How to proofs work in three-valued Kleene logic?

Question

In three-valued logics such as Kleene logic, there is a third truth value U, which represents "undefined", or "who knows?". It behaves like "either true or false", and truth tables for not, and and or can be filled out appropriately:

What does a proof look like in this logic? When does $B$ (semantically) follow from $A$? I can think of a few possibilities:

1. We say $A \vDash B$ if whenever $A$ is T, $B$ is T.

2. We say $A \vDash B$ if whenever $A$ is T, $B$ is either T or U.

3. We say $A \vDash B$ if whenever $A$ is T or U, $B$ is either T or U.

Presumably, once we know what $\vDash$ means, we can define an appropriate proof theory of deduction rules ($\vdash$) which agree with $\vDash$. But I am unsure of which of (1), (2), or (3) is correct. Do they all produce different logics, or is one of (1), (2), (3) equivalent to another via some sort of transformation?

Answer 1

Consider the assertions [i] $C\models D\lor\neg D$ and [ii] $D\land\neg D\models C$. Note that assertion [i] is valid according to (3) (given that $D\lor\neg D$ is never assigned the value $\mathsf{F}$) but is not valid according to (1) (check what happens when you assign the value $\mathsf{T}$ to $C$ and the value $\mathsf{U}$ to $D$). With assertion [ii] its the other way around.

The trouble with definitions such as (2) is that they do not in general give you a Tarskian notion of entailment (they might fail transitivity of entailment).

Other definitions you might want to try next:

4. Say that $A\models B$ if whenever $A$ is $\mathsf{T}$ or $\mathsf{U}$, $B$ is $\mathsf{T}$.

5. Have the values in the truth-table ordered, say, by letting $\mathsf{F}<\mathsf{U}<\mathsf{T}$, and say that $A\models B$ if $v(A)\leq v(B)$, where $v(X)$ denotes the value you assign to $X$.

Definitions such as (4) also do not in general give you a Tarskian notion of entailment (they might fail reflexivity of entailment --- check what happens when $A$ is equal to $B$ and you assign the value $\mathsf{U}$ to $A$).

Definition (5) give you a Tarskian notion of entailment, but obviously fail both assertions [i] and [ii]. Some idea of relevance, in fact, is thereby introduced.

Answer 2

See Many-Valued Logic : we choose a set of designated truth degrees and then we say that a well-formed formula $A$ of a propositional language counts as valid under some valuation $v$ (which maps the set of propositional variables into the set of truth degrees) iff it has a designated truth degree under $v$.

And $A$ is logically valid or a tautology iff it is valid under all valuations.

For Kleene's Three-Valued Logic, see :

• Merrie Bergmann, An Introduction to Many-Valued and Fuzzy Logic (2008), page 71-on :

As in classical logic, we define a tautology in a three-valued logical system to be a formula that always has the value T — there is no assignment on which it has either the value F or the value N [not defined].We define a contradiction in a three-valued logical system to be a formula that always has the value F — that is, it never has the value T or N. It turns out that there are no tautologies or contradictions in [Kleene's Three-Valued Logic].

Thus, the Law of Excluded Middle is not a tautology in [Kleene's Three-Valued Logic] (nor is its negation a contradiction).

We will say that a set $\Gamma$ of formulas entails a formula $P$ in three-valued logic if, whenever all of the formulas in $\Gamma$ are true $P$ is true as well (there is no truth-value assignment on which all the formulas in $\Gamma$ have the value T while $P$ has the value F or N), and an argument is valid in three-valued logic if the set of premises of the argument entails its conclusion.