Truth table
[1] In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
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 truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values.
Clearly, for the Boolean functions, the output belongs to a binary set, i.e. F = {0, 1}.
Each element in the domain represents a combination of input values for the variables A and B.
Rather than a list (set) given above, the truth table then presents these input-output pairs in a tabular format, in which each row corresponds to a member of the domain paired with its corresponding output value, 0 or 1.
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.
[2] Such a system was also independently proposed in 1921 by Emil Leon Post.
Peirce appears to be the earliest logician (in 1883) to devise a truth table matrix.
[4] From the summary of Anellis's paper:[4] 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.
It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John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An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to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.
Here is a truth table that gives definitions of the 7 most commonly used out of the 16 possible truth functions of two Boolean variables P and Q: where T means true and F means false 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: 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.
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.
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.
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.
This truth table is read left to right: 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.
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: Regarding the guide columns[5] 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.
[6] Lee Archie, a professor at Lander University, recommends this procedure, which is commonly followed in published truth-tables: This method results in truth-tables such as the following table for "P ⊃ (Q ∨ R ⊃ (R ⊃ ¬P))", produced by Stephen Cole Kleene:[7] 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.
The function table for this would look like: Similarly, a 4-to-1 multiplexer with select imputs
The truth table associated with the logical implication p implies q (symbolized as p ⇒ q, or more rarely Cpq) is as follows: The truth table associated with the material conditional if p then q (symbolized as p → q) is as follows: p ⇒ q and p → q are equivalent to ¬p ∨ q.
The truth table for p NOR q (also written as p ↓ q, or Xpq) is as follows: The negation of a disjunction ¬(p ∨ q), and the conjunction of negations (¬p) ∧ (¬q) can be tabulated as follows: Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments p and q, produces the identical patterns of functional values for ¬(p ∧ q) as for (¬p) ∨ (¬q), and for ¬(p ∨ q) as for (¬p) ∧ (¬q).
This explains why Tractatusrow in the table given here does not point to the same Truthvaluesrow as in the Tractatus.
