What Are You Trying to Infer? – Part 2

Here is the solution to the puzzle.   The premises and rules of inference used at each step are provided after each conclusion.  The conclusions are numbered 1-4.   How did you do?

For those who would like a more detailed explanation for why the converse and inverse forms are invalid, I have provided truth tables illustrating how both forms allow true premises to lead to false conclusions.

Puzzle
You are about to leave for school in the morning and discover that you don’t have your glasses. You know the following statements are true:

a)      If my glasses are on the kitchen table, then I saw them at breakfast.
b)      I was reading the newspaper in the living room or I was reading the newspaper in the kitchen.
c)       If I was reading the newspaper in the living room, then my glasses are on the coffee table.
d)      I did not see my glasses at breakfast.
e)      If I was reading my book in bed, then my glasses are on the bed table.
f)       If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table.

Where are my glasses?

Solution

  1. The glasses are not on the kitchen table.              (a,d, and modus tollens)
  2. I did not read the newspaper in the kitchen.       (f, 1, and modus tollens)
  3. I read the newspaper in the living room.               (b, 2, and elimination)
  4. My glasses are on the coffee table.                         (c, 3, and modus ponens)


A little more about validity…

Explaining validity requires a truth table.   Let’s look at the truth table for the converse and inverse argument forms.

Converse Error Form

If P, Then Q         (Premise 1)
Q.                            (Premise 2)
Therefore, P.     (Conclusion)

Premise 1 Premise 2 Conclusion
P       Q If  P, Then Q Q Therefore, P
1 T       T T T T
2 T       F F F T
3 F       T T T F
4 F        F T F F

An argument form is valid if it always produces a True conclusion for every row in the truth table for which all premises are True.   In this table, both premises are True in rows 1 and 3.  However, notice that in row 3, P (the conclusion) is actually false.  Therefore, this truth table demonstrates that when both premises are true, the converse argument form may yield a false conclusion. This means the converse form is not a valid argument form.

Inverse Error Form

If P, Then Q        (Premise 1)
~ P.                        (Premise 2)   (the symbol ~ means NOT)
Therefore, ~Q.  (Conclusion)

Premise 1 Premise 2 Conclusion
P     Q    ~P     ~Q If  P, Then Q ~ P Therefore, ~ Q
1 T     T      F         F T F F
2 T     F      F         T F F T
3 F     T      T         F T T F
4 F     F      T         T T T T

In this table, both premises are True in rows 3 and 4.  However, notice that in row 3, ~Q (NOT Q), the conclusion,  is actually false.  Therefore, this truth table demonstrates that when both premises are true, the inverse argument form may yield a false conclusion. Thus, like the converse form, the inverse form is not a valid argument form.

Now that you’ve had a brief course on valid argument forms and rules of inference, try putting them to use. You will find that logic errors occur in discussions far more often than one would expect. Remember: When presented with an argument, verify the form and check the facts.

 

Facebooktwitterpinterestlinkedinmail

Leave a Reply

Your email address will not be published. Required fields are marked *