Logical Proof and Deductive inference in classical and fuzzy logic
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’
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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,
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 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))
Test Your Knowledge:
Hypothesis:
- If you score 95% and above, your success is represented 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 the given hypothesis using logical proof?
Please post your answer / query / feedback in comment section below !