Logic and Conditional Sentences

Date: 10/04/2005 at 03:36:36
From: Cintia
Subject: How to determine is a conditional sentence is truth

I have a question about conditional statements.  I am having a hard
time understanding why two false statements in a conditional makes it
true.

I tried to use different statements to create a truth table but I get 
stuck on the same concept.  I tried a sentence like "If a polygon is a 
square, then the sides are equal."  If I assume a rectangle, then it 
seems to me that the statement is undefined.  Being true or false does 
not even apply.

Date: 10/04/2005 at 15:31:39
From: Doctor Achilles
Subject: Re: How to determine is a conditional sentence is truth

Hi Cintia,

Thanks for writing to Dr. Math.

This is the most difficult part of basic logic, and this exact 
question gave me headaches when I was first learning logic.

The first thing to remember is that logic has only two possible 
answers: true or false.

Let's take the following sentences and say that all of them are true:

  1) I am hungry

  2) I will eat soon

  3) I am not thirsty

  4) I will not drink soon

We can make a few conditional statements out of these (and some
related sentences):

  A) If I am hungry, then I will eat soon.

This sentence has the form "if TRUE, then TRUE".  I think we can agree
that a sentence of this form is TRUE.

  B) If I am hungry, then I will drink soon.

This has the form "if TRUE, then FALSE".  The fact that I am hungry
does not cause me to drink.  Furthermore, I am in fact hungry, and I
won't be drinking anything anytime soon.  Hopefully we can agree that
conditionals like this are FALSE.

  C) If I am thirsty, then I will eat soon.

This has the form "if FALSE, then TRUE".  This gets a little
difficult.  However, the fact of the matter is I will eat soon.  And
even if I were thirsty (in addition to being hungry), I would still
eat soon.  Therefore, the statement "if I were thirsty, I would eat
soon" is still TRUE.

  D) If I am thirsty, then I will drink soon.

This has the form "if FALSE, then FALSE".  Here is where things can
get downright difficult.  I am not thirsty.  However, if I were, then
I would go get something to drink.  So it is true that if I were
thirsty, I would drink soon.  So the conditional is TRUE.

3 of the 4 types of condionals came out TRUE:

  "If TRUE, then TRUE" comes out TRUE
  "If FALSE, then TRUE" comes out TRUE
  "If FALSE, then FALSE" comes out TRUE

The only conditional that comes out false is:

  "If TRUE, then FALSE" comes out FALSE

The reason that a conditional always comes out true where the first
part (the antecedent) is false is because you haven't disproven it.

You can think of it sort of like a burden of proof.  I could say
something like, "If I were 7 feet tall, I would be a professional
basketball player".  Because I am not 7 feet tall, you cannot prove
that that conditional is false.

There are some difficulties with this system.  For example, the
following two conditionals are both true:

  "If I were 7 feet tall, I would be a professional basketball player"

  "If I were 7 feet tall, I would not be a professional basketball       
   player"

Some logicians have attempted to create different types of logic where
the only way a conditional is true is if it is of the form

  "If TRUE, then TRUE"

Still, the only way for it to be false is for it to have the form:

  "If TRUE, then FALSE"

The other two forms (where the antecedent is false) are assigned some
other value, such as "untestable", or "N/A".

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

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