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

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

**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 *in*validity

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

**the negation is false the statement is true**

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.

**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.