# Classical and fuzzy logic: Connective, tautology and contradiction

Classical and logic deal with crisp sets and fuzzy sets, respectively

## What is logic?

It is a method for developing a quantitative reasoning process that can be duplicated and altered using mathematical proofs.

It is the ability of humans to think rationally about something using suitable evidence and inferences.

### Proposition:

It is a collection of declarative statements that have either a truth value as **true **or a truth value as **false**. **Example:** Man is mortal

The veracity (truth) of an element in the proposition P can be assigned a binary truth value, **called T (P)**

All elements u in the universe U that are true for the proposition set P are called the **truth set **of P

All elements u in the universe U that are false for the proposition set P are called the **falsity set **of P

For a universe X and a null set ϕ, we will define the truth value as: T(U) = 1 and T(ϕ) = 0

## Connectives:

It connects the propositional statements together to form new propositions

If P and Q are two propositions in the same universe, they can be combined using 5 logical connectives:

- Disjunction (∨)
- Conjunction (∧)
- Negation (-) or (~)
- Implication (→)
- Equivalence (⟷)

### Connective: Disjunction

The disjunction connective or “logical or” is a term used to represent inclusively or.

P ∨ Q: x ∈ A or x ∈ B

Hence, T(P ∨ Q) = max(T(P), T(Q))

### Connective: Conjunction

If both the propositions P and Q are true, then the compound proposition will be true. It performs logical ‘and’ operation

P ∧ Q: x ∈ A and x ∈ B

Hence, T(P ∧ Q) = min(T(P), T(Q))

### Connective: Negation

The negation of a proposition P is false when P is true and vice versa

if T(P) = 1, then T(P’) = 0

if T(P) = 0, then T(P’) = 1

### Connective: Implication

P→Q: ”**p entails q**” means that it can never happen that p is true and q is not true.

It is also written as “**If P then Q**”, where P is called the premise/hypothesis and Q is called as a conclusion

(P → Q): x ∉ A or x ∈ B

T(P → Q) = T(P’ ∪ Q)

In other words,

**Example: **If you score 90% or above in this subject, then you will get an A grade

### Connective: Equivalence

The equivalence connective is generally used when we have a dual implication for propositions P and Q

i.e. if P → Q and Q → P, then we can say that P ⟷ Q

All five connectives are summarized in the following table:

## Watch on YouTube: Classical and fuzzy logic

## Tautology:

In classical logic, it is useful to consider compound propositions that **are always true**, irrespective of the truth values of the individual simple propositions.

Classical logical compound propositions with this property are called *tautologies**.*

Tautologies are useful for deductive reasoning, proving theorems, and for making **deductive inferences.**

- Modus Ponens: (P ∧ (P → Q)) → Q
- Modus Tollens: (Q’ ∧ (P → Q)) → P’

### Modus Ponens:

**Proof:**

Let us prove that modus ponens is tautology with the help of step-by-step derivation

(P ∧ (P → Q)) → Q

(P ∧ (P’ ∪ Q)) → Q [Implication]

((P ∧ P’) ∪ (P ∧ Q)) → Q [Distributivity]

(ϕ ∪ (P ∧ Q)) → Q [Excluded Middle]

(P ∧ Q) → Q [Identity]

(P ∧ Q)’ ∪ Q [Implication]

(P’ ∪ Q’ ) ∪ Q [De Morgan’s Law]

P’ ∪ (Q’ ∪ Q) [Associativity]

P ∪ X [Excluded Middle]

X [Identity]

T(X)=1

**Proof using the tabular method:**

S = (P ∧ (P → Q)) → Q

To check if the given statement is a tautology or not, we shall derive a truth table and check if all the entries have result T in the last column. If so then the statement is a tautology, else it is not.

### Modus Tollens:

**Proof:**

Let us demonstrate that modus ponens is a tautology via a step-by-step derivation.

(Q’ ∧ (P → Q)) → P’

(Q’ ∧ (P’ ∪ Q)) → P’ [Implication]

(Q’ ∧ P’) ∪ (Q’ ∧ Q )) → P’ [Distributivity]

(Q’ ∧ P’) ∪ ϕ) → P’ [Excluded Middle]

Q’ ∧ P’→ P’ [Identity]

(Q’ ∧ P’)’ ∪ P’ [Implication]

(Q ∪ P) ∪ P’ [De Morgan’s Las]

Q ∪ (P ∪ P’) [Associativity]

Q ∪ X [Excluded Middle]

X [Identity]

T(X) = 1

**Proof using the tabular method:**

S = (Q’ ∧ (P → Q)) → P’

To determine whether a given statement is a tautology, we will create a truth table and see if all of the entries have the outcome T in the last column. If so, the statement is a tautology; otherwise, it is not.

## Contradiction:

A proposition that is always false regardless of the truth value of individual simple propositions constituting that compound proposition.

A ∩ A’ = ϕ

A ∩ ϕ = ϕ

**Example: **Prove that (P ↔ Q) ∧ P’ ∧ Q is a contradiction

Scholarly Reading: Performance evaluation of classical and fuzzy logic control techniques for brushless DC motor drive is studied in this article

## Test Your Knowledge:

- A). Check if (P → Q) ∨ (Q → P) is a tautology or not
- B). Check if (P XOR Q) ^ (P XNOR Q) is a contradiction or not

**Please post your answer / query / feedback in comment section below !**

A) not tautology

Its Tautology Danishta

Hello, Could you please share the slides from the videos with me?