1
Validity is a property of form
The argument is valid when (premises) → conclusion is a tautology. What the propositions mean never enters.
Lesson · Logic and Proof · Proof
Rules of Inference
An argument is valid when its premises cannot all be true with its conclusion false. Modus ponens, modus tollens, syllogism, resolution — and the two invalid forms that look identical until you check.
modus ponens p → q, p ⊢ q VALID
modus tollens p → q, ¬q ⊢ ¬p VALID
affirming the consequent p → q, q ⊢ p INVALID
denying the antecedent p → q, ¬p ⊢ ¬q INVALID
Two valid, two not
Core ideas
What this lesson establishes
Validity is about form, not content. An argument can have false premises and still be valid, and true premises with a true conclusion and still be invalid.
2
Sound is valid plus true premises
A valid argument from false premises proves nothing about the world; it guarantees only that the conclusion follows.
3
The two fallacies are conditional confusions
Affirming the consequent uses the converse; denying the antecedent uses the inverse. “The Conditional and Its Relatives” already showed neither is equivalent to the original.
The material
Valid inference
The definition, the standard rules, and how to check any candidate rule yourself.
Definition · Valid argument
An argument with premises P₁, …, Pₙ and conclusion C is valid when (P₁ ∧ … ∧ Pₙ) → C is a tautology — equivalently, when no assignment makes every premise true and the conclusion false.
Definition · Sound argument
An argument is sound when it is valid and its premises are actually true. Validity is a matter of logic; soundness adds a claim about the world.
The standard rules
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
Each of these is a tautology when written as a conditional, and each can be checked with a truth table in under a minute. They are worth knowing by name because a proof that cites them is a proof someone else can follow.
Resolution deserves a note: it is the single rule automated theorem provers and SAT solvers are built on. Given a formula in CNF, repeatedly resolving clauses either derives the empty clause — proving unsatisfiability — or exhausts the possibilities.
The two fallacies
Theorem · Affirming the consequent is invalid
From p → q and q, nothing follows about p. Take p = F, q = T: both premises hold and the conclusion p is false.
Theorem · Denying the antecedent is invalid
From p → q and ¬p, nothing follows about q. Take p = F, q = T: both premises hold and the conclusion ¬q is false.
Example · Both fallacies in one sentence
"If it rains, the ground is wet." The ground is wet — so it rained? No: a sprinkler wets the ground. It did not rain — so the ground is dry? No: same sprinkler. Both errors come from treating the conditional as a biconditional.
Rules for quantifiers
universal instantiation ∀x P(x) ⊢ P(c) any c in the domain
existential instantiation ∃x P(x) ⊢ P(c) for a NEW name c
universal generalisation P(c) for arbitrary c ⊢ ∀x P(x)
existential generalisation P(c) ⊢ ∃x P(x)
The two with conditions attached are where care is needed. Existential instantiation may only introduce a fresh name — you know something exists, not which thing it is, so it must not be confused with anything already named. Universal generalisation requires that c was genuinely arbitrary: nothing was assumed about it.
Interactive
Truth table
Every assignment, evaluated
Method
Checking an argument
Form first. The content is a distraction until the form is settled.
1
Strip it to symbols
Assign letters to the atomic propositions and write the premises and conclusion. Content that feels persuasive is the enemy here.
2
Try to make premises true and conclusion false
That single assignment, if it exists, proves invalidity. If it cannot exist, the argument is valid.
3
Name the rule, or the fallacy
If it matches a standard form, say which. If it matches a fallacy, say which — and produce the counterexample assignment.
4
Separate validity from truth
A valid argument with a false premise establishes nothing. Say which of the two you are claiming.
Valid or fallacious
Score 0 / 0Choose the answer you can justify.
Is this argument valid?
"If the program has a bug, the test fails. The test failed. Therefore the program has a bug."
p = the program has a bug q = the test fails
premises: p → q, q
conclusion: p
Try p = F, q = T:
p → q = F → T = T premise 1 holds
q = T premise 2 holds
p = F conclusion FAILS
INVALID — affirming the consequent.
The counterexample is not artificial: a test can fail because the test itself is wrong. The valid inference available here is modus tollens — the test passed, therefore no bug of the kind that test detects.
Common mistakes
1
Confusing validity with truth"That conclusion is false, so the argument is invalid" does not follow. A valid argument can carry a false premise to a false conclusion.
2
Reusing a name in existential instantiationFrom ∃x P(x) and ∃x Q(x) you may not conclude ∃x (P(x) ∧ Q(x)). The two witnesses need not be the same object, so each needs a fresh name.
3
Generalising from a non-arbitrary elementUniversal generalisation needs c about which nothing was assumed. If the argument used a property of c, the conclusion does not extend to all x.
Check your understanding
✓
Finish when you can name the fallacy AND produce its counterexample assignment.
Given any two-premise argument about a conditional, you should be able to say in one line whether it is modus ponens, modus tollens, or one of the two fallacies — and for the fallacies, state the row that breaks it.
Note: The four quantifier rules are stated here without full justification because the conditions on them (fresh names, arbitrary elements) are what a proof must respect. “Direct Proof”, “Contraposition and Contradiction” and “Cases, Counterexamples, and Writing a Proof” use them in prose, which is how they appear in practice.