Blythe G. answered • 10/08/19
Philosophy PhD, ABD. Taught undergrad logic.
The first step to set up a truth table is to identify all the atomic sentences. Those are the capital letters. You have P, Q, R. You'll need 2n rows on the table, where n is the number of atomic statements. You have three atomic statements; 23 is 8 (2 x 2 x 2). Plan for 8 horizontal rows of truth values.
Make three vertical columns for the three atomic statements, in alphabetical order. The rule is that in the first atomic statement column you fill in half the rows straight with Ts, then half with Fs. In the next column it's half as many Ts in a row before doing that many Fs, then repeating. Continue until all your atomic statements are filled out; the last column will always alternate Ts and Fs every row. Depending on your instructor, you may have to reverse the order (first atomic column alternates TF every row, last column has a solid block of half T then half F). This pattern will give you a unique combination of Ts and Fs in every row.
So: Fill out the P column with four Ts, then four Fs (TTTTFFFF). In the Q column, alternate two Ts, two Fs (TTFFTTFF). In the R column, alternate Ts and Fs. (TFTFTFTF).
In a sentence with multiple connectives, you have to evaluate each part in turn to get the whole. Start inside the parenthesis with (P v Q). The disjunct "or" symbol is true when one or both of the two halves is true. It is only false when both are false.
So: Add a vertical column for P v Q. Look for every horizontal row of the truth table where P is F AND Q is F. There will be two; mark P v Q false in those two rows and true in every other row.
NOW you're ready to evaluate the whole sentence. A conditional "if" symbol is true EXCEPT when the left side (antecedent) is true and the right side (consequent) is false. The left side is P v Q. The right side is R. The easiest way to fill in the truth values for a conditional is to find the cases where it's FALSE and then mark all the other rows as true.
So: Add a vertical column for the conditional (P v Q) ⊃ R. We're looking for all the horizontal rows with both a T in the P v Q column and an F in the R column. Find those rows, and mark an F in the conditional column for each of them. For all other rows, mark T.
