Determining truth and falsity in PL
With our trusty rules of truth in hand, we can, for any wff in PL, determine whether it is true or false, if we know the truth-value of the simple sentences which occur in it.
For example, suppose we know that P is true, Q is false, and R is true.
What is the truth value of:
| ((P v Q) & ~R) |
First let's describe this wff. The main operator is the ampersand. The left conjunct is a disjunction of sentence letters and the right conjunct is a negated sentence letter. We know that a conjunction is true just in case both conjuncts are true. Let's look at the right conjunct. R is true, so ~R is false. ~R is a conjunct, and it's false, so the whole conjunction is false. (Remember: the bad apple principle.)
We can construct a partial truth-table which illustrates the use of the rules of truth for this wff.
| P | Q | R | (P v Q) | ~R | ((P v Q) & ~R) | |
| T | F | T | T | F | F |
The left three columns list the given truth-values for P,Q, and R. The next three columns compute the truth value for the conjucts, and finally for the whole wff. The table should make it evident that the truth-value of the whole wff depends on the truth-value of the parts.
Exercises:
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