Gödel’s ontological proof for the existence of God – as an inference from the level of rational, logical structure to the level of being – was strongly influenced by Leibniz’s monadology, which is based on the Ur-Monade – God – conceived as absolutely infinite. Gödel defines God as the maximum of positive properties and he characterizes this maximum (of being) with his postulates implicitly as an ultrafilter structure. Using maximality as criterion we have compared the set of positive properties in Scott’s version of Gödel’s theistic theory with modified variants of it to see whether a maximum is still achieved, or not, and what effect this has on modal collapse. The motivation for these, typically weakened, variants of Gödel’s theistic theory has been to avoid modal collapse and the limited determinism associated with it, and to show that the necessary existence of God can still be proved with less strong axioms. A weakening of the maximality criterion of the ultrafilter structure imposed by Gödel’s (and Scott’s) postulates have thus been applied to the set of positive properties in order to allow for alternative models in which modal collapse does not hold. In these alternative models, however, the maximality criterion for the basic quantity of positive properties is abandoned, so that in them God as a “maximum” is not reached. We argue that this is contrary to Gödel’s intention, and that for his conception of God, maximality and absolute infinity are non-negotiable. What Gödel had in mind, as we argue here, was a holistic – absolutely infinite – view of the world, i.e. with modal collapse, but without limited determinism.

What does a Cretan mean when he says that all Cretans are liars? What is his intention? While formal logic only relates to the truth values of the Liar paradox, we relate to its normative and social aspects. We argue that such utterances are used to imply that certain behaviors, even if despicable, constitute local norms. One may posit such claims either to point out that he has transcended the local culture, to socialize others into local customs, or to deflect from being caught lying. This paradox exemplifies group self-deprecation, a communicative practice intended to get us to disagree, rather than agree, with the disparaging claim and blunt the negative consequences of poor behavior. Its rhetoric relies upon \emph{tu quoque}, \emph{secundum quid} and naturalistic fallacies.

The paper “Tautology elimination, cut elimination, and S5” published in this journal presents a novel method for establishing by proof analysis the admissibility of the rule of tautology elimination for certain sequent calculi. Since tautology elimination will typically imply the admissibility of cut, the method promises a new path to show the admissibility of cut for cut-free calculi on which the standard techniques within structural proof theory seem inapplicable. This paper shows that the method as presented involves an error.

In this paper, we analyze Fitch’s paradox of knowability in the framework of fusions of epistemic and alethic modal logics. The paradox arises from accepting the knowability principle, which states that all truths are knowable. However, this leads to the unacceptable conclusion that all truths are known. We introduce a logical system that incorporates all assumptions used by Fitch in his original reasoning, including the knowability principle. We present a natural semantics for this logic, proving the soundness and completeness theorem. Additionally, we present a new semantic proof of the knowability paradox, demonstrating that the problematic conclusion can be derived independently of Fitch’s original proof and showing that the knowability principle itself is the source of the paradox. Using the formal tools introduced, we conduct a semantic analysis of the paradox, which allows us to identify the root cause of its occurrence. Finally, we propose a weakened version of the knowability principle that avoids paradoxical conclusions.

The concept of a causally connected set of point-events plays a crucial role in the point-eventistic definitions of a thing and a process formulated by Zdzisław Augustynek. Unfortunately, Augustynek’s approach to causal connectivity is open to a serious objection that has so far gone unnoticed, stemming from the way causal interactions are viewed in contemporary physics. This finding can hardly be considered favorable for an advocate of the ontology of point-eventism. The aim of this paper is, therefore, firstly, to discuss this objection in detail and, secondly, to share some ideas on how to deal with the problem.

According to quantifier generalism, all facts about the world can be expressed in a language devoid of proper names, whose only referential expressions are variables bound by quantifiers. This paper considers and repels some of the recently raised objections against this position. The central part of the paper presents a critical analysis of the claim advanced by Ted Sider that quantifier generalism is inevitably holistic and therefore requires unusually strong expressive resources when applied to infinite domains. Using an example of arithmetic, it is shown that there is a simple generalistic description of natural numbers that does not resort to any infinitary conjunctions or quantifiers. Such generalistic accounts also exist in many cases involving continua (such as descriptions of matter distribution in a continuous space). Moreover, these accounts are arguably superior to their individualistic counterparts due to their parsimony. In addition to that, Sider’s argument alleging that generalism cannot account for the difference between non-isomorphic models of arithmetic is repelled.

An expression occurs essentially in a formula (or sentence) when it occurs in every formula equivalent to the given formula, taking equivalence as logical equivalence relative to the logic in play in the discussion. Setting aside various niceties, this amounts to provable equivalence if that logic is presented via some proof system, and to valid equivalence if the salient characterization is couched in semantic terms. This notion of essential occurrence, or an informal analog thereof, has found its way into numerous philosophical discussions over the past seventy or more years, and here we tease out some issues of specifically logical interest it presents, stretching that description somewhat so as to subsume under it the frequently mooted connection between the essential occurrence of a singular term in a sentence and that sentence’s being genuinely about what the term denotes. This connection, stressed originally by Nelson Goodman, is touched on in several sections in the main body of the paper, but especially in §4, where it is contrasted with an alternative suggestion due to R. Demolombe and L. Fariñas del Cerro. Some issues raised by this and other parts of the discussion are also treated in several longer notes (referred to by means of letters A, B, . . . , K) which are postponed to an Appendix (§5) of roughly the same length as the main body of the paper. This enables readers with a special interest in one or more topics to consult them selectively, while allowing those with no such interest to avoid involvement with the further details supplied in the associated longer note(s).

This paper presents the history of the world’s first school of paraconsistency: the Torunian School of Paraconsistent Logic. Its founding father is the Polish logician Stanisław Jaśkowski, who first formulated a system of paraconsistent logic. Both the approach he presented and the work of subsequent generations of logicians allow us to speak of the entire school. In this article, we present the characteristics of this school, comparing them with those of other national, but later, schools of paraconsistency. In addition to extensive factual information concerning Jaskowski’s discovery and subsequent development of this topic, we also refer to and comment on the latest works presented in this volume.

The paper addresses Øhstrøm and Hasle’s argument against regarding Jerzy Łoś’s “axiomatization of a fragment of the physical language” as first temporal logic. It is pointed out that the arguments are insufficient to establish their claim.

The Doxastic Arrogance Paradox, DAP, states that the belief that a proposition is an item of knowledge implies that the proposition holds/is true. Some possible sources of DAP, different from the negative introspection principle for knowledge, are pointed out. The emplacement of DAP among related paradoxes of epistemic-doxastic logics is characterized. Finally, some profits of identification of sources of DAP for the philosophical analysis of knowledge and belief are pointed out.