A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
A truth table has one column for each input variable (for example, A and B), and one final column showing the result of the logical operation that the table represents (for example, A XOR B). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values.
A proposition's truth table is a graphical representation of its truth function. The truth function can be more useful for mathematical purposes, although the same information is encoded in both.
Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published in 1921. Such a system was also independently proposed in 1921 by Emil Leon Post.
History Irving Anellis's research shows that C.S. Peirce appears to be the earliest logician (in 1883) to devise a truth table matrix.
From the summary of Anellis's paper: In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional.
Applications Truth tables can be used to prove many other logical equivalences. For example, consider the following truth table:
+
This demonstrates the fact that is logically equivalent to .
Truth table for logic gates Here is a truth table that gives definitions of each of the 7 basic logic gate functions of two Boolean variables P and Q:
Name(function) NOT(negation) AND(conjunction) OR(disjunction) NAND(non-conjunction) NOR(non-disjunction) XOR(non-equivalence) XNOR(equivalence)where means true and means false
Condensed truth tables for binary operators For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example, Boolean logic uses this condensed truth table notation:
T F T F T F T F T F T F T F T F T F T F T F T F
This notation is useful especially if the operations are commutative, although one can additionally specify that the rows are the first operand and the columns are the second operand. This condensed notation is particularly useful in discussing multi-valued extensions of logic, as it significantly cuts down on combinatoric explosion of the number of rows otherwise needed. It also provides for quickly recognizable characteristic "shape" of the distribution of the values in the table which can assist the reader in grasping the rules more quickly.
Truth tables in digital logic Truth tables are also used to specify the function of hardware look-up tables (LUTs) in digital logic circuitry. For an n-input LUT, the truth table will have values (or rows in the above tabular format), completely specifying a Boolean function for the LUT. By representing each Boolean value as a bit in a binary number, truth table values can be efficiently encoded as integer values in electronic design automation (EDA) software. For example, a 32-bit integer can encode the truth table for a LUT with up to 5 inputs.
When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating a bit index k based on the input values of the LUT, in which case the LUT's output value is the kth bit of the integer. For example, to evaluate the output value of a LUT given an array of n Boolean input values, the bit index of the truth table's output value can be computed as follows: if the ith input is true, let , else let . Then the kth bit of the binary representation of the truth table is the LUT's output value, where
Truth tables are a simple and straightforward way to encode Boolean functions, however given the exponential growth in size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations and binary decision diagrams.
Applications of truth tables in digital electronics In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table:
+ Binary addition
where A is the first operand, B is the second operand, C is the carry digit, and R is the result.
This truth table is read left to right: Value pair (A, B) equals value pair (C, R). Or for this example, A plus B equal result R, with the Carry C.
This table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.
With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic operation.
In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of types of values one can have on the inputs increases, the size of the truth table will increase.
For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 22, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 33, or nine possible outputs.
The first "addition" example above is called a half-adder. A full-adder is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a full adder's logic:
A B C* | C R 0 0 0 | 0 0 0 1 0 | 0 1 1 0 0 | 0 1 1 1 0 | 1 0 0 0 1 | 0 1 0 1 1 | 1 0 1 0 1 | 1 0 1 1 1 | 1 1
Same as previous, but.. C* = Carry from previous adder
Methods of writing truth tables Regarding the guide columns to the left of a table, which represent propositional variables, different authors have different recommendations about how to fill them in, although this is of no logical significance.
Alternating method Lee Archie, a professor at Lander University, recommends this procedure, which is commonly followed in published truth-tables:
Write out the number of variables (corresponding to the number of statements) in alphabetical order. The number of lines needed is 2n where n is the number of variables. (E. g., with three variables, 23 = 8). Start in the right-hand column and alternate 's and 's until you run out of lines. Then move left to the next column and alternate pairs of 's and 's until you run out of lines. Then continue to the next left-hand column and double the numbers of 's and 's until completed.
This method results in truth-tables such as the following table for , produced by Stephen Cole Kleene:
Combinatorial method Colin Howson, on the other hand, believes that "it is a good practical rule" to do the following:to start with all Ts, then all the ways (three) two Ts can be combined with one F, then all the ways (three) one T can be combined with two Fs, and then finish with all Fs. If a compound is built up from n distinct sentence letters, its truth table will have 2n rows, since there are two ways of assigning T or F to the first letter, and for each of these there will be two ways of assigning T or F to the second, and for each of these there will be two ways of assigning T or F to the third, and so on, giving 2.2.2. …, n times, which is equal to 2n.
This results in truth tables like this table "showing that and are truth-functionally equivalent", modeled after a table produced by Howson:
Size of truth tables If there are n input variables then there are 2n possible combinations of their truth values. A given function may produce true or false for each combination so the number of different functions of n variables is the double exponential 22n.
n 2n 22n 0 1 2 1 2 4 2 4 16 3 8 256 4 16 65,536 5 32 4,294,967,296 ≈ 4.3 6 64 18,446,744,073,709,551,616 ≈ 1.8 7 128 ≈ 3.4 8 256 ≈ 1.2
Truth tables for functions of three or more variables are rarely given.
Function Tables It can be useful to have the output of a truth table expressed as a function of some variable values, instead of just a literal truth or false value. These may be called "function tables" to differentiate them from the more general "truth tables". For example, one value, , may be used with an XOR gate to conditionally invert another value, . In other words, when is false, the output is , and when is true, the output is . The function table for this would look like: F T
Similarly, a 4-to-1 multiplexer with select inputs and , data inputs , , and , and output (as displayed in the image) would have this function table:
F F F T T F T T
Sentential operator truth tables Overview table Here is an extended truth table giving definitions of all sixteen possible truth functions of two Boolean variables p and q:Information about notation may be found in , , and .
F F T T T, F T F F F T T T, F T F
where
T = true. F = false. The Com row indicates whether an operator, op, is commutative – . The Assoc row indicates whether an operator, op, is associative – . The Adj row shows the operator op2 such that . The Neg row shows the operator op2 such that . The Dual row shows the dual operation obtained by interchanging T with F, and AND with OR. The L id row shows the operator's left identities if it has any values such that . The R id row shows the operator's right identities if it has any values such that .The operators here with equal left and right identities (XOR, AND, XNOR, and OR) are also commutative monoids because they are also associative. While this distinction may be irrelevant in a simple discussion of logic, it can be quite important in more advanced mathematics. For example, in category theory an enriched category is described as a base category enriched over a monoid, and any of these operators can be used for enrichment.
Wittgenstein table
In proposition 5.101 of the Tractatus Logico-Philosophicus, Wittgenstein listed the table above as follows: {| class="wikitable" style="margin:left margin:1em auto 1em auto; text-align:left;" |- ! scope=col | ! scope=col | Truthvalues ! scope=col | ! scope=col colspan="2" | Operator ! scope=col | Operation name ! scope=col | Tractatus |- | 0 ||(F F F F)(p, q)|| ⊥ || false || Opq || Contradiction || p and not p; and q and not q |- | 1 ||(F F F T)(p, q)|| NOR || p ↓ q || Xpq || Logical NOR || neither p nor q |- | 2 ||(F F T F)(p, q)|| ↚ || p ↚ q || Mpq || Converse nonimplication ||q and not p |- | 3 ||(F F T T)(p, q)|| ¬p, ~p || ¬p || Np, Fpq || Negation || not p |- | 4 ||(F T F F)(p, q)|| ↛ || p ↛ q