Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

This Section: 5. Rules of Inference

4. Tautological Implications and Tautological Equivalences Section 5 Exercises 6. Arguments and Proofs Main Logic Page "Real World" Page

5. Rules of Inference

In the last section, we wrote out all our tautologies in what we called "argument form." For instance, Modus Ponens [(pq)p]q was represented as

pq
p
q

In words, this says the following:

In other words, if the first two lines of the argument are known to be true statements, then Modus Ponens assures us that the last line is also a true statement.

According to our convention, small letters stand for atomic statements. Now there is no real reason to restrict modus ponens to apply only to such statements. It seems perfectly reasonable to have Modus Ponens apply to all statements, whether or not they are atomic. As an example, we would like to say the following:

If today is either Monday or Wednesday I will go jogging.
Today is either Monday or Wednesday.
I will go jogging.

In symbols, this is:

(pq)r
pq
r

In other words, we would like to replace Modus Ponens with the more general and hence more usable form:

AB
A
B

where A and B are any statements we care to choose-atomic or not.

In this form, Modus Ponens becomes our first rule of inference:

(and we will still have a list of true statements).


In general, a rule of inference is just an instruction for obtaining additional true statements from a list of true statements. Now if you were studying logic as a mathematics or philosophy major, this might be the only rule of inference you would be given to work with. You would then have to justify the use of the other rules of inference from these. We are not going to be quite so demanding. We'll give you lots of rules of inference to work with from the beginning. Think of them as tools for constructing new statements from old ones; the more tools you have at your disposal, the easier your task becomes. In fact, we are going to allow you to use any of the tautologies listed at the end of Section 4 as rules of inference. (That is why we listed the "argument form" for all of them.)

Rule of Inference T1

Any tautology that appears on the list at the end of the last section can be used as a rule of inference.



Caution

Modus Ponens tells us that, if AB appears on the list, and if A also appears on the list, then we can add B to the list of true statements. If AB appears on the list, but if A does not appear on the list, then we cannot add B to the list. Put another way, if A implies B is true, then we cannot conclude that B is true until we know that A is true.



So far, all the rules of inference that we have been permitted to use come from our list of tautologies. These are not the only kinds of rules of inference we shall allow. Here are two more kinds:

Rule of Inference T2

We can add any tautology that appears in the list of tautologies at the end of the last section as a new line in our list of true statements.



The next two rules are rules we shall use less often than the T1 and T2 rules, but they are sometimes necessary.

Rule of Inference S (Substitution)

We can use a tautological equivalence to replace any part of a compound statement with an equivalent statement.

For instance, we can replace the statement p[~(qr)] with p[(~q)(~r)] by using De Morgan's law, since ~(qr)(~q)(~r). Notice that this is the same as the mathematical rule of substitution: in any equation, if part of an expression is equal to something else, then we can replace it by that something else.

Rule of Inference C (Conjunction)

If A and B are any two lines on our list, then we can add the new line AB.

All this is saying is that, if A and B are both true statements, then so is AB..

Question Are we done yet?

Answer Not quite. What we are doing is giving rules for what lines can be written in the proof of a given argument. We have already been using one rule with gay abandon, and we should write it down:

Rule of Inference P (Premises)

We can write down any premise at any time.

Of course, this does not entitle us to make up premises as we go along; we will always be told what the premises are before we start, and Rule P applies only to those. Also, it is traditional to write down all the premises at once at the beginning, although some people like to write them down only as they are needed.

To summarize, Here are all the rules of inference we shall be using.

Rules of Inference

T1:
Any tautology that appears on the list at the end of the last section can be used as a rule of inference.
T2:
We can add any tautology that appears in the list at the end of the last section as a new line in our list of true statements.
S (Substitution):
We can use a tautological equivalence to replace any part of a compound statement with an equivalent statement.
C (Conjunction):
If A and B are any two lines on our list, then we can add the new line AB.
P (Premises):
If We can write down any premise at any time.


4. Tautological Implications and Tautological Equivalences Section 5 Exercises 6. Arguments and Proofs Main Logic Page "Real World" Page

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble