# What is Inference?

Mathematical logic is often used for logical proofs that are valid arguments that determine the true value of a mathematical statement.

An argument is a sequence of statements and all its preceding statements are called premises.

(therefore) is placed before the conclusion.

A valid argument is one where the conclusion follows from the truth of the premises.

# Rules of Inference: -

# Addition/ Disjunctive syllogism: -

If pis a premise, we can use addition/Disjunctive syllogism rule to derive P∨Q

PP Q

# Example:-

let P be the proposition “He studies very hard” is True,

Therefore “Either he studies very hard or he is a very bad student.”

Here Q is the proposition “he is a very bad student”.

# Conjunction: -

If P and Q are two premises. we can use conjunction rules to derive P^Q

P Q P Q

Example :-

let P -“He studies very hard”

Let Q -“He is the best boy in the class”

Therefore, “He studies very hard and he is the best boy in the class”

# Simplification: -

If P^Q is a premise, we can use simplification rule to derive P.

PQP

Example :-

“He studies very hard and he is the best boy in the class”(P^Q)

Therefore, “He is studies very hard”

# Modus Ponens: -

If P and Q (P→Q) are two premises, we can use modus ponens to derive Q.

PQ P Q

Example: -

“If you a password then you can log on to Facebook” (P→Q)

You have a password (P)

Therefore, “you can log on to Facebook”.

# Modus Tollens: -

If P→Q and ¬Q are two premises, we can use modus Tollens, to drive ¬P.

PQ Q P

## Example:-

“If you have a password, then you log on to Facebook” (P→Q)

“You can not log on to Facebook” (¬Q)

Therefore, you don’t have a password”

# 6. Disjunctive Syllogism: -

If ¬P and P∨Q are two premises, we can use Disjunctive syllogism to derive Q

P PQ Q

## Example: -

“The Ice cream is not vanilla flavoured”.

“The ice cream is either vanilla flavoured pr chocolate flavoured” (P∨Q)

Therefore, “the ice cream is chocolate flavoured”.

# 7.Hypothetical syllogism: -

If P→Q and Q→R are two premises can use hypothetical syllogism to derive P→R

PQ QP PR

## Example: -

“If it rains, I shall not go to school” (P→Q)

“If I don’t go to school “I want need to homework”.

Therefore, if it rains, I Want need to home work.

# 8.Constructive Dilemma: -

If (P→Q) ^ (R→S) and P∨R are two premises, we can use constructive Dilemma to derive Q∨S.

(PQ)(RS) PR QS

## Example: -

“If it rains, “I will take a leave”

“If its hot outside, I will go for a shower” (R→S)

(P∨R) either it will rain or it is hot outside.

Therefore, I will take a leave or I will go for shower.

# 9. Destructive Dilemma: -

If (P→Q) ∧ (R→S) and (¬Q∨¬S are two premises, we can use destructive dilemma to derive (¬P∨¬R)

(PQ) (RS) (QS ) (PR)

## Example: -

“If it rains. I will take a leave” (P→Q)

“If it is hot outside, I will go for a shower” (R→S)

“Either I will not take or I will not go for a shower”.

Therefore, either it does not rain or it is not hot outside”

# Predicate Logic: -

A Predicate is a expression of one or more variables defined on some specific domain. A predicate with Variables can be made a proposition by either assigning a value or by quantifying.

### The following are some examples of predicate.

Let, E(X,Y ) denote “X=Y”

Let, X (a, b, c) denote “a=b+ c=o”

Let, M (X, Y) denote “X Married to Y”

There exist (∃)

For all (∀)

# WFF (Well Formed Formula ): -

Well formed formula is a predicate holding any of the following

All propositional constants and propositional variables are WFFS.

If X is a variable and Y is WFF ∀XY and ∃XY are also WFF

Truth value and False formula is a WFF.

Each automatic formula is a WFF.

All connectives connecting WFFS are WFFs

# Quantifier: -

The variable of predicates is quantified by quantifier. There are two types of predicate logic.

1.Existential Quantifier

2.Universal Quantifier.

# Existential Quantifier: -

If P(x) is a proposition over the universe x then it is denoted as ∃ x P(X) and read as” there exist at least one value in the universe of variable X such that p(X) is true” the quantifier ∃ is called existential quantifier.

There are serval ways to write a proposition, with an existential quantifier,

i.e:-

x∈A) p(x)

∃x∈A such that p(X)

( ∃x ) P(x)

P(xc) is true for some x∈A.

Example: -

Some people are dishonest “Can be transformed into propositional for ∃ x p(x)

Where P(x)is the predicate which dishonest X is dishonest and the universe of discourse is some people.

# Universal Quantifier: -

If p(x) is a proposition over the universe as ∀x, p(x) and read as “For every value x∈A,p(X) is true”. The quantifier ∀ is called the universal quantifier.

There are several ways to write a proposition, with an universal quantifier

i.e,

1. ∀x∈A, p(x) or

2. P(x),∀x∈A or

3. ∀x, p(x) or

4. P(x) is true for all x ∈A

“Man is Mortal” can be transform ed into the propositional from ∀x ,P (x) where p(x) is the predicate which denoted x is Motal and the universe of discourse is all men.

# Negation Quantified Proposition: -

Where we negation a Quantified proposition that is when a universally quantified position that is when a universal quantified proposition is negated. We obtain an extentially quantified proposition. When an extentially quantified proposition is negated w obtain a universally quantified proposition.

The two rules for negations of quantified proposition are as fallows. These are also called De-Morgan law’s .

## Example: -

∀xp(x)^∃yq(Y)

(~∀x p(x)^ ∃ y q(y)x))

~∀x p(x)∨~∃y q(y) [~(p ^q)=~p∨~q]

(∃x∈∪)(x+6)=25)

≅∀x∈∪~ ((x+6=25)

≅∀x∈∪(x+6)≠25

~(∃x p(x)∨ ∀y q(y))

≅ ~∃x p(x) ∨~∀y q(y)

≅ ∀x~p(x)∨ ∃y~q(y)

# Preposition With Multiple Quantifiers: -

The proposition having more than one variable can be quantified with multiple quantifiers. the multiple universal quantified can be arranged in any order without attiring the meaning of the resulting proposition, Also the multiple exist entail quantifier can be arranged in any order without quantifier can be arranged in any order without altering the meaning of the proposition.

The proposition which contains both universal and exertional quantifier, the order of those quantifiers cannot be exchange without attiring the meaning of the propositional .That is

## Example: -

∃x∀y p (x, y)

There exits some x such that p(x ,y) is true for every y

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