Quantified Logic Flashcards
An argument form is invalid iff _________
there are some instances of it that are invalid
If an argument has an invalid form, can it still be valid?
Yes
What are the 4 components of Quantified Logic
1) Names
2) Predicates
3) Quantifiers
4) Variables
What is a Name, in QL?
Meaningful subsentential symbol
What do names in QL refer to?
particular things (often proper nouns)
How are Names represented in QL?
Lowercase letters from a-w (sometimes with subscripts)
Are names in QL wffs?
No
What is a Predicate in QL?
A way to describe a Name, expresses a property
How are Predicates represented in QL?
Capital letters with optional numerical subscripts
What is a monadic predicate?
One argument place
What is a dyadic predicate?
two-place predicate
Ex - ______ is funnier-looking than _______
Fdm
A monadic predicate requires ____ name to make a proposition
1
A diadic predicate requires _____ names to make a proposition
2
3-place predicates require _____ names to make a proposition.
3
What is an SL atomic wff in QL?
A zero-place predicate (requires zero names to express a proposition)
Do all sentence predicate properties of particular individuals?
No, thats why we need quantifiers
words like “every” and “some” are what in QL
Quantifiers
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.
Fc ⊃ ∀xFx
c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y
Someone is funny but not nice.
▶ ∃x(Fx & ¬Nx)
c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y
Calvin has a wife.
▶ ∃xWcx
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.
▶ ∃x(Wcx & ¬Nx)
c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y
Betty doesn’t have a wife.
¬∃xWbx
c Calvin
b Betty
Fx x is funny
Nx x is nice
Wxy x’s wife is y
Everyone nice is funny.
∀x(Nx ⊃ Fx)
Wxy
x’s wife is y
In english, this means
there is some y such that x’s wife is y.
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
