@article {PD:embed, title = {Nice Embedding in Classical Logic}, journal = {Studia Logica}, year = {2016}, pages = {47-78}, abstract = {

It is shown that a set of semi-recursive logics, including many fragments of CL (Classical Logic), can be embedded within CL in an interesting way. A logic belongs
to the set iff it has a certain type of semantics, called nice semantics. The set includes
many logics presented in the literature. The embedding reveals structural properties of the embedded logic. The embedding turns finite premise sets into finite premise sets. The partial decision methods for CL that are goal directed with respect to CL are turned into partial decision methods that are goal directed with respect to the embedded logics.

}, doi = {10.1007/s11225-015-9622-3}, author = {Verd{\'e}e, Peter and Batens, Diderik} } @article {DeBal2006, title = {A new approach to classical relevance.}, journal = {Studia Logica}, volume = {82}, year = {2015}, pages = {1{\textendash}31}, abstract = {

n this paper we present a logic that determines when implications in a classical logic context express a relevant connection between antecedent and consequent. In contrast with logics in the relevance logic literature, we leave classical negation intact - in the sense that the law of non-contradiction can be used to obtain relevantly implications, as long as there is a connection between antecedent and consequent. On the other hand, we give up the requirement that our theory of relevance can define a new standard of deduction. We present and argue for a list of requirements that such a logical theory of classical relevance needs to meet and go on to formulate a system that respects each of these requirements. The presented system is a monotonic and transitive logic that extends the relevance logic ℜ with a richer relevant implication that allows for Disjunctive Syllogism and similar rules. This is achieved by interpreting the logical symbols in the antecedents in a stronger way than the logical symbols in consequents. A proof theory and an algebraic semantics are formulated and interesting metatheorems (soundness, completeness and the fact that it satisfies the requirements for classical relevance) are proven. Finally we give a philosophical motivation for our non-standard relevant implication and the asymmetric interpretation of antecedents and consequents.

}, author = {De Bal, Inge and Verd{\'e}e, Peter} } @article {verdee2013non, title = {Non-monotonic set theory as a pragmatic foundation of mathematics}, journal = {Foundations of science}, volume = {18}, number = {4}, year = {2013}, pages = {655{\textendash}680}, abstract = {

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by G{\"o}del’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.

}, doi = {10.1007/s10699-012-9296-5}, author = {Verd{\'e}e, Peter} }