The Strong Kleene truth-tables are essentially a 3-valued generalizations of the more familiar 2-valued truth-tables from classical logic. As such, they have proven to be an exceedingly useful tool for the analysis and assessment of concepts / arguments that prima facie appear to resist classical analysis. The truth-tables in question are as follows:

$\begin{array}{c|ccc} \land & 1 & 1/2 & 0 \\ \hline 1 & 1 & 1/2 & 0 \\ 1/2 & 1/2 & 1/2 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}$

$\begin{array}{c|ccc} \lor & 1 & 1/2 & 0 \\ \hline 1 & 1 & 1 & 1 \\ 1/2 & 1 & 1/2 & 1/2 \\ 0 & 1 & 1/2 & 0 \\ \end{array}$

$\begin{array}{c|c} \neg & \\ \hline 1 & 0 \\ 1/2 & 1/2 \\ 0 & 1 \\ \end{array}$

(The material implication and bi-implication may be defined in the usual ways)

If we order the 3 truth-values like so: $1$ - $1/2$ - $0$, we can see that just like in classical logic the truth-value of a conjunction is the minimum value of the conjuncts whilst the truth-value of a disjunction is the maximum value of the disjuncts. As well, negation flips $1$ and $0$, with the $1/2$ acting as a fixed-point.

This is all well and good, but a crucial question which needs to be addressed is how we are to define validity using these truth-tables. When it comes to the validity of an argument in classical logic, there are a number of different ways to phrase how this is actually to be captured. In particular, we might phrase the classical validity condition in any of the following ways:

1. There is no model in which all the premises are $1$ and the conclusion is not $1$
2. There is no model in which all the premises are not $0$ and the conclusion is $0$
3. There is no model in which the lowest value of the premises is greater than the value of the conclusion
4. There is no model in which all the premises are $1$ and the conclusion is $0$
5. There is no model in which all the premises are not $0$ and the conclusion is not $1$

A few moments reflection will reveal that these are all equivalent ways of stating the definition of logical consequence in classical logic; but in the presence of the Strong Kleene truth-tables they all turn out to be non-equivalent and in fact they define different logics. Going through the definitions one-by-one, we arrive at the following logics (the numbers for the logics match up with the corresponding validity conditions given above):

1. $K_3$ - This is one of the more famous paracomplete logics, i.e. a logic in which lmplosion fails: $B \vdash A \lor \neg A$
2. $LP$ - On the other hand, this is one of the main paraconsistent systems, i.e. a logic in which Explosion fails: $A \wedge \neg A \vdash B$
3. $FDRM$ - This logic is the first-degree fragment of the semi-relevant logic R-Mingle. In other words, this is the logic we get when we take all the implicational theorems of R-Mingle that don’t have nested implications and replace $\rightarrow$ by $\vdash$. Another way of looking at this logic is that it is the intersection of $K_3$ and $LP$, i.e. it consists of only those arguments that are valid in both of these logics (which is in contrast to how classical logic is the union of $K_3$ and $LP$). This fact explains the validity of the inference that J. Michael Dunn calls “Safety”: $A \wedge \neg A \vdash B \lor \neg B$
4. $ST$ - This is the so-called “Strict-Tolerant Logic”. An interesting fact about this system is that all the classically valid inferences rules are valid in it also. But it doesn’t satisfy all the classical meta-rules. In particular, the transitivity of consequence fails in this logic; making it what’s sometimes calls a “weakly classical logic”.