- 265 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Propositional and predicate logic ' - HarrisCezar

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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.

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

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

Negation is represented by the symbol ¬

if P is a proposition,

then not P is represented by: ¬P

And is represented by the symbol

The or operator is represented by the symbol

Implication is represented by the symbol

Equivalence is represented by the symbol .

P : Physics is easy

Q : Chemistry is interesting

¬PQ

“Physics is not easy and chemistry is interesting”

P : Physics is easy

Q : Chemistry is interesting

¬(PQ)

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

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

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

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

F

Contradictions

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

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.

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: dA

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

Predicatelogicis a powerful way for us to reason about sets.

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

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

- By substitution (giving a particular value to the variable)
- By Quantification

C( x )

Studies( x , y )

Prime( x )

Simba ): Simba is a cat

Olawale, physics ): Olawale studies physics

3 ): 3 is a prime number

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 !

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)

this reads:

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

Or

All cats chase mice.

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)

this reads:

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.

Download Presentation

Connecting to Server..