Formal Proofs for Validity

Truth tables are primarily tools for determining the validity of arguments. However, they are thus limited because although they can demonstrate to us that an argument is valid, they cannot make the progression from premise to premise and from premise to conclusion altogether clear. It is possible to know that an argument is valid while still not understanding how it is valid. Thus, proving an argument's validity rather than simply testing it is vital not only to a full understanding of the argument, but it is also useful for persuasion. In order to make this clearer, we can develop proofs for any argument. These proofs use the Nine Rules of Inference to demonstrate how one premise implies another. In order to work on proofs, we must thus have a firm foundation in some of the basic laws of logic: the Rules of Inference.

The Nine Rules of Inference

The Rules of Inference are basic laws because they are definitionally true. Thus they make good tools for proving arguments, sincethey can connect the premises in necessarily valid ways. Here are the Nine Rules of Inference in symbolic form:
 
 
 
Modus Ponens
p » q
p
q
Modus Tollens
p » q
~q
~p
Hypothetical Syllogism
p » q
q » r
p » r
Disjunctive Syllogism
p v q
~p
q
Constructive Dilemma
(p » q) * (r » s)
p v r
q v s
Conjunction
p
q
p * q
Absorption
p » q
p » (p * q) 
Simplification
p * q
p
Addition

p v q

We use these nine rules to establish intermediate conclusions between premises. It is often hard to see how one gets from the premise p * q to the premise p v q, but with simplification, and then subsequently addition, we can create two more premises in between which show how the relationship works.

Proofs:

To construct a proof for an argument, we first translate the argument into symbolic form, and place the premises in any convenient order. The typical procedure is to place the premises in numbered lines, and to include the conclusion with the final premises. There are different approaches to this, so you should learn to recognize the principles rather than the organization, however. Consider the following argument:

"If I go to the movies, I will spend needed money. However, if I do not go to the movies, I will be able to do homework. I must either go to the movies or not go to the movies. Thus, I must either spend needed money, or stay home and do homework."

It is most convenient to symbolize this using the initial letters of the primary terms. The first premise becomes "M » S", the second "~M » H", the third "M v ~M", and the conclusion "S v (~M * H)". Once arranged, we have the argument --
 
 
1. M»S
2.  ~M»H
3. M v ~M   / :.  S v (~M * H)

We now have an argument, and we can test its validity using the truth tables. If we do, we will find that it is indeed a valid argument. However, it is difficult to comprehend or demonstrate how the premises imply one another or imply the conclusion without further intermediate conclusions. So the next step is to find examples of the Rules of Inference within the premises of the argument so that we may use the basic and valid Rules to imply the same conclusion which the unproved premises imply. Simply put, we try to use the argument in combination with the Rules of Inference to prove the conclusion.

So how exactly is this done? We simply look through the premises of the original argument and see if any one or two will lead to any of the nine rules. This is initially difficult for the student, but after some drill, he or she will be able to more easily recognize the forms. Whenever we come across an applicable rule, we simply do the operation and add the resulting premise to the argument. Then we begin again with the new expanded argument and look for more rules.

In the above argument, we can only perform the operation of Absorption on one of the first two premises. Here a dilemma arises. Is it useful to try both possibilities, or will just one be sufficient to lead to the next premise? Normally, it is good for the student to try to work through all possible options. However, some will invariably lead nowhere, so if you can recognize part of the conclusion or part of another premise in the result of an operation, then it is best to pursue that operation. If we were to perform absorption on the first premise, we would have an unnecessary new premise. However, if we were to perform absorption on the second premise, the resulting premise would contain part of the conclusion. Thus, we pursue this operation.

4. ~M » (~M * H)  ---(2, Abs.)

(Note: A negative symbol "~" can be added to any part a Rule of Inference as long as it remains constant on any uses of that term, as above.) Thus we now have a fourth premise which contains part of the conclusion. We can now look at further instances of the rules using all four premises. In this case, we know that any new operation will probably include the fourth premise, since we used the only useful operations in the first three premises. Again, there are several possibilities, so try pursuing several. One operation that will render productive results is by combining premises 1 and 4 in a Conjunction. The resulting premise is this:

5. (M » S) * [~M » (~M * H)] ---(1, 4 Conj.)

(Note: Just as in mathematics, when we are including a larger premise that includes a set of parentheses, be sure to place brackets around the larger premise, as above.) Now, we may seem hopelessly lost, but there is simply one more operation to do. As the frazzled student looks down the list of rules in a panic, his or her eye lights upon the Constructive Dilemma, and the world becomes right once more. By simply performing the operation of C.D. on premises 3 and 5, we arrive at the conclusion:

6. S v (~M * H)  ---(3,5 C.D.)

When we arrange all the premises in order, we have a completed proof.


1. M » S
2.  ~M » H
3. M v ~M   / :.  S v (~M * H)
4. ~M » (~M * H)  ---(2, Abs.)
5. (M » S) * [~M » (~M * H)] ---(1, 4 Conj.)
6. S v (~M * H)  ---(3,5 C.D.)
               Q.E.D

(Note: It is customary to end proofs with "Q.E.D" which stands for the Latin phrase "quod erat demonstrandum," meaning "what was to be demonstrated." It is simply a way of noting that we have arrived at the conclusion we set out to prove.)

Rules of Replacement:

In the next few weeks we will be working with more complex proofs, so the nine rules are often not sufficient to make the process clear. Thus, we have an additional set of Rules of Replacement that help to clear away the excess baggage that often finds itself in proofs. We can use these to simplify premises or to shorten the process of proofs. There are ten such rules, and they correspond roughly to rules in mathematics.


1. De Morgan's Theorem (De M.) ~(p * q) = (~p v ~q)
~(p v q) = (~p * ~q)
2. Commutation (Com.) (p v q) = (q v p)
(p * q) = (q * p)
3. Association (Assoc.) [p v (q v r)] = [(p v q) v r]
[p * (q * r)] = [(p * q) * r]
4. Distribution (Dist.)  [p * (q v r)] = [(p * q) v (p * r)]
[p v (q * r)] = [(p v q) * (p v r)]
5. Double Negation (D.N.) p = ~~p
6. Transportation (Trans.) (p » q) = (~p » ~q)
7. Material Implication (Impl.) (p » q) = (~p v q)
8. Material Equivalence (Equiv.) (p = q) = [(p » q) * (q » p)]
(p = q) = [(p * q) v (~p * ~q)]
9. Exportation (Exp.) [(p * q) » r] = [p » (q » r)]
10. Tautology (Taut.) p = (p v p)
p = (p * p)


These ten rules are a great help in proving arguments but they are also numerous enough to complicate the issue. The key to understanding them is to treat them just as you treat the nine Rules of Inference. They serve the same function in proofs, so they can fill the role of any Rule of Inference. We will begin with simpler proofs, however, and it will be later on when the student is more practiced that these will be used in full force.





Last updated on 01/10/2003 08:33:33.

Contents of this page Copyright 2002 by Abraham P. Ahern
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