Multi-valued logic—an introduction

Extending classical logic to a larger realm of truth, we shall investigate the various multi-valued logics with their extraordinary truth values beyond truth and falsity.

In classical propositional logic, every statement is either true or false. Do you find this reassuring? Or perhaps you find it inflexible or confining. In multi-valued logic we shall extend beyond those two classical truth values and enter a larger realm of truth. We shall introduce various intermediate or supplementary truth values and investigate the logical systems that result. Surprisingly, it turns out that there are many ways to go about it, even when we would seek to add just one new truth value, and various logicians, beginning with Łukasiewicz (1917), but also Tarski, Post, Kleene, Gödel and others, have all proposed different inequivalent three-valued truth-functional logics. And there is a further hierarchy of multi-valued logics with more than three values, many with infinitely many intermediate truth values. Let us explore!

Three-valued logics

Consider a three-valued logic in which we have not only true and false, but also a third truth-value, which we shall denote by the hash symbol # and pronounce as hash. Perhaps we want to regard # as standing for an unknown truth value; or perhaps it is a place-holding truth value in the case of missing information; or perhaps # is meant to indicate that the statement is only contingently true; or perhaps # means the truth value is somehow under-determined, neither true nor false; or it is perhaps over-determined, indicating both true and false.

There are a variety of metaphors like this that can lead one to particular ways of defining a three-valued logic, and these different motivations can lead to fundamentally different logics. Meanwhile, it must be said that although the metaphors often inspire the preliminary definitions, in some instances the founding ideas do not actually hold up well as the logic is naturally developed further, and ultimately one can criticise a logic when it no longer reflects the motivating conceptions.

One way to define a particular multi-valued truth-functional logic is simply to specify the truth values of the fundamental logical connectives, such as ∧, ∨, ¬, →, ↔, which we can do by simply providing truth tables for them in the multi-valued setting. We may then recursively calculate the truth tables for compound assertions just as in classical logic.

Kleene logic

Kleene logic, for example, is defined by the following truth table.