Vacuous Truth and Triangular Falsehood Quiz

Answer: True

The premise ("this picture shows a triangle") is false so the statement is vacuously true. It doesn't matter that the conclusion is false.

Answer: False

The premise is true but the conclusion is false, making the statement untrue.

Of the four possibilities for an "if [premise] then [conclusion]" statement, a true premise and false conclusion is the only way that the statement is "False". In Boolean logic, this might be represented with a "truth table" of possible "True"/"False" values for the premise, conclusion and overall statement.

Answer: True

The premise is false so the statement is vacuously true - we don't even need to read the conclusion (which happens to be true).

Answer: True

If we selected "False" then the premise is true but the conclusion is false, making the statement false and the answer a contradiction. However, when we select "True", the premise is false, making the statement vacuously true.

It is not a paradox, as it is not the case that both choices ("True" and "False") lead to a contradiction - only "False" does.

Answer: True

Since no question contains three white circles, the statement is vacuously true. All of the zero questions that meet the premise (containing three white circles) meet the conclusion (answer "True").

If this question had three white circles in its picture, we would get an unusual choice: either click "True", making the statement true, or click "False", making the statement false. This is not a paradox as there is no contradiction, just ambiguity as to which answer is correct, but it would break the quiz.

Answer: False

Even though there are no circles, this is not phrased as a conditional that is vacuously true.

Instead, we count the number of circles - zero - and the number of triangles - two - and observe that double zero is not two.

Answer: True

We can answer this question without knowing whether there are five "True" answers or not. If there are five "True" answers then (given the ten total answers of either "True" or "False") that makes five "False" answers, so the statement is true. If there are not five "True" answers then the statement is vacuously true.

In fact it is necessary to answer this question before we know whether there are five "True" answers - which there turn out to be. Question 10 (and in turn question 9) rely on knowing that there are at least five "True" answers in the quiz, including this one.

Answer: False

The premise refers to question 2, which has a "False" answer, but its picture is of a white triangle. A true premise and false conclusion makes the overall statement false.

Answer: False

We have to answer question 10 first, as a "True" answer there would immediately make this statement true.

Question 10 has the answer "False" and the most recent true answer is to question 7. So, the last "True" answer is either question 7 or question 9 (if we select "True" here). Either way, the question number is odd, making this statement false (and the latest "True" answer match up with question 7).

Answer: False

With ambiguity at this point over whether question 9 and 10 have the answer "True", we can say that there are five, six or seven "True" answers in the quiz. There are only four triangles, so we can be confident this answer is "False".

In turn this lets us answer question 9 as "False" and observe that the five "True" answers just barely exceed the four total triangles.