Truth value

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

The truth value is one of the two values, "true" $(T)$ or "false" $(F)$, that can be taken by a given logical formula in an interpretation (model) considered. Sometimes the truth value $T$ is denoted in the literature by $1$ or $t$, and $F$ by $0$ or $f$. If the truth values of elementary formulas are defined in a model $\def\fM{ {\mathfrak M} }$, then the truth value $||A||$ of any formula $A$ can be inductively determined in the following way (for classical logic):

$$||B\& C||=T \iff ||B||=T \text{ and } ||C||=T,$$

$$||B \vee C||=T \iff ||B||=T \text{ or } ||C||=T,$$

$$||B \supset C||=T \iff ||B||=F \text{ or } ||C||=T.$$

$$||\neg B||=T \iff ||B||=F,$$

$$||\forall xB(x)||=T \iff \text{ for all } a\text{ in }\fM:||B(a)|| = T,$$

$$||\exists xB(x)||=T \iff \text{ for some } a\text{ in }\fM:||B|| = T,$$ One sometimes considers interpretations in which logical formulas may take, besides $T$ and $F$, other "intermediate" truth values. In such interpretations, the truth values of formulas may be, e.g., elements of Boolean algebras (so-called Boolean-valued models for classical logic, cf. Boolean-valued model), elements of pseudo-Boolean algebras (also known as Heyting algebras, cf. Pseudo-Boolean algebra) or open sets in topological spaces (for intuitionistic logic), or elements of topological Boolean algebras (for modal logic $S4$) (cf. [RaSi]). In a Boolean-valued model, if the truth values of elementary formulas are defined, then the truth values of compound formulas can be determined as follows

$$||B\& C||=||B||\cap||C||,$$

$$||B\vee C||=||B||\cup||C||,\qquad ||B\supset C||=\overline{||B||}\cup||C||,$$

$$||\neg B||=\overline{||B||},\qquad \forall xB(x) = \bigcap_{a\in\fM} ||B(a)||,$$

$$||\exists xB(x)||=\bigcup_{a\in\fM} ||B(a)||,$$ where $\overline{||B||}$ is the complement to the element $||B||$. For example, in topological models for intuitionistic logic, the truth values of compound formulas can be determined as follows:

$$||B\& C||=||B||\cap||C||, \qquad ||B\vee C||=||B||\cup||C||,$$

$$||B \supset C|| =\text{ Int }(\overline{||B||}\cup||C||), \quad ||\neg B||=\text{ Int }(\overline{||B||},$$

$$||\forall xB(x)|| = \text{ Int }\big( \bigcap_{a\in \fM} ||B(a)||\big),$$

$$||\exists xB(x)|| = \bigcup_{a\in \fM} ||B(a)||,$$ where $\text{ Int }(X)$ denotes the interior of the set $X$.

References

[No]	P.S. Novikov, "Elements of mathematical logic", Oliver & Boyd and Acad. Press (1964) (Translated from Russian) MR0164868 Zbl 0113.00301
[RaSi]	E. Rasiowa, R. Sikorski, "The mathematics of metamathematics", Polska Akad. Nauk (1963) MR0163850 Zbl 0122.24311