Steps for Constructing a Truth-Table
#1: Write in the formula.
#2: Draw the table.
Note: There should be enough columns for all the components of the wff in TL; there should be enough rows for all the possible truth-values of the letter constants.
Rule: If there is one letter constant, there should be three rows; if there are two constants, there should be 5 rows; 3 constants, 9 rows; etc.
#3: Fill in all possible truth-values for the sentence constants.
Rule: If there are two letter constants, input T-T-F-F for the first then T-F-T-F for the second; if there are three letter constants, input T-T-T-T-F-F-F-F for the first, then T-T-F-F-T-T-F-F for the second, then T-F-T-F-T-F-T-F for the third.
#4: Find the main operator.
#5: Calculate the values under the connectives with the smallest scope.
Note: If there is a tie for which is the smallest scope, compute the leftmost operator first.
#6: Calculate the value of the main operator, i.e., the final column.
Rule for Validity Test:
If there is any row on the truth-table that contains all true premises (or premise), but a false conclusion, then the argument is invalid.
If the table contains no row showing true premise(s) and a false conclusion, the argument is valid.
Rule for Consistency Test:
Given two (or more) formulas side by side on top of a table, if there is at least one row where the main operator for all the formulas is true, then the sentences are consistent. If there is no row on which the main operators are true, the sentences are inconsistent.
Rule for Equivalence Test:
If two formulas on top of a table have matching final columns, then they are equivalent. If the final columns do not match, then the formulas are not equivalent.
Use truth-table analysis to assess the following for validity.
P ⊃ Q; ~P; ∴ ~Q
P ⊃ Q; Q; ∴ P
~(P & Q); P; ∴ ~Q
P ⊃ Q; P; ∴ Q
P v Q; ~P; ∴ Q
P ⊃ Q; ~Q; ∴ ~P