What Are You Trying to Infer?

I have the unfortunate habit of reading comments on news sites.   Unlike the articles that precede them, comments are a free-for-all where logic rarely makes an appearance.   Interestingly, it really doesn’t matter whether the article is celebrity gossip or a research report, logic loses.   It’s easy to understand why. Logic is not a required subject for most college majors; therefore, most people wing it.

In logic, an argument is a sequence of statements that leads to a conclusion1–not what ensues from discussing whether or not Alabama should have played LSU for the BCS Championship. A valid argument is one in which the conclusion follows necessarily from the preceding statements.  That is, if the premises are true, a valid argument form will yield a true conclusion.

Validity can be determined using a truth table,  and it addresses only the form of an argument. It does not address the content of the statements that make up the argument.   In other words, validity cannot be used to determine if the statements (i.e., premises and conclusions) that make up an argument are factual. Validity only asserts that, if the statements are factual, then the conclusion drawn from them should be true.  Bottom line: When presented with an argument, verify its form and check the facts.

Arguments consist of one or more premises followed by a conclusion, and have the general form:

Premise 1
Premise 2
Premise n…

Below are inference rules1 that, if followed, assure that an argument’s form is valid.

Modus Ponens
Everyone uses modus ponens, though few know its formal name.    In Latin, It means “method of affirming.”   It has the form:

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

If it is raining, Then I wear boots.
It is raining.
Therefore, I wear boots.

Modus Tollens
Modus tollens means “method of denying” in Latin and has the form:

If P, Then Q
Therefore, NOT P

If it is raining, Then I wear boots.
I am NOT wearing boots.
Therefore, it is NOT raining.

Modus ponens and modus tollens are valid argument forms.  However, there are two argument forms, inverse and converse, that should be avoided because they do not reliably yield valid arguments.  Here are examples of inverse and converse forms. Avoid them!

Inverse Error Form

If P, Then Q
Therefore, NOT Q

If it is raining, Then I wear boots.
It is NOT raining.
Therefore, I am NOT wearing boots.

This conclusion is suspect because it goes beyond the limits of the initial premise, which states that boots are worn EVERY time it rains.   We have no information on what happens when it is NOT raining.  For example, there may be other reasons for wearing boots — when washing a car, fishing, or cleaning out a flooded basement.

Converse Error Form

If P, Then Q
Q is true
Therefore, P

If it is raining, Then I wear boots.
I am wearing boots.
Therefore, it is raining.

Like the inverse conclusion, the converse  conclusion is suspect because it assumes that there are no other reasons for wearing boots besides rain.   From the initial premise, all we know  is that EVERY time it rains, boots are worn.  Concluding that boots are worn ONLY when it rains is, at best, speculation.  The initial premise does not justify the converse conclusion.   Converse and inverse errors are easy to make and easy to swallow.  Watch out for them.

Two additional useful inference rules are elimination and transitivity.

Elimination is very familiar.  Given two choices, either one, but not both, may apply.

Therefore, P

I will travel to Washington by plane OR by car.
My car needs repairs.
Therefore, I will travel by plane.

Transitivity relies on the relationship among a series of implications.

If P, Then Q; If Q, Then R; If P, Then R.

If it rains, Then I wear boots.  If I wear boots, Then I take an umbrella.   Therefore, if it rains, Then I take an umbrella.

Now that you have a basic knowledge of inference rules, try them on this puzzle. See if you can determine which rules are used at each step as you solve it.

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?


1.            Epp SS.  Discrete Mathematics with Applications: Third Edition. Belmont: Brooks/Cole; 2004. All definitions, as well as, the puzzle are taken from the text.    Used by permission.


The solution, along with truth tables illustrating how one determines if an argument form is valid, will appear in the Wednesday post.



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