Classically, a logic is two-valued if every proposition (without free variables) is either true or false and none is both; that is, the logic is consistent and every proposition is decidable. Being two-valued logic is a key feature of classical logic; any logic that is not two-valued is ipso facto nonclassical.
We do not expect that a predicate (a statement with free variables) is either true or false (although it will become true or false if applied to a term with no free variables). We can call a context two-valued if every proposition in that context (every predicate with free variables as given in that context) is either true or false; a logic is two-valued iff its global context is two-valued.
Famously, intuitionistic logic is not two-valued, although it may be two-valued in a weaker sense. Let a logic be intuitionistically two-valued if every proposition is false if and only if it is not true. On the other hand, we can call a logic paraconsistently two-valued if every proposition is true if and only if it is not false.
Applying topos theory to logic, we call a topos two-valued if every global element of its subobject classifier is false if and only if it is not true. More generally, any coherent category (just how general can we make this?) is two-valued if every subterminal object is initial if and only if it is not terminal. Note that we have phrased the definition in the intuitionistic fashion; that way, the theorem that a well-pointed topos is two-valued survives even in constructive mathematics.
Although being two-valued and being boolean are both key features of classical logic and are closely related, they are not the same thing. It is easy to find multi-valued logics which are boolean in that they satisfy the law of excluded middle; indeed, any boolean algebra provides a model. Conversely, a logic may be two-valued in an external sense without being able to prove it of itself.
Similarly, a topos may be two-valued without being Boolean, and for purposes of doing mathematics in the topos it is the latter property which determines whether the internal logic satisfies excluded middle. However, a well-pointed topos (even working in a constructive metatheory, where such toposes might not be boolean or two-valued) is two-valued (in the strong classical sense) if and only if it is boolean.
Generally speaking, two-valuedness is an external property, and booleanness is an internal property.
Last revised on October 17, 2018 at 11:22:35. See the history of this page for a list of all contributions to it.