Classical and fuzzy logic: Connective, tautology and contradiction
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.
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
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 (⟷)
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))
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))
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
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
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:
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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’
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]
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.
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]
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.
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 !