A False Statement Implies Any Statement

Date: 09/25/97 at 20:16:57
From: Kemp
Subject: Logic

Dear Dr.Math,

My name is Megan and I'm 15 yrs old. I'm doing Logic in math right
now, and were learning about Quantifiers and Laws of Inference. I was
wondering if you knew any ways to make learning these and remembering
them easier? If you have any suggestions, I'd be very grateful.

Thank you,
Megan

Date: 09/26/97 at 13:47:34
From: Doctor Mike
Subject: Re: Logic

Dear Megan,  
   
I'll send you the e-mail I sent to my own daughter who is doing 
similar stuff in a college course, and asked me to explain why a False 
statement implies Any statement. This will not cover all you ask 
about, but typically this idea often puzzles people for a while. If 
you still want to know more after this, please write back with a 
question that is more specific, so we will know better what we are
trying to answer. Or, you could check out other books in the library, 
and compare explanations. So, here goes :  
    
"... In case you still are wondering about why a False implies
anything, try this explanation on for size.  It may help. It's just a 
little thing .... just HALF of a truth table.
   
      P  |  Q  |  If P, then Q   (same as P --> Q)
 -------------------------------------------------
         |     |           
      F  |  T  |    T
         |     |           
      F  |  F  |    T      
         |     |           
   
I call it HALF of a truth table because it only has the "F" cases 
for P.  An example of such a false P sentence is "The moon is made of 
green cheese." The truth table just says that if the first part is 
something like that, it doesn't matter what the second part is.  Here 
are two examples that illustrate the two rows of that truth table.
   
   IF the moon is made of green cheese, then you will find
   a $20 bill on the sidewalk sometime this week.
   
   IF the moon is made of green cheese, then you will NOT find
   a $20 bill on the sidewalk sometime this week.
   
I capitalized IF because it is a "big IF"; the first part of those two 
sentences is not going to happen.  To prove either of those two 
sentences false in a court of law, you would have to convince the jury 
about the green cheese, and only then consider the second part. You 
will have to actually wait out the entire next week to find out about 
what really happens concerning the twenty dollars. BUT, no matter what 
you happen to find on the sidewalk, both of those above sentences are 
True. They are True, because they didn't really promise anything, and 
they didn't promise anything because the P-part is false. Think about 
it. It may take some time to sink in."

   
As for the "Quantifiers" part of your question, notice the similarity 
to the word "Quantity," which means "how many."  Logic is not usually 
interested in specific quantities like "113" of "several dozen". The 
important quantities to concentrate on are ALL and SOME : 
    
1. ALL, which usually involves sentences starting out like
   "For all" or "For every," or which could be written that
   way. Consider the sentence "Every police officer wears 
   a badge."  You could also say, "For every police officer,
   that officer wears a badge." In symbols, that could come
   out "For every police officer P, B(P)" where I am using 
   B(P) to stand for P wears a badge. By the way, the
   sentence we are talking about is probably false, because
   of undercover detectives and the like.  
        
   It's sort of a predictable thing that whenever a
   mathematician or logician is turned loose without much
   supervision, very soon some symbolism is introduced (grin).
   Like, how many times have you heard a math teacher start
   working on a word problem by saying "Let X be the unknown"!
   
2. SOME, which usually involves sentences starting out like
   "For some" or "There exists a," or which could be written that
   way. Consider the sentence "Some police officer enjoys 
   listening to Mozart." You could also say, "There exists a
   police officer, such that that officer enjoys Mozart."  In
   symbols again, "There exists a police officer P such that M(P)"
   where I am using M(P) to stand for P enjoys Mozart.    
    
The ALL-type phrases are called Universal Quantifiers because they are 
claiming something is true for all things in a certain collection (or 
universe) of things, as all the police in Brooklyn, or all flute 
players studying at the Eastman School of Music.  
   
The SOME-type phrases are called Existential Quantifiers because they 
are claiming that something exists within a group, such as a course in 
Japanese among the offerings at your local High School.
   
One more thing. When you negate (say "It is false that ...) any 
Universally quantified sentence, you get an Existentially quantified
sentence, and vice versa. An example. Saying that "All police
wear badges" is False, is the same as saying "Some police do NOT
wear badges" is True.   
   
I hope this helps.    
    
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    

Date: 09/08/2004 at 14:41:21
From: Jay
Subject: Why is "false implies true/false" always "true"?

Why are the logical statements "false implies true" and "false 
implies false" always considered "true"?

I've read the previous note, but could you please give a more 
"formal" explanation?

Thank you,

Jay

Date: 09/08/2004 at 15:54:44
From: Doctor Schwa
Subject: Re: Why is "false implies true/false" always "true"?

Hi Jay,

Great question!  More formally, I'd say "implies" means the same
as "subset" in set theory.  That is, when you say
 
  if it rains, then the ground gets wet

you mean 

  the set of times when it rains is a subset of
  the set of times when the ground gets wet.

So, since the empty set is a subset of any set, a false statement
implies any statement.

I hope that helps clear things up!

- Doctor Schwa, The Math Forum
  Check out our web site!  http://mathforum.org/dr.math/

Date: 09/09/2004 at 05:29:20
From: Jay
Subject: Why is "false implies true/false" always "true"?

That's exactly what I wanted to know. Thank you very much!