Logic Laws

Date: 03/04/2003 at 20:36:25
From: Jessica
Subject: Logic laws

We are learning about logic. I do not understand the laws of 
inference, simplification, disjunctive inference, and disjunctive 
addition.

Date: 03/06/2003 at 23:11:23
From: Doctor Achilles
Subject: Re: Logic laws

Hi Jessica,

Thanks for writing to Dr. Math.

For a "crash course" in symbolic logic, see the Dr. Math FAQ:

   http://mathforum.org/dr.math/faq/symbolic_logic.html 

The "crash course" doesn't use the same terms to refer to all of the 
laws that you're asking about. I do recommend that you read it to 
give you background in logic and show you a systematic way to think 
about logic.

With regard to the specific laws you asked about:

1) "inference."  This refers to what the crash course calls "->elim" 
   or what some people call "modus ponens."  The rule is this:

If you have:

  (A -> B)

And you have:

  A

Then you can conclude:

  B

This follows almost directly from the meaning of "->" or "if..., 
then."  The statement

  (A -> B)

means that if A is true, then B must also be true.  It doen't say 
anything about whether A or B actually is true, it just gives a 
relationship between them. However, if we also know

  A

that is, we also know that A is true, *then* we can conclude that B 
is true as well.

2) "simplification."  This is called "^elimination" in the crash 
   course.  The law is this:

If you have:

  (A ^ B)

Then you can conclude:

  A

And/or you can conclude:

  B

The reason is fairly easy to see from the definition of "^" or "and."  
We know for starters that

  (A ^ B)

is a true statement. This means that *both* A and B *have* to be true.  
So we can conclude A is true from that, or if we prefer we can 
conclude B is true, or if we like we can make both of those 
conclusions.

3) "disjunctive inference". This is also sometimes called "disjunctive 
syllogism." The rule states:

If you have:

  (A v B)

And you have:

  ~A

Then you can conclude:

  B

[As a side note, the rule also works if you have (A v B) and you have 
~B, then you can conclude A.]

This one is a little trickier (it's one of the advanced rules in the 
crash course).  Let's take this step by step.

We start off with

  (A v B)

This means that *either* A is true, or B is true, or both are true.  
So there are three possible ways that (A v B) could be true:

First, A is true and B is false.

Second, A is false and B is true.

Third, A is true and B is true.

So far we have no idea which of these three is actually the case.

But then we see that we also have

  ~B

This means that B is false. Let's review are three possible scenarios:

First, A is true and B is false. (This one still may be correct.)

Second, A is false and B is true. (Since we just found out that B is 
false, this *cannot* be correct.)

Third, A is true and B is true. (Since we just found out that B is 
false, this *cannot* be correct.)

So the only scenario that can possibly be correct is: A is true and B 
is false.  We can take that and make a logical statement out of it:

  (A ^ ~B)

And by simplification we can get from that

  A

So A must be true if we start with (A v B) and ~B.

4)  "disjunctive addition." This is called "vaddition" in the crash 
course. It's a fairly short rule, but it seems like cheating so it 
is hard to feel comfortable using it.

If you have:

  A

Then you are allowed to conclude:

  (A v B)

It doesn't matter what B is. B could be something completely new that 
you just made up. B could be something that appears as part of another 
line in the same problem. B could even be something that you know is 
false!

How can that work? How can we go around just willy-nilly adding 
letters to the end of sentences?

Well, let's think about this.  We started off with

  A

So we know that A is true. No matter what we do with the rest of the 
problem, we can be sure that A will always be true.

What if we know already that B is true? Well, in that case (A v B) 
will come out true because the disjunction of 2 truths will be true.

What if we know already that B is false? Well, in that case (A v B) 
will still come out true because the disjunction of a truth with a 
falsehood still comes out true.

What if we don't know whether B is false? What if we saw it earlier, 
but never could prove one way or the other? What if we just made B up?  
In that case, we don't know whether B is true or false, but *either 
way* (A v B) will be true (we just proved that in the first two cases 
above).

Basically, as long as we start off with

  A

and we know that A is true, we can be sure that (A v B) will be true, 
no matter what B is.


One final note: I have used A and B for every example, this (of 
course) works for any sentence letters, it also works for more 
complicated sentences so if you have something like:

  [(A v B) -> (B ^ C)]

and

  (A v B)

then you can do inference on it and conclude

  (B ^ C)


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/