2.3. Failed proofs and counterexamples

2.3.0. Overview

Derivations can also be used to tell when a claim of entailment does not follow from the principles for conjunction.

2.3.1. When enough is enough
A derivation is stopped only when no more rules can be applied. When that is so, any open gap has reached a dead end.

2.3.2. Dividing gaps
The active resources of any dead-end gap can be divided from its goal. To put it another way, we have enough rules to develop further any gap whose proximate argument cannot be divided.

2.3.3. Validity through the generations
If we describe as descendents of a gap the gaps that result from developing and perhaps branching it, the validity of the proximate argument of a gap rests on the validity of the proximate arguments of its descendents.

2.3.4. Sound and safe rules
The derivation rules are designed so that, if a gap can be divided, so can at least one descendent at every stage and, moreover, all of its ancestors.

2.3.5. Presenting counterexamples
Because we have enough rules and the ones we have are well-behaved, any gap that reaches a dead end shows us how to divide the premises of the initial argument from its conclusion.

2.3.6. Reaching decisions
A derivation will always reach a point where we must stop either because all gaps are closed or because there is an open gap to which no more rules can be applied.

2.3.7. Soundness and completeness
The properties of this system of derivations combine to show that it establishes the validity of no argument that is not valid and does establish the validity of all that are.

2.3.8. Formal validity
The sort of validity we test with derivations is the general validity of arguments with a given form. An argument that is not valid in virtue of a given form could be valid nonetheless, and its validity may be recognized by a deeper analysis of its form.