The valid patterns of reasoning that proofs are built from, each with its form, an everyday example and a truth table check, followed by the fallacies that look like them and a proof written step by step.
An argument is a list of statements, the premises, followed by a conclusion, marked with ∴ ("therefore"). It is valid when the conclusion is true in every case where all the premises are true. Validity is about form, not content: replace the letters with any statements you like and a valid form stays valid.
A rule of inference is a short valid form that is worth a name. Each one below was checked by listing every row of its truth table when this page was built. Written as a single statement, each is also a tautology: modus ponens becomes (p → q) ∧ p → q, true in every row.
| Rule | Premises | Conclusion |
|---|---|---|
| Modus ponens | p → q, p | ∴ q |
| Modus tollens | p → q, ¬q | ∴ ¬p |
| Hypothetical syllogism | p → q, q → r | ∴ p → r |
| 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 |
| Constructive dilemma | p → q, r → s, p ∨ r | ∴ q ∨ s |
| Destructive dilemma | p → q, r → s, ¬q ∨ ¬s | ∴ ¬p ∨ ¬r |
Also called affirming the antecedent, →-elimination
From a conditional and its antecedent, conclude its consequent.
✓ Valid: checked on all 4 rows. See the truth table
Also called denying the consequent
From a conditional and the negation of its consequent, conclude the negation of its antecedent.
✓ Valid: checked on all 4 rows. See the truth table
Also called chain rule, transitivity
Two conditionals that link up chain into one.
✓ Valid: checked on all 8 rows. See the truth table
Also called elimination
Rule out one side of an OR and the other side must hold.
✓ Valid: checked on all 4 rows. See the truth table
Also called disjunction introduction
Anything true stays true with an OR attached, whatever the other side says.
✓ Valid: checked on all 4 rows. See the truth table
Also called conjunction elimination
If both parts of an AND are true, each part is true on its own.
✓ Valid: checked on all 4 rows. See the truth table
Also called conjunction introduction
Two statements that are each true can be joined with AND.
✓ Valid: checked on all 4 rows. See the truth table
Two ORs that disagree about p cancel it out. This one rule drives automated theorem provers and SAT solvers.
✓ Valid: checked on all 8 rows. See the truth table
Two conditionals and an OR of their antecedents give an OR of their consequents.
✓ Valid: checked on all 16 rows. See the truth table
Modus tollens applied to two conditionals at once.
✓ Valid: checked on all 16 rows. See the truth table
A formal fallacy is an argument form that looks like a rule but is not valid: some row makes every premise true and the conclusion false. One such row, a counterexample, is enough to sink it.
Often mistaken for modus ponens
The grass could be wet from a sprinkler: q can be true for reasons other than p.
✗ Invalid: counterexample p false, q true. See the truth table
Often mistaken for modus tollens
The conditional says nothing about what happens when p is false.
✗ Invalid: counterexample p false, q true. See the truth table
Often mistaken for disjunctive syllogism
The OR of logic is inclusive, so both sides can be true.
✗ Invalid: counterexample p true, q true. See the truth table
Every rule above as premises and a conclusion, with the three fallacies marked invalid, on one sheet for printing or revision.
Click to download: Rules of inference chart
A truth table can check any argument, but it doubles in size with every letter. A proof reaches the conclusion in a few lines instead: list the premises, then add one line at a time, each following from earlier lines by a rule of inference or by replacing part of a line with something logically equivalent. The proof ends when the conclusion appears.
Take this argument:
t: the train is running; b: I take the bus; s: it is snowing; l: I am late; c: I call ahead.
| Line | Statement | Justification |
|---|---|---|
| 1 | t ∨ b | Premise |
| 2 | s ∧ ¬t | Premise |
| 3 | b → l | Premise |
| 4 | l → c | Premise |
| 5 | ¬t ∧ s | Commutative law, 2 |
| 6 | ¬t | Simplification, 5 |
| 7 | b | Disjunctive syllogism, 1, 6 |
| 8 | b → c | Hypothetical syllogism, 3, 4 |
| 9 | c | Modus ponens, 8, 7 |
| 10 | s | Simplification, 2 |
| 11 | c ∧ s | Conjunction, 9, 10 |
Line 11 is the conclusion, so the argument is valid. Every step was checked when this page was built: the lines it cites entail it, and it has the exact shape of the rule it names. Line 5 uses the commutative law to swap the sides of ∧, since simplification here keeps the left part. The full truth table agrees, over all 32 rows.
Some habits that make proofs easier to find:
Resolution deserves a word of its own. Rewrite every premise and the negation of the conclusion as ORs of letters and negated letters (conjunctive normal form), then resolve pairs of clauses until the empty clause, a contradiction, appears. That one rule is complete for proving that a set of clauses is unsatisfiable, which is why it, and descendants of it, run inside theorem provers and the SAT solvers used to verify chips.
Rules of inference are valid argument forms: patterns such as "p → q and p, therefore q" whose conclusion is true whenever all the premises are true, whatever statements the letters stand for. A proof is a chain of steps, each justified by one of these rules applied to earlier lines.
Both start from a conditional p → q. Modus ponens adds p and concludes q: it affirms the antecedent. Modus tollens adds ¬q and concludes ¬p: it denies the consequent. They are the two valid ways to use a conditional; the two mirror images, affirming the consequent and denying the antecedent, are fallacies.
Modus ponens is Latin for "the way that affirms", short for modus ponendo ponens, "the way that affirms by affirming". Modus tollens is "the way that denies", from modus tollendo tollens. The names describe what the second premise does to the conditional.
Check its truth table: in every row where all the premises are true, the conclusion must be true too. Equivalently, (premises joined with ∧) → conclusion must be a tautology. Every rule on this page was checked that way when the page was built, and the calculator will check any argument you type.
A rule of inference works in one direction on whole lines: from p ∧ q you may infer p, but not the reverse. A logical equivalence works in both directions and may replace any part of a statement, because both sides always have the same truth value. Proofs use both.
No. Validity is about form: if the premises are true, the conclusion must be. "If the moon is cheese, it is edible; the moon is cheese; so it is edible" is valid modus ponens with a false premise. An argument that is valid and has true premises is called sound, and its conclusion is true.