The aim of this paper is to formalize what the authors consider a common practice in science, philosophy and reasoning in general: to ignore contradictory information and reason always with consistent sets of information. The paper focus on implementing this leading idea in logics formulated as axiomatic (i.e. Hilbert-style) formal systems (proof-theoretical point of view) and valuation structures (semantical point of view) by using the key notion of “paradeduction” (Section 4).
The main results are:

1.

Let \(\vdash_{S}\) (resp., \(\vDash_{S}\)) be the deductive proof-theoretical (resp., semantical) relation of a given axiomatic system (resp., valuation structure) and \(\vdash_{S}^{P}\) (resp., \(\vDash_{S}^{P}\)) the “paradeductive” (resp., “paraconsequence”) restriction of \(\vdash_{S}\) (resp., \(\vDash_{S}\)). We have: if \(\vdash_{S}\) is sound and complete w.r.t. \(\vDash_{S}\), then \(\vdash_{S}^{P}\) is also sound and complete w.r.t. \(\vDash_{S}^{P}\) (Theorem 4.3).

2.

If an axiomatic formal system has certain properties, then its paradeducibility relation is paraconsistent (Theorem 5.1).

Both in §1 and §6 the authors compare their method with other similar procedures to be found in the literature. Also, some open problems are remarked in section 6 (Conclusion) of the paper.