THE FOUNDATIONS: LOGIC AND

PROOFS•Propositional Logic•Propositional Equivalences•Predicates and Quantifiers

SYMBOLS [Exclusive Or] [disjunction(or)] [conjunction(and)] [not] [if…then…] [if and only if]

PROPOSITIONAL LOGIC Proposition – Sentence that declare fact

either true or false but not both. Examples:1. F and S are alphabet. 2. They are beautiful. (is not proposition because is not

declarative)3. 2 + 1 = 10 4. x + y = 6 (is not proposition because it is neither

true or false)

/

x

x/

PROPOSITIONAL VARIABLES We use letters to represent propositions such as p,q,r,s…

Truth Table -display relationship between truth values of propositionsTypes of truth tables:

1. the Negation of a Proposition.

2. Conjunction of Two Propositions.

3. Disjunction of Two Propositions.

4. Exclusive Or of Two Propositions.

5. Conditional Statement p → q.

6. Biconditional p ↔ q.

7. The Truth Table of (p ∨￢ q) → (p q).∧8. Precedence of Logical Operators.

TYPES OF TRUTH TABLES Table 1: Truth table for negationOf a proposition

p p

TF

FT

Table 2: Truth table for the Conjunction of two Propositions

p q p q

T TT FF TF F

TFFF

Table 3: Truth table for the Disjunction of two Propositions

p q p q

T TT FF TF F

TTTF

Table 4: Truth table for the Exclusive Or of two Propositions

p q p q

T TT FF TF F

FTTF

Table 5: Truth table for the Implication pq

p q p q

T TT FF TF F

TFTT

Table 6: Truth table for The Biconditional p q

p q p q

T TT FF TF F

TFFT

Table 7: Precedence of Logical Operators

Operator Precedence

1

23

45

PROPOSITIONAL EQUIVALENCE Tautology : a compound proposition that is always

true no matter what the truth values of the propositional occurs in it.

Example: pp Contradiction : a compound proposition that is

always false. Example: pp Contingency : a compound proposition that is

neither a tautology nor a contradiction.

EXAMPLE OF TAUTOLOGY AND CONTRADICTIONp p pp pp

T F T F

F T T F

pp is always true

pis always false

LOGICAL EQUIVALENCES Compound propositions that have the

same truth values in all possible cases Example: p q = -p -q

a) If it rains then, I stay at homeb) If I don’t stay at home, then it thus not

rain

PREDICATES A predicate or propositional function is a

description of the property (or properties) a variable or subject may have.

A proposition may be created from a propositional function by either assigning a

value to the variable or by quantification.

In general, the set of all x in the universe of discourse having the atribute P(x) is

  called the truth set of P(x). That is, the truth set of P(x) is :             { x ϵ U |P(x) }

where the truth value can be whether true or false...

See more at: http://weartificialintelligence.blogspot.com/#sthash.DyhdBMPq.dpuf

EXAMPLE Suppose P(x) is the predicate x + 2 = 2x, and the

universe of discourse for x is the set f1;2;3g. Then...

xP(x) is the proposition “For every x in {1,2,3} x + 2 = 2x." This proposition is false.

xP(x) is the proposition “There exists x in {1,2,3} such that x + 2 = 2x. This proposition is true.

Example 1 :

The propositional function P(x) is given by "x > 0".The universe of discourse for x is the set of integers.To create a proposition from P, we may assign a value for x. For example, 

setting x = -3, we get P(-3): "-3 > 0", which is false. setting x = 2, we get P(2): "2 > 0", which is true.

Example 2 :

Suppose P(x) is the predicate “ x has fur ". The universe of discourse for x is the set of all animals. P(x) is a true statement if,    x is a cat.  P(x) is false if,     x is an alligator.

Example 3 :

There also involve 2 or more variable. consider "x =  y + 3" and we can denote it as Q(x , y). Q is a predicate and the question is what is the truth value for Q(1,2) and Q(3,0) ?

Answer for Q(1,2),set x = 1 and y = 2 and substitute into "x =  y + 3",which u get false. Same as Q(3,0),which u get true.

Answer must be shown in table.

QUANTIFIERS

A quantifier turns a propositional function into a proposition without assigning specific values for the variable.

There are primarily two quantifiers:

the universal quantifier the existential quantifier

UNIVERSAL QUANTIFIER

The universal quantification of P(x) is the proposition

 “P(x) is true for all values x in the universe of discourse.”

Notation : "For all x P(x)" or "For every x P(x)"

is written  ∀xP(x).

example 1 :

Let P(x) be the statement “x+1>x.” What is the truth value of the Quantification ∀xP(x)

where the universe of discourse consist of all real numbers ? 

solution : Since P(x) is true for all real numbers x, the quantification  ∀xP(x)

 is true. x x+1>x ∀x P(x)

1 2>1 true

2 3>2 True

3 4>3 true

example 2 :

What is the truth value of ɏx P(x), where P(x) is the statement "x² < 10" and the universe discourse consist of the positive integers not exceed 4 ?

Solution: the statement  ɏx P(x) is the same as the conjunction 

P(1) ^ P(2) ^ P(3) ^ P(4), since the universe discourse consist of

the integers 1,2,3 n 4. Since P(4) which the statements "4² < 10 " is false, it follows that  ɏx P(x) is false.

EXISTENTIAL QUANTIFIER

The existential quantification of P(x) is the proposition

"There exists an element x in the universe of discourse such that P(x) is true."

Notation: "There exists x such that P(x)" or "There is at least one x such that

P(x)" is written  Ǝ x P(x).

Example 1 : Let P(x) denote the statement "x >3" What is the

truth value of the quantification ƎxP(x), where the universe discourse consist of all real numbers ?

solution : Since "x>3" is true--for instance ,

when x = 4--the existancial

qualification of P(x),which

is  ƎxP(x) is true.

x x > 3 Ǝ x P(x)

1 1 > 3 true

2 2 > 3 true

4 4 > 3 false

Example 2 :

What is the truth value of  ƎxP(x), where P(x) is the statement "x² > 10" and the universe discourse consist of the positive integers not exceed 4 ?

Solution: Since the universe of discourse is

{1,2,3,4}, the proposition ƎxP(x) is the same as the disjunction

P(1) v P(2) v P(3) v P(4), Since P(4) which the statements "4² > 10

" is true, it follows that  ƎxP(x) is true.

TABLE 1  QUANTIFIERS

  Statement When True ? When False ?

ɏ x P(x)

Ǝ x P(x)

P(x) is true for every x.

There is an x for which P(x) is true.

There is an x for which P(x) is false.

P(x) is false for every x.

EXAMPLES OF USING QUANTIFIERS IN REALITY Hard versus soft science

The ease of quantification is one of the features used to distinguish hard and soft sciences from each other.

Social sciences

In the social sciences, quantification is an integral part of economics and psychology.