Two Approaches to Constructing Direct Proofs of Validity

There are several means by which one can approach the steps in constructing proofs of validity in sentence logic. There are two simple and straightforward approaches that make it a little simpler for those of you who are having some difficulty with them. These are the mechanical and "reasoned" approaches.

I. The Mechanical Approach.

This is the simplest of the two approaches to proofs, though sometimes more time-consuming since, when you use it, you may produce many lines in a proof that are unnecessary. But if all you are trying to do is get the answer, it usually works. In general, do this:

a. Look at the premises and determine whether you can apply one line to another, i.e., see whether you can, for example, use MP, MT, HS, DS, or any of the other rules of inference (not replacement) immediately. If you can, do it.

b. If it was possible to apply one line to another, and if you performed whatever operation was appropriate, then try to use the line you just constructed. Try, again, to apply one line to another. (In very simple proofs, this usually works quite well in deriving the conclusion directly).

c. If step 1 did not work for you, then look at the premises to determine whether there is one of the rules of replacement that can be applied to a line such that you can, after altering it, use one line against another (such as in step 1).

d. If you try step 3 and it works, now apply the line you have just constructed to some already existing line in the premises. Now, you will try to use the line you just constructed in order to produce another line of the proof with one of the rules of inference. Keep doing this until you reach the conclusion.

e. Unfortunately, this method is sometimes less than efficient. The reason that this is the case appears below, implied by the second method of constructing a proof of validity.

II. The Thoughtful Approach.

a. Attempt to determine where the conclusion appears, or what components of the conclusion appear, in which premises. That is, for example, if you need "-A" in the conclusion, look for a premise with "A" or "-A" in it.

b. Now, determine which rule(s) of inference or replacement would be useful in obtaining the "target" sentence in step 1, above.

c. It is not always so simple as this, however, since you will usually have to start altering other lines in the premises so that they can be applied to each other, or simplified, etc., in order that your original idea for deriving the conclusion will work.

d. Once you have determined how to alter a line, try to apply the newly altered line to another line in the argument. Then, try to use that new line against some other, etc.

e. If the conclusion is an implication, try to determine whether the premises can be re-cast as implications, too. If they can, you are probably going to use hypothetical syllogism to derive the conclusion (though this is certainly not always the case). If the conclusion is a single term, such as "A", "B", etc., you might want to see whether MT, MP or DS will work. They are very commonly used. If the conclusion is a disjunction, try CD or HS with IMPL. If the conclusion is a biconditional, re-word the conclusion (symbolically) with the two means by which material equivalence can be re-stated, then determine where the components of those re-statements appear in the premises. If the conclusion is a disjunction, another useful means of determine how to derive the conclusion is to think of it by using ADD. For example, if the conclusion is "A v B", you might want to try to derive either "A" or "B", then add the missing sentence to the sentence you have already derived.

f. Always keep in mind what you are trying to do. You are trying to derive the conclusion. So make sure that you do not "mindlessly" derive lines, derive the conclusion, but don't notice it, and keep on going.

III. In general, here's what you ought to do:

a. Try to apply one line to another.

b. If that does not work, alter an existing line so that one line will apply to another.

c. Re-write the conclusion in a few different ways using the rules of replacement so that you can see different ways in which the conclusion could appear as you construct the proof.

d. Develop the habit of using DeM whenever you see a negation on the outside of a set of parentheses. Remember, however, that DeM only works with "." and "v", so if you see, for example, "-(A --> B), you cannot use DeM directly,but you must alter the inside of the parentheses such that you end with "-(-A v B)." Now you can use DeM.

e. If you are lost, and you see a premise that is a conjunction, simplify it. Now try to use one or the other of the conjuncts against some other line.

f. If there is a premise that is a biconditional, re-state it with one, the other, or both of the alternate statements of the rule EQUIV.

g. If there is a premise that is an implication, but you cannot seem to use it, change it to a disjunction using IMPL.

h. If the conclusion is a disjunction, you might try to derive one half of the disjunction, then add the other half.

i. If the conclusion is a conjunction, you may derive both halves, then use CONJ at the end of the proof.

j. If the conclusion is an implication, try HS.

k. If the conclusion is a single sentence, try MP, MT, or DS on the premises, or alter them so that one of those rules might apply.

Remember that these are only general guidelines to get started in a proof. They do not always work. They only give you a starting point in many cases. The only substitute for studying is studying, so work LOTS of problems.

Remember also that the most commonly used rules are: DeM, IMPL, MP, MT, HS, ADD, and SIMP. They are not only commonly used, they can usually get you out of a mess in a proof.