Abstract Aristotle’s Analytica Priora contains three systems of syllogistic proof, what are known as ‘figures’ 1–3. Aristotle states that all syllogistic inferences reduce to the first one but does not carry that through. It turns out that this reduction hides up to three nested instances of indirect proof. Syllogistic derivations are given as two-dimensional derivation trees—the only addition of substance to Aristotle’s proofs—with the result that nested indirect proofs can be resolved into a single such step, in perfect analogy to steps of normalization in natural deduction.

Abstract We take a close look at what is required to show the uniqueness of connectives in something very like Belnap’s sense. We find that negation demands more than conjunction, disjunction and the conditional. However, what it demands is quite in line with how we think of negation in a natural language but perhaps not in a formal language.

Abstract The representation theorem for odd or even involutive FL $$_e$$ e -chains via bunches of layer groups, as presented in [12], has been refined to reveal a more direct constructional relationship between these chains and bunches of layer groups, entirely circumventing the intermediary notion of layer algebras. Leveraging this refinement, it is demonstrated that both the variety of semilinear odd involutive FL $$_e$$ e -algebras and its idempotent symmetric subvariety admit densification. Finally, by applying the algebraic techniques developed in [14], the strong standard completeness of Involutive Uninorm Logic with Fixed Point ( $$\textbf{IUL}^{fp}$$ IUL fp ) is established, thereby strenghtening the main result of [11].

Abstract This paper is concerned with the properties of the prime spectrum of MV-algebras, namely the space of its prime ideals with the Zariski topology which is an important topological invariant of MV-algebras. The prime spectrum is a complete invariant in Boolean algebras, but in MV-algebras this is not the case. Nevertheless knowledge of the prime spectrum gives a lot of information: There are many properties of MV-algebras which depend only on the spectrum. We call these properties “topological” and we study them. For instance, we prove that simplicity and strong simplicity are topological as well as the class of hyperarchimedean MV-algebras and the class of regular MV-algebras, while the class of perfect MV-algebras and algebraically closed are not.

Abstract Models employing hyperreal probabilities can be used to represent epistemic agents that have both credences (represented by propositions whose probability has a non-unit standard part) and full beliefs (represented by propositions with a probability at most infinitesimally smaller than 1). This article introduces operations of belief contraction in such models. Contraction is a process in which a proposition that was initially fully believed becomes a credence. We investigate two operations for such probability-based belief contraction. The first operation, which employs Jeffrey revision, gives rise to an operation that is close to the AGM operation of full meet contraction. The second, somewhat more complex, operation is of a more general type, and has AGM partial meet contraction as a special case. In both models, standard relations of epistemic entrenchment can be constructed from the probability function, which reveals a possibly somewhat surprising connection between probability and epistemic entrenchment.