402,981 research outputs found
Formal logic: Classical problems and proofs
Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational
Deductively Sound Formal Proofs
Could the intersection of [formal proofs of mathematical logic] and [sound deductive inference] specify formal systems having [deductively sound formal proofs of mathematical logic]?
All that we have to do to provide [deductively sound formal proofs of mathematical logic] is select the subset of conventional [formal proofs of mathematical logic] having true premises and now we have [deductively sound formal proofs of mathematical logic]
From Logical Calculus to Logical Formality—What Kant Did with Euler’s Circles
John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations
Developing an ontology of mathematical logic
An ontology provides a mechanism to formally represent a body of knowledge. Ontologies are one of the key technologies supporting the Semantic Web and the desire to add meaning to the information available on the World Wide Web. They provide the mechanism to describe a set of concepts, their properties and their relations to give a shared representation of knowledge. The MALog project are developing an ontology to support the development of high-quality learning materials in the general area of mathematical logic. This ontology of mathematical logic will form the basis of the semantic architecture allowing us to relate different learning objects and recommend appropriate learning paths. This paper reviews the technologies used to construct the ontology, the use of the ontology to support learning object development and explores the potential future use of the ontology
Unpacking the logic of mathematical statements
This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61students in six small sections of a “bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or evaluate their validity
The Informal Logic of Mathematical Proof
Informal logic is a method of argument analysis which is complementary to
that of formal logic, providing for the pragmatic treatment of features of
argumentation which cannot be reduced to logical form. The central claim of
this paper is that a more nuanced understanding of mathematical proof and
discovery may be achieved by paying attention to the aspects of mathematical
argumentation which can be captured by informal, rather than formal, logic. Two
accounts of argumentation are considered: the pioneering work of Stephen
Toulmin [The uses of argument, Cambridge University Press, 1958] and the more
recent studies of Douglas Walton, [e.g. The new dialectic: Conversational
contexts of argument, University of Toronto Press, 1998]. The focus of both of
these approaches has largely been restricted to natural language argumentation.
However, Walton's method in particular provides a fruitful analysis of
mathematical proof. He offers a contextual account of argumentational
strategies, distinguishing a variety of different types of dialogue in which
arguments may occur. This analysis represents many different fallacious or
otherwise illicit arguments as the deployment of strategies which are sometimes
admissible in contexts in which they are inadmissible. I argue that
mathematical proofs are deployed in a greater variety of types of dialogue than
has commonly been assumed. I proceed to show that many of the important
philosophical and pedagogical problems of mathematical proof arise from a
failure to make explicit the type of dialogue in which the proof is introduced.Comment: 14 pages, 1 figure, 3 tables. Forthcoming in Perspectives on
Mathematical Practices: Proceedings of the Brussels PMP2002 Conference
(Logic, Epistemology and the Unity of the Sciences Series), J. P. Van
Bendegem & B. Van Kerkhove, edd. (Dordrecht: Kluwer, 2004
Wittgenstein on Pseudo-Irrationals, Diagonal Numbers and Decidability
In his early philosophy as well as in his middle period, Wittgenstein holds a purely
syntactic view of logic and mathematics. However, his syntactic foundation of logic
and mathematics is opposed to the axiomatic approach of modern mathematical logic.
The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers
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