Abstract
In the previous chapters, LFIs have been approached exclusively from the propositional viewpoint. This is justified by the fact that the main notions and issues of paraconsistency in general, and LFIs, in particular, occur at the propositional level, related to their main connectives, namely, paraconsistent negation, consistency and inconsistency operators. This chapter gives a full account of LFIs for first-order languages, taking into account that quantified versions of LFIs are essential for mathematical applications such as set theory, and also for applications in computer science, such as databases and logic programming.
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Notes
- 1.
That is: \(\bot _\beta {~\vdash _{\mathbf{QmbC}}} \psi \) for every \(\psi \).
- 2.
As usual, if \(\Theta \) is a subsignature of another signature \(\Theta '\), then any structure \(\mathfrak {A}\) over \(\Theta '\) can be seen as a structure over \(\Theta \), by ‘forgetting’ the interpretation of the symbols in \(\Theta '\) that do not belong to \(\Theta \). Such a structure over \(\Theta \) is called the reduct of \(\mathfrak {A}\) to \(\Theta \) (see [13]).
- 3.
Namely, an additional clause must be added: if t is a variable \(x_i\) then \(t^{\mathfrak {P}}_{\vec x}[\vec a]\) is \(a_i\).
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Carnielli, W., Coniglio, M.E. (2016). First-Order LFIs. In: Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-33205-5_7
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