Rules of Inference: Valid Logical Arguments Complete Guide

A rule of inference is a guaranteed-valid pattern for building arguments: if the premises match the pattern, the conclusion must be true. Modus ponens, for instance, says that from "if p then q" and "p" you may conclude "q".

Rules of inference are the legal moves of proof: a proof is a chain of these steps, each applying a rule to established facts. Learning them by name lets you recognize valid reasoning and spot the famous invalid patterns.

Key Concepts

• Argument: a sequence of premises followed by a conclusion, written premises ∴ conclusion.
• Valid argument: it is impossible for all premises to be true while the conclusion is false.
• Modus ponens: p → q, p ∴ q.
• Modus tollens: p → q, ¬q ∴ ¬p.
• Hypothetical syllogism: p → q, q → r ∴ p → r (implications chain together).
• Disjunctive syllogism: p ∨ q, ¬p ∴ q.
• Addition: p ∴ p ∨ q. Simplification: p ∧ q ∴ p. Conjunction: p, q ∴ p ∧ q.
• Resolution: p ∨ q, ¬p ∨ r ∴ q ∨ r.

How It Works

Apply a rule of inference in three steps:

1. Match your known facts to the premises of a rule, substituting your actual statements for p, q, r.
2. Check that the match is exact: modus ponens needs the same p in both "p → q" and "p".
3. Write down the licensed conclusion; it is now established and can feed the next step.

Chain rules together and you get a proof: each line is either a premise, an assumption, or the result of a rule applied to earlier lines.

Worked Example

Show that "If it rains, the ground gets wet; the ground is not wet; therefore it did not rain" is valid. With p = "it rains" and q = "the ground gets wet", the premises are p → q and ¬q, the conclusion ¬p: exactly modus tollens. Verify by truth table:

• Row (T, T): ¬q = F, so a premise fails.
• Row (T, F): p → q = F, so a premise fails.
• Row (F, T): p → q = T, ¬q = F. Premise fails.
• Row (F, F): both premises true, and ¬p = T.

The only row with both premises true also has a true conclusion, so the argument is valid.

Common Misconceptions

• Affirming the consequent is valid: from p → q and q, concluding p is a fallacy. Wet ground does not prove rain; the row (p false, q true) has true premises and a false conclusion.
• Denying the antecedent is valid: from p → q and ¬p, concluding ¬q is a fallacy. No rain does not guarantee dry ground.
• "Valid" means "the conclusion is true." Validity is about the form: true premises force a true conclusion. An argument can be valid with false premises, and a true conclusion does not make a bad argument good.

Real-World Applications

• Mathematical proofs: every proof step is modus ponens, modus tollens or another rule in disguise.
• Debugging: "if the bug were in module A, the test would fail; the test passes; so the bug is not in module A" is modus tollens.
• Automated reasoning: theorem provers apply these rules mechanically to derive conclusions from knowledge bases.

Rules of inference are the verified legal moves of reasoning: modus ponens, modus tollens, hypothetical and disjunctive syllogism, addition, simplification, conjunction and resolution. Match the pattern exactly and never fall for affirming the consequent or denying the antecedent.