Constructing Complete Truth-tables
We have been focusing on partial truth-tables, tables where we are just looking at one possible assignment of truth and falsity to sentence letters, and then determining the truth value of the entire proposition with that assignment. But for the purposes of Chapter 4, where we will look at the semantic features of propositions and arguments, it will be necessary to construct complete truth tables.
Let's approach this task as we usually do, with examples. Suppose we want to construct a complete truth table for the wff:
| ((P v Q) & ~R) |
Remember that in truth-tables we compute the truth values of
the whole wff by looking at the truth values of the parts. We start by writing
down the sentence letters and the component wffs along the top row. Notice that
we move left to right from the simplest components to the whole wff:
| P | Q | R | (P v Q) | ~R | ((P v Q) & ~R) | |
Next we assign truth values to the simplest parts, the sentence letters. This is shown in yellow below:
| P | Q | R | (P v Q) | ~R | ((P v Q) & ~R) | |
| T | T | T | ||||
| T | T | F | ||||
| T | F | T | ||||
| T | F | F | ||||
| F | T | T | ||||
| F | T | F | ||||
| F | F | T | ||||
| F | F | F |
Since there are 3 distinct sentence letters, there are 23 = 8 possible combinations, and we've listed them in the 8 rows. (Notice that we begin by alternating true and false by n/2 then by n/4, then by n/8. If you do this for n rows, until you get to n/n, you'll cover all the possible combinations.)
Now to the right of the solid black vertical, we've listed the next-largest parts of the wff. Since the wff is a conjunction, we've listed the parts of the conjunction. We compute the truth values of the parts, i.e. the conjuncts, shown in gray:
| P | Q | R | (P v Q) | ~R | ((P v Q) & ~R) | |
| T | T | T | T | F | ||
| T | T | F | T | T | ||
| T | F | T | T | F | ||
| T | F | F | T | T | ||
| F | T | T | T | F | ||
| F | T | F | T | T | ||
| F | F | T | F | F | ||
| F | F | F | F | T |
Finally, we use the rule for conjunction to compute the truth value of the whole wff in the final column:
| P | Q | R | (P v Q) | ~R | ((P v Q) & ~R) | |
| T | T | T | T | F | F | |
| T | T | F | T | T | T | |
| T | F | T | T | F | F | |
| T | F | F | T | T | T | |
| F | T | T | T | F | F | |
| F | T | F | T | T | T | |
| F | F | T | F | F | F | |
| F | F | F | F | T | F |
Since the whole wff is a conjunction, it comes out true when both conjuncts are true, otherwise false. The final column is what we're after, as noted in green.
You should now do the Complete Truth-table Exercise.
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