Rules of Inference

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 In logic and mathematics, rules of inference (also known as inference rules or transformation rules) are a set of logical forms that allow us to derive conclusions from premises. They provide a systematic way to build arguments and construct proofs.

Key Concepts

  • Premise: A statement assumed to be true.
  • Conclusion: A statement derived from the premises using rules of inference.
  • Valid Argument: An argument where the conclusion logically follows from the premises.
  • Sound Argument: A valid argument where all the premises are true.

Common Rules of Inference

Here are some of the most common rules of inference:

  1. Modus Ponens (MP): If P, then Q. P. Therefore, Q.

    • Example: If it rains, then the ground gets wet. It is raining. Therefore, the ground is wet.

  2. Modus Tollens (MT): If P, then Q. Not Q. Therefore, not P.

    • Example: If it rains, then the ground gets wet. The ground is not wet. Therefore, it is not raining.

  3. Hypothetical Syllogism (HS): If P, then Q. If Q, then R. Therefore, if P, then R.

    • Example: If it rains, then the ground gets wet. If the ground gets wet, then the grass grows. Therefore, if it rains, then the grass grows.

  4. Disjunctive Syllogism (DS): P or Q. Not P. Therefore, Q.

    • Example: It is either sunny or cloudy. It is not sunny. Therefore, it is cloudy.

  5. Simplification (Simp): P and Q. Therefore, P.

    • Example: I have a red pen and a blue pen. Therefore, I have a red pen.

  6. Conjunction (Conj): P. Q. Therefore, P and Q.

    • Example: I have a red pen. I have a blue pen. Therefore, I have a red pen and a blue pen.

  7. Addition (Add): P. Therefore, P or Q.

    • Example: I have a red pen. Therefore, I have a red pen or a blue pen.

  8. Resolution (Res): P or Q. Not P or R. Therefore, Q or R.

    • Example: It is either raining or snowing. It is not raining or it is windy. Therefore, it is snowing or it is windy.

Using Rules of Inference

Rules of inference are used to construct formal proofs. A proof is a sequence of statements, where each statement is either a premise or follows from previous statements using a rule of inference.

Example

Let's prove that if it rains, then the grass grows, given the following premises:

  1. If it rains, then the ground gets wet.
  2. If the ground gets wet, then the grass grows.

Proof:

  1. If it rains, then the ground gets wet. (Premise)
  2. If the ground gets wet, then the grass grows. (Premise)
  3. If it rains, then the grass grows. (Hypothetical Syllogism: 1, 2
By understanding and applying rules of inference, you can construct logical arguments and build valid proofs.

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