two-valued logic

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.

Revised on August 26, 2010 23:00:56
by Toby Bartels
(75.88.93.39)