When we think about various ways we can define words like rectangle and trapazoid, we in logic and math sometimes use language differently than we do when we are being ordinary humans. I have found some of the work following up on what H.P. Grice called conversational implicature helpful.
As a logician, I happily say that all four-side triangles are purple because I know there is no counterexample, no four-sided triangle that is not purple. As an ordinary person, I think that anyone who talks about four-sided triangles knowing that there are none is a blithering idiot.
Logicians in ancient Greece worried about this problem, calling it existential import. I am having a senior moment and can?t remember who it was who looked at this problem in the light of Grice, but I found the view very helpful.
If I tell you that Bob broke his finger but know he was seriously injured in a car crash, then I have not lied. But since I could easily have said he was seriously injured, I have violated the expectation that, in a good faith conversation, I will be as informative as possible and useful in the number of words I am using,
If I see a square and tell you it is a rectangle, I violate this principle. But in a proof I may want to apply a theorem about rectangles to squares. So I stay a logician, but am aware that lay people may find my way of using language a little unnatural sometimes.
And I am wondering how we should help preschool children come to be comfortable with the idea that a square is a rectangle.