Markus Gabriel’s pluralism and Graham Priest’s monism can be considered new forms of skepticism in contemporary philosophy. Gabriel considers ‘emptiness’ and Priest ‘nothingness’ as safe havens for logic and philosophy. In the same way, traditional skeptics also considered nothing secure. Furthermore, we observe in this book that whilst both Priest and Gabriel present rather eloquent theories, they nevertheless lack more substantial proofs, much like many important theories in philosophy, such as Leibniz’s monadology.

In this paper, we present two ways of modelling every epistemic formal conditional commitment that involves (at most) three key epistemic attitudes: acceptance, rejection and neither acceptance nor rejection. The first one consists of adopting the plurality of every mixed Strong Kleene logic (along with an epistemic reading of the truth-values), and the second one involves the use of a unified system of six-sided inferences, named 6SK, that recovers the validities of each mixed Strong Kleene logic. We also introduce a (...) sequent calculus that is sound and complete with respect to both approaches. We compare both accounts, and finally, we suggest that the plurality of Strong Kleene logic as well as the general framework 6SK are linked to formal epistemic norms via bridge principles. (shrink)

This note clarifies the significance of the proof of Gödel’s first disjunct obtained through the formalisation of Penrose’s second argument within the DTK system. It analyses two formulations of the first disjunct – one general and the other restricted – and dwells on the demonstration of the restricted version, showing that it yields the following result: if by F we denote the set of propositions derivable from any formalism and by K the set of mathematical propositions humanly knowable, then, given (...) certain conditions, F necessarily differs from K. Thus it is possible that K surpasses F but also, on the contrary, that F surpasses K. In the latter case, however, the consistency of F is humanly undecidable. (shrink)

We present a uniform characterisation of three-valued logics by means of a bisequent calculus (BSC). It is a generalised form of a sequent calculus (SC) where rules operate on the ordered pairs of ordinary sequents. BSC may be treated as the weakest kind of system in the rich family of generalised SC operating on items being some collections of ordinary sequents, like hypersequent and nested sequent calculi. It seems that for many non-classical logics, including some many-valued, paraconsistent and modal logics, (...) the reasonably modest generalisation of standard SC offered by BSC is sufficient. In this paper, we examine a variety of three-valued logics and show how they can be formalised in the framework of BSC. We present a constructive syntactic proof that these systems are cut-free, satisfy the subformula property, and allow one to prove the interpolation theorem in many cases. (shrink)

This paper will examine the Küng-Armstrong trilemma against Class Nominalism. We will see that combining Class Nominalism and Zermelo-Fraenkel set theory (ZF) can provide us with a sophisticated version of Class Nominalism, namely ZF-Class Nominalism, which successfully addresses the objection and leads to a moderate version of ineffabilism about the putative set-membership relation.

Accounting for the epistemic contribution of deduction has been a pervasive problem for logicians interested in deduction, such as, among others, Jakko Hintikka. The problem arises because the conclusion validly deduced from a set of premises is said to be “contained” in that set; because of this containment relation, the conclusion would be known from the moment the premises are known. Assuming this, it is problematic to explain how we can gain knowledge by deducing a logical consequence implied by a (...) set of known premises. To address this problem, we offer an alternative account of the epistemic contribution of deduction as the process required to deduce a conclusion or a theorem, understanding such a process not only in terms of the number of steps in the derivation but also, more importantly, in terms of the reason for or justification for every step. That is, we do not know a proposition unless we have a justification or proof of that proposition. With this goal in mind, we develop a justification logic system which exhibits the epistemic contribution of a deductive derivation as the resulting justified formula. (shrink)