Brouwer's Life, Art and Mysticism is the ideological manifesto of one of the greatest mathematical philosophers of this century. It is a seemingly contradictory declaration of romantic rebellion against rationalism and science by a man who brought constructivist rigor to mathematical and logical practice; the emotional plea of a fanatical environmentalist for a return to `nature', a defiant call to reject the formal trappings of society arising from a deep resentment of authority and of the intellectual and social aspects of the human make-up.

A translation by Walter P. van Stigt

Whitehead, in his famous book Process and Reality, proposed a definition of point assuming the concepts of "region" and "connection relation" as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper, proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.

The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To obtain a more precise mapping, both LJ and LDJ are extended by adding a "pseudo-difference" operator ∸ which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems ${\bf LJ}^{∸}$ and ${\bf LDJ}^{∸}$, and a simply consistent but $\omega$-inconsistent Set Theory with Unrestricted Comprehension Schema based on LDJ is sketched.

Given a classical theory T, a Kripke structure ${\bf K} = (K, \leq, (A_{\alpha})_{\alpha \in K})$ is called T-normal (or locally T) if for each $ \alpha \in K $, $ A_{\alpha} $ is a classical model of T. It has been known for some time now, thanks to van Dalen, Mulder, Krabbe, and Visser, that Kripke models of HA over finite frames $ (K, \leq) $ are locally ${\bf PA}$. They also proved that models of ${\bf HA}$ over the frame $ (\omega, \leq) $ contain infinitely many Peano nodes. We will show that such models are in fact ${\bf PA}$-normal, that is, they consist entirely of Peano nodes. These results are then applied to a somewhat larger class of frames. We close with some general considerations on properties of non-Peano nodes in arbitrary models of ${\bf HA}$.

This paper presents a constructive interpretation for the proofs in classical logic of $\Sigma^0_1$ -sentences and for a witness extraction procedure based on Prawitz's reduction rules.

We develop a set-theoretic semantics for Cocchiarella's second-order logical system ${\bf RRC^\ast}$. Such a semantics is a modification of the nonstandard sort of second-order semantics described, firstly, by Simms and later extended by Cocchiarella. We formulate a new second order logical system and prove its relative consistency. We call such a system ${\bf \equiv RRC^\ast}$ and construct its set-theoretic semantics. Finally, we prove completeness theorems for proper normal extensions of the two systems with respect to certain notions of validity provided by the semantics.

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by providing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fraïssé type theorem. We also give characterizations of elementary classes in equality-free logic. As a by-product we characterize the sentences that are logically equivalent to an equality-free one.