Truth function
The truth values of functions play in formal logic a central role, since they indicate the ( extensional ) form of the logic of a combination of components is uniquely determined, and can be interpreted as connectives compound statements such as gates in compositions of switching elements.
Example
The truth value of the entire set " Peter comes and the Queen is " depends on the truth values of the subsets " Peter is coming" (p) and " the Queen is coming" (q).
The sentence " p and q " is true when both p and q are true, false otherwise.
As a model for the here and expressed by the conjunction that is, a function with two arguments (p, q ) are used, the
The example can be generalized now 16 different 2-digit truth functions defined by each of the four 2- tuple - these are:
With this definition, a particular representation of all four 2- tuple - for example:
This also possible connectives can be interpreted as a truth-function; this characterizes the classical propositional logic and sets them down for example from the modal propositional logic.
By virtue of the mapping w → 1 and f → 0 ( or, alternatively, w → 0 and f → 1, see logic level ) corresponds to every truth value function of a Boolean function which can be represented in a Boolean algebra.
Counterexample
The truth value of the sentence " Peter 's because the Queen is coming" is not a function of the truth values of its constituent sentences - because even if both subsets are true, so that 's not yet certain that Peter 's because the Queen is, for this very reason. This causality can not be represented as a truth-functional combination of the subsets. Therefore, for the causal justification it needs a wider context.
Truth Tables
A simple way to define a truth value function for a finite number of truth values , the truth table.
The following table indicates all the 1- digit bivalent truth-functions.
A truth function always maps all tuples of its domain - here both 1 -tuple
The following table shows the 16 possible assignment pattern 2 -digit bivalent truth values of functions by the values 0 and 1 discussed above et function or AND here is the function; the eq function or XNOR is the function.
Furthermore, the aut - or XOR function; is the vel - function or OR; is the Peirce - or NOR function; is the Sheffer function or NAND; is the seq function and corresponds to the conditional or the material implication.
And are constant functions, which always deliver the same value for all possible inputs: or; they are interpreted as a tautology or a contradiction (and therefore occasionally called verum or falsum ).
Less clear is the possible occupancy patterns would be show trivalent truth-values functions. The statement ( p) would be next to " w" and "f", a third value can be assigned to - for example, "u" for undetermined - and the same applies to the possible function values . This results in 33 = 27 different 1- digit trivalent truth-values functions. To specify a 2-digit trivalent need the two columns p and q = 9 then 32 rows are removed instead of 22 = 4. In the following columns 39 = 19,683 possible variations of the truth values would be to tabulate for all 2- digit trivalent truth-functions ( as opposed to 16 all 2- digit divalent ).
- Logic