Logic & Critical Reasoning Truth-Table Analysis

Transcript:Logic & Critical Reasoning Truth-Table Analysis:
Logic & Critical Reasoning Truth-Table Analysis What is a Truth-Table A truth-table is a table that shows the distribution of truth-values, T and F, over a set of compound formulas. The distribution is determined by a fixed set of truth-values as inputs, the definitions of the truth functions involved in the compound formulas, and the determination of dependency. The fixed set of inputs are under ‘P’ and ‘Q’ at the left. The final value of the formula is under the main connective in bold. Computation goes from subordinate operators to the main connective Rules for Constructing a Truth-Table The number of rows that a truth-table needs is determined by the number of basic statement letters involved in the set of formulas that will be involved in the computation. The formula for the rows is 2n where n = the number of basic statement letters involved. The most common instances are where n = 1, 2, and 3; in those cases one needs 2, 4, and 8 rows respectively. The number of columns that a truth-table needs is determined by the number of basic statement letters and formulas involved. There should be one column for each basic statement letter and formula. For each formula the computation proceeds from weakest scope to widest scope. The operation in the widest scope is computed last. And if two sub-formulas of a a formula have no greater weight in scope, then it does not matter which one is computed first. Distributing Basic Truth-Values in the Rows A truth-table is a computation based on an initial set of input truth-values. These values are fixed. The number of basic rows is given by the number of basic statement letters, each of which can be either true or false. Thus, in making a truth-table, one needs to account for every possible combination of how T and F can be distributed amongst the basic statement letters prior to computing the compound cases via the definition. The easiest way to make sure all possible distributions of T and F have been accounted for, one can use the procedure of halving and alternating Ts and Fs, explained by the case in which one has 3 basic statement letters, P, Q, and R, and thus 8 rows, by the formula 2n. Set the first column under P to 4Ts, followed by 4Fs. Set the second column under Q to 2Ts, 2Fs, 2Ts,