2.3.7. Soundness and completeness

Our current system is sufficient, conservative, and decisive, and it therefore provides a decision procedure. But we can cut up its properties in another way. Because it is decisive as well as accurate in its answers, we can say both of the following about any derivation:

The if parts of these together say that the system is accurate, and we have seen that they follow from its conservativeness (along with sufficiency in the case of the second statement). The only if parts follow from the if parts given decisiveness. (For example, if the ultimate argument is valid, it must be the case that all gaps close because otherwise, given decisiveness, we would reach a dead-end gap and the ultimate argument would not be valid.) Moreover, the only if parts of the two claims above together imply decisiveness because an argument will always be either valid or invalid, so they tell us that eventually either all gaps close or we reach a dead-end gap.

But these two claims, like the properties of soundness and safety, are not of equal importance. The first is closely tied to the use of derivations to establish validity while the second is similarly related to their use to find counterexamples and establish invalidity. The first is of special interest also because it can be established in some cases where decisiveness fails, and we will take it as the key property of our system of derivations in chapters 7 and 8 when we must abandon decisiveness.

It is standard to give different names to the two parts of the first statement:

When we can be sure that (1a) is true, we say that the system is sound. We have seen that a system will be sound in this sense if all its rules are sound. When we can be sure that (1b) is true, we say the system is complete because such a system provides a proof for each valid argument.

We can show that a system is complete if we know (i) that its rules are safe and the system as whole is sufficient and we know also that (ii) any derivation whose ultimate argument is valid eventually reaches an end. Property (ii) is not full decisiveness since it applies only to derivations whose ultimate argument is valid. This sort of partial decisiveness is something we will be able to establish for the systems of chapters 7 and 8, for which full decisiveness does not hold. And, because this partial decisiveness is enough to provide completeness, all systems that we will study in the course are both sound and complete.