3.4. Counterexamples to reductios

3.4.0. Overview

All derivations that fail will now end in the failure of a reductio, and this produces some small changes in what we say about the failure of derivations.

3.4.1. When reductios fail
Changes in the arguments used to show the sufficiency, conservativeness, and decisiveness of the system of derivations correspond to changes in the way we confirm counterexamples.

3.4.2. Some examples of consistency
When a reductio fails, we know that its premises are not inconsistent, so derivations that fail will now lead us to consistent sets of sentences.