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Theory of Inference for the Statement Calculus
To deduce new statements from the statements whose truth we already know, Rules of Inference are used.
What are Rules of Inference for?
Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements.
An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypotheses). The symbol "∴" (read "therefore") is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises.
Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.
Important Definitions
- Argument − A sequence of premises that ends with a conclusion.
- Validity − An argument is valid if and only if whenever the premises are true, the conclusion cannot be false.
- Fallacy − An incorrect reasoning resulting in invalid arguments.
Argument Structure
An argument structure consists of premises and a conclusion −
Premises: P1, P2, P3, ..., Pn Conclusion: Q If P1 ∧ P2 ∧ P3 ∧ ... ∧ Pn → Q is a tautology, then the argument is VALID, otherwise INVALID.
Table of Rules of Inference
| Rule | Name | Structure |
|---|---|---|
| P ∴ P ∨ Q | Addition | From P, conclude P ∨ Q |
| P, Q ∴ P ∧ Q | Conjunction | From P and Q, conclude P ∧ Q |
| P ∧ Q ∴ P | Simplification | From P ∧ Q, conclude P |
| P → Q, P ∴ Q | Modus Ponens | If P implies Q and P is true, then Q |
| P → Q, ¬Q ∴ ¬P | Modus Tollens | If P implies Q and Q is false, then ¬P |
| P ∨ Q, ¬P ∴ Q | Disjunctive Syllogism | From P ∨ Q and ¬P, conclude Q |
| P → Q, Q → R ∴ P → R | Hypothetical Syllogism | Chain two implications |
| (P → Q) ∧ (R → S), P ∨ R ∴ Q ∨ S | Constructive Dilemma | Two implications with disjunction |
| (P → Q) ∧ (R → S), ¬Q ∨ ¬S ∴ ¬P ∨ ¬R | Destructive Dilemma | Deny consequents to deny antecedents |
Example
Consider the following statements −
- If it rains, I shall not go to school.
- If I don't go to school, I won't need to do homework.
Let's identify the propositions −
P = It rains Q = I shall go to school R = I need to do homework Hypothesis 1: P → ¬Q (If it rains, I don't go to school) Hypothesis 2: ¬Q → ¬R (If I don't go to school, no homework)
Using Hypothetical Syllogism (P → ¬Q, ¬Q → ¬R ∴ P → ¬R) −
Conclusion: P → ¬R "If it rains, I won't need to do homework."
The argument is valid because (P → ¬Q) ∧ (¬Q → ¬R) → (P → ¬R) is a tautology.
Conclusion
Rules of Inference are fundamental tools for constructing valid logical proofs. They provide standard templates like Modus Ponens, Modus Tollens, and Hypothetical Syllogism to derive new true statements from existing premises in a mathematically rigorous way.
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