Logical Proof and Deductive inference in classical and fuzzy logic

by codecrucks · Published 21/08/2021 · Updated 08/03/2023

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

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

Each step of the argument follows the laws of logic.

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

Example: Logical proof

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

Hypothesis:

Scientists are mathematicians
Logical thinkers do not believe in magic
Mathematicians are logical thinkers

Conclusion:

Scientists do not believe in magic

Proof:

We will assign a capital letter to represent each hypothesis as a 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 a compound predicate by joining the individual predicates in the hypothesis using different connectives.

Scientists are mathematicians: P → Q
Logical thinkers do not believe in magic: R → S’
Mathematicians are logical thinkers: Q → R
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

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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Implication:

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

χ_R(x, y) = max⁡((min⁡(χ_A(x), χ_B(y))), min((1 – χ_A(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

χ_R(x, y) = max⁡((min⁡(χ_A(x), χ_B(y))), min((1 – χ_A(x)), χ_C(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 in universe X and B is a set defined in 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))

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