\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\TextOrMath }[2]{#2}\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\toprule }[1][]{\hline }\)
\(\let \midrule \toprule \)
\(\let \bottomrule \toprule \)
\(\def \LWRbooktabscmidruleparen (#1)#2{}\)
\(\newcommand {\LWRbooktabscmidrulenoparen }[1]{}\)
\(\newcommand {\cmidrule }[1][]{\ifnextchar (\LWRbooktabscmidruleparen \LWRbooktabscmidrulenoparen }\)
\(\newcommand {\morecmidrules }{}\)
\(\newcommand {\specialrule }[3]{\hline }\)
\(\newcommand {\addlinespace }[1][]{}\)
\(\def \LWRpagenote {1}\)
\(\newcommand {\pagenote }[2][\LWRpagenote ]{{}^{\mathrm {#1}}}\)
\(\require {colortbl}\)
\(\let \LWRorigcolumncolor \columncolor \)
\(\renewcommand {\columncolor }[2][named]{\LWRorigcolumncolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\let \LWRorigrowcolor \rowcolor \)
\(\renewcommand {\rowcolor }[2][named]{\LWRorigrowcolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\let \LWRorigcellcolor \cellcolor \)
\(\renewcommand {\cellcolor }[2][named]{\LWRorigcellcolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\require {cancel}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {mathtools}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand {\approxcolon }{\approx \vcentcolon }\)
\(\newcommand {\Approxcolon }{\approx \dblcolon }\)
\(\newcommand {\simcolon }{\sim \vcentcolon }\)
\(\newcommand {\Simcolon }{\sim \dblcolon }\)
\(\newcommand {\dashcolon }{\mathrel {-}\vcentcolon }\)
\(\newcommand {\Dashcolon }{\mathrel {-}\dblcolon }\)
\(\newcommand {\colondash }{\vcentcolon \mathrel {-}}\)
\(\newcommand {\Colondash }{\dblcolon \mathrel {-}}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\Newextarrow \xLongleftarrow {10,10}{0x21D0}\)
\(\Newextarrow \xLongrightarrow {10,10}{0x21D2}\)
\(\let \xlongleftarrow \xleftarrow \)
\(\let \xlongrightarrow \xrightarrow \)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\LWRnicearrayarray }[1]{\begin {array}{#1}}\)
\(\def \LWRnicearrayarrayopt #1[#2] {\begin {array}{#1}}\)
\(\newenvironment {NiceArray}[2][]{\ifnextchar [{\LWRnicearrayarrayopt {#2}}{\LWRnicearrayarray {#2}}}{\end {array}}\)
\(\newcommand {\LWRnicearraywithdelimtwo }[2][]{\ifnextchar [{\LWRnicearrayarrayopt {#2}}{\LWRnicearrayarray {#2}}}\)
\(\newenvironment {NiceArrayWithDelims}[2]{\def \LWRnicearrayrightdelim {\right #2}\left #1\LWRnicearraywithdelimtwo }{\end {array}\LWRnicearrayrightdelim }\)
\(\newenvironment {pNiceArray} {\begin {NiceArrayWithDelims}{(}{)}} {\end {NiceArrayWithDelims}} \)
\(\newenvironment {bNiceArray} {\begin {NiceArrayWithDelims}{[}{]}} {\end {NiceArrayWithDelims}} \)
\(\newenvironment {BNiceArray} {\begin {NiceArrayWithDelims}{\{}{\}}} {\end {NiceArrayWithDelims}} \)
\(\newenvironment {vNiceArray} {\begin {NiceArrayWithDelims}{\vert }{\vert }} {\end {NiceArrayWithDelims}} \)
\(\newenvironment {VNiceArray} {\begin {NiceArrayWithDelims}{\Vert }{\Vert }} {\end {NiceArrayWithDelims}} \)
\(\newenvironment {NiceMatrix}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {pNiceMatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bNiceMatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {BNiceMatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vNiceMatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {VNiceMatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newcommand {\LWRnicematrixBlock }[1]{#1}\)
\(\def \LWRnicematrixBlockopt <#1>#2{#2}\)
\(\newcommand {\Block }[2][]{\ifnextchar <\LWRnicematrixBlockopt \LWRnicematrixBlock }\)
\(\newcommand {\diagbox }[2]{\begin {array}{l}\hfill \quad #2\\\hline #1\quad \hfill \end {array}}\)
\(\let \hdottedline \hdashline \)
\(\newcommand {\Hline }[1][]{\hline }\)
\(\newcommand {\CodeBefore }{}\)
\(\newcommand {\Body }{}\)
\(\newcommand {\CodeAfter }{}\)
\(\newcommand {\line }[3][]{}\)
\(\newcommand {\RowStyle }[2][]{}\)
\(\newcommand {\LWRSubMatrix }[1][]{}\)
\(\newcommand {\SubMatrix }[4]{\LWRSubMatrix }\)
\(\newcommand {\OverBrace }[4][]{}\)
\(\newcommand {\UnderBrace }[4][]{}\)
\(\newcommand {\HBrace }[3][]{}\)
\(\newcommand {\VBrace }[3][]{}\)
\(\newcommand {\ShowCellNames }{}\)
\(\newcommand {\tabularnote }[2][]{}\)
\(\newcommand {\cellcolor }[3][]{}\)
\(\newcommand {\rowcolor }[3][]{}\)
\(\newcommand {\LWRrowcolors }[1][]{}\)
\(\newcommand {\rowcolors }[4][]{\LWRrowcolors }\)
\(\newcommand {\rowlistcolors }[3][]{\LWRrowcolors }\)
\(\newcommand {\columncolor }[3][]{}\)
\(\newcommand {\rectanglecolor }[4][]{}\)
\(\newcommand {\arraycolor }[2][]{}\)
\(\newcommand {\chessboardcolors }[3][]{}\)
\(\newcommand {\ldots }[1][]{\dots }\)
\(\newcommand {\Cdots }[1][]{\cdots }\)
\(\newcommand {\Vdots }[1][]{\vdots }\)
\(\newcommand {\Ddots }[1][]{\ddots }\)
\(\newcommand {\Iddots }[1][]{\mathinner {\unicode {x22F0}}}\)
\(\newcommand {\Hdotsfor }[1]{\ldots }\)
\(\newcommand {\Vdotsfor }[1]{\vdots }\)
\(\newcommand {\AutoNiceMatrix }[2]{\text {(AutoNiceMatrix #1)}}\)
\(\let \pAutoNiceMatrix \AutoNiceMatrix \)
\(\let \bAutoNiceMatrix \AutoNiceMatrix \)
\(\let \BAutoNiceMatrix \AutoNiceMatrix \)
\(\let \vAutoNiceMatrix \AutoNiceMatrix \)
\(\let \VAutoNiceMatrix \AutoNiceMatrix \)
\(\newcommand {\tcbset }[1]{}\)
\(\newcommand {\tcbsetforeverylayer }[1]{}\)
\(\newcommand {\tcbox }[2][]{\boxed {\text {#2}}}\)
\(\newcommand {\tcboxfit }[2][]{\boxed {#2}}\)
\(\newcommand {\tcblower }{}\)
\(\newcommand {\tcbline }{}\)
\(\newcommand {\tcbtitle }{}\)
\(\newcommand {\tcbsubtitle [2][]{\mathrm {#2}}}\)
\(\newcommand {\tcboxmath }[2][]{\boxed {#2}}\)
\(\newcommand {\tcbhighmath }[2][]{\boxed {#2}}\)
This exam has 36 questions, for a total of 100 points. Place your name on the answersheet (last page). Place proofs on the blank space on the answersheet.
Q1. An interpretation of RL 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 RL it assigns that name one and only one item in \(\mathcal {D}\)
-
4. *for each \(n\)-place predicate term in RL assigns, it assigns that predicate term a set of \(n\)-tuples composed of elements from \(\mathcal {D}\)
Q2. What is a derivation of \(\psi \) from \(\Gamma \) using \(\mathbf {RD}\)?
-
1. *a finite string of formulas from a set \(\Gamma \) of RL 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.
-
2. finite string of wffs starting with some premises \(\Gamma \) and ending with \(\psi \).
-
3. a finite string of wffs starting with some premises \(\Gamma \) or assumptions and ending with \(\psi \).
-
4. an infinite string of wffs starting with some premises \(\Gamma \) or assumptions and ending with \(\psi \).
Q3. What does the following mean: \(\Gamma \vdash \mathbf {Q}\)
-
1. *\(\Gamma \vdash \mathbf {Q}\) says \(\mathbf {Q}\) is a syntactic consequence of \(\Gamma \) (meaning that there is a derivation of \(\mathbf {Q}\) from \(\Gamma \)).
-
2. \(\Gamma \vdash \mathbf {Q}\) says \(\mathbf {Q}\) is a semantic consequence of \(\Gamma \) (there is no model such that the wffs of \(\Gamma \) are true and \(\mathbf {Q}\) is false).
-
3. \(\Gamma \vdash \mathbf {Q}\) says \(\mathbf {Q}\) is a hypostatic abstraction from \(\Gamma \)
-
4. \(\Gamma \vdash \mathbf {Q}\) says \(\mathbf {Q}\) intuitively follows from \(\Gamma \). That is, if you imagine \(Q\) in a proof, you can reason to \(\Gamma \).
Q4. Which of the following symbols are RL names (indicate all that apply)?
-
1. \(\forall \)
-
2. *\(b\)
-
3. *\(c\)
-
4. \(x\)
Write any free variables in the following wffs on the line. If there are no free variables, write ``none''.
Q5. \(Px\) --- Answer: \(x\)
Q6. \(Gab\) --- Answer: none
Q7. \((\forall x)(\forall y)Gyx\) --- Answer: none
Determine whether the following wffs are true or false by using the following model: \(\mathcal {D}= \{1,2,3,4,5\}\), \(\mathscr {I}(a)=1\), \(\mathscr {I}(b)=2\), \(\mathscr {I}(c)=3\), \(\mathscr {I}(d)=4\), \(\mathscr {I}(e)=5\), for all other names
\(\alpha \), \(\mathscr {I}(\alpha )=4\), \(\mathscr {I}(N)=\{1,2,3,4,5\}\), \(\mathscr {I}(G)=\{\langle 2,1 \rangle , \langle 3,2 \rangle , \langle 3,1\rangle , \langle 5, 1\rangle \}\), \(\mathscr {I}(I)=\{\}\), \(\mathscr {I}(E)=\{2, 4\}\),
\(\mathscr {I}(O)=\{1,3, 5\}\)
Q8. \(Ea\) --- Answer: F
Q9. \(Od\) --- Answer: F
Q10. \((\forall x)Ex\) --- Answer: F
Q11. \(\neg (\exists y)Oy\) --- Answer: F
Q12. \((\exists x)(Nx\wedge \neg Ox)\) --- Answer: T
Determine whether the following wffs are true or false by using the following model: \(\mathcal {D}= \{Tek, Shinji, Lain\}\), \(\mathscr {I}(t)=Tek\), \(\mathscr {I}(s)=Shinji\), \(\mathscr {I}(l)=Lain\), for all other names \(\alpha \), \(\mathscr
{I}(\alpha )=Shinji\), \(\mathscr {I}(Lxy)=\{\langle Tek, Tek \rangle , \langle Tek, Lain \rangle , \langle Lain, Lain \rangle \}\), \(\mathscr {I}(Hx)=\{Tek\}\)
Q13. \(\neg Hs\) --- Answer: T
Q14. \(Llt\) --- Answer: F
Q15. \((\forall x)\neg (Lxs)\) --- Answer: T
Q16. \((\forall x)\neg (Lsx)\) --- Answer: T
Q17. \((\exists x)Hx\) --- Answer: T
Q18. \((\forall x)(Hx\rightarrow Lxx)\) --- Answer: T
Translate the wffs below into English using the following key: \(\mathcal {D}= \{Tek, Shinji, Lain\}\), \(t=Tek\), \(s=Shinji\), \(l=Lain\), for all other names \(\alpha \), \(\alpha =Shinji\), \(Lxy=\) \(x\) loves \(y\), \(Hx=\) \(x\) is happy. \(Rx=\) \(x\) is rich.
Q20. \(\neg Hs\) --- Answer: Shinji is not happy.
Q21. \(Llt\) --- Answer: Lain loves Tek.
Q22. \((\forall x)Lxl\) --- Answer: Everyone loves Lain.
Q23. \(\neg (\exists x) Lxs\) --- Answer: No one loves Shinji. Also: It is not the case that someone exists that loves Shinji.
Q24. \((\exists x)(Hx\wedge Lxx)\) --- Answer: Someone who is happy loves themselves.
Q25. \((\forall x)(Rx\rightarrow Lxx)\) --- Answer: All rich people love themselves.
Writing the abbreviation for the single derivation rule that is represented in the following:
Q27. \((\forall z)(Qz\rightarrow Qz)\vdash Qc\rightarrow Qc\) --- Answer: \(\forall E\)
Q28. \(Ca\wedge Ca\vdash (\exists z)(Cz\wedge Cz)\) --- Answer: \(\exists I\)
Q29. \(\neg Qab \vdash (\exists z)\neg Qxb\) --- Answer: \(\exists I\)
Q30. From \(Bb\) to \((\forall y)By\) provided (1) \(b\) is not in a premise or in an assumption of an active subproof and (2) \(b\) is not in \((\forall y)By\)? --- Answer: \(\forall I\)
Q31. Starting from \((\exists x)Fx\), suppose \(Fa\) is assumed. Next, suppose \(\chi \) is derived in the subproof starting with \(Fa\). Finally, suppose \(\chi \) is deprived using \((\exists x)Fx\) and the entire subproof. --- Answer: \(\exists E\)
Q32. \(\neg (\forall z)Zz\vdash (\exists z)\neg Zz\) --- Answer: \(QN\)
Provide proofs for the following:
Q33. \(Sb, (\forall x)Qx\vdash (\exists x)Qx\)
--- Answer: \(Sb, (\forall x)Qx\vdash (\exists y)Qy\)
Q34. \(\neg Fbb\wedge Fba \vdash (\exists x)(\exists y)\neg Fxy\)
--- Answer: \(\neg Fbb\wedge Fba \vdash (\exists x)(\exists y)\neg Fxy\)
Q35. \(Qa\wedge Pa, (\forall x)(Px\rightarrow Mx), (\forall x)Px \vdash (\forall x)Mx\)
--- Answer: \(Qa\wedge Pa, (\forall x)(Px\rightarrow Mx), (\forall x)Px \vdash (\forall x)Mx\)
Q36. \((\exists x)\neg Wx\vdash (\exists x)(Mx\vee \neg Wx)\)
--- Answer: \((\exists x)\neg Wx\vdash (\exists x)(Mx\vee \neg Wx)\)
