TThat stories may teach us some important lessons, e.g. about morals, is not new; what is a recent topic of discussion, however, is whether stories can be equally useful for logic. Can we learn something valuable about logical validity from stories? We address this problem in this paper. We first examine two opposing positions on that matter, with a positive and a negative answer to the question of whether stories may teach us something about logic. We shall then suggest that those positions both labor under the assumption that logic has a descriptive role aiming to correctly describing validity simpliciter. We argue that as a result of the use of stories in logic, such an assumption must be abandoned in favor of a more local view of logical consequence. The view makes sense of the use of stories in logic, and also accommodates the claim that the plurality of stories may be seen as leading us to no universally applicable logic at all.

We discuss how to formulate and understand contradiction. After emphasizing the importance of a correct formulation for a notion as important as the notion of contradiction, we present a variety of formulations of the proposition corresponding to “p and ¬p”, which is often considered as expressing contradiction. We then discuss the standard example of contradiction in classical logic and the way Wittgenstein defines contradiction in the Tractatus, not using negation. After that, we point out the variety of connectives which are nowadays called negations and often denoted by the same symbol, underlying that negations obeying neither the law of non-contradiction, nor the excluded middle are considered part of the family. We then recall how contradiction is defined within the theory of oppositions, drawing attention to the fact that this theory is against considering that the pair p and ¬p forms a contradiction if we take into account the whole family of negations, including paraconsistent negations. At the end of the journey, we set up a list of notions involving negation and contradiction and propose a terminology that may be useful to dispel confusion and promote understanding.

This paper offers an account of the ship of Theseus paradox along the lines of the so-called nonstandard primitivism about vagueness. This account is inspired by a model of the ship of Theseus paradox offered by Dinis that considers near-equality, in the context of Nonstandard Analysis, as the proper way to model the `same as' relation. The output is a class of models which unifies the semantic account of vague gradable adjectives recently proposed by Dinis and Jacinto with that of the `'same as' relation. It does so by taking both paradoxes to arise from a confusion between relations of marginal difference between vague degrees and ``"small'' precise relations between the things that have those degrees.

I In this paper, we investigate the nature of empirical hypotheses used in scientific reasoning and the act of formulating hypotheses. This is achieved through a novel logical framework in which we provide specific semantics for two types of hypotheses: a strong and a weak sense of hypothesis, each characterized by different logical structures. This framework enables us to better characterize certain aspects of hypothetical reasoning in scientific practice, especially when we attempt to rationally deny the content of an empirical hypothesis.

This paper compares two logical conditionals which are strengthenings of the strict conditional and avoid the paradoxes of strict implication. The logics of both may be viewed as extensions of KT, and the two conditionals are interdefinable in KT. The implicative conditional requires that its antecedent and consequent be both contingent. The consequential conditional may be viewed as a weakening of the implicative conditional, insofar as it also admits the case in which the antecedent and the consequent are strictly equivalent (either both necessary or both impossible). The two conditionals share a number of properties, among them Transitivity, Contraposition, Aristotle’s Thesis, Weak Boethius’ Thesis and Aristotle’s Second Thesis. They also share some restricted principles such as Possibilistic Monotonicity, Possibilistic Simplification and Possibilistic Right Weakening. They differ in relation to Identity, which is validated by consequential implication, while the implicative conditional only validates the restricted principle of Possibilistic Identity. The relations between the two conditionals are represented by two Aristotelian cubes of opposition, one involving the contrariety between If A, then B and If A, then ¬B, according to Weak Boethius’ Thesis, and the other the contrariety between If A, then B and If ¬A, then B, according to Aristotle’s Second Thesis. We also explore the relations between the two logical conditionals and natural language conditionals, emphasizing the dependence of the latter on the context, and the need to distinguish natural language conditionals which may be viewed as consequential or implicative, on one side, and concessive and some other types of conditionals, on the other.

It is possible to understand the expressive power of a logic as issuing from its capacity to express properties of its models. There are some ways to formally capture whether a property of models is expressible, among them is one based on the notion of definability, and one based on the notion of discrimination. If the logics to be compared are defined within the same class of models, one can employ the notions of definability and discrimination directly to obtain formal conditions for relative expressiveness. This paper studies generalizations of these formal conditions to cases where the compared logics are defined within different classes of models. There have been proposed in the literature formal conditions of two main kinds: with forward and with backward model-mappings. It is shown that none of them is adequate, despite their initial reasonableness. Moreover, we argue that general and reasonable formal conditions for relative expressiveness involving forward mappings are not likely to be found, given that they turn out to be highly dependent on specific features of the compared logics. On the other hand, it will be argued that there is a reasonable formal condition involving backward model-mappings.

It is an unusual property for a logic to prove a formula and its negation without ending up in triviality. Some systems have nonetheless been observed to satisfy this property: one group of such non-trivial negation inconsistent logics has its archetype in H. Wansing’s constructive connexive logic, whose negation-implication fragment already proves contradictions. N. Francez and Y. Weiss subsequently investigated relevant subsystems of this fragment, and Weiss in particular showed that they remain negation inconsistent. In this note, we take a closer look at this phenomenon in the systems of Francez and Weiss, and point out two types of necessary conditions, one proof-theoretic and one relevant, which any contradictory formula must satisfy. As a consequence, we propose a nine-fold classification of provable contradictions for the logics.

In the 1950s Peter Strawson analyzed the works of Bertrand Russell regarding fundamental definitions of meaning, sentences and truth value. The debate between them uncovered many issues that Fregean, truth-functional logics have when defining concepts from natural language. To reconcile the Fregean paradigm with the reality of language use, Strawson proposed the concept of presuppositions  necessary preconditions for the truth of other sentences. We believe that his proposition stemmed primarily from the problem caused by the fact that Fregean, truth-functional logics are not sensitive to the contents of sentences and reduce them to their logical values. This is bound to produce a mismatch between the way logic models reasoning and the way language users reason since real-life reasoning is performed on the contents of sentences and not their logical values. Inspired by the ideas of Strawson and Roman Suszko, who initiated the paradigm of non-Fregean logics, we propose a new solution to the debate between Strawson and Russell. In our solution, the content implication connective is used to express content relations between sentences. We move away from truth and falsehood as the sole two semantic correlates of sentences and instead work in a system where the contents of sentences are their semantic correlates.

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.

The principle of explosion asserts that any formula can be derived from any pair of other contradictory formulas. Paraconsistent logic is typically regarded as a logic in which the universal validity of this principle is questioned. Therefore, a key point is determining when the validity can be considered universal to classify a logic as paraconsistent. A pertinent example to illustrate this point is the calculus CB1 that admits the principle but only for negated formulas, i.e., from any set {α, ∼α} any other formula follows if and only if α is of the form ∼γ. Another example is Sette’s calculus P1, which is paraconsistent at the level of variables but not complex formulas. Both serve as compelling examples of the so-called borderline cases. In this paper, we examine several calculi expected to be paraconsistent at the level of literals. It means that a pair of formulas, α and ∼α, can yield any β if, and only if α is neither a propositional variable nor is its iterated negation. Furthermore, it is assumed that in some calculi presented here, β must adhere to specific restrictions. Once these conditions are satisfied, we refer to calculus as paraconsistent in a “controllable manner”.