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A set of assertions is consistent provided they can all be true at the same time. Naive individuals could prove consistency using the formal rules of a logical calculus, but it calls for them to fail to prove the negation of one assertion from the remainder in the set. An alternative procedure is for them to use an intuitive system (System 1) to construct a mental model of all the assertions. The task should be easy in this case. However, some sets of consistent assertions have no intuitive models and call for a deliberative system (System 2) to construct an alternative model. Formal rules and mental models therefore make different predictions. We report three experiments that tested their respective merits. The participants assessed the consistency of temporal descriptions based on statements using “during” and “before.” They were more accurate for consistent problems with intuitive models than for those that called for deliberative models. There was no robust difference in accuracy between consistent and inconsistent problems. The results therefore corroborated the model theory.

We propose using rule languages to encode complex reasoning algorithms in a declarative way. This approach—which follows the classical slogan “Algorithm = Logic + Control”—promises to turn high-level specifications of logical calculi as systems of inference rules into declarative rule-based models that can be executed on state-of-the-art rule engines. Simple rule languages suffice for simple logics, and we review our results on using Datalog rules to reason in the description logic $$\mathcal {EL}$$ . For more expressive logics, a suitably expressive yet implementable rule language often seems to be missing. To fill this gap, we consider an extension of Datalog with sets, Datalog(S), that can be executed by modern existential-rule reasoners, and we use it to present a rule-based reasoning calculus for the expressive description logic $$\mathcal {ALC}$$ .

In the present paper, the small article by George Boole, published the next year after the publication of his famous work The Mathematical Analysis of Logic, is analyzed. We discuss to what extent it can be considered as an adequate and representative overview of his logical views. The mathematical, extralogical and philosophical foundations of Boole’s logic are explored. It is pointed out that the mathematical basis of his logical system was his research in the field of linear differential equations. Moreover, his pioneering discoveries were based on arbitrary philosophical and methodological assumptions that were not directly related to the field of logic and mathematics. Some of the original psychological theory of mind, its philosophy of mathematics, which presupposes the existence of a universal mathematical calculus, and some provisions of Immanuel Kant’s logico-philosophical doctrine should be recognized as such assumptions., Проводится разбор и анализ небольшой статьи Джорджа Буля, опубликованной на следующий год после выхода его знаменитой работы «Математический анализ логики». Исследуется вопрос о том, насколько ее можно рассматривать как адекватный и репрезентативный обзор его логической концепции. Также исследуются математические, экстралогические и философские основания логики Буля. Указывается, что математическим основанием его логической системы послужили его исследования в области линейных дифференциальных уравнений. Кроме того, в основании его пионерских открытий лежали произвольные философско-методологическим допущения, которые напрямую не относились к области логики и математики. Такими допущениями следует признать некоторую оригинальную психологическую теорию сознания, его философию математики, предполагающую существование универсального математического исчисления, и некоторые положения логико-философской концепции И. Канта.

McCulloch and Pitts were the first to use and Alan Turing’s notion of computation to understand neural, and thus cognitive, activity. McCulloch and Pitts’s contributions included (i) a formalism whose refinement and generalization led to the notion of finite automata, which is an important formalism in computability theory, (ii) a technique that inspired the notion of logic design, which is a fundamental part of modern computer design, (iii) the first use of computation to address the mind–body problem, and (iv) the first modern computational theory of cognition, which posits that neurons are equivalent to logic gates and neural networks are digital circuits.

The introduction of safety cases into the practice of safety assurance has revolutionized safety engineering. Via a ‘safety argument’, a safety case aims to explicate, and to provide some structure for, the kind of reasoning involved in demonstrating that a system is safe. To date, there are several notations for writing down safety arguments. These notations suffer from not having a well-founded semantics, making them deficient w.r.t. the requirements of a serious approach to engineering. We consider that a well-founded semantics for safety arguments ought to be based on logical principles in the form of a logical calculus. Logic is the basis for reasoning in mathematics, philosophy, and science, and the same should be true for safety reasoning. With this goal in mind, we take some steps towards constructing a logical calculus for safety arguments by exploring some of the features of this calculus. Moreover, we look into the essential role that evidence plays in safety arguments. Evidence sets apart safety arguments from their traditional logical counterpart, as assumptions in safety arguments must be grounded on (i.e., justified by) data from the empirical world. We present our thoughts on these matters, and illustrate them by means of examples. We consider that our work establishes a framework for discussing safety arguments in a more rigorous manner.

We analyze the McCulloch–Pitts seminal 1943 paper on a logical calculus for neural networks from the point of view of contemporary mathematical logic. Originally presented as a fragment of Carnap’s Language II, the cited calculus is regarded in this chapter as a simple propositional temporal language with past tense modalities. We rewrite proofs of main results of the McCulloch–Pitts paper using this language.

Mereology, the doctrine of the relations of part to whole and of parts to parts, has so far awoken the interest of only a small number of Leibniz’s scholars. Since the publication of the pioneering paper of Hans Burkhardt and Wolfgang Degen (Topoi 9(1):3–13, 1990), entirely devoted to Leibniz’s mereology, very few works have been published on the same topic. Moreover, these works tend to consider mereology in the general setting of Leibniz’s metaphysics and do not pay due attention to those essays where Leibniz systematically develops a mereological calculus. In the years 1686–1690, indeed, Leibniz wrote a series of essays, concerning the so-called ‘plus-minus calculus’, where a very interesting mereological doctrine is developed.

In spite of maintaining rigorously the formalistic position, this chapter considers the logical calculus not as a language but as a formal construction (of logicians and mathematicians). We make the following distinction: A calculus is exposed in a language, but we are calculating with something which is denoted by part of a language. An expression is a finite sequence of typographical signs. A language is considered as a set of expressions. Certain sub-sets of a language which only consist of theorems are called "systems." The symbols of the usual logical systems will be classified principally into symbols of variables and symbols of constants. The symbols of variables denote variables, whereas we understand by "variable" that which (in different connections) is identifiable with different fixed formal objects (the constants) that are the values of the variables. We classify now the symbols of variables and constants as propositional symbols, connectives, individual symbols, p-functional (or predicative) symbols, and operators.