The Short-Cut Truth Table Method
and Forking
Peter Suber, Philosophy Department, Earlham College

In the full truth table method for testing validity, we must construct all the rows of the table even if only some of them turn out to be relevant for the test of validity. In the short-cut truth table method, we construct only those relevant to testing validity. This is a great savings in time and labor.

In a nutshell, in the short-cut method we try to invalidate the argument we're testing. If we succeed, then it is invalid. If we fail, then it is valid. More precisely, if we fail in a special way, then we can conclude that the argument is valid. We must necessarily fail (find our path barred by contradiction); otherwise our failure to invalidate the argument may be due to our lack of ingenuity, not to the actual validity of the argument.

An "invalidating row" of a truth table (also called a "counterexample" to the argument) is a row in which every premise is true and the conclusion false. In the full truth table method, we construct all the possible rows and then look to see whether there are any invalidating rows. (If there is one or more, the argument is invalid; if there are none, it is valid.) In the short-cut method, we try to make an invalidating row or counterexample.

The short-cut method is usually simple and easy. It only becomes more difficult when we are unable to finish making the hypothetical invalidating row without experimenting (as Copi puts it at p. 62) with "various 'trial assignments'." This is called forking, and Copi does not explain how to make it systematic. The chief purpose of this hand-out is to provide directions for forking.

Forking is easier to describe roughly than with perfect precision and particularity. Here is a rough description. Afterwards I try to give all the precision you might ever need.

Short description

If assuming invalidity doesn't give you enough information to finish making truth-value assignments to all the simple statements in the argument, then you must fork. Take one of the simple statements to which you have so far assigned no truth-value. Assume it to be true, and write down somewhere that you are doing so. Three outcomes are now possible. (1) If you can finish making truth-value assignments and find a contradiction among them, then try the other fork; that is, assume the statement to be false which you just assumed to be true. (2) If you can finish but find no contradiction, then you may stop without checking the other fork; the argument is invalid. (3) You can't finish; fork again. Keep going until you have found contradictions on both forks of all "forking assumptions" (sign of a valid argument) or until you have found one non-contradictory fork (sign of an invalid argument), whichever comes first.

Long description

If you want more precision, here are the steps to take:

  1. Assume invalidity. That is, make every premise true and the conclusion false.
  2. Carry out the assumption of invalidity. That is, make other truth-value assignments based on the initial ones.
  3. If you finish, then interpret what you've done.
  4. If you cannot finish, then you must fork.

Since forking always assigns a truth-value to a previously unvalued simple statement, repeated forking will eventually assign truth-values to all the simple (and hence, compound) statements. Hence, this process cannot go on forever; it will always allow you to "finish" and decide the validity of the argument. The maximum amount of work you'll do will be to create all the rows of a full truth-table --plus the overhead of keeping track of your forks. In practice you'll almost never have to fork more than once or twice.

The short-cut truth table method is a direct proof of invalidity, and an indirect proof of validity. It proves invalidity directly because it produces an invalidating row. Or, as logicians would say, it produces a counterexample to the argument's validity. The form may appear valid, but the counterexample shows us what kind of statements (by truth-value, not by content) could be substituted for the argument's variables to reveal its invalidity.

It proves validity indirectly by proving that the assumption of invalidity leads to a contradiction. You might ask: is it legitimate to infer validity from the fact that the assumption of invalidity leads to a contradiction? Test this for yourself. Let "V" stand for the proposition, "The argument in question is valid." The short-cut method, then, relies on this principle:

[~V (A · ~A)] V

Or: If ~V leads to a contradiction (A · ~A), then ~V is false, which is to say that V is true. If you have any doubts about this principle, construct a truth table for it. You'll see that it is a tautology.

Here are some common pitfalls in using the short-cut truth table method.

  1. To forget how to start, namely, to assume invalidity.
  2. To forget what it means to assume invalidity, namely, make each premise true and the conclusion false.
  3. To assign the the correct truth-value to a statement, but not to put it under the statement's main connective. This might lead you later in the proof to mistake which statement has that truth-value.
  4. Because contradictions are "bad", to infer invalidity from the presence of a contradiction. Remember that contradictions show that the assumption of invalidity is bad, not that the argument is bad.
  5. To guess at truth-values when you lack information to deduce them.
  6. To carry out forking badly. To stop with the first fork when testing the second fork is necessary.
  7. To overlook contradictory truth-value assignments when they are actually present.

Here are some arguments which require forking, if you want to practice.

  1. p (q · r)
    p (q · r)
    q · r

  2. p q
    q r
    r s
    p · s

  3. ~(p q)
    ~(p q)

  4. p q
    q r
    p · r

  5. p q
    q r
    p r

  6. (p · ~q) (~p · q)
    ~(p q)


Here's how to use the short-cut method to test whether a compound statement is a tautology, contradiction, or contingency.

  1. Assume that the statement is false. (Put F under its main connective.) Call this the first assumption.
  2. Carry out the first assumption until finished, if you can.
  3. If the first assumption leads to a contradiction, then the statement cannot be false; it is a tautology.
  4. If the first assumption can be carried out consistently, then reverse it and assume that the statement is true. (Put T under its main connective.) Call this the second assumption.
  5. If you cannot finish either the first or second assumptions, then fork.


Yes, forking is confusing. A future hand-out on truth trees will turn adversity to advantage and show us a way to organize forking so that it clarifies the test of validity rather than obstructing and confusing it.

This file is an electronic hand-out for the course, Symbolic Logic.

Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.

Ribbon] Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A. Copyright © 1997, Peter Suber.