Question 1

select the best translation into predicate logic.
-All cities and settlements are nicely placed and productive.

Accepted Answer

The statement "All cities and settlements are nicely placed and productive" can be translated into predicate logic using the universal quantifier (∀) and conditional (⊃) meaning "if...then" as follows:

(∀x)[(Cx ∨ Sx) ⊃ (Nx • Px)]

This reads: "For all x (cities and settlements), if x is a city or settlement, then x is nicely placed and productive."

Choice D is the best option because it correctly uses the conditional connective (⊃) to capture the "if...then" relationship between the two properties and the disjunction (∨) to allow for both cities and settlements to satisfy the statement. Choice A is incorrect because it uses conjunction (∧) instead of conditional, which would require both properties to always hold together. Choice B is also incorrect because it uses conjunction instead of conditional and does not allow for settlements to be included. Choice C is incorrect because it uses disjunction instead of conditional, which would allow for cities and settlements to have only one of the two properties.

Question 2

select the best translation into predicate logic.
-If Tortuga is a city, then some settlements are not nicely placed.

Accepted Answer

C) Ct ⊃ (∃x)(Sx • ∼Nx) correctly translates the statement. "Ct" represents "Tortuga is a city," "⊃" represents "implies," "∃x" represents "there exists some x," "Sx" represents "x is a settlement," "•" represents "and," and "∼Nx" represents "x is not nicely placed." This structure captures the meaning that if Tortuga is a city, then there exists at least one settlement that is not nicely placed.

Question 3

select the best translation into predicate logic.
-Either only cities are nicely placed or some settlements are not productive.

Accepted Answer

The answer of select the best translation into predicate logic.
-Either...

Question 4

select the best translation into predicate logic.
-All cities are productive if and only if they are both nicely placed and not on the water.

Accepted Answer

The correct translation is A) (∀x){Cx ⊃ [Px ≡ (Nx • ∼Wx)]}, which reads as "For all x, if x is a city, then x is productive if and only if x is nicely placed and not on the water." This accurately captures the given statement.

Question 5

select the best English interpretation of the given statements of predicate logic. 
f: Fifi
g: Gigi
Px: x is a poodle Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(Pf • Pg) • [(Rf • Rg) • (Sf • ∼Sg)]

Accepted Answer

(Pf • Pg) means both Fifi and Gigi are poodles. 
(Rf • Rg) means both of them are loved. 
(Sf • ∼Sg) means Fifi will fetch balls while Gigi will not. 
So the entire statement can be interpreted as: Fifi and Gigi are poodles, and both are loved, though Fifi will fetch balls while Gigi will not.

Question 6

select the best English interpretation of the given statements of predicate logic. 
f: Fifi
g: Gigi
Px: x is a poodle Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(∀x)[(Px • Qx) ⊃ ∼Rx]

Accepted Answer

The statement translates to "For all x, if x is a poodle and x is abused, then x is not loved," which means no abused poodles are loved.

Question 7

select the best English interpretation of the given statements of predicate logic. 
f: Fifi
g: Gigi
Px: x is a poodle Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(∃x)[(Px • Rx) • (Sx • Tx)]

Accepted Answer

The statement says that there exists an x such that x is a poodle and x is loved, and x will fetch balls and sticks. This can be translated to "Some loved poodles will fetch balls and sticks."

Question 8

select the best English interpretation of the given statements of predicate logic. 
f: Fifi
g: Gigi
Px: x is a poodle Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(∀x)(Px ⊃ Qx) ⊃ (∃x)(Px • ∼Rx)

Accepted Answer

The correct interpretation of the given statement is "If all poodles are abused, then there exists at least one poodle that is not loved." This matches choice C.

Question 9

select the best English interpretation of the given statements of predicate logic. 
f: Fifi
g: Gigi
Px: x is a poodle Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(∀x){(Px • Qx) ⊃ [(Rx ⊃ (∼Sx ⊃ Tx)]}

Accepted Answer

The statement can be translated as "For all x, if x is a poodle and x is abused, then if x is loved, then if x will not fetch balls, then x will fetch sticks." Therefore, all abused poodles, if they are loved, will fetch sticks if they will not fetch balls.

Question 10

select the best English interpretation of the given statements of predicate logic. 
f: Fifi
g: Gigi
Px: x is a poodle Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(∃x)[Px • (∼Qx • Rx)] ⊃ (∀x)[(Px • Rx) ⊃ Tx)]

Accepted Answer

The statement is a conditional statement where the antecedent is (∃x)[Px • (∼Qx • Rx)] which translates to "There exists a poodle that is not abused and loved". The consequent is (∀x)[(Px • Rx) ⊃ Tx)] which translates to "For all poodles that are loved and not abused, they will fetch sticks". Therefore, the best interpretation is "If some poodles are not abused but are loved, then all loved poodles will fetch sticks", which is logically equivalent to the given statement.

Question 11

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $ \lor $ Dx)]
-Which wffs below are not in the scope of '(∃x)'?

Accepted Answer

The scope of the existential quantifier '(∃x)' includes everything that follows it in the formula, which means all the wffs (well-formed formulas) listed are within its scope.

Question 12

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $ \lor $ Dx)]
-Which variables are bound by the '(∃x)'?

Accepted Answer

The '(∃x)' quantifier at the beginning of the formula binds all instances of 'x' within the scope of the formula, making all the variables 'x' in the formula bound by it.

Question 13

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $ \lor $ Dx)]
-Is the formula open or closed?

Accepted Answer

The formula is closed because it contains a quantifier (∃x) that binds the variable x, meaning all instances of x in the formula are accounted for by this quantifier, leaving no free variables.

Question 14

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $ \lor $ Dx)]
-Which of the following variables in the formula are free?

Accepted Answer

In the given formula, the variable x is bound by the existential quantifier (∃x) at the beginning of the formula. This means that all instances of x within the scope of this quantifier are considered bound, not free.

Question 15

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

Accepted Answer

The main operator of the formula is the existential quantifier (∃x), as it indicates that there exists at least one object that satisfies the conditions inside the brackets.

Question 16

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-Which wffs below are not in the scope of '(∀x)'?

Accepted Answer

None of the wffs are outside the scope of '(∀x)'.

Question 17

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-Which variables are bound by the '(∀x)'?

Accepted Answer

The '∀x' quantifier binds all instances of 'x' within its scope, which includes the x following E, F, and G, making all of them bound variables.

Question 18

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-Is the formula open or closed?

Accepted Answer

The formula is closed because it contains no free variables (all variables are bound by a quantifier).

Question 19

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-Which of the following variables in the formula are free?

Accepted Answer

The variable x is bound by the universal quantifier (∀x) at the beginning of the formula, meaning all instances of x within the scope of this quantifier are considered bound. The variable d in Hd is not quantified and thus would be the only free variable, but it is not an option given.

Question 20

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-Which is the main operator of the formula?

Accepted Answer

The answer of refer to the following formula: (∀x)[(Ex \(...

Quiz 4: Monadic Predicate Logic

select the best translation into predicate logic. -All cities and settlements are nicely placed and productive.

select the best translation into predicate logic. -If Tortuga is a city, then some settlements are not nicely placed.

select the best translation into predicate logic. -Either only cities are nicely placed or some settlements are not productive.

select the best translation into predicate logic. -All cities are productive if and only if they are both nicely placed and not on the water.

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(Pf • Pg) • [(Rf • Rg) • (Sf • ∼Sg) ]

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x) [(Px • Qx) ⊃ ∼Rx]

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∃x) [(Px • Rx) • (Sx • Tx) ]

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x) (Px ⊃ Qx) ⊃ (∃x) (Px • ∼Rx)

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x) {(Px • Qx) ⊃ [(Rx ⊃ (∼Sx ⊃ Tx) ]}

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∃x) [Px • (∼Qx • Rx) ] ⊃ (∀x) [(Px • Rx) ⊃ Tx) ]

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx $\lor$ Dx) ] -Which wffs below are not in the scope of '(∃x) '?

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx $\lor$ Dx) ] -Which variables are bound by the '(∃x) '?

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx $\lor$ Dx) ] -Is the formula open or closed?

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx $\lor$ Dx) ] -Which of the following variables in the formula are free?

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

refer to the following formula: (∀x) [(Ex $\lor$ Fx) ⊃ (Gx • Hd) ] -Which wffs below are not in the scope of '(∀x) '?

refer to the following formula: (∀x) [(Ex $\lor$ Fx) ⊃ (Gx • Hd) ] -Which variables are bound by the '(∀x) '?

refer to the following formula: (∀x) [(Ex $\lor$ Fx) ⊃ (Gx • Hd) ] -Is the formula open or closed?

refer to the following formula: (∀x) [(Ex $\lor$ Fx) ⊃ (Gx • Hd) ] -Which of the following variables in the formula are free?

refer to the following formula: (∀x) [(Ex $\lor$ Fx) ⊃ (Gx • Hd) ] -Which is the main operator of the formula?

Quiz 4: Monadic Predicate Logic

select the best translation into predicate logic. -All cities and settlements are nicely placed and productive.

select the best translation into predicate logic. -If Tortuga is a city, then some settlements are not nicely placed.

select the best translation into predicate logic. -Either only cities are nicely placed or some settlements are not productive.

select the best translation into predicate logic. -All cities are productive if and only if they are both nicely placed and not on the water.

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(Pf • Pg) • [(Rf • Rg) • (Sf • ∼Sg) ]

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x) [(Px • Qx) ⊃ ∼Rx]

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∃x) [(Px • Rx) • (Sx • Tx) ]

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x) (Px ⊃ Qx) ⊃ (∃x) (Px • ∼Rx)

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x) {(Px • Qx) ⊃ [(Rx ⊃ (∼Sx ⊃ Tx) ]}

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∃x) [Px • (∼Qx • Rx) ] ⊃ (∀x) [(Px • Rx) ⊃ Tx) ]

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx ∨ \lor ∨ Dx) ] -Which wffs below are not in the scope of '(∃x) '?

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx ∨ \lor ∨ Dx) ] -Which variables are bound by the '(∃x) '?

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx ∨ \lor ∨ Dx) ] -Is the formula open or closed?

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx ∨ \lor ∨ Dx) ] -Which of the following variables in the formula are free?

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

refer to the following formula: (∀x) [(Ex ∨ \lor ∨ Fx) ⊃ (Gx • Hd) ] -Which wffs below are not in the scope of '(∀x) '?

refer to the following formula: (∀x) [(Ex ∨ \lor ∨ Fx) ⊃ (Gx • Hd) ] -Which variables are bound by the '(∀x) '?

refer to the following formula: (∀x) [(Ex ∨ \lor ∨ Fx) ⊃ (Gx • Hd) ] -Is the formula open or closed?

refer to the following formula: (∀x) [(Ex ∨ \lor ∨ Fx) ⊃ (Gx • Hd) ] -Which of the following variables in the formula are free?

refer to the following formula: (∀x) [(Ex ∨ \lor ∨ Fx) ⊃ (Gx • Hd) ] -Which is the main operator of the formula?