Question 1

1. (∃x)(Ax • Bx) ⊃ (∀x)Dx
2. ∼Da
-Which of the following propositions is derivable from the given premises in M?

Accepted Answer

A)  (∀x)(Ax • Dx) 
B)  (∃x)(Ax • Dx) 
C)  (∀x)(Ax ⊃ Bx) 
D)  (∀x)(Ax ⊃ ∼Bx) 
E)  ∼(∃x)Dx 
A)  (∀x)(Ax • Dx) 
B)  (∃x)(Ax • Dx) 
C)  (∀x)(Ax ⊃ Bx) 
D)  (∀x)(Ax ⊃ ∼Bx) 
E)  ∼(∃x)Dx

Question 2

Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant.
-Which of the following propositions can also be proved from the premises of the above argument?

Accepted Answer

A)  (∀x)Nx $ \lor $ (∀x)Px 
B)  (∀x)(Nx ≡ Cx) 
C)  (∀x)(Nx $ \lor $ Px) 
D)  (∀x)(Nx $ \lor $ Cx) 
E)  (∀x)[Nx ⊃ (∼Px • Cx)] 
A)  (∀x)Nx $ \lor $ (∀x)Px 
B)  (∀x)(Nx ≡ Cx) 
C)  (∀x)(Nx $ \lor $ Px) 
D)  (∀x)(Nx $ \lor $ Cx) 
E)  (∀x)[Nx ⊃ (∼Px • Cx)]

Question 3

determine whether the given formula is a logical truth of M or not. If it is not a logical truth, select a false valuation.
-(∃x)(Ix • Jx) ⊃ [(∀x)Ix ⊃ (∀x)Jx]

Accepted Answer

A)  Logical truth 
B)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: False Ja: True
Ib: True Jb: False 
C)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: True Ja: True
Ib: True Jb: False 
D)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: True Ja: True
Ib: False Jb: False 
E)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: True Ja: False
Ib: True Jb: False 
A)  Logical truth 
B)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: False Ja: True
Ib: True Jb: False 
C)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: True Ja: True
Ib: True Jb: False 
D)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: True Ja: True
Ib: False Jb: False 
E)  Not a logical truth. False valuation in a domain of 2 members, in which: Ia: True Ja: False
Ib: True Jb: False

Question 4

1. (∃x)Qx ⊃ (∀x)(Rx ⊃ Sx)
2. (∀x)∼Qx ⊃ (∃x)Sx
3. (∀x)Rx / (∃x)Sx
-Which of the following propositions is not a likely last line of the indented sequence for an indirect proof of the given argument?

Accepted Answer

A)  Rx • ∼Rx 
B)  Qx • ∼Qx 
C)  Rb • ∼Rb 
D)  Qb • ∼Qb 
E)  Sx • ∼Sx 
A)  Rx • ∼Rx 
B)  Qx • ∼Qx 
C)  Rb • ∼Rb 
D)  Qb • ∼Qb 
E)  Sx • ∼Sx

Question 5

select the best translation into predicate logic.
-All idealists are apriorists, but not theists.

Accepted Answer

A)  (∀x)[(Ix ⊃ Ax) • ∼Tx] 
B)  (∀x)[Ix ⊃ (∼Ax • Tx)] 
C)  (∀x)(Ix ⊃ Ax} $ \lor $ (∀x)(Tx ⊃ ∼Ax) 
D)  (∀x)[Ix ⊃ (Ax • ∼Tx)] 
A)  (∀x)[(Ix ⊃ Ax) • ∼Tx] 
B)  (∀x)[Ix ⊃ (∼Ax • Tx)] 
C)  (∀x)(Ix ⊃ Ax} $ \lor $ (∀x)(Tx ⊃ ∼Ax) 
D)  (∀x)[Ix ⊃ (Ax • ∼Tx)]

Question 6

1. (∀x)(Jx ⊃ Kx)
2. ∼(∀x)Kx
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Accepted Answer

A)  (∀x)∼Kx 
B)  Ja ⊃ Kb 
C)  ∼(∃x)Kx 
D)  (∃x)∼Kx 
E)  ∼(∃x)∼Kx 
A)  (∀x)∼Kx 
B)  Ja ⊃ Kb 
C)  ∼(∃x)Kx 
D)  (∃x)∼Kx 
E)  ∼(∃x)∼Kx

Question 7

Translate each sentence into predicate logic, using the given translation keys.
Ax: x is an athlete
Dx: x has determination
Px: x plays professional sports 
Sx: x receives a scholarship 
Tx: x is tall
Wx: x works hard
-Only tall athletes play professional basketball.

Accepted Answer

The answer of Translate each sentence into predicate logic, using...

Question 8

1. (∃x)(Mx • Ox) ⊃ (∃x)Nx
2. (∃x)(Px • Mx)
3. (∀x)(∼Px $ \lor $ Ox)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Accepted Answer

A)  Px • Mx 
B)  Px $ \lor $ Ox 
C)  (Ma • Ob) ⊃ (∃x)Nx 
D)  Pc • Mc 
E)  ∼Pn $ \lor $ Ox 
A)  Px • Mx 
B)  Px $ \lor $ Ox 
C)  (Ma • Ob) ⊃ (∃x)Nx 
D)  Pc • Mc 
E)  ∼Pn $ \lor $ Ox

Question 9

(∃x)(Px • Qx) ⊃ [(∃x)Px • (∃x)Qx]
-Which of the following propositions is also derivable in M given that same assumption?

Accepted Answer

A)  Qx 
B)  Px • Qx 
C)  (∀x)(Px • Qx) 
D)  (∀x)(Px ⊃ Qx) 
E)  Qb 
A)  Qx 
B)  Px • Qx 
C)  (∀x)(Px • Qx) 
D)  (∀x)(Px ⊃ Qx) 
E)  Qb

Question 10

select the best translation into predicate logic.
-No athletes who play professional sports and have determination are neither tall nor work hard.

Accepted Answer

A)  (∀x){[(Ax • Px) • Dx] ⊃ ∼(Tx $ \lor $ Wx)} 
B)  (∀x){[(Ax • Px) • Dx] ⊃ ∼∼(Tx $ \lor $ Wx)} 
C)  (∀x){[(Ax • Px) • Dx] ⊃ (∼Tx $ \lor $ ∼Wx)} 
D)  ∼ (∀x){[(Ax • Px) • Dx] ⊃ (Tx $ \lor $ Wx)} 
A)  (∀x){[(Ax • Px) • Dx] ⊃ ∼(Tx $ \lor $ Wx)} 
B)  (∀x){[(Ax • Px) • Dx] ⊃ ∼∼(Tx $ \lor $ Wx)} 
C)  (∀x){[(Ax • Px) • Dx] ⊃ (∼Tx $ \lor $ ∼Wx)} 
D)  ∼ (∀x){[(Ax • Px) • Dx] ⊃ (Tx $ \lor $ Wx)}

Question 11

select a counterexample for the given invalid argument.
-1. (∀x)(Kx $ \lor $ Lx)
2) (∃x)∼Kx / (∀x)Lx

Accepted Answer

A)  Counterexample in a domain of 2 members, in which: Ka: True La: True
Kb: False Lb: True 
B)  Counterexample in a domain of 2 members, in which: Ka: True La: False
Kb: True Lb: True 
C)  Counterexample in a domain of 2 members, in which: Ka: True La: True
Kb: True Lb: True 
D)  Counterexample in a domain of 2 members, in which: Ka: False La: False
Kb: False Lb: False 
E)  Counterexample in a domain of 2 members, in which: Ka: True La: False
Kb: False Lb: True 
A)  Counterexample in a domain of 2 members, in which: Ka: True La: True
Kb: False Lb: True 
B)  Counterexample in a domain of 2 members, in which: Ka: True La: False
Kb: True Lb: True 
C)  Counterexample in a domain of 2 members, in which: Ka: True La: True
Kb: True Lb: True 
D)  Counterexample in a domain of 2 members, in which: Ka: False La: False
Kb: False Lb: False 
E)  Counterexample in a domain of 2 members, in which: Ka: True La: False
Kb: False Lb: True

Question 12

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof.
-1. (∀x)(Fx ⊃ ∼Gx)
2. (∃x)(Hx • Gx) / (∃x)(Hx • ∼Fx)

Accepted Answer

The answer of derive the conclusions of each of the...

Question 13

If some journalists are not respectable, then all journalists on the web have work to do. Some journalists on the web lack assignments. It is not the case that something without an assignment has work to do. So all journalists are respectable.
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Accepted Answer

A)  (Jx • Ix) • ∼Ax 
B)  (∀x)∼(∼Ax • Wx) 
C)  ∼Aa • Wa 
D)  (Ja • ∼Ra) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] 
E)  (∃x)(Jx • ∼Rx) ⊃ ∼(∀x)∼[(Jx • Ix) ⊃ Wx] 
A)  (Jx • Ix) • ∼Ax 
B)  (∀x)∼(∼Ax • Wx) 
C)  ∼Aa • Wa 
D)  (Ja • ∼Ra) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] 
E)  (∃x)(Jx • ∼Rx) ⊃ ∼(∀x)∼[(Jx • Ix) ⊃ Wx]

Question 14

select a counterexample for the given invalid argument.
-1. (∀x)(Hx ⊃ ∼Ix)
2) (∀x)(Ix ⊃ Ux) / (∀x)(Hx ⊃ ∼Ux)

Accepted Answer

A)  Counterexample in a domain of 1 member, in which: Ha: True Ia: False Ua: True 
B)  Counterexample in a domain of 1 member, in which: Ha: False Ia: False Ua: True 
C)  Counterexample in a domain of 1 member, in which: Ha: True Ia: False Ua: False 
D)  Counterexample in a domain of 1 member, in which: Ha: False Ia: False Ua: False 
E)  Counterexample in a domain of 1 member, in which: Ha: False Ia: True Ua: False 
A)  Counterexample in a domain of 1 member, in which: Ha: True Ia: False Ua: True 
B)  Counterexample in a domain of 1 member, in which: Ha: False Ia: False Ua: True 
C)  Counterexample in a domain of 1 member, in which: Ha: True Ia: False Ua: False 
D)  Counterexample in a domain of 1 member, in which: Ha: False Ia: False Ua: False 
E)  Counterexample in a domain of 1 member, in which: Ha: False Ia: True Ua: False

Question 15

select a counterexample for the given invalid argument.
-1. (∃x)Sx
2) (∀x)[Sx ⊃ (Tx ⊃ ∼Ux)]
3) Ua • Ub
4) (∃x)∼Ux / (∃x)(Sx • ∼Tx)

Accepted Answer

A)  Counterexample in a domain of 3 members, in which:
Sa: True Ta: False Ua: True
Sb: False Tb: False Ub: True
Sc: True Tc: True Uc: False 
B)  Counterexample in a domain of 3 members, in which:
Sa: False Ta: False Ua: True
Sb: True Tb: True Ub: False
Sc: True Tc: True Uc: False 
C)  Counterexample in a domain of 3 members, in which:
Sa: True Ta: False Ua: True
Sb: True Tb: True Ub: True
Sc: True Tc: True Uc: False 
D)  Counterexample in a domain of 3 members, in which:
Sa: False Ta: False Ua: False
Sb: False Tb: True Ub: False
Sc: True Tc: True Uc: False 
E)  Counterexample in a domain of 3 members, in which:
Sa: False Ta: False Ua: True
Sb: False Tb: True Ub: True
Sc: True Tc: True Uc: False 
A)  Counterexample in a domain of 3 members, in which:
Sa: True Ta: False Ua: True
Sb: False Tb: False Ub: True
Sc: True Tc: True Uc: False 
B)  Counterexample in a domain of 3 members, in which:
Sa: False Ta: False Ua: True
Sb: True Tb: True Ub: False
Sc: True Tc: True Uc: False 
C)  Counterexample in a domain of 3 members, in which:
Sa: True Ta: False Ua: True
Sb: True Tb: True Ub: True
Sc: True Tc: True Uc: False 
D)  Counterexample in a domain of 3 members, in which:
Sa: False Ta: False Ua: False
Sb: False Tb: True Ub: False
Sc: True Tc: True Uc: False 
E)  Counterexample in a domain of 3 members, in which:
Sa: False Ta: False Ua: True
Sb: False Tb: True Ub: True
Sc: True Tc: True Uc: False

Question 16

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $ \lor $ Dx)]
-Which is the main operator of the formula?

Accepted Answer

A)  ∃x 
B)  $ \lor $ 
C)  ∼ 
D)  • 
E)  None of the above. 
A)  ∃x 
B)  $ \lor $ 
C)  ∼ 
D)  • 
E)  None of the above.

Question 17

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof.
-1. (∃x)(Gx • ∼Hx) ≡ (∃x)(Hx • ∼Ix)
2. (∀x)(Hx ⊃ Ix) / (∀x)(Gx ⊃ Hx)

Accepted Answer

The answer of derive the conclusions of each of the...

Question 18

Some websites have open comments which are not anonymous. Any website is either anonymous or requires a login. So something with open comments requires a login.
-Which of the following is the best translation into M of this argument?

Accepted Answer

A)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax $ \lor $ Lx)] / (∀x)(Ox • Lx) 
B)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax $ \lor $ Lx)] / (∃x)(Ox • Lx) 
C)  (∃x)[Wx • ∼(Ox • Ax)]
(∀x)(Ax $ \lor $ Lx) / (∃x)(Ox • Lx) 
D)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax • Lx)] / (∃x)(Ox • Lx) 
E)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)(Ax $ \lor $ Lx) / (∃x)(Ox • Lx) 
A)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax $ \lor $ Lx)] / (∀x)(Ox • Lx) 
B)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax $ \lor $ Lx)] / (∃x)(Ox • Lx) 
C)  (∃x)[Wx • ∼(Ox • Ax)]
(∀x)(Ax $ \lor $ Lx) / (∃x)(Ox • Lx) 
D)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax • Lx)] / (∃x)(Ox • Lx) 
E)  (∃x)[Wx • (Ox • ∼Ax)]
(∀x)(Ax $ \lor $ Lx) / (∃x)(Ox • Lx)

Question 19

∼(∃x)(Px • Qx) ≡ (∀x)(Px ⊃ ∼Qx)
-Consider assuming '∼(∃x)(Px • Qx)' for a conditional proof of one direction of the biconditional in the above logical truth. Which of the following propositions is a legitimate second step in that proof?

Accepted Answer

A)  (∀x) ∼ (Px • Qx) 
B)  Px ⊃ ∼Qx 
C)  Px • Qx 
D)  Pc • Qc 
E)  ∼(∀x)(Px ⊃ ∼Qx) 
A)  (∀x) ∼ (Px • Qx) 
B)  Px ⊃ ∼Qx 
C)  Px • Qx 
D)  Pc • Qc 
E)  ∼(∀x)(Px ⊃ ∼Qx)

Question 20

1. (∀x)[(Px $ \lor $ Qx) ⊃ (Rx • ∼Sx)]
2. (∀x)[Rx ⊃ (Tx ⊃ ∼Sx)]
-Which of the following propositions is derivable in M from the given premises?

Accepted Answer

A)  (∀x)[Px ⊃ (Sx ⊃ ∼Tx)] 
B)  (∀x)[Px ⊃ (Qx • ∼Tx)] 
C)  (∀x)Qx 
D)  (∀x)[Px ⊃ (Qx • ∼Sx)] 
E)  (∀x)[Px ⊃ (Qx ⊃ Tx)] 
A)  (∀x)[Px ⊃ (Sx ⊃ ∼Tx)] 
B)  (∀x)[Px ⊃ (Qx • ∼Tx)] 
C)  (∀x)Qx 
D)  (∀x)[Px ⊃ (Qx • ∼Sx)] 
E)  (∀x)[Px ⊃ (Qx ⊃ Tx)]

1. (∃x)(Ax • Bx) ⊃ (∀x)Dx 2. ∼Da -Which of the following propositions is derivable from the given premises in M?

Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant. -Which of the following propositions can also be proved from the premises of the above argument?

determine whether the given formula is a logical truth of M or not. If it is not a logical truth, select a false valuation. -(∃x)(Ix • Jx) ⊃ [(∀x)Ix ⊃ (∀x)Jx]

1. (∃x)Qx ⊃ (∀x)(Rx ⊃ Sx) 2. (∀x)∼Qx ⊃ (∃x)Sx 3. (∀x)Rx / (∃x)Sx -Which of the following propositions is not a likely last line of the indented sequence for an indirect proof of the given argument?

select the best translation into predicate logic. -All idealists are apriorists, but not theists.

1. (∀x)(Jx ⊃ Kx) 2. ∼(∀x)Kx -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Translate each sentence into predicate logic, using the given translation keys. Ax: x is an athlete Dx: x has determination Px: x plays professional sports Sx: x receives a scholarship Tx: x is tall Wx: x works hard -Only tall athletes play professional basketball.

1. (∃x)(Mx • Ox) ⊃ (∃x)Nx 2. (∃x)(Px • Mx) 3. (∀x)(∼Px $\lor$ Ox) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

(∃x)(Px • Qx) ⊃ [(∃x)Px • (∃x)Qx] -Which of the following propositions is also derivable in M given that same assumption?

select the best translation into predicate logic. -No athletes who play professional sports and have determination are neither tall nor work hard.

select a counterexample for the given invalid argument. -1. (∀x)(Kx $\lor$ Lx) 2) (∃x)∼Kx / (∀x)Lx

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof. -1. (∀x)(Fx ⊃ ∼Gx) 2. (∃x)(Hx • Gx) / (∃x)(Hx • ∼Fx)

select a counterexample for the given invalid argument. -1. (∀x)(Hx ⊃ ∼Ix) 2) (∀x)(Ix ⊃ Ux) / (∀x)(Hx ⊃ ∼Ux)

select a counterexample for the given invalid argument. -1. (∃x)Sx 2) (∀x)[Sx ⊃ (Tx ⊃ ∼Ux)] 3) Ua • Ub 4) (∃x)∼Ux / (∃x)(Sx • ∼Tx)

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $\lor$ Dx)] -Which is the main operator of the formula?

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof. -1. (∃x)(Gx • ∼Hx) ≡ (∃x)(Hx • ∼Ix) 2. (∀x)(Hx ⊃ Ix) / (∀x)(Gx ⊃ Hx)

Some websites have open comments which are not anonymous. Any website is either anonymous or requires a login. So something with open comments requires a login. -Which of the following is the best translation into M of this argument?

∼(∃x)(Px • Qx) ≡ (∀x)(Px ⊃ ∼Qx) -Consider assuming '∼(∃x)(Px • Qx)' for a conditional proof of one direction of the biconditional in the above logical truth. Which of the following propositions is a legitimate second step in that proof?

1. (∀x)[(Px $\lor$ Qx) ⊃ (Rx • ∼Sx)] 2. (∀x)[Rx ⊃ (Tx ⊃ ∼Sx)] -Which of the following propositions is derivable in M from the given premises?

Introducing Logic

Propositional Logic: Syntax and Semantic

Inference in Propositional Logic

Full First-Order Logic

Filters

Exam 4: Monadic Predicate Logic