[The following is summary. Boldface 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
3.2 Truth Tables for Propositions
We can construct truth-table evaluations for all truth value assignments of a proposition. First we establish all possible value assignment combinations for the individual terms. Then we fill-out the atomic formula values within the proposition. Next we determine the values for the operators, working from those with the least scope progressively to the one with the greatest scope, which gives the value for the whole proposition.
We can make truth tables to evaluate all “the different ways in which truth values can be assigned to propositional letters” (69). We will follow the two steps we used before:
(1) Write the appropriate truth value underneath each proposition.
(2) Starting with the truth-functional operator with the least scope and proceeding to the truth- functional operator with the most scope, use the appropriate truth-functional definition to determine the truth value of the complex proposition.
Only this time we will follow this procedure for all possible truth value assignments. [I will skip to the second example. The first example can be found on pp.69-71.] We will evaluate
We do so by creating a truth table for it with the different combinations of values for P and Q.
Then we begin with the atomic formulas. We will start by replicating the values for the P’s.
The we will copy the values for Q.
We start with the negation, because it has the least scope.
Then the disjunction has the next least scope.
Now finally we do the conditional, which has the next least-scope.
Now, this operator is the main operator, so its values are the values for the whole proposition.
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.