Chapter 9: Quantifiers
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Variables and atomics wffs (well formed formulas)
o Behave syntactically like names
 They appear in the same places that names appear.
 1) Cube(d)
 2) FrontOf(a,b)
 3) Adjoins(c,e)
 4) Cube(x)
 5) FrontOf(x,y)
 6) Adjoins(c,x)
 7) Adjoins(y,e)
o All of these are wffs.
o 1-3 are sentences AND wffs
o constants a,b,c,d,e,f…
o variables u,v,w,x,y,z.
o So what’s the difference between a sentence and a well
formed formula?
 The answer is 9.3
• For now: a sentence is a wff that has no
unbound variables. (There is no variable that
does not have a quantifier applying to it.)
9.2 Quantifiers ∀, ∃
- We use quantifiers, along with variables, to turn wffs into sentences by binding
the unbound variables.
(∀)—universal
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- ∀x: read that as “for every object x…)
ie: Every object is a cube
o ∀xCube(x)
Everything is either a cube or a tet
o ∀x(Cube(x) ∨Tet(x))
Every tet is small
o ∀x(Tet(x) → Small(x))
(∃)—existential
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∃x: read that as “for at least one object x…”
o Ie: there is at least one tet.
 ∃ xTet(x)
o Some tet is small
 ∃x(Tet(x) ^ Small(x))
o There is at least one cube in front of b
 ∃x(Cube(x) ^ FrontOf(x,b))
9.3 – Wffs and Sentences
- wffs: an atomic wff is any n-ary predicate followed by n individual
symbols
- any variable that occurs in a wff is unbound/free
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Sentences:
-
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If P is wff, so is ¬P
If P1….Pn are wffs, so is (P1 ∧ P2 ∧…PN)
o (applies to every Boolean connective)
If P is wff and x is a variable, the ∀xP is a wffs and any occurance of x
in ∀xP is said to be bound.
A sentence is a wff with no unbound variables.
o Two ways to convert wffs into sentences
1) Binding with a quantifier
o broad
2) substituion
o replace any free variables in a wff with constants
Scope
-
Be careful how parentheses are used.
o ∃x(Doctor(x) ∧ Smart(x))
 Some doctor is smart
o ∃xDoctor(x) ∧Smart(x)
 There are doctors and x is smart
9.4 -- Semantics
- Existential
o ∃xP(x) is true iff at least one object satisfies P(x)
• ∃xCube(x)
- Universal
o ∀xP(x) is true iff every object satisfies P(x)
• ∀xTet(x)
All vs. Only
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“All” can be a quantifier
o (ie: all freshman…)
“Only” can also be a quantifier
o (ie: only freshman…)
All freshmen are eligible for the scholarship
o Tells us that being a freshman is a sufficient condition for
eligibility
o Ax(Freshman(x)  Eligible(x))
Only freshmen are eligible for the scholarship
o Tells us that being a freshmen is a necessary condition for
eligibility
 You’re eligible only if you’re a freshmen
 Ax(Eligibile(x)  Freshman(x))
All 16 year olds can get a drivers license
∀x(16(x)  Lic(x))
Only 16 year olds can get a drivers lic.
∀x(Lic(x)  16(x))
9.5 – The Four Aristotelian Forms
- Aristotle codified what are called categorical propositions into 4
distinct forms.
o Categorical propositions are propositions that express the
relationship between two classes or categories.
 IE: All dogs are mammals
- The four forms are:
o (A) Complete Inclusion – All dogs are mammals -- ∀x(Dog(x)
→ Mammals(x))
o (E) Complete Exclusion – No dogs are cats
 Two versions: ∀x(Dog(x) → ¬Cats(x)) OR ¬∃x(Dog(x)
∧ Cat(x))
o (I) Partial Inclusion – Some dogs are collies -- ∃x(Dog(x) ∧
Collies(x))
 ** Note: this SHOULD NOT be translated as ∃x(Dog(x)
→ Collies(x)))
o (O) Partial Exclusion – Some dogs are not collies -- ∃x(Dog(x)
∧ ¬Collies(x))
9.6 – Complex Noun Phrases
- Read the text closely here.
- The main point to take away is that you’ll often be using multiple
conjunctions to complete express all the nouns and properties in a
sentence.