Discrete Structures

# Lecture Notes, 20 Sep 2010

 Last Update: 20 September 2010 Note: or material is highlighted

## §1.3: Predicates & Quantifiers (cont'd)

1. (Recursive) Definition of Well-Formed Proposition of FOL:

Remember:

• "terms" (or NPs) name or describe objects in the domain;
• "predicate" (or VPs) name properties of objects or relations among ≥2 objects
• "variables" (or pronouns) are like variables in programming languages

1. Base Cases:

1. Atomic propositions (p, q, r, …) of propositional logic are well-formed (atomic) propositions of FOL.

2. If t1,…,tn are terms (NPs)
& if R is an n-place predicate,
then R(t1,…,tn) is a WF (atomic) proposition of FOL (a "subatomic" proposition)

2. Recursive Cases:

• If A, B are WF (atomic or molecular) propositions of FOL,
& if v is a variable,
then:

1. ¬A
2. (AB)
3. (AB)
4. (AB)
5. (AB)
6. (AB)
7. v[A]
8. v[A]

are WF (molecular) propositions of FOL

2. Definition of Term of FOL:

1. If v is a variable, then v is a term of FOL.

E.g., x, y, z, … are terms

2. If c is a constant, then c is a term of FOL.

E.g., "fred", "evelyn", 1, √2, … are terms

3. (another clause to be presented later, maybe…)
4. Nothing else is a term.

• In particular, no predicate is a term!
I.e., if R is a predicate, then R is not a term.

So: predicates can never appear in term-position;
i.e., in FOL, predicates cannot be terms of other predicates;
i.e., predicates cannot appear inside other predicates

3. Syntax & Semantics of the Quantifiers:

NAME SYNTAX ENGLISH E.G. SEMANTICS
universal
quantifier
v[A],
where v is any variable
& where A is any proposition,
including:
all of the "old" ones
& any of these "new" ones;

if A is atomic,
the brackets can be omitted;

e.g., ∀xR(x)
Everything in the domain satisfies A.

For all objects v in the domain, A is the case.

All humans are mortal ≡

x[Human(x) → Mortal(x)]

Note the use of →!

tval(∀vA)=T
iff
all objects in the domain are such that tval(A)=T

e.g., tval(∀xR(x))=T
iff
all objects in the domain have the property named by R

existential
quantifier
v[A],
as above

e.g., ∃xR(x)

Something satisfies A

There is (or: there exists) an object v in the domain for which A is the case.

Some human is mortal ≡

x[Human(x) ∧ Mortal(x)]

Note the use of ∧!!

tval(∃vA)=T
iff
at least one object in the domain is such that tval(A)=T

e.g., tval(∃xR(x))=T
iff
at least one obj in the domain has the property named by R

4. The truth value of a quantified proposition depends on the domain:

propositionDomain:
W={1,2,3,…} N={0,1,2,3,…} Z={…–3,–2,–1,0,1,2,3,…}
x[x > 0] TF
(because ¬(0>0))
F
x[x ≥ 0] TTF

Notes:

"∀x[x > 0]" can also be written (slightly more grammatically) as: ∀x[>(x, 0)]

"∀x[x ≥ 0]" can also be written (slightly more grammatically) as: ∀x[>(x, 0) ∨ =(x, 0)]

5. Finite domains:

Consider the domain be {0,1,2,3}.
Then tval(∀xR(x)) = T iff, for all values of x in the domain, tval(R(x)) = T
iff tval(R(0))=T & tval(R(1))=T & tval(R(2))=T & tval(R(3))=T

∴ ∀xR(x) ≡ R(0) ∧ R(1) ∧ R(2) ∧ R(3)

Similarly, we can show that: ∃xR(x) ≡ R(0) ∨ R(1) ∨ R(2) ∨ R(3)

• xR(x) is like a for-loop!
• xR(x) is like a search procedure!

6. De Morgan's Laws for Quantifiers (or: How to Negate a Quantifier):

In a finite domain, say {0,1}:

xP(x) ≡ P(0) ∧ P(1).

∴ ¬∀xP(x) ≡ ¬(P(0) ∧ P(1))
≡ (¬P(0) ∨ ¬P(1)), by DeMorgan for the negation of ∧
≡ ∃x[¬P(x)]

Not only is this also true in an infinite domain, but so is this:

¬∃xP(x) ≡ ∀x[¬P(x]

and these:

xP(x) ≡ ¬∀x¬P(x)
xP(x) ≡ ¬∃x¬P(x)