Inductive and Deductive Examples

Date: 11/21/2001 at 12:39:48
From: Mehrdad Khademi
Subject: LOGIC 7th grade: Is this question in logic valid?

Hi Dr. Math,

I have a math background but had only one course in Logic as an 
undergrad a long time ago. My son's Life sciences teacher gave them a 
question in Logic, and I am pretty certain that:
 a) The statement is not a sound argument, and
 b) as a result of (a) it cannot be proved inductively or deductively.

The question was:
Read the following situation and then show how you would solve the 
problem with deduction and then with induction.

Bob wants to figure out what his teacher wants for his birthday, but 
he cannot ask his teacher directly. How does he pick the perfect 
present?


I told my son that the statement is not a sound logical argument, 
because:

  a) The conclusion of the argument that he wants to prove is not an 
     objective statement (e.g., an opinion, recommendation, etc.).

  b) Even if the statement were a valid argument, it would be a "non 
     sequitur" (arguments whose conclusion cannot follow from the 
     premise(s)). If we take there to be either 2 simple premises: 
    [1] Bob wants to figure out what his teacher wants for his 
        birthday and 
    [2] He cannot ask his teacher directly 
    (or take them as one combined premise), then the conclusion will 
    not follow (and is not objective) from the premise(s). 

My son got full credit for his absurd answer to the question  
(certainly not logical statements). But I don't want him to learn 
logic incorrectly. I would like to point this out to his teacher and 
provide him with a few simple situations that do make sense for 7th 
graders as sound logical arguments.

First Request: I appreciate your comments/modifications about my 
remarks.

Second Request: Could you please provide me with few English 
statements that the teacher can use in his class to introduce his 
students to logic and to demonstate the use of deductive and inductive 
reasoning (proofs) for these English statements (not puzzles or 
mathematical statements, since I do have plenty of those myself).

Thanks a lot,
Mehrdad (Parent of a 7th grader)

Date: 11/30/2001 at 14:30:17
From: Doctor Achilles
Subject: Re: LOGIC 7th grade: Is this question in logic valid?

Hi Mehrdad,

Thanks for writing to Dr. Math.

This is an interesting question that required some thinking.

You are certainly right that the question does not, by itself, give 
enough information to either induce or deduce an answer. However, I 
think that it does provide an avenue for learning about those 
concepts.

Let me first start by proposing a few other (fairly simple) example 
variations on the question that I feel would demonstrate induction and 
deduction (I changed the teacher to a female for ease in pronoun use):

-------Start of Deduction variant -----------

Read the following situation and then show how you would solve the 
problem with deduction.

Bob wants to figure out what his teacher wants for her birthday, but 
he cannot ask his teacher directly. How does he pick the perfect 
present?

These are the things Bob knows:

 1) She has a "Far Side" calender that she always laughs at. (premise)
 2) She drinks coffee every morning from a plain mug. (premise)
 3) She owns no decorated mug. (premise)
 4) All people who have "Far Side" calendars like any other "Far Side"  
    items. (premise)
 5) All people who drink coffee like to drink from decorated mugs.  
    (premise)
 6) All people who do not have something they like would like to get 
    that thing. (premise)
 7) The perfect present to get someone is something that that person 
    would like. (premise)

-------End of deduction variant-----

(It is left to the student to fill out the appropriate conclusions 
from these premises).

A similar induction variant can be created with appropriate inductive 
premises.

The way the question as presented differs from my variant is that the 
student is left to come up with his/her own premises AND then make a 
deductive conclusion from those premises. I believe that if a student 
is able to generate premises from which one could validly infer a 
statement of the form:

  "The perfect birthday present for Bob's teacher is x."

where x is some potential present, AND THEN that student were able to 
go through the premises and make the proper valid inferences to lead 
up to the valid conclusion, that student will have learned about 
deduction and demonstrated his/her knowledge of the subject.

The way that these should be graded, in my opinion, is:

  a) Did the student set up premises that allowed for a valid  
     inference to be made?

  b) Did the student clearly show the steps in making that valid 
     inference?

A similar grading criterion can be used for the inductive variant.  
I.e. did the student start from a collection of facts to induce a 
conclusion? Was the inductive reasoning explained well, etc.

One other note: I have a fair amount of practice doing logic, so my 
premises were many and elaborate. Someone just learning may not have 
as many or as elaborate premises (although it is still important that 
the inference be valid). I don't know what your son turned in, so I'm 
not sure what logical holes, if any, it might have had.

If you would still like to find examples of logical statments, I'd 
suggest you search our archives at:

   http://www.mathforum.com/mathgrepform.html   

or look at our Logic area in the archives:

   http://mathforum.org/dr.math/tocs/logic.high.html   

If that's not helpful, you can write back and I'll try to come up with 
some for you.

I hope this has been helpful for you.  If you'd like to discuss this 
issue some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/02/2001 at 15:28:25
From: Mehrdad Khademi
Subject: Re: LOGIC 7th grade: Is this question in logic valid?

Hello Dr. Achilles,

Thank you so much for your compehensive answer. I don't think most
7th graders will come up with the premises that you came up with.
(Actually I know none of them did, but most of them got full credit!)
I came up with a conclusion to your variant statement, which I wanted
to run by you. I will refer to your statements as P1, P2,..,P7.

from P1 to P7 one can infer:
The perfect gift for Bob's teacher is either a "decorated mug" or a
"Far Side" item. This is a logical inference and the conclusion is:
Bob should pick either a decorated mug or a far side item for his
teacher's birthday. I think both the "logical inference" and the 
conclusion are necessary. Do you agree with me or not?

2) On the inductive version of your variant. One trivial way is to
change all the "ALL"s in your premises P4, P5 and P6 to something
like "MOST." Is there a better way to do it?

3) I went to the sites that you recommended and could not find
any other English example (I don't mean puzzles). If you can forward
me some more English examples I will much appreciate it.

4) In your opinion is it a good idea to make some universal 
assumptions as accepted truth (So that 7th graders do not have to come 
up with a lot of premises) and use those, and a "lax" form of rules of 
inference to come up with questions that are closer to a sound 
argument than the question the teacher gave them? An example of this 
could be:

Given:
Defn   D1: Good grade means a grade above 3.75 (on a 4.0 scale).
Defn   D2: A good college is one of the top 100 colleges in the USA.
Axiom A1: Studying hard produces good grades.
Axiom A2: A college degree with a good grade (an honor degree) will 
produce a successful person.

Show how you would solve this problem using "deduction":
If you study hard, you will be successful.
I will use Ij for inference #j. Note that I1 is not a sound argument 
and I may need your help on this.
  I1: For Good grades mean that you will get into a good college.
  I2: By axiom A1 you will get good grades in a good college.
  I3: For good grades mean you will get an honor's degree
  from a good college (I know this is neither an axiom nor a
  valid inference).
  I4: By Axiom A2 An honor's degree implies you'll be a
  successful person.

5) I think the question above is perhaps better than something
   like the following, which is a valid logical inference. But please
   note that to a 7th grader it is like proving the "obvious."

Use deduction to show that:
Given premises:
P1 : No women are men  P2: Some doctors are women
Deduce "Some doctors are not men".

I1:  All women are nonmen.       Applying Obversion to P1.
I2:  Some doctors are nonmen     Follows from P2 and I1
I3:  Some doctors are not men.   Apply Obversion to I2

Thanks a lot for and looking forward to your response.
Mehrdad Khademi

Date: 12/03/2001 at 16:45:56
From: Doctor Achilles
Subject: Re: LOGIC 7th grade: Is this question in logic valid?

Hi again Mehrdad,

Thanks for writing back.

I should note that a simpler deduction could be created. Just a simple 
syllogism:

  A)  For all people who have a desk, a pen is the perfect gift.
  B)  Bob's teacher has a desk
  etc.

1) You actually spotted a logical hole in my premises. I should've 
added:

  There exist mugs decorated with "Far Side" cartoons.

2) Changing the "all"s to "most" would make it inductive. Another 
thing is just to say something like:

   Bob has given all of his teachers apples for Christmas, and each 
   time it has been the perfect gift.
   Induce from there that that will hold true for this teacher as 
   well.

3) (See below)

4) I chose to use universals in my premises because they lend 
themselves much more easily to elementary deductions.  I think that 
they are good ways to show students just learning about logic the idea 
of deduction. They also make very clear the idea that in a valid 
deduction it is IMPOSSIBLE for the premises to all be true and the 
conclusion to be false.

Generalizations (e.g. most, some) are perhaps better for induction.  
However, it is possible to deduce conclusions from non-universal 
statements (almost always generalized, or non-universal, conclusions).

I think your grades example needs another two axioms:

  A3: Studying hard will get you into a good college.
  A4: Good grades will get you an honors degree.

Then, however, the inferences are perhaps too easy.

5) This brings up an interesting question about teaching introductory 
logic to 7th graders. Most of the examples I had in mind would seem 
like "proving the obvious" to a 7th grader. There are really a couple 
of issues here:

First, in 7th grade, one is usually only given examples that take one 
or two steps.

Second, deductive logic is designed so that each individual step is 
always obvious.

From that you can easily deduce that:

  7th grade logic examples will appear to be obvious.

There are, however, some classic rules and logic examples that are 
easy for 7th graders (most appear obvious), but they are the 
foundation for more involved logic.

Finally, I owe you a few examples (the first few are obvious, the 
later ones I hope are not quite so):

I) The syllogism

  A) All men are mortal. (major or universal premise)
  B) Socrates is a man. (minor or particular premise)
  C) Socrates is mortal. (conclusion)

II) Modus ponens (this seems a lot like a syllogism, but the logical 
forms are different)

  A) If it is raining, then it is wet.
  B) It is raining.
  C) It is wet.

III) Modus tollens (this one is less obvious, and it is a good 
exercise to have someone explain WHY it is a valid inference)

  A) If it is raining, then it is wet.
  B) It is not wet.
  C) It is not raining.

IV) Fallacy of affirming the concequent (this is INVALID; having 
someone explain why is a good exercise)

  A) If it is raining, then it is wet.
  B) It is wet.
  C) It is raining.

V) DeMorgan's rule's (this one is tricky to do formally, but an 
informal (i.e. in prose) proof would show a good grasp of mid high-
school level logic):

  Given: notA and notB
  Prove: not(A or B)

  Given: not(A or B)
  Prove: notA and notB

  Given: notA or notB
  Prove: not(A and B)

  Given: not(A and B)
  Prove: notA or notB

VI) Learning how to deduce from an "or" statement (here and elsewhere 
I use an "inclusive or" which means "A or B" is true if A is true or 
if B is true or if both are true).

  Given: (P and Q) or (Q and R)
  Prove: Q

Rule: Given an "or", assume the left and derive something.
      Then, assume the right and derive THAT SAME THING AGAIN.
      Then, you are entitled to keep what you got from both sides.

Steps to take:

  a) P and Q  (assumption)
  b) Q        (simplification of a)
  c) Q and R  (assumption)
  d) Q        (simplification of b)
  e) Q        (for keeps, now that we've proved it from both halves of 
               the "or")

Figuring out why this is valid shows what I would call "mastery" of 
logic for even a high-schooler.

As far as induction goes:

Good examples can be found in any classic science conclusion. Take any 
relation F = ma, PV = constant, etc. from classic science and start 
with observations such as:

   Force          mass             acceleration
    1               1                  1
    2               2                  1
    2               1                  2
etc.

or:

    Pressure     Volume (of a gas)
       1            1
       2            0.5
       3            0.33333
etc.

I hope all of this is helpful to you. If you'd like to talk about it 
some more, write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/