Geometry / Logic

[Porridge]

Question for those who have some knowledge of mathematical logic, bonus if you have some familiarity with Aristotelian logic.

DS is using Derek Owens for Geometry this year. DO uses Jacob’s Geometry.

We’re both a bit perplexed / annoyed at the sections about syllogisms and logic. I know they’re trying to introduce basic logic to the students, but the way it is presented is confusing. They present syllogisms in multiple different ways (categorical, hypothetical)  without explaining the differences between the types. And they draw Euler diagrams in a way that would work for a special type of syllogism called a sorites (I know this term only because I’m going through Kreeft’s Socratic Logic with DD), but doesn’t really work for categorical syllogisms. 

I’m sure this is a nit-picky quibble that’s more noticeable to me because I’m going through logic with DD right now. But even DS, who isn’t doing Socratic Logic, pointed out that the course  presents syllogisms in different formats as if they are interchangeable, which is confusing.

I never studied formal logic in the context of Geometry or math, so perhaps the logic sections presented in Jacobs (by DO) are perfectly valid in the mathematical logic world.  Perhaps our confusion comes from trying to reconcile two different systems of logic that use similar terminology but in different ways and with different conventions?  Or are DO and Jacobs just legitimately vague and confusing?  Is there a mathematical logic for dummies primer that anyone can point me to so I can try to reconcile these things in my mind?

It’s a minor thing, and I don’t have a lot of time to sink into it right now, but if there is an easy resource, I’d like to take a look at it.

[cintinative]

Does DO's geometry actually use the text? Because most of his other math classes have their own text "Based on" some original text, but there could be differences.

If the class doesn't use the text, I would take a look at this portion in the original Jacob's text (you can use archive.org for this) and see how it is presented there.

We used Jurgenson's and I don't remember if logic was covered or not.  If it was, we didn't do it, I don't think. 

You could change the title of this thread to Jacobs Geometry and Logic and perhaps that would get more informed opinions than mine.  😃

[daijobu]

Could you post some photos or screenshots from the portions of the textbook that illustrate the issue?  Or reproduce the statements here?  

[idnib]

2 hours ago, cintinative said:

We used Jurgenson's and I don't remember if logic was covered or not.  If it was, we didn't do it, I don't think. 

 

Jurgensen's Chapter 2 is Induction and Deduction. Topics include: Meaning of Induction and Intuition, Perpendicular Lines and Circles, Spheres, Deductive Reasoning, Deduction and Logic, Conditionals (If-Ten), The Law of Detachment, and Converses, Inverses, and Contrapositives.

 

Not sure if this is helpful, but I had the book right next to me.

[cintinative]

15 minutes ago, idnib said:

Jurgensen's Chapter 2 is Induction and Deduction. Topics include: Meaning of Induction and Intuition, Perpendicular Lines and Circles, Spheres, Deductive Reasoning, Deduction and Logic, Conditionals (If-Ten), The Law of Detachment, and Converses, Inverses, and Contrapositives.

 

Not sure if this is helpful, but I had the book right next to me.

Clearly I don't remember that! LOL. Thank you. Is symbolic logic covered? I think that is what the OP is referring to.

[lmrich]

3 hours ago, cintinative said:

Does DO's geometry actually use the text? Because most of his other math classes have their own text "Based on" some original text, but there could be differences.

If the class doesn't use the text, I would take a look at this portion in the original Jacob's text (you can use archive.org for this) and see how it is presented there.

We used Jurgenson's and I don't remember if logic was covered or not.  If it was, we didn't do it, I don't think. 

You could change the title of this thread to Jacobs Geometry and Logic and perhaps that would get more informed opinions than mine.  😃

DO uses the textbook - he hands out in his in-person classes.  There are assignments in the textbook. 

[Porridge]

DO does use the Jacobs textbook for this course. We are only on chapter 2. He has the student read the text first, then watch the videos which so far pretty much repeat the text. 

Edited September 5, 2023 by Porridge

[Porridge]

Sounds like Jacobs doesn’t go into as much logic as Jurgensen. 
 

The table of contents shows 6 logic topics in chapter 2

 conditional statements

 definitions

 direct proof

 indirect proof 

deductive system

 some famous theorems of geometry

[idnib]

3 hours ago, cintinative said:

Clearly I don't remember that! LOL. Thank you. Is symbolic logic covered? I think that is what the OP is referring to.

At a pretty basic level, along with truth tables.

[Porridge]

Sounds like symbolic logic is what I need to look into.

[cintinative]

11 hours ago, Porridge said:

Sounds like symbolic logic is what I need to look into.

Here is a textbook we used after we covered some of the Aristotelian logic. It's free online.  http://home.iitk.ac.in/~avrs/PH142/Books/Patrick2012.pdf

[daijobu]

The photos you have posted seem okay to me from a mathematical perspective.  A statement is equivalent to its contrapositive.  A statement is not necessarily equivalent to its converse.  

I prefer to read the statement   as "p implies q" or "if p then q."  

The example I learned in high school is to let

p = "you are human"

q = "you have a heart"

The statement  reads "If you are human then you have a heart."  

The converse,  reads "If you have a heart then you are human" and is false because dogs and other animals also have hearts.  

However the contrapositive  reads "If you do not have a heart then you are not human" is true.

I had not seen those Venn style diagrams in the margins before, but they make sense to me.  

Sometimes it is easier to evaluate whether a statement is true if you can rewrite it in terms of variables as in the AMC problem here:

https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_11

I'm not familiar with how Aristotle handles logic.  What is his approach to these sorts of statements?  

Edited September 6, 2023 by daijobu

[daijobu]

It also makes sense to me that a definition would have a double arrow.  Triangle and 3-sided polygon are the same thing, so one implies the other and vice versa.  

What specifically is your objection?  

[daijobu]

Here are some more fun logic problem for practice if your students are having difficulty:

https://artofproblemsolving.com/wiki/index.php/1974_AHSME_Problems/Problem_13

https://artofproblemsolving.com/wiki/index.php/1986_AJHSME_Problems/Problem_22

https://artofproblemsolving.com/wiki/index.php/1985_AJHSME_Problems/Problem_25

[daijobu]

For fun I looked up some of my own logic exams from high school.  Note the handwritten and barely legible purple mimeographed print straight from the 1980s!  And yes, students who scored >90% earned a smiley face on their papers.  In high school.  I'm not ashamed to admit that I lived for those smiley faces.  

  

[Porridge]

On 9/6/2023 at 9:53 AM, daijobu said:

It also makes sense to me that a definition would have a double arrow.  Triangle and 3-sided polygon are the same thing, so one implies the other and vice versa.  

What specifically is your objection?  

Apologies for the delay in responding -  it ended up being a very busy week!

No specific objection - more confusion. Seeing your old papers was super helpful (it looks like it was a great class!) It made me realize that this course (unlike yours) doesn't go into as much depth with regard to logic as it could. They're just giving the bare bones of logic as a prelude to geometry. I do think we were confused in part because we're trying to reconcile two different systems of logic that use similar terminology but in slightly different ways and with slightly different conventions. I don't think the systems contradict - I think we just need to study them a bit more, but Jacobs and DO don't go into very much depth.

Ds was confused because in DO's notes, he presents syllogisms in more than one format - in one place as a hypothetical (if, then), later as a categorical (all x is y, etc.) In one place he throws in an argument with more than 2 premises in the middle of a page where he's talking about syllogisms (which can only have 2 premises). But I think we've sorted it out.  DS intuitively understands the concepts - for instance, that the converse of a statement is not necessarily equivalent, but that the contrapositive (DO doesn't go into this level of depth) is valid. 

 

I loved smileys in red pen, too 🙂

 

ETA

The more I think about it, I really think DO and Jacobs don't need to introduce the idea of a syllogism at all. Very few Geometry proofs fit into the format of a syllogism. Throwing in the term is confusing if it's not going to be used in any practical way in the course. One of the homework probs was to write a syllogism - Ds wrote one that was logically sound but didn't fit the proper form. The form wasn't taught, but DS was docked points until we pointed out that the form hadn't been taught. I think they want the students to get the idea that the predicate of one premise can serve as the subject of another, and that you can then logically connect a chain of ideas to arrive at a conclusion. They also want the kids to examine whether the converse of a statement can be true.  You can introduce those ideas without the concept of a syllogism.

This is a super minor quibble. I will probably take DS through a formal logic course sometime in high school. DD and I have gotten a lot out of Socratic Logic. I may encourage her to take a symbolic logic course as well.

Edited September 12, 2023 by Porridge

[Not_a_Number]

I've definitely seen geometry classes be used as a crash course on logic . . . basically, because you need logic for proofs, and geometry is often where proofs are introduced, and a surprising number of people really can't see that "A implies B" is not the same as "B implies A" and all sorts of other nonsense. 

In my experience, you don't need former logical training. But it's useful to think through the concepts. And I find that seeing the concepts in concrete situations is more helpful than anything else. 

On a totally unrelated note, I've been teaching younger kids logic in my math class by playing Minesweeper 😂. It's surprising what a good logic workout that game is if you make kids explain WHY they think certain squares are safe and certain square are not. A lot of the times, kids come up with something like "There's a 1 here and a 2 here so it's safe!" As you can see, making their logic make more sense takes some work . . .