Method of Truth Tables
== Proof Technique == The '''method of truth tables''' is a technique for determining the validity of propositional formulas with respect to boolean interpretations. In particular, for discerning if a propositional formula is a tautology for boolean interpretations. To start with, we establish the characteristic truth tables of the various logical connectives. We write one row for each boolean interpretation of the set of variables that we are concerned with. From Count of Rows of Truth Table this amounts to $2^n$ rows if there are $n$ variables. There are therefore two rows in the truth table for the only non-trivial unary connective: $\begin{array}{|c||c|} \hline p & \neg p \\ \hline F & T \\ T & F \\ \hline \end{array}$ ... and four rows in the truth tables for the binary connectives (the usual subset of which is given below): $\begin{array}{|cc||c|c|c|c|c|c|c|c|} \hline p & q & p \land q  & p \lor q & p \implies q & p \iff q & p \impliedby q & p \oplus q & p \uparrow q & p \downarrow q \\ \hline F & F & F & F & T & T & T & F & T & T \\ F & T & F & T & T & F & F & T & T & F \\ T & F & F & T & F & F & T & T & T & F \\ T & T & T & T & T & T & T & F & F & F \\ \hline \end{array}$ There are various sorts of proof this technique can be put to, as follows. These will be illustrated by various examples. == Proof of Tautology == {{:Method of Truth Tables/Proof of Tautology}} == Proof of Interderivability == {{:Method of Truth Tables/Proof of Interderivability}} == Proof of Logical Implication == {{:Method of Truth Tables/Proof of Logical Implication}} == Indirect Technique == {{:Method of Truth Tables/Indirect Technique}} == Notational Convenience == It is not actually necessary to include the truth values of the variables themselves (as we have done in the leftmost columns). One is equally justified to write this: $\begin{array}{ccccccc} ((p & \implies & q) & \implies & p) & \implies & p \\ \hline F & T & F & F & F & T & F \\ F & T & T & F & F & T & F \\ T & F & F & T & T & T & T \\ T & T & T & T & T & T & T \\ \end{array}$ and it serves just as well. However, it can help to clarify the derivation, as well as making the truth table easier to construct, if they ''are'' included. It's a matter of personal taste. == Also known as == The '''method of truth tables''' is also sometimes referred to as the '''method of matrices'''. However, it can be argued that the term '''truth table''' is more specific than '''matrix''' and hence preferable. == Comment == Note that solution by truth table is valid only for Aristotelian logic, as it takes for granted the Law of Excluded Middle and the Principle of Non-Contradiction. Within that context, it is a completely mechanical procedure and about as exciting as a strip-tease artist who starts the performance naked. == Sources == * {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski|ed = 2nd|edpage = Second Edition|prev = Hypothetical Syllogism|next = Definition:Logical Not/Notational Variants}}: $\S \text{II}.13$: Symbolism of sentential calculus * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition|prev = Definition:Therefore|next = Method of Truth Tables/Proof of Tautology}}: $\S 3.3$: The Construction and Application of Truth-Tables * {{BookReference|Logic: Techniques of Formal Reasoning|1964|Donald Kalish|author2 = Richard Montague|prev = Definition:Truth Table|next = Definition:Proper Name}}: $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 6$ * {{BookReference|Beginning Logic|1965|E.J. Lemmon|prev = Definition:Biconditional/Truth Table/Matrix Form|next = Method of Truth Tables/Proof of Tautology}}: $\S 2.3$: Truth-Tables * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Modus Tollendo Ponens/Sequent Form/Case 1/Proof 2|next = Modus Ponendo Ponens}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Definition:Conditional/Truth Table|next = Method of Truth Tables/Proof of Tautology}}: $\S \text{I}: 4$: Using the Definitions * {{BookReference|Logic for Mathematicians|1988|Alan G. Hamilton|ed = 2nd|edpage = Second Edition|prev = Bottom-Up Specification of Propositional Logic/Examples/Example 1|next = Rule of Material Implication/Formulation 1/Proof}}: $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Truth Table|next = Definition:Tautology/Formal Semantics}}: $\S 1.6$: Truth Tables and Tautologies * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Count of Rows of Truth Table|next = Definition:Boolean Interpretation}}: $\S 1.4.1$: The meaning of logical connectives