Rules of Logical Inference

Core Idea

Rules of inference are patterns of reasoning that guarantee a true conclusion when applied to true premises. They preserve truth: if the premises are true, the conclusion must be true. These rules form the foundation of valid deductive proof.

How It's Best Learned

Learn a few key rules deeply (modus ponens, modus tollens, hypothetical syllogism) rather than trying to memorize many. Understand why each rule works using truth tables or logical intuition.

Common Misconceptions

• Confusing valid reasoning with true conclusions (validity requires true premises).
• Applying rules backwards (e.g., affirming the consequent).
• Assuming common-sense reasoning is logically valid.

Explainer

A rule of inference is a template for constructing valid arguments. You know from your study of conditional statements that "P → Q" means "if P then Q," and you know what it means for a statement to be a tautology — true under every truth assignment. Rules of inference are exactly those argument forms whose logical structure guarantees truth-preservation: whenever the premises are true, the conclusion must be true. That guarantee is what makes a proof valid, as opposed to merely persuasive.

The most important rule is modus ponens: from P and P → Q, conclude Q. In natural language: "It is raining" and "If it rains, the ground gets wet" together imply "The ground gets wet." Verify this with a truth table — the only row where both premises are true forces Q to be true. Modus tollens runs the same conditional in reverse: from ¬Q and P → Q, conclude ¬P. If the ground is dry, and rain would wet it, then it hasn't rained. Notice what is not valid: from Q and P → Q, you cannot conclude P. That error is called affirming the consequent — it is the classic confusion between "if P then Q" and "if Q then P."

Hypothetical syllogism chains conditionals: from P → Q and Q → R, conclude P → R. This is transitivity of implication, and it is what allows multi-step proofs — each step passes the truth forward until you reach the desired conclusion. Disjunctive syllogism handles or-statements: from P ∨ Q and ¬P, conclude Q. If at least one of P, Q is true and P is false, Q must be true. This is the logical engine behind proof by cases and proof by elimination.

The overarching principle is validity vs. truth. An argument is valid if the conclusion follows necessarily from the premises — regardless of whether the premises are actually true. "All fish can fly; salmon are fish; therefore salmon can fly" is valid (the structure is correct) but unsound (a premise is false). In mathematics, where the axioms are stipulated to be true, validity and soundness coincide — but the distinction reminds you to check both the logical structure of your proof and the truth of your starting assumptions. Rules of inference give you the structural half; your domain knowledge gives you the truth half.