Logical Proof:

Logical proof is an argument that establishes the validity of a proposition. Although proofs may be based on inductive logic, in general the term proof connotes a rigorous deduction. In

– Britanica

Logic is a way to quantitatively develop a reasoning process that can be replicated and manipulated with mathematical proofs

Logic is the ability of humans to reasonably think about something and make a decision based on some proofs

A proof is an argument from a hypothesis to a conclusion.

Each step of the argument follows the laws of logic.

  1. Analyze a given compound proposition which is given in the form of a linguistic statement
  2. Decompose this statement into single propositions
  3. Express the statement algebraically with all its logical connectives in proper place
  4. Verify the truth value of the statement with a truth table

Example: Logical proof

let us try to infer the conclusion from given set of hypothesis

Hypothesis:

  1. Scientists are mathematicians
  2. Logical thinkers do not believe in magic
  3. Mathematicians are logical thinkers

Conclusion:

  • Scientists do not believe in magic

Proof:

We will assign a later to represent each hypothesis as predicate.

  • P: A person is a scientist
  • Q: A person is a mathematician
  • R: A person is a logical thinker
  • S: A person believes in magic

Let us create compound predicate by joining the individual predicates in hypothesis using different connectives.

  1. Scientists are mathematicians: P → Q
  2. Logical thinkers do not believe in magic: R → S’
  3. Mathematicians are logical thinkers: Q → R
  4. Scientists do not believe in magic: P → S

((P → Q) ∧ (R → S’) ∧ (Q → R)) → (P → S’)

We cant use the disjunction connective here because in the case of disjunction, not all the statements in the hypothesis are required to prove the conclusion.

To check the truth of a given statement it is crucial to prove the entire hypothesis and the conclusion as true

logical proof

Proof by contradiction:

A statement is true only if its alternate statement is false

((P → Q) ∧ P) → Q

((P’ ∨ Q) ∧ P) → Q

(((P’ ∨ Q) ∧ P))’ ∨ Q

Alternate statement: ((((P’ ∨ Q) ∧ P))’ ∨ Q)’ = ((P’ ∨ Q) ∧ P) ∧ Q’

Logical Proof using cotradiction
Proof by contradiction

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Deductive inference

Implication:

The rule IF A, THEN B, i.e. (P → Q) = R = (A × B) ∪ (A’ × Y), is defined in function-theoretic terms as,

χR(x, y) = max⁡((min⁡(χA(x), χB(y))), min((1 – χA(x)), 1))

The compound rule IF A, THEN B, ELSE C can also be defined in terms of a matrix relation as

R = (A × B) ∪ (A’ × C)

where the membership function is determined as

χR(x, y) = max⁡((min⁡(χA(x), χB(y))), min((1 – χA(x)), χC(y) ))

Example:

Suppose we have 2 universes for a thermistor problem and it is described by a collection of elements:

X = {1, 2, 3}, Y = {1, 2, 3, 4, 5}.

A crisp set A is defined on universe X and a crisp set B on universe Y as follows: A = {1, 3}, B = {2, 4, 5}

The inference given is If A then B: R=(A × B) ∪ (A’ × Y)

Deductive Inference:

Also called approximate reasoning

The ultimate goal of fuzzy logic is to form the theoretical foundation for reasoning about imprecise propositions; such reasoning has been referred to as approximate reasoning [Zadeh, 1976, 1979].

Suppose we have a rule of the form IF A, THEN B, where A is a set defined on universe X and B is a set defined on universe Y. As discussed before, this rule can be translated into a relation between sets A and B

R = (A × B) ∪ (A’ × Y)

The modus ponens deduction is used as a tool for making inferences in rule-based systems. A typical if–then rule is used to determine whether an antecedent (cause or action) infers a consequent (effect or reaction).

Suppose a new antecedent, say A’, is known. Can we use modus ponens  deduction to infer a new consequent, say B’, resulting from the new antecedent? That is, can we deduce, in rule form, IF A’, THEN B’?

Yes, through the use of the composition operation.

B′ = A′ ∘ R = A′ ∘ ((A × B) ∪ (A’ × Y))


Test Your Knowledge:

Hypothesis:

  • If you score 95% and above, your success is represent by a good grade
  • 1.You score 95% and above

Conclusion:

  • Your success is represented by a good grade

Can we derive the above conclusion from given hypothesis using logical proof?

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