An expanded high school logic unit
Hey you wanna be a total pal and outline (as briefly as you need to) how you handle a logic unit? It’s the end of the year and I’ve been kind of bouncing around between argument and vocabulary without an exit strategy. Any help would be great.
Sure, and you (or anyone else), feel free to ask me to expand on my sometimes cryptic shorthand. What you do depends on how far you want to go, and in what direction.
(the good stuff’s beneath the fold –>)
- symbolize statements (not objects)
- negation
- and, or, if… then…
- logical equivalence (aka biconditional); operation or relation?
- truth tables (intro, for each operation)
- more complicated truth tables (and more complicated, etc)
- using truth tables to show logical equivalence: material implication
- Venn diagrams (demorgans laws)
- Euler diagrams (conditional and contrapositive; converse and inverse)
- Rules of inference: (these go by various names, so I will list them by what they do)
- p -> q; p : q
- p -> q; ~q: ~p
- p v q; ~p: q
- p -> q; q -> r: p -> r
- p; q: p & q
- p: p v q
- p&q: p
- Deductive proofs (symbolized, and from English)
- Indirect proofs
- Testing for validity
(I do a bit more, those are the most of the biggies)
I don’t handle all of these in exactly this way. You could pick your way through what’s interesting. You might want a whole lot of translation from whole language stuff (negation of English statements is tricky). Or you might want to get to Venn Diagrams or Euler Diagrams or Truth Tables, and just stop there and up the level of challenge. (Kids like that place). Or you might race to the rules of inference, and give them things like:
Jack likes Jill. Jill likes Steven or Jill likes Jack. Candy likes Bob and Jack. It is not true that Steven likes Candy and Jill. If Steven doesn’t like Candy, then Candy doesn’t like Bob. Jill likes Steven or Steven likes Jill. What are the chances of Jack and Jill hooking up?
I’ve done different things at different times. These days, with the luxury of a full term course, and 5 three-day weeks for this unit, and having already well-established what validity is in whole language context, I race through the start, hit tables and both sorts of diagrams, and get to proofs kind of fast, dwelling on both direct and indirect, as well as validity testing.
JD2718 
Nice! Thanks a mil for the speedy response. Planning just got a lot easier this weekend.
Uf. I just did some Venn diagramming and I’m pretty sure I should’ve scaffolded the worksheet a little better. If you got a worksheet in soft copy, I’d be obliged if you could send it my way.
I am not quite certain what scaffolding means. And I have no work sheets this year (so far). How’s this:
Over the last week the kids learned to make truth tables. Yesterday I had them make a table for (~p v ~q) = ~(p & q) (I should latex better looking symbols, but you get it…) Then I offered them a diagram with a universe and “a” and “b” (overlapping circles) inside it. You know the picture. Together we built, by shading, a, then ~b. We moved on to a & b, then a v b… It was helpful that, when originally discussing negation, disjunction, and conjunction, that we had linked them to complement, union, and intersection, respectively.
Finished by asking them to create diagrams for a, b, ~a, ~b, a v b, ~(a v b), ~a v ~b, ~(~a v ~b), a & b, ~(a & b), ~a & ~b, ~(~a & ~b). They started in class, will complete them for homework, and Tuesday we will be ready to prepare some rules of inference and replacement for proofs.
Tuesday we will take DeMorgan’s Laws from their diagrams (they already that we are headed there). We already took “material implication”, a->b = a v ~b from a truth table. Before we started Venn Diagrams we took “hypothetical syllogism” (in NY they call this the chain rule), ie, [(a->b) & (b->c)]->(a->c) from an Euler Diagram (essentially, concentric circles).
With the term rapidly drawing to a close, and with all sorts of missed days coming up, I am in a hurry to get them to a few proofs and some testing for validity. Spring is in the air, and senior attendance has become a bit, er, spotty, so I really need to move fast now.
This feels good. Quick, but good. I started with my Geometry students by defining some sets like A = { freshmen in this class }, B = { girls in this class }, C = { students whose first name starts with B } and then running through some intersections / unions / complements. Pretty sure that neglected some of the theory necessary to move into DeMorgan’s Laws. Since we just started, I may backtrack. This is a better blueprint for next year in any case, so thanks.
Homework due today was a pile of Venn Diagrams. On the AM quiz one question was to show that ~a & ~b and ~(a & b) were not the same thing. I got 50% truth tables, 50% Venn Diagrams, and they looked good.
Today I moved into a few valid forms / rules of inference. I presented
a -> b / a / therefore b and
a -> b / ~b / therefore ~a and the invalid forms
a -> b / b and
a -> b / ~a
The students used Euler Diagrams to prove Modus Ponens and Modus Tollens valid, and to prove the other two invalid. And then we wrote a few simple proofs.
The year is almost up, but I desperately want to reach indirect proofs. Last year I heard “so why didn’t you show it to us this way in Geometry?” Of course they are mathematically more mature, but I hope that other kids get that ‘second bite of the apple’ feeling.
Proofs by contradictions in curriculum feels little funny to me. When I teach my students they tend to like them to the point that they don’t want to use anything else. But from high school geometry teachers I hear that they hardly ever teach it. Or should I say they spend a tiny bit of time on it. They say the problem is that proofs by contradiction need to be written in paragraph format. Some of them said that’s too hard for students. Other said that they don’t accept anything that’s not in two column form. I know you don’t like to teach geometry. But sometimes you do teach it. Why don’t you teach proof by contradiction in geometry? Also, I still have to harp on this because I love geometry :) I do think that I asked several times what it is about teaching geometry that you don’t like. I keep looking for an answer, thinking that maybe I had forgotten it, but can’t find it. I then looked through your mathematics tagged posts, and virtually none of them are geometry. Maybe it’s not just teaching it, maybe the problem is geometry.
Damn. I spoke too soon. I just found a whole bunch of geometry puzzles. Sorry! I wish I could buy some patience. My yoga instructor said we’d work on it, but I don’t think we’re very successful.
In my school we teach proof by contradiction in geometry. Nothing special there, it works well in the inequality chapter, already in the textbook. However, we take pains to introduce the concept in algebra, a year earlier. Not that the kids do proofs by contradiction at that stage, but they do have some exposure.
So now I have seniors (and a few juniors) in logic who are getting a second bite at the apple. They will, generally, do better this time than they did 2 or 3 years ago.
I’m about to teach an intro to philosophy class in my high school for the first time. Of course, I want to start with some serious logic, but don’t have any resources beyond my college notes. Any suggestions on free stuff or stuff I could reasonably afford myself (since it’s too late for the school to get anything for me)?
Do you have an outline?
I started with a textbook (for logic, not philosophy), and stuck fairly close to it, at least my first time through. However, my principal asked for a course description, and instead of reinventing the wheel, I googled for course outlines from the equivalent, college course. Some of them were quite detailed, some came close to what I was doing, and I meshed them into a nice description of my course.
What I am suggesting is that you review the syllabi and descriptions for intro philosophy classes from a few colleges. I’m sure you will be able to google up plenty. Some will also include readings, questions, etc. The only thing that you won’t find are high school-like activities, if you like to teach with such things. You could also of course go with straight lecture or with discussion, in which case you’ll probably be able to find all you need.
By the way, who will your students be? Motivated, ready to read lots?
Thanks for your suggestion! I’ll keet at it. I think my students will be motivated in class time, as this is a first time elective many were happy to get into. But as for independent reading, I don’t think so….
Hey, this is my attempt at solving the Jack likes Jill puzzle. I would love some feedback.
Jack likes Jill is represented as JkJl, and so on.
Jack likes Jill. Jill likes Steven or Jill likes Jack. Candy likes Bob and Jack. It is not true that Steven likes Candy and Jill. If Steven doesn’t like Candy, then Candy doesn’t like Bob. Jill likes Steven or Steven likes Jill. What are the chances of Jack and Jill hooking up?
1. JkJl= A
2. JlSt V JlJk C V B
3. CnBb & CnJk D & E
4. -(StCn & StJl) -(F & L)
5. [[-StCn V –StJl]] -F V -L (by equivalence)
6. -StCn –> -CnBb -F –>-D
7. JlSt V StJl C V L
6 and 3 by modus tollens: conclusion: F
F and 5 by disjunctive syllogism: concl: -L
-L and 7 by disjunctive syllogism: concl: C
Jill likes Stan but Stan doesn’t like Jill. So Stan’s not a viable distraction.
Since both B and –B are possible, and you need A & B for them to hook up, the chances are 50%.
What do you think? Did I miss something here?
I threw that out when I wrote the post as an example of what a problem would look like. I had never worked it out. The logic part of what you’ve done is good. But there are two points:
Stan (Steve?) — we’ll just call him St — “St would not be a viable distraction” involves some interpretation. I don’t know the rules of this soap opera, and would prefer not to guess at them.
More clearly, your conclusion should be “possible” rather than 50%.
Actually, maybe we could take your “distraction” idea, make some “pairing up” rules, and have a nice problem set. Tell you what, if I produce something, I’ll post it. Thanks.
thanks for your feedback
one change I will make to this problem for my students is to change the second sentence to “Steven likes Jill or Jill likes Jack” That way the chance they’d hook up is 100%, since Jack likes Jill and Jill likes Jack.
truth table for: p v(~q ^r)
False for
pqr
FTT
FTF
FFF
true for all others.
Do you know how the rules work?
no
then me giving you answers (which I did) probably won’t help – if you can read what I wrote.
Why don’t we back up. Is there a simpler problem you can do? Can you write the truth table for ~p v q ?
thanks
we were given this for homework and told to put it in colmn form, but we were not told how to do it. The book is not much help.
The wikipedia article is not a bad place to start.
I appreciate your help, thank you.