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### A False Statement Implies Any Statement

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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

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!
```
Associated Topics:
High School Logic

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