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Montgomery Blair High School, Farm Secruity Administration, Library of CongressPhilosophy 103: Introduction to Logic
How to Construct a Truth Table

Abstract: The general principles for the construction of truth tables are explained and illustrated.

  1. How to construct the guide columns:

    1. Write out the number of variables (corresponding to the number of statements) in alphabetical order.

    2. The number of lines needed is 2n where n is the number of variables. (E. g., with three variables, 23 = 8).

    3. Start in the right-hand column and alternate T's and F's until you run out of lines.

    4. Then move left to the next column and alternate pairs of T's and F's until you run out of lines.

    5. Then continue to the next left-hand column and double the numbers of T's and F's until completed.

  2. Example: construct a  truth table for p   ( q   r )

     

    Line
    No.
     

    I, A variables in alphabetical 
    order
     
    III, A
    First 
    line
    all T

    p q r  r p (q  r)
    1 T T T T T
    I, B
    number of lines 
    = 2n

    III, D
    half T's
    half F's
    2 T T F T T
    3 T F T T T
    4 T F F F T
    5 F T T T T
    6 F T F T T
    7 F F T T T
    8 F F F F F
    III, B
    last
    line
    all F


    I, E

    alternate
    double of previous
    T and F

    I, D

    alternate
    pairs of
    T and F


    I, C

    alternate
    T and F




  3. As check that the guide columns were done correctly:

    1. The first horizonal line will be all T's.

    2. The last horizonal line will be all F's.

    3. The left-most column will be evenly divided; the first half all T's and the second half all F's.

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