Quantified Logic Flashcards by Flora Mendes

Q

An argument form is invalid iff _________

A

there are some instances of it that are invalid

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Q

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

A

Yes

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Q

What are the 4 components of Quantified Logic

A

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

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Q

What is a Name, in QL?

A

Meaningful subsentential symbol

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Q

What do names in QL refer to?

A

particular things (often proper nouns)

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Q

How are Names represented in QL?

A

Lowercase letters from a-w (sometimes with subscripts)

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Q

Are names in QL wffs?

A

No

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Q

What is a Predicate in QL?

A

A way to describe a Name, expresses a property

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Q

How are Predicates represented in QL?

A

Capital letters with optional numerical subscripts

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Q

What is a monadic predicate?

A

One argument place

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Q

What is a dyadic predicate?

A

two-place predicate

Ex - ______ is funnier-looking than _______
Fdm

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Q

A monadic predicate requires ____ name to make a proposition

A

1

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Q

A diadic predicate requires _____ names to make a proposition

A

2

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Q

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

A

3

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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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Q

Do all sentence predicate properties of particular individuals?

A

No, thats why we need quantifiers

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Q

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

A

Quantifiers

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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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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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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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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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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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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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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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What is the scope of the quantifier?

Part of the formula it affects

Can quantfiers be main connectives?

Yes

A quantifier binds __________ within its scope

the variables of its kind

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

Negation

If the quantifier is the main connective then

its scope is the entire rest of the sentence.

What are the symbols involved in QL? (6)

▶ 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 (∃, ∀)

What is an expression of QL?

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

Does QL differentiate between a wff and a sentence?

Yes

Does SL differentiate between a wff and a sentence?

No

What is a term in QL?

A name or a variable

What is an atomic formula in QL?

An n-place predicate followed by n terms

Atomic formulas are _____

wffs

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 _______

in another wff.

Are wffs with free/unbound variables sentences in QL?

No

What is a sentence in QL?

a wff with no unbound variables.

Fx is a

wff but not a sentence

∃xFx is a

wff and a sentence

What does the sentence ∃xFx mean?

Something has property F

What does the sentence ∀xFx mean?

Everything has property F

What does Universal quantifier ∀ mean?

every instance is true

What does Existential quantifier ∃ mean?

at least one instance is true.

Is ∃x(Fx & ∀xGx) a setnence?

No You can't overlap a quantifier. In this case, both ∃ and ∀ bind x

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

identifies the objects that the quantifiers quantify over.

Fx x is funny Px x is a person Translate “everyone is funny" if the UD does not limit to just people

∀x(Px ⊃ Fx)

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.

∃x(Fx & Lax) ⊃ Ha

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.

∃x((Mx & ∃y(My & Lxy) & ¬Hx)

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.

∀x(Fx ⊃ Vxb)

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.

∀x(Hx ⊃ Vxd) & ∀x(Tx ⊃ Vxd)

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

conjunction

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

conditional

What does ∀x(Fx ⊃ Gx) mean

everything that is F is G

What does ∃x(Fx & Gx) mean

something is F and G

X |= Φ iff there is ____________

no model that satisfies X and falsifies Φ

Models in SL are ______________

assignments of atomic sentences to truth conditions.

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

universe of discourse (UD)

I(a) = Mama Bear means that _______________

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

Another word for interpretation is

referent

Can a monadic predicate categorize every object the same?

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

A two-place predicate has _____ in its extension

Ordered pairs

A three-place predicate will have ______ in its extension

ordered triples

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

in the extension of I(F ).

True or false: Lab |= ¬Lba?

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

A universal formula is true iff ________

each instance is true.

A QL model includes (3)

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

An existentially quantified formula is true iff _______________

at least one of its instances is true.

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

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

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

(¬Sa & Mac) (¬Sb & Mbc) (¬Sc & Mcc)

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

conjunction

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

disjunction

∀x (Px ⊃ Rxa) is equivalent to

(Pa ⊃ Raa) & (Pb ⊃ Rba) & (Pc ⊃ Rca)

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

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

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

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

∀x (Bx ⊃ Sx ) Every instance is true; so every _______ is true.

conditional

Existentials almost never govern

Conditionals

X |= Ψ means

very interpretation satisfying X satisfies Ψ .

Tree rules for a 3-object domain (4)

▶ 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

The tree methods tells you whether the root is _________.

satisfiable

If all branches are closed, the root is __________.

unsatisfiable

If any open branch is complete, the root is ____________

satisfiable (and the branch tells you how).

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

Φ that uses a new name.

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

whatever instance of Φ you like.

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

whatever instance of Φ you like.

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

an instance of ¬Φ that uses a new name.

A QL branch is complete iff (2)

▶ 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.

What is a general formula in QL trees?

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

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

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

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

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

For general rules, only take a new name if

there are no names in the branch already

For QL trees, what do you start with?

Main connectives

Better to branch _____ when possible

later

Are infinite trees satisfiable?

Yes

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

infinite branch