Two adaptive logics of norm-propositions

Devising the set of abnormalities for a given defeasible rule

“…Even finding out whether, in general, a predicative set of statements is inconsistent or not, or whether two predicative sets are incompatible with each other or not requires defeasible reasoning [Batens, Meheus, 2000;Meheus, Provijn, 2004]. Further examples concern the logic of questions [De Clercq, Verhoeven, 2004;Meheus, 2001], handling deontic conflicts [Beirlaen, Straßer, 2013a;Beirlaen, Straßer, 2013b;Goble, 2014;Meheus et al, 2010a;Meheus et al, 2010b;Straßer, 2010;Straßer et al, 2012;Van De Putte et al, in press; Van De Putte, Straßer, 2012] and many more. A whole different family are corrective adaptive logics, like the one for handling inconsistency, started in the 1980s [Batens, 1985;Batens, 1986;Batens, 1989] and having resulted in too many papers to refer to in the present context, and those handling ambiguity [Batens, 2002;Vanackere, 1999a;Vanackere, 1999b;Vanackere, 2000;Vanackere, 2001].…”

Adaptive Logic Characterizations of Input/Output Logic

“…7 More precisely,R is the maximal superset of R such that for all sets of I/O-pairs G, applying R to G results in the same set of I/O-pairs as applyingR to G. [35,36] which extend on results in [41,42]. 8 By G R we denote the closure of G under R. 9 The general form of the syntactic construction is given by Definition 1. Table 1 shows, among others, how the I/O operations out 1 to out 4 and out + 1 to out + 4 from [29] are obtained by combinations of the rules defined above.…”

“…Our reconstructions trivialize inconsistent sets of facts while the corresponding original I/O logics do not trivialize them, except when rules such as (F) are derivable. 9 As usual, the closure of a set X under a set of rules R is the smallest superset of X that is closed under applications of rules in R. 10 Here is why: (⇒) is trivial. When giving examples in this paper, we will only use the operation of simple-minded output out 1 which allows us to focus on the formal novelties that are introduced in the current paper.…”

“…A similar approach can be used to model weak or negative permissions [9,31]. The idea is that A is permitted whenever we can deduce that there is no obligation to ¬A.…”

“…The idea is that A is permitted whenever we can deduce that there is no obligation to ¬A. Arguably, in many applications this notion of weak permissions is too restrictive and a stronger principle seems more adequate which allows us to derive the permission to bring about A whenever out¬A is not derivable (and not just if ¬out¬A is derived, see also [9]). This can be modelled by ¬out¬A where the negation with the wide scope (the one preceding 'out') is a default negation (i.e., it is interpreted as a 'negation as failure to derive').…”