unicode{x03B9}}\) \(\def\upkappa{\unicode{x03BA}}\) \(\def\upvarkappa{\unicode{x03F0}}\) \(\def\uplambda{\unicode{x03BB}}\) \(\def\upmu{\unicode{x03BC}}\) \(\def\upnu{\unicode{x03BD}}\) \(\def\upxi{\unicode{x03BE}}\) \(\def\upomicron{\unicode{x03BF}}\) \(\def\uppi{\unicode{x03C0}}\) \(\def\upvarpi{\unicode{x03D6}}\) \(\def\uprho{\unicode{x03C1}}\) \(\def\upvarrho{\unicode{x03F1}}\) \(\def\upsigma{\unicode{x03C3}}\) \(\def\upvarsigma{\unicode{x03C2}}\) \(\def\uptau{\unicode{x03C4}}\) \(\def\upupsilon{\unicode{x03C5}}\) \(\def\upphi{\unicode{x03D5}}\) \(\def\upvarphi{\unicode{x03C6}}\) \(\def\upchi{\unicode{x03C7}}\) \(\def\uppsi{\unicode{x03C8}}\) \(\def\upomega{\unicode{x03C9}}\) \(\def\italpha{\unicode{x1D6FC}}\) \(\def\itbeta{\unicode{x1D6FD}}\) \(\def\itvarbeta{\unicode{x03D0}}\) \(\def\itgamma{\unicode{x1D6FE}}\) \(\def\itdigamma{\mathit{\unicode{x03DD}}}\) \(\def\itdelta{\unicode{x1D6FF}}\) \(\def\itepsilon{\unicode{x1D716}}\) \(\def\itvarepsilon{\unicode{x1D700}}\) \(\def\itzeta{\unicode{x1D701}}\) \(\def\iteta{\unicode{x1D702}}\) \(\def\ittheta{\unicode{x1D703}}\) \(\def\itvartheta{\unicode{x1D717}}\) \(\def\itiota{\unicode{x1D704}}\) \(\def\itkappa{\unicode{x1D705}}\) \(\def\itvarkappa{\unicode{x1D718}}\) \(\def\itlambda{\unicode{x1D706}}\) \(\def\itmu{\unicode{x1D707}}\) \(\def\itnu{\unicode{x1D708}}\) \(\def\itxi{\unicode{x1D709}}\) \(\def\itomicron{\unicode{x1D70A}}\) \(\def\itpi{\unicode{x1D70B}}\) \(\def\itvarpi{\unicode{x1D71B}}\) \(\def\itrho{\unicode{x1D70C}}\) \(\def\itvarrho{\unicode{x1D71A}}\) 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{x2A08}}\limits }\) \(\newcommand {\bigtimes }{\mathop {\unicode {x2A09}}\limits }\) \(\newcommand {\modtwosum }{\mathop {\unicode {x2A0A}}\limits }\) \(\newcommand {\sumint }{\mathop {\unicode {x2A0B}}\limits }\) \(\newcommand {\intbar }{\mathop {\unicode {x2A0D}}\limits }\) \(\newcommand {\intBar }{\mathop {\unicode {x2A0E}}\limits }\) \(\newcommand {\fint }{\mathop {\unicode {x2A0F}}\limits }\) \(\newcommand {\cirfnint }{\mathop {\unicode {x2A10}}\limits }\) \(\newcommand {\awint }{\mathop {\unicode {x2A11}}\limits }\) \(\newcommand {\rppolint }{\mathop {\unicode {x2A12}}\limits }\) \(\newcommand {\scpolint }{\mathop {\unicode {x2A13}}\limits }\) \(\newcommand {\npolint }{\mathop {\unicode {x2A14}}\limits }\) \(\newcommand {\pointint }{\mathop {\unicode {x2A15}}\limits }\) \(\newcommand {\sqint }{\mathop {\unicode {x2A16}}\limits }\) \(\newcommand {\intlarhk }{\mathop {\unicode {x2A17}}\limits }\) \(\newcommand {\intx }{\mathop {\unicode {x2A18}}\limits }\) \(\newcommand {\intcap }{\mathop {\unicode {x2A19}}\limits }\) \(\newcommand {\intcup }{\mathop {\unicode {x2A1A}}\limits }\) \(\newcommand {\upint }{\mathop {\unicode {x2A1B}}\limits }\) \(\newcommand {\lowint }{\mathop {\unicode {x2A1C}}\limits }\) \(\newcommand {\bigtriangleleft }{\mathop {\unicode {x2A1E}}\limits }\) \(\newcommand {\zcmp }{\mathop {\unicode {x2A1F}}\limits }\) \(\newcommand {\zpipe }{\mathop {\unicode {x2A20}}\limits }\) \(\newcommand {\zproject }{\mathop {\unicode {x2A21}}\limits }\) \(\newcommand {\biginterleave }{\mathop {\unicode {x2AFC}}\limits }\) \(\newcommand {\bigtalloblong }{\mathop {\unicode {x2AFF}}\limits }\) \(\newcommand {\arabicmaj }{\mathop {\unicode {x1EEF0}}\limits }\) \(\newcommand {\arabichad }{\mathop {\unicode {x1EEF1}}\limits }\) \(\let \symsf \symsfup \) \(\def\Alpha{\unicode{x0391}}\) \(\def\Beta{\unicode{x0392}}\) \(\def\Gamma{\unicode{x0393}}\) \(\def\Digamma{\unicode{x03DC}}\) \(\def\Delta{\unicode{x0394}}\) \(\def\Epsilon{\unicode{x0395}}\) \(\def\Zeta{\unicode{x0396}}\) \(\def\Eta{\unicode{x0397}}\) \(\def\Theta{\unicode{x0398}}\) \(\def\Vartheta{\unicode{x03F4}}\) \(\def\Iota{\unicode{x0399}}\) \(\def\Kappa{\unicode{x039A}}\) \(\def\Lambda{\unicode{x039B}}\) \(\def\Mu{\unicode{x039C}}\) \(\def\Nu{\unicode{x039D}}\) \(\def\Xi{\unicode{x039E}}\) \(\def\Omicron{\unicode{x039F}}\) \(\def\Pi{\unicode{x03A0}}\) \(\def\Varpi{\unicode{x03D6}}\) \(\def\Rho{\unicode{x03A1}}\) \(\def\Sigma{\unicode{x03A3}}\) \(\def\Tau{\unicode{x03A4}}\) \(\def\Upsilon{\unicode{x03A5}}\) \(\def\Phi{\unicode{x03A6}}\) \(\def\Chi{\unicode{x03A7}}\) \(\def\Psi{\unicode{x03A8}}\) \(\def\Omega{\unicode{x03A9}}\) \(\def\alpha{\unicode{x1D6FC}}\) \(\def\beta{\unicode{x1D6FD}}\) \(\def\varbeta{\unicode{x03D0}}\) \(\def\gamma{\unicode{x1D6FE}}\) \(\def\digamma{\mathit{\unicode{x03DD}}}\) \(\def\delta{\unicode{x1D6FF}}\) \(\def\epsilon{\unicode{x1D716}}\) \(\def\varepsilon{\unicode{x1D700}}\) \(\def\zeta{\unicode{x1D701}}\) \(\def\eta{\unicode{x1D702}}\) \(\def\theta{\unicode{x1D703}}\) \(\def\vartheta{\unicode{x1D717}}\) \(\def\iota{\unicode{x1D704}}\) \(\def\kappa{\unicode{x1D705}}\) \(\def\varkappa{\unicode{x1D718}}\) \(\def\lambda{\unicode{x1D706}}\) \(\def\mu{\unicode{x1D707}}\) \(\def\nu{\unicode{x1D708}}\) \(\def\xi{\unicode{x1D709}}\) \(\def\omicron{\unicode{x1D70A}}\) \(\def\pi{\unicode{x1D70B}}\) \(\def\varpi{\unicode{x1D71B}}\) \(\def\rho{\unicode{x1D70C}}\) \(\def\varrho{\unicode{x1D71A}}\) \(\def\sigma{\unicode{x1D70E}}\) \(\def\varsigma{\unicode{x1D70D}}\) \(\def\tau{\unicode{x1D70F}}\) \(\def\upsilon{\unicode{x1D710}}\) \(\def\phi{\unicode{x1D719}}\) \(\def\varphi{\unicode{x1D711}}\) \(\def\chi{\unicode{x1D712}}\) \(\def\psi{\unicode{x1D713}}\) \(\def\omega{\unicode{x1D714}}\)

Exam version: su26a

David W. Agler

June 23, 2026

This exam has 11 questions, for a total of 100 points and X bonus points. Place your name on the answersheet (last page). Place proofs on the blank space on the answersheet. Good luck!

Writing the abbreviation (e.g., \(\forall I\)) for the single derivation rule that is represented in the following:

Q1. Starting from \((\exists x)Fx\), suppose \(Fa\) is assumed. Next, suppose \(\phi \) is derived in the subproof starting with \(Fa\). Finally, suppose \(\phi \) is deprived using \((\exists x)Fx\) and the entire subproof. --- Answer: \(\exists E\)

Q2. \(\neg Qab\wedge Pa \vdash (\exists z)(\neg Qzb\wedge Pz)\) --- Answer: \(\exists I\)

Q3. \((\forall z)(Fz\wedge \neg Fz)\vdash Fd\rightarrow \neg Fd\) --- Answer: \(\forall E\)

Q4. 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: \(\forall I\)

Q5. \(\neg (\forall z)Fz\vdash (\exists z)\neg Fz\) --- Answer: \(QN\)

Provide proofs for the following:

Q6. \((\exists x)Fx\rightarrow (\forall y) By, Fa\vdash Ba\)

--- 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.)

Q7. \(Faa, (\forall x)(\forall y)Lxy\)
\(\vdash (\exists z)(\exists x)Lzx\)

--- 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.)

Q8. \((\exists x)\neg Mx\vdash (\exists y)(My\lor \neg Py)\)

--- 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.)

Q9. \(\vdash (\forall x)(Lxx\rightarrow (Gx\rightarrow Lxx))\) --- Answer: \(\vdash (\forall x)(Lxx\rightarrow (Gx\rightarrow Lxx))\)

(A natural deduction proof with four lines. Line 1: Pa, assumption for conditional introduction. Line 2: Pa, from line 1 by reiteration. Line 3: Pa implies Pa, from lines 1 through 2 by conditional introduction. Line 4: for all x

Px implies Px, from line 3 by universal introduction.)

Q10. \(\vdash (\forall x)(Lxx\rightarrow (Gx\rightarrow Lxx))\)

--- 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.)

Translate the following arguments and then provide a proof of the conclusion from the premises. In some cases, part of the argument will be provided for you.

Q11. Some Penn State students are good logicians (\((\exists x)(Px\land Gx)\)). All good logicians are smart. Therefore, some Penn State students are smart.

--- Answer: Translation: \((\exists x)(Px\land Gx), (\forall x)(Gx\to Sx)\vdash (\exists x)(Px\land Sx)\)

(A natural deduction proof with ten lines. Line 1: there exists an x such that Px and Gx, premise. Line 2: for all x Gx implies Sx, premise. Line 3: Pa and Ga, assumption for existential elimination. Line 4: Ga implies Sa, from

line 2 by universal elimination. Line 5: Pa, from line 3 by conjunction elimination. Line 6: Ga, from line 3 by conjunction elimination. Line 7: Sa, from lines 4 and 6 by conditional elimination. Line 8: Pa and Sa, from lines 5 and 7 by
conjunction introduction. Line 9: there exists an x such that Px and Sx, from line 8 by existential introduction. Line 10: there exists an x such that Px and Sx, from line 1 and lines 3 through 9 by existential elimination.)

Bonus questions. Totally optional.

Bonus Question: Let \(=\) be a new operator in QL such that \(\alpha =\alpha \) is a wff, where \(\alpha \) is a QL name. Now let \(=E\) be a derivation rule such that from \(\alpha =\beta \) and a wff \(\phi \) containing \(\alpha \) (or \(\beta \)), you may substitute \(\alpha \) for \(\beta \) in \(\phi \) or \(\beta \) for \(\alpha \). In other words, \(a=b, \phi \vdash \phi (a/b)\) or \(a=b, \phi \vdash \phi (b/a)\). With this in mind, prove \(a=b, b=c, Pa\vdash Pc\). Bonus Question: Free point. If you are reading this, it was very nice to have you in my class.

Derivation rules

Definition 1: Conjunction Introduction \(\wedge I\).

\(\phi , \psi \vdash \phi \wedge \psi \) or \(\phi , \psi \vdash \psi \wedge \phi \)

Definition 2: Conjunction Elimination (\(\wedge E\)).

\(\phi \wedge \psi \vdash \phi \) or \(\phi \wedge \psi \vdash \psi \)

Definition 3: Conditional Introduction (\(\rightarrow I\)).

(Proof)

Definition 4: Conditional Elimination (\(\rightarrow E\)).

\(\phi \rightarrow \psi , \phi \vdash \psi \)

Definition 5: Reiteration (R).

\(\phi \vdash \phi \)

Definition 6: Negation Introduction (\(\neg I\)).

(Proof)

Definition 7: Negation Elimination (\(\neg E\)).

(Proof)

Definition 8: Disjunction Introduction (\(\vee I\)).

\(\phi \vdash \phi \vee \psi \) or \(\phi \vdash \psi \vee \phi \)

Definition 9: Disjunction Elimination (\(\vee E\)).

(Proof)

Definition 10: Biconditional Introduction (\(\leftrightarrow I\)).

(Proof)

Definition 11: Biconditional Elimination (\(\leftrightarrow E\)).

\(\phi \leftrightarrow \psi , \phi \vdash \psi \) or \(\phi \leftrightarrow \psi , \psi \vdash \phi \)

Definition 12: Disjunctive Syllogism (DS).

\(\phi \vee \psi , \neg (\psi ) \vdash \phi \) or \(\phi \vee \psi , \neg (\phi ) \vdash \psi \)

Definition 13: Modus Tollens (MT).

\(\phi \rightarrow \psi , \neg (\psi ) \vdash \neg (\phi )\)

Definition 14: Hypothetical Syllogism (HS).

\(\phi \rightarrow \psi , \psi \rightarrow \chi \vdash \phi \rightarrow \chi \)

Definition 15: Double Negation (DN).

\(\phi \dashv \vdash \neg \neg (\phi )\)

Definition 16: De Morgan's Laws (DeM).

\(\neg (\phi \vee \psi ) \dashv \vdash \neg (\phi )\wedge \neg (\psi )\) or \(\neg (\phi \wedge \psi ) \dashv \vdash \neg (\phi )\vee \neg (\psi ) \)

Definition 17: Implication (IMP).

\(\phi \rightarrow \psi \dashv \vdash \neg (\phi ) \vee \psi \)

Definition 18: Universal Elimination (\(\forall E\)).

\((\forall x)\phi (x_1\ldots x_n) \vdash \phi (\alpha _1\ldots \alpha _n / x_1\ldots x_n)\) where \(x\) is not in \(\phi (\alpha _1\ldots \alpha _n)\)

Definition 19: Existential Introduction (\(\exists I\)).

\(\phi (\alpha _i) \vdash (\exists x)\phi (x_n / \alpha _n)\) where \(x\) is not in \(\phi (\alpha _i)\)

Definition 20: Universal Introduction \((\forall I)\).

\(\phi (\alpha _1 \ldots \alpha _n)\vdash (\forall x)\phi (x_1 \ldots x_n / \alpha _1, \ldots \alpha _n)\) where the name \(\alpha \) does not occur as premise, as an assumption in an open subproof, or in \((\forall x)\phi (x_1 \ldots x_n/\alpha _1, \ldots \alpha _n)\) and where \(x\) is not in \(\phi (\alpha _1 \ldots \alpha _n)\)

Definition 21: Existential Elimination (\(\exists E\)).

(Proof)

Definition 22: Quantifier Negation \((QN)\).

\(\neg (\forall x)\phi \dashv \vdash (\exists x)\neg \phi \) or \(\neg (\exists x)\phi \dashv \vdash (\forall x)\neg \phi \)