This open access book makes a case for extending logic beyond its traditional boundaries, to encompass not only statements but also also questions. The motivations for this extension are examined in ...detail. It is shown that important notions, including logical answerhood and dependency, emerge as facets of the fundamental notion of entailment once logic is extended to questions, and can therefore be treated with the logician’s toolkit, including model-theoretic constructions and proof systems. After motivating the enterprise, the book describes how classical propositional and predicate logic can be made inquisitive—i.e., extended conservatively with questions—and what the resulting logics look like in terms of meta-theoretic properties and proof systems. Finally, the book discusses the tight connections between inquisitive logic and dependence logic.
This open access book is the first ever collection of Karl Popper's writings on deductive logic. Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His ...philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics. This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work.
CORE TYPE THEORY Abstract van Dijk, Emma; Ripley, David; Gutierrez, Julian
Bulletin of the Section of Logic,
01/2023, Letnik:
52, Številka:
2
Journal Article
Recenzirano
Neil Tennant's core logic is a type of bilateralist natural deduction system based on proofs and refutations. We present a proof system for propositional core logic, explain its connections to ...bilateralism, and explore the possibility of using it as a type theory, in the same kind of way intuitionistic logic is often used as a type theory. Our proof system is not Tennant's own, but it is very closely related. The difference matters for our purposes, and we discuss this. We then turn to the question of strong normalization, showing that although Tennant's proof system for core logic is not strongly normalizing, our modified system is.
One of the nice properties of the first-order logic is the compactness of satisfiability. It states that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in ...fuzzy logics will pose various kinds of compactness in these logics. In this article, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in H'{a}jek Basic logic, some new results are given around this issue. It will be shown that there are topologies on $0,1$ and $0,1^2$ for which the interpretation of all logical connectives of the Basic logic is continuous. Furthermore, a topology on first-order structures will be introduced for any similarity relation as well. Then by the same ideas as in continuous logic, the results around the compactness of satisfiability will be extended for Basic logic.
To endow computers with common sense is one of the major long-term goals of Artificial Intelligence research. One approach to this problem is to formalize commonsense reasoning using mathematical ...logic. Commonsense Reasoning is a detailed, high-level reference on logic-based commonsense reasoning. It uses the event calculus, a highly powerful and usable tool for commonsense reasoning, which Erik T. Mueller demonstrates as the most effective tool for the broadest range of applications. He provides an up-to-date work promoting the use of the event calculus for commonsense reasoning, and bringing into one place information scattered across many books and papers. Mueller shares the knowledge gained in using the event calculus and extends the literature with detailed event calculus solutions to problems that span many areas of the commonsense world. * Covers key areas of commonsense reasoning including action, change, defaults, space, and mental states. * The first full book on commonsense reasoning to use the event calculus. * Contextualizes the event calculus within the framework of commonsense reasoning, introducing the event calculus as the best method overall. * Focuses on how to use the event calculus formalism to perform commonsense reasoning, while existing papers and books examine the formalisms themselves. * Includes fully worked out proofs and circumscriptions for every example.
This book provides comprehensive and accessible coverage of the disciplines of philosophy of mathematics and philosophy of logic. After an introduction, the book begins with a historical section, ...consisting of a chapter on the modern period, Kant and his intellectual predecessors, a chapter on later empiricism, including Mill and logical positivism, and a chapter on Wittgenstein. The next section of the volume consists of seven chapters on the views that dominated the philosophy and foundations of mathematics in the early decades of the 20th century: logicism, formalism, and intuitionism. They approach their subjects from a variety of historical and philosophical perspectives. The next section of the volume deals with views that dominated in the later twentieth century and beyond: Quine and indispensability, naturalism, nominalism, and structuralism. The next chapter in the volume is a detailed and sympathetic treatment of a predicative approach to both the philosophy and the foundations of mathematics, which is followed by an extensive treatment of the application of mathematics to the sciences. The last six chapters focus on logical matters: two chapters are devoted to the central notion of logical consequence, one on model theory and the other on proof theory; two chapters deal with the so-called paradoxes of relevance, one pro and one contra; and the final two chapters concern second-order logic (or higher-order logic), again one pro and one contra.
The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, ...we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
What is the theory without power set? Gitman, Victoria; Hamkins, Joel David; Johnstone, Thomas A.
Mathematical logic quarterly,
08/2016, Letnik:
62, Številka:
4-5
Journal Article
Recenzirano
Odprti dostop
We show that the theory
, consisting of the usual axioms of
but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, ...replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of
in which ω
1
is singular, in which every set of reals is countable, yet ω
1
exists, in which there are sets of reals of every size
, but none of size
, and therefore, in which the collection axiom fails; there are models of
for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of
for which the Gaifman theorem fails, in that there is an embedding
of
models that is Σ
1
‐elementary and cofinal, but not elementary; there are elementary embeddings
of
models whose cofinal restriction
is not elementary. Moreover, the collection of formulas that are provably equivalent in
to a Σ
1
‐formula or a Π
1
‐formula is not closed under bounded quantification. Nevertheless, these deficits of
are completely repaired by strengthening it to the theory
, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach
.
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