SHORTENED TRUTH TABLE TECHNIQUE

You have at this point learned two separate techniques for proving arguments to be valid:

a) truth tables, and

b) formal proof using the 19 rules plus CP (and IP)

Each has its advantages and disadvantages: Advantages                                                 Disadvantages

                                Truth Tables:

entirely mechanical                                     long and tedious
                                                                 for arguments of
Gives Y/N answer:                                     >4 propositions

Valid or invalid                                         Careless errors easy
 

                                    Formal Proof:

shorter for arguments                             requires strategy
of >4 propositions                             gives only Yes answer

careless errors less likely                     failure to construct a
                                                          formal proof of validity
                                                        does not = proof of invalidity

What is the Shortened Truth Table Technique?

The shortened truth table technique offers a third possibility for proof which has the advantage of truth tables in that it can give a Y/N answer: the argument is shown to be valid or invalid, but it short cuts the long and tedious nature of the full blown truth table.

What do I use the Shortened Truth Table Technique to Prove?

The great advantage of the Shortened Truth Table Technique is that it can be used to prove either validity or invalidity -just like any truth table. Therefore -unlike formal proofs- this technique can prove both the validity and the invalidity of arguments. In this way it avoids the primary disadvantage of formal proof.

What is the goal in using this technique?

You can think of the shortened truth table technique as like a game with permitted and forbidden moves. The objective of the game is to find a row out of all the rows in a full truth table which has all true premisses and a false conclusion. Such a row, if it exists, would, of course, show the argument to be invalid. Thus the objective of the game is to prove the argument is invalid. Therefore "success" in this game means the argument is invalid, and failure in this game means the argument is valid.

The greatest single problem students have with this technique is remembering that "success" means the argument is INVALID!

 How do I use the Shortened Truth Table Technique?

Whether you are attempting to prove validity or invalidity you ALWAYS play the game the same way: You always begin by assuming the premisses are all true and the conclusion is false, i.e. that the argument is invalid. You then "reason backwards" from the assumed truth of the premisses and falsity of the conclusion to try to find a set of truth values to assign to the individual propositions which would make the premisses all come out true and yet the conclusion to be false. If you succeed in finding such a set of truth values (a "row" of a full table) that shows it is possible for the premisses to be true and the conclusion false, so the argument is invalid.

However, if every attempt to find such a set of T values ends in a contradiction, then the game cannot succeed because there is no set of truth values which will make all the premisses true and the conclusion false, so in other words there is no such row in the full table. Therefore the argument must be valid. The is a proof by "reductio ad absurdum."

How do I actually show this?

1. Write out the argument in a spacious fashion, skipping lines between premisses and conclusion. On the side write a "score box" to keep track of the assigned truth values of the statements, with one box for each proposition in the argument.

2. Write an "F" above the main relation of the conclusion, and a "T" above the main relation in each of the premisses. (Note that if the statement is a denial, the truth value is written above the negation symbol.

3. Proceed to reason "backwards" (from whole to part) to determine the truth values of the components of the premisses and conclusion. You write a "T or and "F" above each letter or symbol as long as that truth value is forced.

When is a truth value "forced"?

A truth value is "forced" when knowledge of the truth value of a whole statement determines the truth values of its components. This occurs in the following cases:

A disjunction is known to be false: both disjuncts are forced to be F

A conjunction is known to be true: both conjuncts are forced to be T

An implication is known to be false: the antecedent must be T and the consequent must be F The negation of a statement always forces the negated statement to have the opposite truth value: if the negation is T, the statement is false, if

the negation is false the statement is true
 
 

All of these cases are "good" in playing the game; other cases are less good.
 
 

4. As long as each truth value assignment is forced keep playing the game, and keep a record in the "score box" on the side. Three possibilities await:

a) you succeed in finding a consistent set of truth values for each single

statement component.

b) you are forced to contradict yourself in assigning truth values to component statements (i.e. you assign a symbol "T in one statement and "F" in another); or c) you run out of forced moves before reaching either a) or b) above.

What does this mean?

If possibility a) occurs you have succeeded in the game of trying to prove the argument invalid: ergo the argument is INVALID. You're done, write your answer.

If possibility b) occurs, your attempt to prove the argument invalid has contradicted itself, your attempt to prove the argument invalid has failed, or in other words it is logically impossible to have all true premisses and a false conclusion; ergo the argument is VALID by reductio ad absurdum. You're done, write your answer.

If possibility c) occurs, go to the next step.

5. When you have come to an end of all possible forced moves (be sure to recheck to see that you have in fact come to an end, nothing further is forced) but have not yet either determined the truth values of all components or found a contradiction, then you have to make a hypothetical truth value assignment. This means that you arbitrarily try out assigning a "T" or "F" to any statement, and then proceed as before looking for further forced moves. Again the three possibilities a), b), and c) above await you.

If a) occurs, the game is over; the argument is INVALID.

However, if b) occurs, you have not (yet) proved the argument is valid by reductio ad absurdum, because your truth value assignments (which led to the contradiction of case b)) could be just because you chose the "wrong" hypothetical truth value to assign. You now must return to the point where you made the hypothetical assignment and try the other alternative. If your first hypothesis was to assign a "T" now you need to try an "F" (or vice versa). Again, a), b), or c) could occur. If a) occurs, the argument is invalid and the game is over, or if b) occurs the argument is now proved valid by reduction ad absurdum; and the game is over. If c) occurs again, then repeat the process (return to step 5) with a second hypothesis, etc.