Logical Implication: The Quartets Task
This task is based on the following research article:
Hub, A., & Dawkins, P. C. (2018). On the construction of set-based meanings for the truth of mathematical conditionals. The Journal of Mathematical Behavior, 50, 90-102.
- Analyzed the logic of statements involving the connectives "and" and "or," as well as determined their negations.
- 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:
- Practice drawing Euler diagrams relating the truth sets of P and Q.
- Determine the validity of familiar logical implications and provide counterexamples when they are false.
- Use their conclusions about familiar logical implications to explore, in general, when a statement of the form P→Q is true and when it is false.
- Explore spatial relationships between the truth sets of P and Q when an implication is true and when it is false. (e.g. a true implication requires the truth set of P to be a subset of the truth set of Q.)
- Abstract the logical structure of a logical implication and its converse, contrapositive, and inverse by noticing the four statements within each quartet all follow the same logical form.
- Conjecture that a logical implication is equivalent to its contrapositive.
- Use Euler diagrams to explain contrapositive equivalence.
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:
- 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?
- When is an implication true? When is it false? How do we know?
- What is the negation of an implication?
- 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.
As you browse the videos below, reflect on the following:
- How students are reasoning with Euler diagrams.
- How might Euler diagrams support students who treat logical implication as an action (like Mary in Video 3) to abstract logic across quartets?
- How might your perspectives about instruction on logical implication be changing?
Student 1 (00:00): [The truth set] of Q would completely exist within the truth set of P or if P, then Q, cuz like I feel like Q is inside completely of P, so there's no outside, there's no exception. The truth set of Q completely exists within the truth set of P.
Student 2 (00:14): Yeah.
Student 3 (00:15): Okay. So then I guess that's what she's saying up there, where it's like, if not Q, then
Student 4 (00:21): So if it doesn't exist within the truth set
Student 3 (00:23): (That doesn't make sense to me)
Student 4 (00:23): of, wait, sorry. You said Q is in P, right?
Student 1 (00:29): If the statement is, if P, then Q.
Student 2 (00:32): So if it doesn't exist,
Student 1 (00:32): Then the truth set of Q completely exists in the truth set of P.
Student 4 (00:37): So if it doesn't exist in P...
Student 2 (00:38): It's not in Q.
Student 4 (00:38): then it's definitely not in Q.
Student 1 (00:40): Yeah, yeah, yeah. So I think that's correct.
Student 4 (00:44): The first one and the third one, I guess we use that for those.
Student 3 (00:49): So up there, that doesn't really make sense to me. If you're using the circle, so if it's like,
Student 1 (00:53): Oh, because
Student 3 (00:54): (I think it's if not P then not Q)
Student 1 (00:54): up there they switched Q and P
Student 3 (00:56): Yeah
Student 1 (00:56): so I think that'd be wrong.
Student 3 (00:59): I think that would be wrong too.
Student 1 (01:00): I think it's supposed to be like
Student 3 (01:01): Ohhh
Student 2 (01:03): So if we're saying it's a multiple of three, but non a multiple of six
Student 1 (01:12): Up there, it's saying
Student 4 (01:13): It's sayign if it's uh a number is a multiple of six,
Student 1 (01:16): Yeah
Student 4 (01:16): then it's a multiple of three. You said that's true, right? That's within here.
Student 3 (01:22): Yeah.
Student 1 (01:22): The one on the board says if it's not a multiple of six, then it's not a multiple of three.
Speaker 2 (00:00): So like even for the false ones. So A1, it would be like if a number is a multiple of three, then it is a multiple of six. But if we're drawing our Euler diagram like this, right? That would just be something in this region, right?
Speaker 3 (00:11): Mmhmm
Speaker 1 (00:11): Mmhmm
Speaker 3 (00:13): So and that kind of shows that if a number is a multiple of three, it doesn't necessarily have to be a multiple of six. There's a lot else that it can be a multiple of.
Speaker 1 (00:20): So that's the counterexamples it would be something in that area.
Speaker 3 (00:23): Yeah.
Speaker 2 (00:24): Oh
Speaker 3 (00:24): That's what I think
Speaker 2 (00:25): Oh yeah yeah
Speaker 3 (00:25): and then for the true statement, if a number is a multiple of six, then it is a multiple of three. That's true. Because every single number, that's a multiple of six, it's in the multiple of three circle.
Speaker 4 (00:36): Should we also ... should we also draw the negated sets? I think not
Speaker 3 (00:42): That's what I was...
Speaker 4 (00:44): Yes, that is what you were saying.
Speaker 3 (00:45): Yeah
Speaker 4 (00:45): Yeah okay
Speaker 3 (00:47): Or like the false sets, maybe not the negations
Speaker 4 (00:49): Yeah
Speaker 1 (00:51): But didn't she say like use the... like draw the Euler diagram or draw the Euler diagram and then use it to like verify if like we got it right?
Speaker 4 (01:01): Yeah.
Speaker 3 (01:01): Yeah. So if we go to three, if a number is not a multiple of six, then it is not a multiple of three. That's true. Cuz then we could just be on the outside. Right?
Speaker 1 (01:11): Wait, which one?
Speaker 4 (01:12): No that's not true
Speaker 2 (01:12): For A3?
Speaker 1 (01:13): If it's not a multiple of three, if it's not a multiple of... What does it start with?
Speaker 3 (01:18): If a number is not,
Speaker 4 (01:20): Are we talking about A3?
Speaker 3 (01:20): yeah, A3. If a number is not a multiple of six
Speaker 4 (01:23): It could still be in three though.
Speaker 3 (01:25): That's true. So maybe that one's true then.
Speaker 4 (01:29): But it could also be outside
Speaker 3 (01:31): It could also be outside
Student 1 (00:00): Compare and contrast your conclusions across each quartet.
Mary (00:03): They're the same.
Student 2 (00:04): They're the same pattern.
Student 1 (00:05): False, true, false, true, false, true, false.
Student 2 (00:07): Yeah. Both of them they're
Student 1 (00:08): Sorry. False true false true...
Student 3 (00:11): Why don't we start talking about rectangles and squares to switch it up a little spice into it?
Mary (00:17): Add a little chicken spice
Student 2 (00:18): So we know that all across the board, basically we have like two things. The two true statements and the two false statements are basically having the same truth sets.
Student 1 (00:33): Yeah
Student 2 (00:34): And that's just the same across the board for all of the quar quartet, right? Because comparing across all of them.
Student 1 (00:42): So A and C had essentially the same Euler diagram.
Student 2 (00:46): Yeah.
Mary (00:47): Yeah
Student 1 (00:47): But B had a completely different Euler diagram, but still somehow got false. True false truth
Mary (00:53): because it was still asking whether or not, so if a triangle is not acute, then it is obtuse. We know that's not true because obtuse and acute don't fall into a "nested egg."
Student 2 (01:03): Yeah
Mary (01:03): They fall into two parts. (or different parts)
Student 2 (01:05): Yeah but even though they are separate, they are still similar in the sense like B and C.
Student 1 (00:00): Oh, oh, here's one conclusion that I might make. The first one, one of the truth sets was nested. One of the spaces, multiples of six was nested within one of the other sets
Student 2 (00:13): The multiples of three.
Student 1 (00:15): Here, neither was nested and there was actually no overlap, so we could never tell that either was anything.
Student 2 (00:23): Mhmm
Student 1 (00:23): That's why they were all false. Then for the last one, for the [muffled] was all quadrilaterals. There was some nesting, again, in that squares have to be rectangles. So
Student 3 (00:38): You could say, well, I think in all of 'em, if I'm not going to word this right, but it's like if P implies Q, then like then ... then... not Q implies not P.
Student 1 (00:55): No, no, that's correct. Yeah. I'm just saying from a spatial understanding
Student 3 (00:59): Yeah, yeah. I'm just trying like the relationship between that
Student 1 (01:03): No no no that sounds right.
Student 2 (01:04): If we think about the Euler diagram, that's basically the same area outside the larger circle, like that doesn't include the area inside the smaller circle.
Student 3 (01:13): Yeah
Student 2 (01:13): I don't know, maybe that was more of a clunky way of saying it
Student 1 (01:15): No, no, that's an excellent way to say that because if you think about it, the space outside the smaller circle can still include space in the
Student 2 (01:23): Yeah
Student 1 (01:23): bigger circle, but the space outside the bigger circle, the one in which the smaller circle is nested
Student 2 (01:29): Yeah
Student 1 (01:29): Cannot, by definition, include any space of the inner circle.
This is a big task for students to take on. Rather than waiting until the end to have whole-class discussion, you may find it more productive to have periodic check-ins with everyone. We share below a few videos from our check-ins.
Video 1: After students have a few minutes to work, Rachel checks in with the class to see how they're drawing their Euler diagrams. This is a great way to:
- Make sure that students who are struggling with Euler diagrams can continue to make progress on the task.
- Prompt students to reflect on whether they need to draw a different Euler diagram for each statement within a quartet. Can all four statements be depicted on the same diagram?
Video 2: Rachel checks in to see what conclusions students were drawing across quartets. One student conjectures contrapositive equivalence. To offer other students a chance to make sense of why this conjecture must be true, Rachel sends students back into groups.
Video 3: After students discuss in groups why P implies Q is equivalent to ~Q implies ~P, Rachel checks back in to see how they are making sense of it. A student explains the relationship using Euler diagrams.
After this task, students are ready to formally summarize and prove their ideas that this activity brought forth. Here is a list of some of the things students might do next:
- Draw an Euler diagram depicting the universally true implication P→Q (for arbitrary statements P and Q).
- Determine the precise conditions for when a logical implication is false and state its negation.
- Explore the vacuous case to determine the validity of P→Q in all cases (via the Law of Excluded Middle). Note: this is in contrast to traditional instruction where the truth table of an implication is given to the students when conditional statements are first defined.
- Define logical equivalence.
- Use Euler diagrams to determine whether two statements are logically equivalent and conjecture general methods for proving two statements are logically equivalent.
- Prove an implication is not logically equivalent to its converse, but is logically equivalent to its contrapositive.
- Define biconditional statements and explore why they are mathematically useful (e.g. The Invertible Matrix Theorem).