Logical Implication: The Quartets Task

• An introduction to representing logical statements with Euler diagrams.
• The definition of a conditional statement.

Productive engagement with the quartets task should provide students with opportunities to:

When we launch this task (see video below), we first highlight some fundamental questions that we'd like to determine the answers to. The questions closely align with the goals of the task. Here is what we display for students:

1. How does an Euler diagram depicting P and Q inform us about whether the implication P → Q is true? Can the same Euler diagram be used to determine the validity of other statements of P and Q?
2. When is an implication true? When is it false? How do we know?
3. What is the negation of an implication?
4. If we know the truth value of an implication, what conclusions can we draw about the validity of other, related statements?

These questions prime students to look for logical ways of reasoning that apply to all implications of the form P → Q regardless of their specific mathematical content. While the task begins by asking students to investigate relationships between specific, familiar statements within the same quartet--and hence mathematical context--students are eventually invited to look for logical similarities across quartets/contexts.

Inviting students to first play with logical implications involving familiar objects (e.g. "If a quadrilateral is a square, then it is a rectangle") allows students to leverage their existing ways of reasoning before transitioning to the abstract setting. Displaying the fundamental questions for students in advance of the task alerts them they should eventually be looking for general logical conclusions they can draw about all statements of the form P → Q.