A valid argument can still have a false conclusion. Validity is about the shape, not the facts.
Judge the argument
"If it rains, the game is cancelled. The game was cancelled. Therefore it rained." Is this valid?
Invalid. This is the fallacy of affirming the consequent: from p → q and q, nothing follows about p. The game might have been cancelled for a typhoon warning, a power cut or an unplayable pitch. The original promised only that rain would cause a cancellation, never that rain was the only possible cause.
Only the contrapositive is guaranteed to match the original's truth. The converse and the inverse can each be false while the original is true.
original: square → four sides
true
converse: four sides → square
FALSE: it could be a rectangle
inverse: not a square → not four sides
FALSE: a rectangle again
contrapositive: not four sides → not a square
true, as it must be
A single rectangle breaks both the converse and the inverse. That is no coincidence: the converse and the inverse are contrapositives of each other, so they always share a fate.
"If it rains the game is cancelled. It rained. So the game was cancelled." — modus ponens. "…The game went ahead. So it did not rain." — modus tollens. Both watertight.
"If it rains the game is cancelled" does not say rain is the only reason for a cancellation. Reasoning from a cancellation back to rain relies on that extra claim — and it was never made.
Denying the antecedent: "If a number ends in 0, it is divisible by 5. 35 does not end in 0. So 35 is not divisible by 5."
p: ends in 0, q: divisible by 5
name the parts
given p → q and ¬p, concluded ¬q
denying the antecedent
invalid: 35 = 5 × 7
a counterexample
The conditional says nothing about numbers that do not end in 0, so no conclusion about them follows.
A chain: "If I finish my homework, I watch TV. If I watch TV, I sleep late. So if I finish my homework, I sleep late."
p → q and q → r
p: homework, q: TV, r: sleep late
therefore p → r
the links join end to end
valid
a hypothetical syllogism
The middle statement q must be the conclusion of one premise and the hypothesis of the other; then it drops out.
An "or" with one side ruled out: "The answer is 12 or 15. It is not 12. So it is 15."
p ∨ q and ¬p
p: it is 12, q: it is 15
therefore q
the only option left
valid
a disjunctive syllogism
Ruling one side out forces the other. Knowing one side is TRUE, however, tells us nothing about the other, because "or" is inclusive.
p → q where p: divisible by 6, q: even
name the parts
given: p → q and q
affirming the consequent
invalid, and 14 proves it
even but not divisible by 6
Validity and truth are separate questions. An argument can be valid with a false conclusion, if a premise was false. And it can reach a true conclusion invalidly, by luck. Only validity is about the shape, and only the shape is what logic checks.
Periodical exams ask for the converse, inverse or contrapositive of a statement, the name of an argument form, or whether a short argument is valid; entrance tests favour the two fallacies dressed in everyday words. Name p and q, write the argument in symbols, and match it to a form. For a 10-second check, look for one situation where every premise is true and the conclusion false — if you find one, the argument is invalid.
Judge the argument
"If a student passes, she gets a certificate. Maria did not get a certificate. Therefore Maria did not pass." Is this valid?
The form is p → q with ¬q, concluding ¬p — modus tollens, which is valid. A would need the second premise to be p, that Maria passed. The giveaway is that the second premise negates the CONCLUSION of the conditional, and reasoning backwards from a denied conclusion is always safe.
Hint: write the conditional as p → q, then ask whether the second premise is p, q, ¬p or ¬q. Only p and ¬q lead anywhere.
