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Propositional and predicate logic Propositional and predicate logic At the end of this lecture you should be able to: distinguish between propositions and predicates ; utilize and construct truth tables for a number of logical connectives ;

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Propositional and predicate logic

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Propositional and predicate logic

Propositional and predicate logic

At the end of this lecture you should be able to:

• distinguish between propositions and predicates;

• utilize and construct truth tables for a number of logical connectives;

• determine whether two expressions are logically equivalent;

• explain the difference between bound and unbound variables;

• bind variables by substitution and by quantification.

Propositions

In classical logic, propositions are statements that are either TRUE or FALSE…..

There are seven days in a week

2 + 4 = 6

London is the capital of France.

The food at UEL tastes nice.

Put 10 into X

Using Symbols..

In mathematics we often represent a proposition symbolically by a variable name such as Por Q.

P: I go shopping on Wednesdays

Q :102.001 > 101.31

Logical connectives..

Negation

Negation is represented by the symbol ¬

if P is a proposition,

then not P is represented by: ¬P

I like dogs

P

I do notlike dogs

¬P

Connectives can be defined by truth tables….

P

¬P

T

F

F

T

The andoperator

And is represented by the symbol 

I like shopping

P

The sun is shining

Q

I like shopping and the sun is shining

PQ

The truth table for 'and'

P

Q

P Q

T

T

T

T

F

F

F

T

F

F

F

F

The oroperator

The or operator is represented by the symbol 

It is raining

P

Today is Tuesday

Q

It is raining or today is Tuesday

PQ

The truth table for ‘or'

P

Q

P Q

T

T

T

T

F

T

F

T

T

F

F

F

The implicationoperator

Implication is represented by the symbol 

It is Wednesday

P

I do the ironing

Q

If it is WednesdayI do the ironing

PQ

The truth table for implication

P

Q

P  Q

T

T

T

T

F

F

F

T

T

F

F

T

The equivalence operator

Equivalence is represented by the symbol .

I’ve passed my exam

P

I’ve passed my coursework

Q

I’ve passed my module

M

I will pass my module if and only if I pass my exam and my coursework.

M

(PQ)

The truth table for equivalence

P

Q

P  Q

T

T

T

T

F

F

F

T

F

F

F

T

Compound statements

P : Physics is easy

Q : Chemistry is interesting

¬PQ

“Physics is not easy and chemistry is interesting”

Compound statements

P : Physics is easy

Q : Chemistry is interesting

¬(PQ)

“It is not true both that physics is easy and that chemistry is interesting.”

Logical equivalence

Two compound propositions are said to be logically equivalent if identical results are obtained from constructing their truth tables;

This is denoted by the symbol .

For example

¬ ¬P P

P

¬P

¬¬P

T

F

T

F

T

F

T

T

T

F

F

T

F

F

Logical equivalence : a demonstration

(P  Q)P Q

P

Q

P  Q

(P  Q)

P

Q

P Q

T

F

F

F

F

F

T

F

T

T

F

T

T

F

T

F

T

T

T

T

T

F

Tautologies

A statement which is always true (that is, all the rows of the truth table evaluate totrue) is called a tautology.

For example, the following statement is a tautology:

P P

This can be seen from the truth table:

P

¬P

P P

F

T

T

T

T

F

A statement which is always false (i.e. all rows of the truth table evaluate tofalse) is called a contradiction.

For example, the following statement is a contradiction:

P P

Again, this can be seen from the truth table:

P

¬P

P P

F

F

T

F

Sets

A set is any well-defined, unordered, collection of objects;

For example we could refer to:

• the set containing all the people who work in a particular office;

• the set of whole numbers from 1 to 10;

• the set of the days of the week;

• the set of all the breeds of cat in the world.

Representing sets

A = {s, d, f, h, k }B = {a, b, c, d, e, f}

the symbol  means "is an element of".

the statement "d is an element of A" is written: dA

the statement "p is not an element of A" is written: pA

Predicatelogicis a powerful way for us to reason about sets.

Predicates

A predicate is a truth valued expression containing free variables;

These allow the expression to be evaluated by giving different values to the variables;

Once the variables are evaluated they are said to be bound.

Examples

C(x):x is a cat

Studies(x,y):x studies y

Prime(n):n is a prime number

Binding Variables

There are two ways in which variables in predicates can be given values.

• By substitution (giving a particular value to the variable)

• By Quantification

Substitution

C( x )

Studies( x , y )

Prime( x )

Simba ):Simba is a cat

Olawale, physics ):Olawale studies physics

3 ):3 is a prime number

Quantification

A quantifier is a mechanism for specifying an expression about a set of values;

There are three quantifiers that we can use, each with its own symbol:

The Universal Quantifier,

The Existential Quantifier

The Unique Existential Quantifier !

The Universal Quantifier, 

This quantifier enables a predicate to make a statement about all the elements in a particular set.;

For example:

If M(x) is the predicate x chases mice, we could write:

x Cats M(x)

For all the x’s which are members of the set Cats, x chases mice

Or

All cats chase mice.

The Existential Quantifier 

In this case, a statement is made about whether or not at least one element of a set meets a particular criterion.

For example

if, P(n) is the predicate n is a prime number, we could write:

n P(n)

There exists an n in the set of natural numbers such that n is a prime number

or

There exists at least one prime number in the set of natural numbers.

The Unique Existential Quantifier !

This quantifier modifies a predicate to make a statement about whether or not precisely one element of a set meets a particular criterion.

For example

If G(x) is the predicate x is green, we could write

!x  Cats  G(x)

this would mean:

There is one and only one cat that is green.