Discrete Structures

# Lecture Notes, 17 Sep 2010

 Last Update: 19 September 2010 Note: or material is highlighted

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

1. In English, every (atomic) sentence consists of a noun phrase (NP) & a verb phrase (VP)

• The sentence "My computer is a Mac" consists of the NP "my computer" and the VP "is a Mac".
• "Fido barks" consists of the NP "Fido" & the VP "barks".

1. The NP names or describes an object;
the VP says something about the object.

2. The NP is called the "subject" of the sentence;
the VP is called the "predicate" of the sentence.

3. The subject is what the sentence is about;
it refers to an object in the "world"
(more generally, it refers to an object in the domain of discourse,
i.e., the world that we are talking about)

4. The predicate says something about the subject;
it refers to a property or relation

2. Propositional functions:

1. "x is a Mac" is not a sentence;
it is neither true nor false.

• Nor is it a "predicate", contrary to what some textbooks say!
• Only "is a Mac" is a predicate.

2. Rather, it is a propositional function

• As a (mathematical) function, its input is a value for x
& its output is a proposition with a truth value

3. Here's another propositional function:

2 + y < 4

1. Its subject is y;
2. its predicate is "2 +     < 4"

• this predicate names the property of "being such that when added to 2, the sum is < 4"

3. We can denote the predicate by a capital letter,
& write it in mathematical-function notation:

• P(y) stands for: 2 + y < 4

Here, y is called P's "variable" or "parameter" or "argument" or "term".

A term is a NP that names or describes an object in the domain

4. We can make a table showing possible input values to this propositional function, its output values, and—because the output is a proposition—we can show the truth value of the output:

yP(y)tval(P(y))
02+0<4T
12+1<4T
22+2<4F
32+3<4F

4. Preds can apply to 1 or more terms:

1. If a predicate applies to two or more terms,
then it names a relation among the terms.

• E.g., in "John bought a book",
there is a 2-place predicate "bought" that has 2 terms: "John" and "a book"

2. E.g., in x + y < 4,
there are 2 terms: x, y
& a 2-place predicate: "    + … < 4"

• This predicate names "the relation between two numbers such that when added together their sum is < 4"
• We can denote a 2-place predicate by a capital letter
& use functional or relational notation:
Q(x, y) can stand for: x + y < 4

• Note that P(y) above stands for the same propositional function as Q(2, y)

3. We can make a table for Q, similar to the one above for P:

x y Q(x,y) tval(Q(x,y))
000+0<4T
636+3<4F

3. (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