[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.3: Truth Tables
3.7 Short Truth Table Test for Invalidity
There is a more efficient way to show the invalidity of an argument than merely filling out the full truth table. This technique is called “forcing”. We first evaluate the conclusion as false and the premises as true. Then, we work backward, finding the assignments for each component term that will make each premise true. If such assignments can be found, then the argument is invalid.
3.6 Truth Table Analysis of Arguments
Previously we used truth tables to test the validity of arguments. This can prove time consuming especially for more complicated arguments. Agler will show us a more efficient method called “forcing”.
Rather than beginning the test by assigning truth values to propositional letters, then using the truth-functional operator rules to determine the truth values of complex propositions, and then analyzing the argument to see if it is valid or invalid, the forcing method begins by assuming that the argument is invalid and working backward to assign truth values to propositional letters.
He will illustrate with an example argument:
P→Q, R∧¬Q ⊢ Q
[We want to see if it is invalid. This requires finding at least one value assignment that will evaluate the premises as true and the conclusion as false. So we can begin our table with those results that we are trying to find.]
Now that we have established these desired values, “we work backward from our knowledge of the truth-functional definition and the truth values assigned to the complex propositions” (91). So consider the part:
This is a conjunction. There is only one way it can be true, namely, if both conjuncts are true. Thus the R and the negation of Q must be true (91).
So we have determined already that v(R) = T.
And also we know that “if v(¬Q) = T, then v(Q) = F” (91).
Since v(Q) = F, we can note it among the assignments.
And we can fill all other instances of Q as F (p.92).
Now, we need the value for P. We know that P→Q cannot be true if the antecedent is true and the consequent is false. Since we already know that the consequent is false, that means for P→Q to be true, v(P) needs to be F.
So we can fill its value in among the assignments.
And we now have assignments for P, Q, and R that make
P→Q, R∧¬Q ⊢ Q
This method began by assuming the argument was invalid (i.e., the premises were assumed to be true and the conclusion to be false). From there, we worked backward using the truth values and the truth-table definitions to obtain a consistent assignment of valuations to the propositional letters.
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.