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. Dead ends and counterexamples
The active resources of any dead-end gap can be separated from its goal. To put it another way, we can run out of ways to develop an open gap only when there is a counterexample to its proximate argument.

2.3.3. Validity through the generations
If we describe as descendents of a gap the gaps that result from developing and perhaps dividing 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 has a counterexample, so does at least one descendent at every stage and, moreover, each of its ancestors.

2.3.5. Confirming 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 separate 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 does not indicate validity for any argument that is not valid and does indicate validity for every argument that is valid.

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 due to features not captured by that analysis.