2

In the Wikipedia article "Boolean-valued model", one reads:

In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra.

This statement about the completeness of the Boolean algebra in which the truth values are defined is echoed by Monk in the SEP article "The Mathematics of Boolean Algebras", section 7.1:

In model theory, one can take values in any complete BA rather than the two-element BA.

Why is the BA in which the propositions take value supposed to be complete? Is it simply because the two-element BA is itself complete and Boolean-valued models are seen as generalisation, or is there something deeper in it?

Beginner
  • 574

1 Answers1

6

The definition of $[\![\exists x\varphi(x)]\!]$ is $\sum_{a\in V}[\![\varphi(a)]\!]$. So you need completeness to guarantee that this has a well-defined truth value.

Otherwise, pick an antichain which has no supremum, $D$, and for each $b\in D$ find some $a_b$ for which $[\![\varphi(a_b)]\!]=b$, then what would be the truth value of $\exists x\varphi(x)$?

Asaf Karagila
  • 393,674