Deck 9: Predicate Logic

Question 1

In predicate logic, Na is a

A) statement.
B) individual constant.
C) statement function.
D) individual variable.

Answer 1

statement function.

Question 2

Which of the following is the best translation into symbols of the sentence "All fickle people are difficult to please"? (Fx: x is a fickle person, and Dx: x is a person difficult to please.)

A) (x)(Fx • Dx)
B) (x)(Fx → Dx)
C) (x)Fx → Dx
D) (x)(Fx ↔ Dx)

Answer 2

(x)(Fx → Dx)

Question 3

In which of the following is the variable "x" free?

A) ($x)Fx • Gy
B) ($x)(y)(Gy → Fx)
C) (x)((Fx • Gx) → Hx)
D) Fx • (x)(Gx → Hx)

Answer 3

Fx • (x)(Gx → Hx)

Question 4

In logicese, (x)(Fx • Gx) says

A) everything is both an F and a G
B) anything that is an F is a G
C) for any x, if x is an F then x is a G
D) all Fs are Gs.

Question 5

In logicese, (x)~Fx says

A) everything is an F
B) nothing is an F
C) something is an F
D) there is at least one F

Question 6

All of the following are well-formed formulas of predicate logic except

A) A.
B) Ab.
C) A~b.
D) ~Ab

Question 7

All of the following are well-formed formulas of predicate logic except

A) (y)(Ay ⋁ ($y)By)
B) ((y)Ay ⋁ ($y)By)
C) (y)(x)(Ay ⋁ Bx)
D) Ay ⋁ Bx

Question 8

All of the following are well-formed formulas of predicate logic except

A) (a)Fa.
B) (w)Pw.
C) ($y)($x)(Ly ↔ Mx).
D) Hx ⋁ Bd.

Question 9

A precisely described and finite procedure for solving a problem is called

A) a logical process.
B) a finite method.
C) a tautology.
D) an algorithm.

Question 10

The finite universe method involves

A) describing a possible situation where the premises of an argument are true and the conclusion of the argument is true.
B) describing a possible situation where the conclusion of the argument is true.
C) describing a possible situation where the premises are false.
D) describing a possible situation where the premises are true and the conclusion is false.

Question 11

When using the finite universe method for a two-object universe (containing a and b), the statement ($x) Fx would be translated as

A) Fa ⋁ Fb.
B) Fa • Fb.
C) Fa.
D) Fb.

Question 12

When using the finite universe method for a two-object universe (containing a and b), the statement (x)Fx would be translated as

A) Fa ⋁ Fb.
B) Fa • Fb.
C) Fa.
D) Fb.

Question 13

The implicational rule of inference that permits us to derive a specific instance from a universal statement is

A) Existential Generalization.
B) Existential Instantiation.
C) Universal Generalization.
D) Universal Instantiation.

Question 14

Which of the following is a correct inference from (x)Fx → Ga?

A) Fa → Ga
B) Fa → Gx
C) ~(x)Fx ⋁ Ga
D) Fx → Gx

Question 15

Which of the following cannot be validly inferred from Ba ⋁ Bb?

A) ($x)Bx ⋁ Bb
B) ($x)(Bx ⋁ Bb)
C) ~~(Ba ⋁ Bb)
D) ($y)(Ba ⋁ By)

Question 16

Which of the following can be validly inferred from Bb • ~Ba by Existential Generalization?

A) ($y)By • ~Ba
B) ($x)Bb • ~Bx
C) ($x)(Bb • ~Bx)
D) ($x)~Bx

Question 17

Which of the following may be validly inferred from (x)Rx?

A) ~(x)~Rx
B) ~($x)~Rx
C) ~($x)Rx
D) ($x)~Rx

Question 18

Which of the following may be validly inferred from ~($x)Fx?

A) (x)~Fx
B) ($x)~Fx
C) ($x)Fx
D) ~(x)~Fx

Question 19

Which of the following is the best symbolization of "Nothing is unnatural"? (Nx: x is natural.)

A) ~($x)Nx
B) (x)~Nx
C) ~(x)~Nx
D) ~($x)~Nx

Question 20

Which of the following is the best symbolization of "Every number is either odd or even"? (Nx: x is a number; Ox: x is odd; Ex: x is even.)

A) (x)(Nx → (Ox • Ex))
B) (x)(Nx ⋁ (Ox • Ex))
C) (x)(Nx → (Ox ⋁Ex))
D) (x)(Nx • (Ox ⋁ Ex))

Question 21

When using CP or RAA, UG may not be employed under which of the following conditions?

A) The instantial constant occurs in the assumption.
B) The instantial constant occurs in any line of the proof.
C) The instantial constant occurs in the last line of the proof.
D) The conclusion of the argument is a conditional.

Question 22

When using reductio ad absurdum to prove (x)Fx ⋁ ($x)Gx, we begin by assuming

A) ~(x)Fx ⋁ ($x)Gx.
B) (x)Fx.
C) ~((x)Fx ⋁ ($x)Gx).
D) (x)Fx • ~($x)Gx.

Question 23

Where Lxy: x loves y, "Everyone loves someone" is best symbolized as

A) (x)(y)Lxy.
B) ($x)(y)Lxy.
C) ($x)(y)Lyx.
D) (x)($y)Lxy.

Question 24

Where Sxy: x is shorter than y, ($x)(y)Sxy says

A) everyone is shorter than someone.
B) someone is shorter than everyone.
C) everyone is such that someone is shorter.
D) someone is such that everyone is shorter.

Question 25

The relation "being married to" is

A) symmetrical.
B) nonsymmetrical.
C) transitive.
D) reflexive.

Question 26

The relation "being a friend of" is

A) irreflexive.
B) asymmetrical.
C) nontransitive.
D) intransitive.

Question 27

The relation "being greater than" is

A) reflexive.
B) transitive.
C) intransitive.
D) nonreflexive.

Question 28

Which of the following is a correct use of UI?

A) inferring (x)Fax from (y)(x)Fxy
B) inferring ($y)Fay from ($y)(x)Fxy
C) inferring ($x)(Fx → Fxa) from ($x)(Fx → (y)Fxy)
D) inferring (y)(Fy → Ga) from (x)(y)(Fy → Gx)

Question 29

Which of the following is an instance of (x)[Lx → ($y)(Ly → Gyx)]?

A) La → ($y)(Ly → Gyx)
B) La → ($y)(Ly → Gyb)
C) La → ($y)(Ly → Gya)
D) Lx → ($y)(Ly → Gyz)

Question 30

From (x)($y)(Fx ↔ Gxy) we may validly infer

A) ($y)(Fa ↔ Gay).
B) (x)(Fx ↔ Gxa).
C) ($y)(Fy ↔ Gyy).
D) ($y)(Fa ↔ Gxy).

Question 31

Assuming "z" is not found earlier in the proof, from ($x)($y)((Px • Py) → Lxy) we may validly infer

A) ($x)((Px • Pb) → Lxb).
B) ($y)((Pa • Py) → Lay).
C) ($y)((Py • Py) → Lyy).
D) ($x)((Pa • Py) → Lay).

Question 32

"Only Oswald assassinated Kennedy" can best be symbolized (o: Oswald; k: Kennedy; Axy: x assassinated y) as

A) Aok • (x)(Axk → x = o).
B) (x)(Axk → x = o).
C) ($x)Axk • (x)(Axk → x = k).
D) Aok.

Question 33

"Something creates everything except itself" can best be symbolized (Cxy: x creates y) as

A) ($x)(y)Cxy.
B) ($x)[(y)Cxy • ~Cxx].
C) ($x)[~Cxx • (y)(~y = x → Cxy)].
D) ($x)(y)Cxy • ($x)~Cxx.

Question 34

"There are at least two people in the presidential race" can best be symbolized (Px: x is a person; Rx: x is running for president) as

A) ($x)($y)((Px • Rx) • (Py • Ry)).
B) ($x)($y)(((Px • Rx) • (Py • Ry)) • ~x = y).
C) (x)(y)(((Px → Rx) • (Py → Ry)) • ~x = y).
D) (x)(y)((Px • Rx) → ~x = y).

Question 35

"There is one and only one President of the United States " can best be symbolized (Pxy: x is a president of y; u: the United States) as

A) ($x)(Pxu • (y)(Pyu → y = x)).
B) (x)(y)(Pxu → ~Pyu).
C) ($x)Pxu • (x)(y)((Pxu • Pyu) → x = y).
D) (x)(Pxu • (y)(~y = x → ~Pyu)).

Question 36

The inference from a = b and Fa to Fb is permitted by

A) symmetry.
B) identity.
C) transitivity.
D) Leibniz's law.

Question 37

Which of the following is an incorrect application of Leibniz's law?

A) inferring Fa ⋁ Fb from Fa ⋁ Fa and a = b
B) inferring (x)(Fx → Fb) from (x)(Fx → Fc) and b = c
C) inferring ($x)(Fx → Fx) from ($x)(Fx → Fy) and x = y
D) inferring (x)Fx → Gb from (x)Fx → Gz and z = b

Question 38

Which of the following can be inferred from ~a = b by symmetry?

A) ~b = a
B) a = ~b
C) b = ~a
D) ~b = ~a

Question 39

The identity rule allows us to

A) infer that if a property belongs to a thing then it belongs to whatever is identical to that thing.
B) enter statements of self-identity as lines in a proof.
C) replace one constant or variable with another that is identical with it.
D) switch the constants or variables in an identity statement (e.g., from a = b to b = a).

Question 40

Given that a = b, which of the following is an incorrect application of Leibniz's law?

A) inferring Ca • Da from Cb • Da
B) inferring Ab → Db from Aa → Da
C) inferring (w)Bww from (w)Bwa
D) inferring Lca → Lac from Lcb → Lbc

Question 41

An individual constant is a capital letter that designates a property.

Question 42

An individual variable is a lowercase letter (v through z) that serves as a placeholder.

Question 43

Gx is a statement.

Question 44

(x)Fx is a universally quantified statement.

Question 45

In ($x)Fx • Gx the variable in Gx falls within the scope of the existential quantifier.

Question 46

The occurrence of the variable "y" in (x)(Fx • Gy) is free.

Question 47

(x)(Hx → ~Ix) is a universal negative statement.

Question 48

(x)(Hx → Mx) is logically equivalent to (x)(~Hx ⋁ Mx).

Question 49

($x)(Dx → Cx) is a particular affirmative statement.

Question 50

~Ab is a well-formed formula.

Question 51

An algorithm is a precisely described finite procedure for solving a problem.

Question 52

The finite universe method is an algorithm for proving that an argument in predicate logic is valid.

Question 53

Every invalid argument in predicate logic can be shown to be invalid by using the finite universe method.

Question 54

In general, the inference from a universal statement to its corresponding particular is invalid when their subject terms denote an empty class.

Question 55

Only the implicational rules of inference from statement logic apply to predicate logic.

Question 56

~Hx is an instance of (x)~Hx.

Question 57

Fa • Ga is an instantiation (by UI) of (x)Fx • Gx.

Question 58

One can correctly move from ~(x)Gx to ~Gb by UI.

Question 59

Existential Instantiation and Universal Instantiation are both rules of equivalence.

Question 60

One can correctly move from ($x)Cx to Cb by EI.

Question 61

Fa • Ga is an instance of ($x)(Fx • Gx).

Question 62

For Existential Instantiation, the constant we instantiate to may not occur earlier in our proof.

Question 63

In applying EI, we instantiate to an individual constant.

Question 64

Correctly using Universal Generalization on Bc → Ba yields (x)Bx → Ba.

Question 65

From Fc • Bc we can validly infer ($x)(Fx • Bc) using Existential Generalization.

Question 66

(x)~Fx is logically equivalent to ~($x)Fx.

Question 67

Quantifier negation is an equivalence rule.

Question 68

(x)(Fx → ($y)~Fy) may be validly inferred from (x)(Fx → ~(y)Fy).

Question 69

"Everyone has a mother" can be symbolized as (x)(Px → (y)(Py • Myx)), where Px: x is a person, and
Mxy: x is the mother of y.

Question 70

A relation R is symmetrical just in case if a bears R to b then b bears R to a.

Question 71

A relation R is transitive when things bear R to themselves.

Question 72

"Being the sibling of" is an irreflexive relation.

Question 73

Logical equivalence is a symmetrical relation.

Question 74

One can correctly move from (y)($z)Lzy to ($z)Lzz by UI.

Question 75

From ($x)(y)Fxy we can validly infer (y)Fay, assuming that "a" is not found earlier in the proof.

Question 76

From ($x)(y)Fxy we can validly infer ($x)Fxa, assuming that "a" is not found earlier in the proof.

Question 77

One can correctly move from (x)(y)Fxy → Ga to (y)Fay → Ga by UI.

Question 78

From ($x)Fxa → Ga we can validly infer ($y)(($x)Fxy → Gy).

Question 79

One can correctly move from Fbx to ($x)Fxx by EG.

Question 80

In (y)s = y • Ly, the scope of the quantifier is s = y.