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Difference between If...then and Suppose...then

Date: 11/25/2002 at 22:02:35
From: Dan
Subject: Proofs: difference between "if..then" and "Suppose..then"

What is the difference between "If X, Then Y" and "Suppose X. Then 
Y."? Here, X and Y are given statements used in proving something.  
Can you explain this with an example?

Thanks so much!
Dan


Date: 11/25/2002 at 23:51:09
From: Doctor Achilles
Subject: Re: Proofs: difference between "if..then" and "Suppose..then"

Hi Dan,

Thanks for writing to Dr. Math.

That is a very interesting and difficult question.

In mathematical logic, there is a notion of a sentence.  A sentence 
is one of three things:

  1)  A simple sentence, given by a letter.
  2)  The negation of a sentence.
  3)  The joining of two sentences by a connective (such 
      as "and," "or," and "if...then")

You can build very complicated sentences by applying (2) and (3) 
reiteratively. You can find a detailed explanation of sentences (and 
some other basic logic) at:

   A Crash Course in Symbolic Logic
   http://www.mathforum.com/dr.math/faq/symbolic_logic.html 

The sentence "If X, then Y" is just a single sentence. It is no more 
interesting logically than the sentence "P and Q" or "A or B" or "Not 
Z." It is simply a statement that X and Y have a certain truth-
relationship to each other (specifically, that X cannot be true 
at the same time that Y is false).

The problem with the English words "if...then" is that they have too 
many connotations associated with them. Informally, in everyday 
language, 'if' and 'then' can be used in dozens of ways. They are 
often used interchangeably in English with phrases such as "supposing 
... conclude." So in English, you can argue that "if...then" is 
equivalent to "suppose...then," but not so in logic.

This is okay for English, but in mathematical logic, we want to have a 
very strict definition of "if...then," that is to say, we want one and 
only one of the English uses to apply. In order to make sure that 
people don't get confused between the rather squishy and malleable 
English words "if...then," logicians use symbols.

The symbol in logic for 'and' is ^.  Thus "P and Q" is written (P^Q).

The symbol in logic for 'or' is v.  Thus "R or S" is written (RvS).

The symbol in logic for 'not' is ~.  Thus "Not A" is written ~A.

The symbol in logic for 'if..then' is ->.  Thus "If X, then Y" is 
written (X->Y).

The use that has been chosen by logicians for the -> symbol is as 
follows:

  For any pair of sentences X and Y, the sentence (X->Y) means 
exactly and only "It is forbidden that X be true at the same time 
that Y is false".


So "if X, then Y" is a single sentence. By contrast "Suppose X. Then 
Y." is actually two sentences. More than that, "Suppose X. Then Y." is 
a pair of commands. Commands are like orders; they tell you to do 
something (like "Sit." or "Do your homework.").

Let's start by looking at the first command: "Suppose X." What is this 
telling us to do? Well, to suppose is basically to pretend for a 
minute that something is true. For instance, if I'm in a courtroom, I 
could say something like "Suppose the criminal is overweight."  In 
doing this, I am asking the court to "play along" with me for a while 
by pretending that the criminal is overweight. This does not mean that 
I have proven that the criminal is overweight, merely that I think we 
can learn something by pretending for a while. Of course, everyone in 
the courtroom knows this, and sooner or later we will all have to stop 
supposing (pretending) and come back to reality.

What about the second command: "Then Y."? I believe (and I hope you 
agree) that this command is implying something like "Then, conclude 
Y." Or, more explicitly "While you are supposing X, conclude Y." What 
this means is, I hope, fairly clear from the expanded command. We 
started with our first command by pretending that X is true. We can 
then do logical work on X (maybe we'll derive a contradiction from it, 
or combine it with things we know are true or other things we may also 
be pretending are true for the time being, etc.).

Let's compare two problems you may see in a logic class:

   I)  Suppose (A^B).  Then, conclude A.

  II) ((A^B)->A)

The first problem gives you a set of directions. It tells you two 
steps that it wants you to do. First, it wants you to suppose (A^B).  
[When I'm doing logic, I always use curly brackets {} to set off 
supposed or pretend sentences - see the crash course cited above for 
more.]

    {                                   Starting a new supposition

    1)  (A^B)                           Assumption (supposition)

Now that we have supposed (A^B), what is the next step?  We have to 
conclude A. To do that we need the help of a rule of logic. This rule 
is called simplification (or, as I like to call it ^elimination).  The 
rule says, if you have a sentence of the form (X^Y), then you are 
entitled to take from that either X or Y or both.

Let's give it a shot:

    {                                   Starting a new supposition

    1)  (A^B)                           Assumption (supposition)

    2)  A                               obtained from line 1
                                        by simplification of the ^
                                        [^elimination on 1]

    }                                   Closing off the supposition

That's it; we have done both parts of problem (I).

Now let's turn to problem (II).  Remember, it was:

  II) ((A^B)->A)

This problem makes no sense by itself.  It might as well have been:

  II') ((A^C)v(B->D))

It's just a sentence in symbolic logic. This is not a legitimate  
problem because we are not told to do anything. We can just as easily 
sit and look at the problem, cross it out, copy it exactly, hold it up 
to a mirror, or whatever we want. We need to know what we are to do 
with this sentence ((A^B)->A).

Maybe the problem appears in a section with the heading "Copy the 
following symbolic logic sentences on another piece of paper."  In 
that case, that is what we are to do.

Maybe it appears in a section with the heading "Construct a truth 
table for the following symbolic logic sentences." Then we should do 
that.

Maybe it appears in a section that says "Give an argument that the 
following sentences are tautologies." A 'tautology' is a sentence that 
is always true, no matter what. If we are told to do that, this is how 
we might proceed:

   The sentence ((A^B)->A) is a tautology because it is impossible 
   for (A^B) to be true without both of the following conditions: 
   (1) A is true and (2) B is true. Therefore, it is impossible for 
   (A^B) to be true at the same time that A is false, and therefore 
   the sentence satisfies the requirements of the -> symbol.

[Notice that I didn't use the English words "if...then" at all in the 
argument.]

So much for the difference between the sentence (X->Y) and the pair 
of commands: "Suppose X.  Then, conclude Y." Despite the fact that 
they are very different, they are also closely related.

It turns out that one way to prove that (X->Y) is a tautology is to 
follow the pair of commands "Suppose X.  Then, conclude Y."  To do 
this we need to go back to our problem (I) above:

    {                                   Starting a new supposition

    1)  (A^B)                           Assumption (supposition)

    2)  A                               obtained from line 1
                                        by simplification of the ^
                                        [^elimination on 1]

    }                                   Closing off the supposition

And add a new rule to our repertoire. The rule is something I call: 
"->introduction". Here is how it goes. If you suppose x, and under 
that supposition conclude Y, then you are entitled to close off the 
supposition and state that (X->Y) is true.

    {                                   Starting a new supposition

    1)  (A^B)                           Assumption (supposition)

    2)  A                               obtained from line 1
                                        by simplification of the ^
                                        [^elimination on 1]

    }                                   Closing off the supposition

3)  ((A^B)->A)                          ->introduction on lines 1-2


So the connection between the pair of commands "Suppose X. Then Y." 
and the sentence "If X, then Y." is that the pair of commands is a set 
of instructions for proving the sentence is true.

However, do not confuse the set of commands with the product they 
produce. To do that is like trying to eat your mother's favorite 
cookie recipe for dessert. The set of commands is merely a set of 
instructions for making the sentence; it is not the sentence itself.  
The problem is that the set of commands is so simple that our 
intelligent minds see directly and immediately through the 
instructions to the conclusion. (And, unlike mother's cookies, the 
sentence tastes just as papery as the instructions.)

I 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/ 
Associated Topics:
High School Logic

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