Logic: Systematic Method for Constructing Truth Tables for $n$ Statements
Systematic Method for Constructing Truth Tables for $n$ Statements
When constructing a truth table for $n$ propositional statements $P_1, P_2, P_3, \dots, P_n$, follow this recursive halving procedure to ensure all unique combinations of truth values ($\text{T}$ and $\text{F}$) are systematically enumerated without repetition or omission.
The General Algorithm
Step 1: Determine Dimensions
Total Rows: For $n$ binary statements, there are $2^n$ unique truth-value combinations.
Total Columns: Prepare $n$ initial columns for the input statements $P_1, P_2, \dots, P_n$ (plus additional columns for composite logical expressions as needed).
Step 2: Fill Column 1 ($P_1$)
Since there are $2^n$ rows, divide the total rows into two equal halves of size $\frac{2^n}{2} = 2^{n-1}$.
Fill the first $2^{n-1}$ rows with $\text{T}$.
Fill the remaining $2^{n-1}$ rows with $\text{F}$.
Step 3: Fill Column 2 ($P_2$)
Halve the block size again to $\frac{2^{n-1}}{2} = 2^{n-2}$.
Alternately fill blocks of size $2^{n-2}$ with $\text{T}$ and $\text{F}$:
Top half (where $P_1 = \text{T}$): First $2^{n-2}$ rows $\text{T}$, next $2^{n-2}$ rows $\text{F}$.