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PHIL 220 > Quantified Logic > Flashcards

Quantified Logic Flashcards

1
Q

An argument form is invalid iff _________

A

there are some instances of it that are invalid

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2
Q

If an argument has an invalid form, can it still be valid?

A

Yes

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3
Q

What are the 4 components of Quantified Logic

A

1) Names
2) Predicates
3) Quantifiers
4) Variables

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4
Q

What is a Name, in QL?

A

Meaningful subsentential symbol

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5
Q

What do names in QL refer to?

A

particular things (often proper nouns)

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6
Q

How are Names represented in QL?

A

Lowercase letters from a-w (sometimes with subscripts)

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7
Q

Are names in QL wffs?

A

No

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8
Q

What is a Predicate in QL?

A

A way to describe a Name, expresses a property

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9
Q

How are Predicates represented in QL?

A

Capital letters with optional numerical subscripts

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10
Q

What is a monadic predicate?

A

One argument place

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11
Q

What is a dyadic predicate?

A

two-place predicate

Ex - ______ is funnier-looking than _______
Fdm

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12
Q

A monadic predicate requires ____ name to make a proposition

A

1

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13
Q

A diadic predicate requires _____ names to make a proposition

A

2

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14
Q

3-place predicates require _____ names to make a proposition.

A

3

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15
Q

What is an SL atomic wff in QL?

A

A zero-place predicate (requires zero names to express a proposition)

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16
Q

Do all sentence predicate properties of particular individuals?

A

No, thats why we need quantifiers

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17
Q

words like “every” and “some” are what in QL

A

Quantifiers

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18
Q

c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y

If Calvin is funny, then everybody is funny.

A

Fc ⊃ ∀xFx

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19
Q

c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y

Someone is funny but not nice.

A

▶ ∃x(Fx & ¬Nx)

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20
Q

c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y

Calvin has a wife.

A

▶ ∃xWcx

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21
Q

c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y

Calvin has a wife who is not nice.

A

▶ ∃x(Wcx & ¬Nx)

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22
Q

c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y

Betty doesn’t have a wife.

A

¬∃xWbx

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23
Q

c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y

Everyone nice is funny.

A

∀x(Nx ⊃ Fx)

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24
Q

Wxy
x’s wife is y

In english, this means

A

there is some y such that x’s wife is y.

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25
Q

What is the scope of the quantifier?

A

Part of the formula it affects

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26
Q

Can quantfiers be main connectives?

A

Yes

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27
Q

A quantifier binds __________ within its scope

A

the variables of its kind

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28
Q

Like quantifiers, what else counts as a connective even though it doesn’t technically connect multiple sentences?

A

Negation

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29
Q

If the quantifier is the main connective then

A

its scope is the entire rest of
the sentence.

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30
Q

What are the symbols involved in QL? (6)

A

▶ Predicates (capital letters with optional subscripts)
▶ Names (lower-case letters a–w, with optional subscripts)
▶ Variables (lower-case letters x, y, z, with optional subscripts)
▶ Sentential connectives (¬, & , ∨ , ⊃ , ≡ )
▶ Brackets ( (, ) )
▶ Quantifiers (∃, ∀)

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31
Q

What is an expression of QL?

A

any string of symbols, whether or not it is
meaningful.

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32
Q

Does QL differentiate between a wff and a sentence?

A

Yes

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33
Q

Does SL differentiate between a wff and a sentence?

A

No

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34
Q

What is a term in QL?

A

A name or a variable

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35
Q

What is an atomic formula in QL?

A

An n-place predicate followed by n terms

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36
Q

Atomic formulas are _____

A

wffs

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37
Q

for any wff, so long as it doesn’t contain a quantifier for
a given variable, putting a quantifier with that variable in front of it
results _______

A

in another wff.

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38
Q

Are wffs with free/unbound variables sentences in QL?

A

No

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39
Q

What is a sentence in QL?

A

a wff with no unbound variables.

40
Q

Fx is a

A

wff but not a sentence

41
Q

∃xFx is a

A

wff and a sentence

42
Q

What does the sentence ∃xFx mean?

A

Something has property F

43
Q

What does the sentence ∀xFx mean?

A

Everything has property F

44
Q

What does Universal quantifier ∀ mean?

A

every instance is true

45
Q

What does Existential quantifier ∃ mean?

A

at least one instance is true.

46
Q

Is ∃x(Fx & ∀xGx) a setnence?

A

No
You can’t overlap a quantifier.

In this case, both ∃ and ∀ bind x

47
Q

In QL, what does a Universe of Discourse do (UD)?

A

identifies the objects that the quantifiers quantify over.

48
Q

Fx x is funny
Px x is a person

Translate “everyone is funny” if the UD does not limit to just people

A

∀x(Px ⊃ Fx)

49
Q

UD: Canadian citizens

a Alice
b Betty
c Calvin
d Doug
Fx x is female
Hx x is homosexual
Mx x is male
Tx x is transgender
Lxy x loves y
Vxy x will vote for y
Pxy x is a parent of y

If Alice loves a female person, then Alice is homosexual.

A

∃x(Fx & Lax) ⊃ Ha

50
Q

UD: Canadian citizens

a Alice
b Betty
c Calvin
d Doug
Fx x is female
Hx x is homosexual
Mx x is male
Tx x is transgender
Lxy x loves y
Vxy x will vote for y
Pxy x is a parent of y

Some males love males without being homosexual.

A

∃x((Mx & ∃y(My & Lxy) & ¬Hx)

51
Q

UD: Canadian citizens

a Alice
b Betty
c Calvin
d Doug
Fx x is female
Hx x is homosexual
Mx x is male
Tx x is transgender
Lxy x loves y
Vxy x will vote for y
Pxy x is a parent of y

Every female citizen will vote for Betty.

A

∀x(Fx ⊃ Vxb)

52
Q

UD: Canadian citizens

a Alice
b Betty
c Calvin
d Doug
Fx x is female
Hx x is homosexual
Mx x is male
Tx x is transgender
Lxy x loves y
Vxy x will vote for y
Pxy x is a parent of y

All homosexuals and transgender people will vote for Doug.

A

∀x(Hx ⊃ Vxd) & ∀x(Tx ⊃ Vxd)

53
Q

If the main connective is an existential, it often has a _________
inside

A

conjunction

54
Q

If the main connective is a universal, it often has a _________ inside

A

conditional

55
Q

What does ∀x(Fx ⊃ Gx) mean

A

everything that is F is G

56
Q

What does ∃x(Fx & Gx) mean

A

something is F and G

57
Q

X |= Φ iff
there is ____________

A

no model that satisfies X and falsifies Φ

58
Q

Models in SL are ______________

A

assignments of atomic sentences to truth conditions.

59
Q

What a formula with quantifiers is talking about depends on the ______

A

universe of discourse (UD)

60
Q

I(a) = Mama Bear means that _______________

A

the interpretation I assigns the name
a to the object Mama Bear.

61
Q

Another word for interpretation is

A

referent

62
Q

Can a monadic predicate categorize every object the same?

A

Yes

Possible for all interpretations to be true, or for all interpretations to be false

63
Q

A two-place predicate has _____ in its extension

A

Ordered pairs

64
Q

A three-place predicate will have ______ in its extension

A

ordered triples

65
Q

For any n-place predicate F and names α1…αn, F α1…αn is true in I
iff <α1, … αn> is ___________

A

in the extension of I(F ).

66
Q

True or false: Lab |= ¬Lba?

A

False

There are possible models that satisfy Lab and ¬¬Lba, so the QL argument form from Lab to ¬Lba is invalid.

67
Q

A universal formula is true iff ________

A

each instance is true.

68
Q

A QL model includes (3)

A

-the UD
-an interpretation of names
-the extensions of predicates.

69
Q

An existentially quantified formula is true iff _______________

A

at least one of its
instances is true.

70
Q

What are the three instances of ∀x ∃y (¬Sy & Myx )

A

∃y (¬Sy & Mya)
∃y (¬Sy & Myb)
∃y (¬Sy & Myc)

71
Q

What are the three instances of y (¬Sy & Myc)

A

(¬Sa & Mac)
(¬Sb & Mbc)
(¬Sc & Mcc)

72
Q

A universal ∀x Φ is equivalent to the _________ of Φ’s x -instances

A

conjunction

73
Q

An existential ∃x Φ is equivalent to the __________ of Φ’s x -instances.

A

disjunction

74
Q

∀x (Px ⊃ Rxa) is equivalent to

A

(Pa ⊃ Raa) & (Pb ⊃ Rba) & (Pc ⊃ Rca)

75
Q

∀x (Px ⊃ ∃y (¬Sy & Myx )) is equivalent to

A

(Pa ⊃ ∃y (¬Sy & Mya)) & (Pb ⊃ ∃y (¬Sy & Myb)) & (Pc ⊃ ∃y (¬Sy & Myc))

76
Q

(Pa ⊃ ∃y (¬Sy & Mya)) & (Pb ⊃ ∃y (¬Sy & Myb)) & (Pc ⊃ ∃y (¬Sy & Myc))

Is equivalent to

A

(Pa ⊃ ((¬Sa & Maa) ∨ (¬Sb & Mba) ∨ (¬Sc & Mca))
&
(Pb ⊃ ((¬Sa & Mab) ∨ (¬Sb & Mbb) ∨ (¬Sc & Mcb))
&
(Pc ⊃ ((¬Sa & Mac) ∨ (¬Sb & Mbc) ∨ (¬Sc & Mcc)))

77
Q

∀x (Bx ⊃ Sx )

Every instance is true; so every _______ is true.

A

conditional

78
Q

Existentials almost never govern

A

Conditionals

79
Q

X |= Ψ means

A

very interpretation satisfying X satisfies Ψ .

80
Q

Tree rules for a 3-object domain (4)

A

▶ Existential: Branch into the three instances
▶ Universal: Linear development into the three instances
▶ Negated Existential: Linear development into the negations of the three
instances
▶ Negated Universal: Branch into the negations of the three instances
Dr. Jonathan Ichikawa PHIL 220: Symbolic Logic 3 / 2

81
Q

The tree methods tells you whether the root is _________.

A

satisfiable

82
Q

If all branches are closed, the root is __________.

A

unsatisfiable

83
Q

If any open branch is complete, the root is ____________

A

satisfiable (and the branch
tells you how).

84
Q

To resolve a formula of the form ∃x Φ, extend any open branch
containing that formula with an instance of _________

A

Φ that uses a new name.

85
Q

To resolve a formula of the form ∀x Φ, extend any open branch
containing that formula with ____________

A

whatever instance of Φ
you like.

86
Q

To resolve a formula of the form ¬∃x Φ, extend any open branch
containing the negation of that formula with

A

whatever instance of Φ
you like.

87
Q

To resolve a formula of the form ¬∀x Φ, extend any open branch
containing that formula with

A

an instance of ¬Φ that uses a new name.

88
Q

A QL branch is complete iff (2)

A

▶ Every resolvable formula has been resolved, and
▶ Every name in the branch (and at least one name) has been substituted
into every general formula in the branch.

89
Q

What is a general formula in QL trees?

A

a sentence whose main connective is a universal or
a negated existential

90
Q

For particular rules (existential, negated universal), you _________

A

take just one
instance, and it must be a new name

91
Q

For general rules (universal, negated existential), you ________________

A

take as many
instances as you like, and you must use every old name in the branch.

92
Q

For general rules, only take a new name if

A

there are no names in the branch already

93
Q

For QL trees, what do you start with?

A

Main connectives

94
Q

Better to branch _____ when possible

A

later

95
Q

Are infinite trees satisfiable?

A

Yes

96
Q

If a branch will never close, then the _________ characterizes an
interpretation that satisfies the root.

A

infinite branch

PHIL 220 (1 decks)

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