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Return to Introduction to Formal Logic Student Resources
Section 5.03 Self-Quiz
Quiz Content
*
not completed
.
Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Lx ⊃ (Oxy ≡ ~Oyx)]
2. (∀x)[(∃y)~Oyx ⊃ ~Mx]
3. (Pa • La) • Oab
La ⊃ (Oay ≡ ~Oya)
correct
incorrect
Pa
correct
incorrect
(∃y)~Oya ⊃ ~Mx
correct
incorrect
La ⊃ (Oya ≡ ~Oay)
correct
incorrect
Lb ⊃ (Oby ≡ ~Oyx)
correct
incorrect
*
not completed
.
Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Lx ⊃ (Oxy ≡ ~Oyx)]
2. (∀x)[(∃y)~Oyx ⊃ ~Mx]
3. (Pa • La) • Oab
~Mb
correct
incorrect
Pa • ~Ma
correct
incorrect
La • Lb
correct
incorrect
Oba
correct
incorrect
~Oba • ~Mb
correct
incorrect
*
not completed
.
Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x){[Mx • (∃y)Pxy] ⊃ Pxo}
2. (∀x)[(∃y)Qxy ≡ Pxa]
3. (∀x)(Mx ⊃ Nx)
4. (∀x)(Nx ⊃ Qxb)
Na ⊃ Qbb
correct
incorrect
(∃y)Qyy ≡ Pya
correct
incorrect
[Mo • (∃y)Poy] ⊃ Pxo
correct
incorrect
(∀x)(Qxo ≡ Pxa)
correct
incorrect
[Mx • (∃y)Pxy] ⊃ Pxo
correct
incorrect
*
not completed
.
Which of the following propositions is derivable from the given premises in F?
1. (∀x){[Mx • (∃y)Pxy] ⊃ Pxo}
2. (∀x)[(∃y)Qxy ≡ Pxa]
3. (∀x)(Mx ⊃ Nx)
4. (∀x)(Nx ⊃ Qxb)
(∀x)[Nx ⊃ (∃y)Pyx]
correct
incorrect
(∀x)[Nx ⊃ Pxo]
correct
incorrect
(∀x)[Mx ⊃ (∃y)Pxy]
correct
incorrect
(∀x)[(Mx • Nx) ⊃ ~(∀y)Pxy]
correct
incorrect
(∀x)[Mx ≡ Pxo]
correct
incorrect
*
not completed
.
Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Ix ⊃ (∃y)Jyx]
2. (∀x)[Jxa ⊃ (Kx ∨ Lx)]
3. (∃x)Ix • (∀x)~Kx
Jxa ⊃ (Ka ∨ La)
correct
incorrect
(∃x)Ix
correct
incorrect
Ix ⊃ (∃y)Jxy
correct
incorrect
~Ka
correct
incorrect
Jaa ⊃ (Kx ∨ Lx)
correct
incorrect
*
not completed
.
Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Ix ⊃ (∃y)Jyx]
2. (∀x)[Jxa ⊃ (Kx ∨ Lx)]
3. (∃x)Ix • (∀x)~Kx
(∀x)Lx
correct
incorrect
La
correct
incorrect
~Ka
correct
incorrect
(∃x)Lx
correct
incorrect
Jaa
correct
incorrect
*
not completed
.
Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(∀y)(Mxy ≡ ~Myx)
2. (∃x)(Kx • Mxa)
3. (∀x)(∀y)(~Mxy ⊃ Mxd)
~Mab ⊃ Mad
correct
incorrect
Mab ≡ ~Mba
correct
incorrect
~Mdd ⊃ Mdd
correct
incorrect
Kb • Mba
correct
incorrect
Kb • Maa
correct
incorrect
*
not completed
.
Which of the following propositions is derivable from the given premises in F?
1. (∀x)(∀y)(Mxy ≡ ~Myx)
2. (∃x)(Kx • Mxa)
3. (∀x)(∀y)(~Mxy ⊃ Mxd)
Mad
correct
incorrect
Mdd
correct
incorrect
Mbb
correct
incorrect
Mda
correct
incorrect
Maa
correct
incorrect
*
not completed
.
Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(Ax ⊃ ~Bx) ⊃ (∃x)Dx
2. (∀x)[Dx ⊃ (∃y)(Ey • Fxy)]
3. (∀x)[(∃y)Fyx ⊃ Gx]
4. ~(∃x)(Ax • Bx)
~(Aa • Ba)
correct
incorrect
(∃y)Fyy ⊃ Gy
correct
incorrect
Dx ⊃ (∃y)(Ey • Fay)
correct
incorrect
Ax ⊃ ~Bx
correct
incorrect
(∀x)~(Ax • Bx)
correct
incorrect
*
not completed
.
Which of the following propositions is derivable from the given premises in F?
1. (∀x)(Ax ⊃ ~Bx) ⊃ (∃x)Dx
2. (∀x)[Dx ⊃ (∃y)(Ey • Fxy)]
3. (∀x)[(∃y)Fyx ⊃ Gx]
4. ~(∃x)(Ax • Bx)
(∀x)(Ex ⊃ Gx)
correct
incorrect
(∃x)(Ex • Gx)
correct
incorrect
(∃x)(Dx • Gx)
correct
incorrect
(∀x)(Dx ≡ Gx)
correct
incorrect
(∃x)(Ex • Dx)
correct
incorrect
*
not completed
.
Consider assuming '(∀x)[Px ⊃ (∃y)(Qy • Rxy)]' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∃x)(∃y)Rxy
Pa ⊃ (Qa • Raa)
correct
incorrect
Pa ⊃ (Qb • Rab)
correct
incorrect
Assume '~(∃x)(∃y)Rxy' for a nested indirect proof.
correct
incorrect
Assume '~(∃x)(∃y)Rxy' for a nested conditional proof.
correct
incorrect
(∃x)(∃y)Rxy
correct
incorrect
*
not completed
.
Which of the following propositions is also derivable in F?
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∃x)(∃y)Rxy
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∀x)(∀y)Ryx
correct
incorrect
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∃x)(∃y)Ryx
correct
incorrect
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∀x)(∀y)~Rxy
correct
incorrect
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ ~(∀x)(∀y)Rxy
correct
incorrect
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ ~(∀x)(∀y)~Rxy
correct
incorrect
*
not completed
.
Consider assuming '(∀x)(∀y)[(Px • Py) ⊃ Qxy]' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)(∀y)[(Px • Py) ⊃ Qxy] ⊃ (∃x)(∃y)Qxy
(∀y)[(Px • Py) ⊃ Qxy]
correct
incorrect
(∀x)[(Px • Py) ⊃ Qxy]
correct
incorrect
Assume '(∃x)(∃y)Qxy' for a nested indirect proof.
correct
incorrect
Px • Py
correct
incorrect
(Px • Py) ⊃ Qxy
correct
incorrect
*
not completed
.
Which of the following propositions is also derivable in F?
(∀x)(∀y)[(Px • Py) ⊃ Qxy] ⊃ (∃x)(∃y)Qxy
(∀x)(∀y)Qxy ⊃ (∃x)(∃y)[(Px • Py) • Qxy]
correct
incorrect
(∀x)(∀y)~Qxy ⊃ (∃x)(∃y)~[(Px • Py) • Qxy]
correct
incorrect
(∀x)(∀y)~Qxy ⊃ (∃x)(∃y)[(Px • Py) • ~Qxy]
correct
incorrect
(∀x)(∀y)Qxy ⊃ ~(∃x)(∃y)[(Px • Py) • Qxy]
correct
incorrect
(∀x)(∀y)Qxy ⊃ (∃x)(∃y)[(Px • Py) ⊃ ~Qxy]
correct
incorrect
*
not completed
.
Consider assuming '(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)]' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∃x)(Px • Qx) ⊃ (∃x)Rxx]
Px
correct
incorrect
Assume '(∃x)(Px • Qx)' for a nested conditional proof.
correct
incorrect
Assume '~(∃x)Rxx' for a nested indirect proof.
correct
incorrect
Assume '(∃x)(Px • Qx) ⊃ (∃x)Rxx' for a nested indirect proof.
correct
incorrect
Pa ⊃ (Qa ⊃ Raa)
correct
incorrect
*
not completed
.
Which of the following propositions is also derivable in F?
(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∃x)(Px • Qx) ⊃ (∃x)Rxx]
(∀x)[(∀y)(Qy ⊃ Rxy) ⊃ Px] ⊃ [(∀x)(~Px ⊃ ~Qx) ∨ (∃x)Rxx]
correct
incorrect
(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∀x)(Px ⊃ ~Qx) ∨ (∃x)Rxx]
correct
incorrect
(∀x)[Px ⊃ ~(∃y)(Qy • Rxy)] ⊃ [(∀x)(Px ⊃ ~Qx) ∨ (∃x)Rxx]
correct
incorrect
(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∀x)(~Px ⊃ Qx) ∨ (∃x)Rxx]
correct
incorrect
(∀x)[Px ⊃ ~(∃y)(Qy • Rxy)] ⊃ [(∀x)(~Px ⊃ Qx) ∨ (∃x)Rxx]
correct
incorrect
*
not completed
.
Consider assuming '(∀x)(∀y)(Pxy ≡ Pyx)' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∃x)Pax ⊃ (∃x)Pxa]
Assume '(∃x)Pax' for a nested conditional proof.
correct
incorrect
(∀x)(Pxy ≡ Pyx)
correct
incorrect
Pab ≡ Pba
correct
incorrect
Assume '~(∃x)Pxa' for a nested indirect proof.
correct
incorrect
Assume ~(∃x)Pax' for a nested indirect proof.
correct
incorrect
*
not completed
.
Which of the following propositions is also derivable in F?
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∃x)Pax ⊃ (∃x)Pxa]
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∀x)Pxa ⊃ (∀x)Pax]
correct
incorrect
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∃x)~Pxa ⊃ (∃x)~Pax]
correct
incorrect
[(∀x)~Pxa ⊃ (∀x)~Pax] ⊃ (∀x)(∀y)(Pxy ≡ Pyx)
correct
incorrect
(∀x)(∀y)(~Pxy ≡ Pyx) ⊃ [(∀x)Pax ⊃ (∀x)Pxa]
correct
incorrect
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∀x)~Pxa ⊃ (∀x)~Pax]
correct
incorrect
*
not completed
.
Which of the following is a good assumption for an indirect proof of the above logical truth?
~[(∀x)Pxa • (∀x)~Pbx]
(∃x)Pxa ∨ (∃x)Pbx
correct
incorrect
Pxa • Pbx
correct
incorrect
~(∀x)Pxa ∨ ~(∀x)~Pbx
correct
incorrect
~(∀x)Pxa
correct
incorrect
(∀x)Pxa • (∀x)~Pbx
correct
incorrect
*
not completed
.
Which of the following is a good assumption for a conditional proof of the above logical truth?
~[(∀x)Pxa • (∀x)~Pbx]
(∀x)Pxa
correct
incorrect
(∃x)Pbx
correct
incorrect
~(∃x)Pbx
correct
incorrect
~(∀x)Pxa
correct
incorrect
(∃x)(Pxa • Pbx)
correct
incorrect
*
not completed
.
Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∃x)Bxa
2. (∃x)Cxa / (∃x)(Bxa • Cxa)
Valid
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Baa: True Caa: False
Bba: False Cba: True
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Baa: False Caa: True
Bba: False Cba: True
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Baa: True Caa: True
Bba: False Cba: False
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Baa: False Caa: False
Bba: True Cba: False
correct
incorrect
*
not completed
.
Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)[Ax ⊃ (∃y)(Ay • Fxy)]
2. Aa • ~Faa / (∀x)~Fxx
Valid
correct
incorrect
Invalid. Counterexample in a domain of one member, in which:
Aa: True Faa: False
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Aa: True Faa: False Fba: True
Ab: True Fab: True Fbb: True
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Aa: True Faa: False Fba: False
Ab: False Fab: True Fbb: False
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Aa: True Faa: False Fba: True
Ab: False Fab: True Fbb: True
correct
incorrect
*
not completed
.
Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)[Px ⊃ (∀y)(Ry ⊃ Txy)]
2. (∀x)[Qx ⊃ (∀y)(Ry ⊃ ~Txy)]
3. (∃x)Rx / ~(∃x)(Px • Qx)
Valid
correct
incorrect
Invalid. Counterexample in a domain of one member, in which:
Pa: False Qa: True Ra: True Taa: False
correct
incorrect
Invalid. Counterexample in a domain of one member, in which:
Pa: True Qa: False Ra: True Taa: True
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Pa: True Qa: True Ra: True Taa: False Tba: True
Pb: False Qb: True Rb: True Tab: False Tbb: False
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Pa: True Qa: False Ra: True Taa: False Tba: True
Pb: False Qb: True Rb: False Tab: True Tbb: False
correct
incorrect
*
not completed
.
Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∃x)Faxb • (∃x)Fabx
2. (∀x)[(Faxb • Fabx) ⊃ Gx] / (∃x)Gx
Valid
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Ga: False Faab: True Faba: False
Gb: False
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Ga: False Faab: False Faba: True
Gb: False
correct
incorrect
Invalid. Counterexample in a domain of three members, in which:
Ga: False Faab: True Fabc: True
Gb: False Faba: False Facb: False
Gc: False Fabb: False
correct
incorrect
Invalid. Counterexample in a domain of three members, in which:
Ga: False Faab: False Fabc: False
Gb: False Faba: True Facb: True
Gc: False Fabb: True
correct
incorrect
*
not completed
.
Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x){(Hx ⊃ (∃y)[Iy • (∀z)(Jz ⊃ Lxzy)]}
2. (∃x)(Hx • Jx)
3. (∀x)[(∃y)Lxxy ⊃ Kx] / (∃x)(Hx • Kx) • (∃x)(Jx • Kx)
Valid
correct
incorrect
Invalid. Counterexample in a domain of one member, in which:
Ha: True Ja: True Laaa: True
Ia: True Ka: False
correct
incorrect
Invalid. Counterexample in a domain of one member, in which:
Ha: True Ja: False Laaa: False
Ia: False Ka: True
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Ha: True Ja: True Laaa: True Lbaa: True
Hb: False Jb: True Laab: False Lbab: True
Ia: True Ka: True Laba: True Lbba: False
Ib: True Kb: False Labb: True Lbbb: False
correct
incorrect
Invalid. Counterexample in a domain of two members, in which:
Ha: True Ja: False Laaa: True Lbaa: False
Hb: True Jb: True Laab: False Lbab: True
Ia: True Ka: True Laba: False Lbba: False
Ib: True Kb: False Labb: True Lbbb: True
correct
incorrect
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