# Fuzzy Logic - Traditional Fuzzy Refresher

Logic, which was originally just the study of what distinguishes sound argument from unsound argument, has now developed into a powerful and rigorous system whereby true statements can be discovered, given other statements that are already known to be true.

### Predicate Logic

This logic deals with predicates, which are propositions containing variables.

A predicate is an 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 to the variable or by quantifying the variable.

Following are a few examples of predicates −

• Let E(x, y) denote "x = y"
• Let X(a, b, c) denote "a + b + c = 0"
• Let M(x, y) denote "x is married to y"

### Propositional Logic

A proposition is a collection of declarative statements that have either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. The propositional variables are dented by capital letters (A, B, etc). The connectives connect the propositional variables.

A few examples of Propositions are given below −

• "Man is Mortal", it returns truth value “TRUE”
• "12 + 9 = 3 – 2", it returns truth value “FALSE”

The following is not a Proposition −

• "A is less than 2" − It is because unless we give a specific value of A, we cannot say whether the statement is true or false.

### Connectives

In propositional logic, we use the following five connectives −

• OR (∨∨)
• AND (∧∧)
• Negation/ NOT (¬¬)
• Implication / if-then (→→)
• If and only if (⇔⇔)

### OR (∨∨)

The OR operation of two propositions A and B (written as A∨BA∨B) is true if at least any of the propositional variable A or B is true.

The truth table is as follows −

A B A ∨ B
True True True
True False True
False True True
False False False

### AND (∧∧)

The AND operation of two propositions A and B (written as A∧BA∧B) is true if both the propositional variable A and B is true.

The truth table is as follows −

A B A ∧ B
True True True
True False False
False True False
False False False

### Negation (¬¬)

The negation of a proposition A (written as ¬A¬A) is false when A is true and is true when A is false.

The truth table is as follows −

A ¬A
True False
False True

### Implication / if-then (→→)

An implication A→BA→B is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true.

The truth table is as follows −

A B A→B
True True True
True False False
False True True
False False True

### If and only if (⇔⇔)

A⇔BA⇔B is a bi-conditional logical connective which is true when p and q are same, i.e., both are false or both are true.

The truth table is as follows −

A B A⇔B
True True True
True False False
False True False
False False True

Well Formed Formula

Well Formed Formula (wff) is a predicate holding one of the following −

• All propositional constants and propositional variables are wffs.
• If x is a variable and Y is a wff, ∀xY and ∃xY are also wff.
• Truth value and false values are wffs.
• Each atomic formula is a wff.
• All connectives connecting wffs are wffs.

## Quantifiers

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

• Universal Quantifier
• Existential Quantifier

### Universal Quantifier

Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀.

∀xP(x) is read as for every value of x, P(x) is true.

Example − "Man is mortal" can be transformed into the propositional form ∀xP(x). Here, P(x) is the predicate which denotes that x is mortal and the universe of discourse is all men.

### Existential Quantifier

Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃.

∃xP(x) for some values of x is read as, P(x) is true.

Example − "Some people are dishonest" can be transformed into the propositional form ∃x P(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people.

## Nested Quantifiers

If we use a quantifier that appears within the scope of another quantifier, it is called a nested quantifier.

Example

• ∀ a∃bP(x,y) where P(a,b) denotes a+b = 0
• ∀ a∀b∀cP(a,b,c) where P(a,b) denotes a+(b+c) = (a+b)+c

Note − ∀a∃bP(x,y) ≠ ∃a∀bP(x,y)