Classical and logic deals 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 as the truth set of P

All elements u in the universe U that are false for the proposition set P are called as 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 on the same universe, they can be combined using 5 logical connectives:

  1. Disjunction (∨)
  2. Conjunction (∧)
  3. Negation (-) or (~)
  4. Implication (→)
  5. Equivalence (⟷)

Connective: Disjunction

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

P ∨ Q: x ∈ A or x ∈ B

Hence, T(P ∨ Q) = max⁡(T(P), T(Q))

disjunction connective
Truth table for disjunction

Connective: Conjuction

If both the proposition 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))

conjunction connective
Truth table for conjunction

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

negation connective
Truth table for negation

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 as the premise/hypothesis and Q is called as conclusion

(P → Q): x ∉ A or x ∈ B

T(P → Q) = T(P’ ∪ Q)

In other words,

implication equation
implication connective
Truth table for implication

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

equivalence connective
Truth table for equivalence

All five connectives are summarized in following table:

Truth table of connectives
Truth table of connectives

Watch on YouTube: Classical and fuzzy logic

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, for 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 tabular method:

S = (P ∧ (P → Q)) → Q

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

Proof of modus ponens
Proof of modus ponens

Modus Tollens:

rules

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 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.

 Proof of modus tollens
Proof of modus tollens

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 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 contradiction or not

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