Rules of Inference

Mathematical Logic · Axiom Academy

Unit 3: Proof Theory - Mathematical Logic An inference rule is a logical pattern that allows us to derive a conclusion from one or more premises. The general form is: The premises appear above the horizontal line, and the conclusion appears below. This notation means: "If the premises are true, then the conclusion must be true." The horizontal line notation (also called a sequent ) is fundamental to formal logic. Above the line are the hypotheses or premises , below the line is the conclusion . An inference rule is sound if it preserves truth: whenever all premises are true, the conclusion must also be true. This is the most important property of any inference rule. The most fundamental inference rule is Modus Ponens (Latin for "mode of affirming"). It states: if we know that P implies Q, and we know P is true, then we can conclude Q is true. Example: If "It is raining" implies "The ground is wet", and we observe "It is raining", then we can conclude "The ground is wet". Modus Tollens (Latin for "mode of denying") works by contraposition: if P implies Q, and Q is false, then P must be false. Example: If "It is raining" implies "The ground is wet", and we observe "The ground is not wet", then we can conclude "It is not raining". 5. Conjunction and Simplification Conjunction allows us to combine two true statements into one compound statement. Simplification allows us to extract individual parts from a conjunction.

This is the written version of the interactive lesson above. See the full Mathematical Logic course.