Truth value function

A truth value function , also known as truth function for short , is a mathematical function that maps truth values to truth values. The domain of an n-place truth function is the set of all n-tuples of truth values, its range of values is the set of truth values. In classical logic , the underlying set of truth values {w, f} only includes the two values "true" (w) and "false" (f); Truth functions on this basis are therefore more precisely called n-place two-valued .

The truth value functions play a central role in formal logic , as they clearly specify the (extensional) form of the logical connection of a combination of components, and can be interpreted as joiners of combined statements as well as gates in combinations of switching elements.

example

The truth value of the entire sentence "Peter is coming and the Queen is coming" depends on the truth values of the partial sentences "Peter is coming" (p) and "The Queen is coming" (q). The sentence "p and q" is true if both p and q are true, otherwise false. A function with two arguments (p, q) that assigns the tuple <w, w> - both arguments are true - the function value w - the proposition is true - and the three can serve as a model for the conjunction expressed here by and other possible 2-tuples each have the value f (as value curve in the scheme: wfff ). This truth function is called AND (p, q) or also et function et (p, q).

Generalizing the example, 16 different 2-digit truth functions can now be defined by adding each of the four 2-tuples - that is: <w, w>, <w, f>, <f, w>, <f, f> - each one of the two truth values is assigned. See the table below .

With this definition, a certain mapping of all four 2-tuples - for example: <w, w>, <f, f> are true, the other two are false (in the scheme: wffw ) - unambiguously a logical link form of two sub-clauses - for example "p if and only if q "in the sentence" Peter comes exactly when the Queen comes "- be assigned. The truth function of the latter example is also called the eq-function eq (p, q), since it corresponds to the (material) equivalence, the biconditional .

This means that possible joiners can also be understood as a truth function; this characterizes classical propositional logic and sets it apart from modal propositional logic, for example .

By means of the assignment w → 1 and f → 0 (or alternatively w → 0 and f → 1, see logic level ), every truth value function corresponds to a Boolean function that can be represented in a switching algebra .

Counterexample

The truth value of the sentence "Peter is coming because the Queen is coming" is not a function of the truth values of its sub-clauses - because even if both sub-clauses are true, so that it is not yet certain that Peter is coming because the Queen is coming, for this very reason. This causality is not to be presented as a truth-functional link between the sub-clauses. A further connection is therefore required for the causal justification.

The paradoxes of the material implication motivated to look for alternatives to classical logic. Either by developing multi-valued logics or by doing without truth functions "in the usual sense" in the semantic justification of a logic calculus (cf. modal logic ).

Truth tables

A simple way of defining a truth value function for a finite number of truth values is the truth table .

The table below shows all 1-digit two-valued truth functions . A truth function always maps all tuples of its domain - here both 1-tuples <w> and <f> in column p of the argument - in the truth value set. Where and are constant functions ; is the identical one- digit truth function; is the negation function non (p), also negation for short . ${\ displaystyle f_ {1} ^ {1}}$ ${\ displaystyle f_ {4} ^ {1}}$ ${\ displaystyle f_ {2} ^ {1}}$ ${\ displaystyle f_ {3} ^ {1}}$

${\ displaystyle {\ begin {array} {| c || c | c | c | c |} \ hline p & f_ {1} ^ {1} \ & f_ {2} ^ {1} \ & f_ {3} ^ {1 } \ & f_ {4} ^ {1} \\\ hline w & w & w & f & f \\ f & w & f & w & f \\\ hline \ end {array}}}$

The following overview shows the 16 possible assignment patterns for 2-digit two-value truth value functions using the values 1 and 0 (with the assignment w → 1 and f → 0). The et function or AND discussed above is the function here ; the eq function or XNOR is the function . ${\ displaystyle f_ {8} ^ {2}}$ ${\ displaystyle f_ {7} ^ {2}}$

Furthermore, the aut function or XOR ; is the vel function or OR ; is the Peirce function or NOR ; is the Sheffer - function or NAND ; is the seq function and corresponds to the conditional or the material implication . ${\ displaystyle f_ {10} ^ {2}}$ ${\ displaystyle f_ {2} ^ {2}}$ ${\ displaystyle f_ {15} ^ {2}}$ ${\ displaystyle f_ {9} ^ {2}}$ ${\ displaystyle f_ {5} ^ {2}}$

${\ displaystyle {\ begin {array} {| c | c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |} \ hline p & q & f_ {1} ^ {2} \ & f_ {2} ^ {2} \ & f_ {3} ^ {2} \ & f_ {4} ^ {2} \ & f_ {5} ^ {2} \ & f_ {6} ^ {2} \ & f_ {7} ^ {2} \ & f_ {8} ^ {2} \ & f_ {9} ^ {2} \ & f_ {10} ^ {2} & f_ {11} ^ {2} & f_ { 12} ^ {2} & f_ {13} ^ {2} & f_ {14} ^ {2} & f_ {15} ^ {2} & f_ {16} ^ {2} \\\ hline 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 \\ 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 \\ 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 \ \ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\\ hline \ end {array}}}$

${\ displaystyle f_ {1} ^ {2}}$ and are constant functions that always return the same value for all possible inputs: or ; they are also interpreted as a tautology or as a contradiction (and therefore sometimes called verum or falsum ). ${\ displaystyle f_ {16} ^ {2}}$ ${\ displaystyle 1}$ ${\ displaystyle 0}$

The possible assignment patterns of trivalent truth value functions would be shown less clearly . The statement (p) could then be assigned a third value in addition to "w" and "f" - for example "u" for indefinite - and the same applies to the possible function values. This results in 3 ³ = 27 different 1-digit three-value truth value functions . For the specification of 2-digit three-valued lines, instead of the 2 ² = 4, 3 ² = 9 lines would have to be entered in the two columns p and q . In the following columns 3 ⁹ = 19,683 possible variations of the truth values would have to be tabulated for all 2-digit three-valued truth functions (compared to the 16 all 2-digit two-valued ones listed above).

The number of 3-digit values truth functions is to divalent base = 2 ⁸ = 256 and trivalent then = 3 ²⁷ = 7.625.597.484.987 (which could be shown here even less clear). ${\ displaystyle 2 ^ {2 ^ {3}}}$ ${\ displaystyle 3 ^ {3 ^ {3}}}$

Individual evidence

↑ Cf. Kuno Lorenz : Truth function , in: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. 2nd Edition. Volume 8: Th - Z. Stuttgart, Metzler 2018, ISBN 978-3-476-02107-6 , p. 386
↑ Hence, possibly also: "It should be possible to state a priori whether I can, for example, be in the position to have to designate something with the sign of a 27-digit relation." ( Ludwig Wittgenstein : Tractatus logico-philosophicus . Kegan Paul, Trench, Trubner & Co., London 1922, number 5.5541 ).

[1] Cf. Kuno Lorenz : Truth function , in: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. 2nd Edition. Volume 8: Th - Z. Stuttgart, Metzler 2018, ISBN 978-3-476-02107-6 , p. 386

[2] Hence, possibly also: "It should be possible to state a priori whether I can, for example, be in the position to have to designate something with the sign of a 27-digit relation." ( Ludwig Wittgenstein : Tractatus logico-philosophicus . Kegan Paul, Trench, Trubner & Co., London 1922, number 5.5541 ).