To determine whether or not a statement is a self-contradiction:

1. Construct a complete truth table
2. Inspect the column under the main operator
3. If the column under the main operator contains all F’s, the statement is a self-contradiction.

To determine whether or not a statement is a contingency:

1. Construct a complete truth table
2. Inspect the column under the main operator
3. If the column under the main operator contains mixed T’s and F’s, the statement is a contingency.  The “T” rows show the conditions under which the statement comes out true; the “F” rows show the conditions under which the statement comes out false.

To determine whether or not two statements are logically equivalent:

1. Construct complete truth tables for both statements.
2. Inspect the columns under the main operators.
3. If the columns under the main operators are identical (same Ts and Fs for every row), the statements are logically equivalent.

To determine whether or not two statements are contradictories:

1. Construct complete truth tables for both statements.
2. Inspect the columns under the main operators.
3. If the columns under the main operators are exactly opposite (different truth values for every row), the statements are contradictories.

To determine whether or not a set of statements is consistent:

1. Construct complete truth tables for all the statements.
2. Inspect the columns under the main operators.
3. If there is a row of all the truth tables in which the main operator of each statement is true, the statements are consistent, i.e., it is possible that they might all be true.  The line(s) in which the statements all come out true shows the conditions under which the statements can be all true.

To determine whether or not a set of statements is  inconsistent:

1. Construct complete truth tables for all statements.
2. Inspect the columns under the main operators.
3. If there is no row of all the truth tables in which the main operator of each statement is true, the statements are inconsistent, i.e., it is impossible that they might all be true.

To determine if an argument is valid:

1. Translate from English if necessary
2. Write the premises and conclusion horizontally, premises on left, conclusion on right.  Order of premises does not matter. 
3. Do complete truth table.  (Or use short cut method.)
4. Inspect the truth table.  Is there any row in which premises all come out true and conclusion comes out false?  If yes, argument is invalid.  If no, argument is valid.