Mathematical Logic

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

Logique, calcul et représentation : du point de vue des monades

by Sylvain Cabanacq

It's just an abstract - Full paper following very soon...

Cardinality quantifiers in MLO over trees

by Vince Bárány

The structure of logical consequence: proof-theoretic conceptions

by Ole Hjortland

PhD Thesis, University of St Andrews, 2009

The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple... more The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alternative approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic semantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources to provide a proof-theoretic semantics for both intuitionistic and classical logic.

We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic.

(2012c) Towards a Geometrical Syntax: a Formalization of Radical Minimalism

by Diego Gabriel Krivochen

Mathematical Logic Tutor-Propositional Calculus

by Antonio Moreno

The Semiring Properties of Boolean Propositional Algebras

by Shrisha Rao

Oral History of Sir Antony Hoare

by Jonathan Bowen

Oral History of Sir Antony Hoare, CHM Reference number: X3698.2007, Computer History Museum, USA. Hoare (Sir Antony, C.A.R.) Oral History, Cambridge, UK, recorded 8 September 2006.

See http://www.computerhistory.org/collections/accession/102658017

Oral History of Sir Antony Hoare.
Interviewed by: Jonathan P. Bowen.
Recorded: September 8, 2006, Cambridge,... more Oral History of Sir Antony Hoare.
Interviewed by: Jonathan P. Bowen.
Recorded: September 8, 2006, Cambridge, United Kingdom
CHM Reference number: X3698.2007.
© 2006 Computer History Museum, USA.

AN INVESTIGATION INTO THE NATURE OF MATHEMATICAL MEANING

by Jonathan Okeke

Invariancia lógica a través de espacios de Chu

by Julio Ostalé

The Manifold Concept of Logical Consequence: Tarski, Etchemendy and more

by Matteo Bianchetti

Draft only

I discuss Etchemendy's hidden assumptions that there is only one proper concept of logical consequence and that the... more I discuss Etchemendy's hidden assumptions that there is only one proper concept of logical consequence and that the main aim of logic is to investigate it. I show that the concept of logical consequence is essentially related to extra-logical ideas (from metaphysics, epistemology and concerning the nature and the aims of logic). I show that the concept of logical consequence is not central in Aristotle's works: his main problem is to show how we can justify a categorical sentence. I show that in the history of logic we find at least two great divisions of the considerations on the concept of logical consequence: following rules (e.g. Frege) and formal truth-preservation (e.g. Bolzano). I conclude with some reflections on this point.

Three levels of translation into many-sorted logic

by Julio Ostalé

On the Interpretation of the Propositional Calculus

by Tristan Haze

A fairly polished draft

The question considered is 'How can formulae of the propositional calculus be brought into a representational relation... more The question considered is 'How can formulae of the propositional calculus be brought into a representational relation with the world?'. Four approaches are discussed: (1) the denotational approach, on which formulae are taken to denote objects, (2) the abbreviational approach, on which formulae and connectives are taken to abbreviate natural-language expressions, (3) the truth-conditional approach, on which truth-conditions are stipulated for formulae, and (4) the modelling approach, on which formulae, together with either valuation- or proof-theory, are regarded as an abstract structure capable of bearing (via stipulation) a representational relation to the world.

The modelling approach is developed here for the first time. The simple technical apparatus used for this is then applied to two issues in the philosophy of logic. (1) I demonstrate a corollary or converse to Carnap's result that certain 'non-normal' valuation-functions can be added to the set of admissible valuations of formulae without destroying the soundness and completeness of standard proof-theories. This sheds considerable light on a recent thread of the inferentialism debate which involves dialectical use of Carnap's result. (2) I show how the approach can be extended to quantification theory, by defining a model-theoretic notion of validity equivalent to the usual one, but making use of a proof-theoretic apparatus in place of the device of assigning values to formulae. This sheds light on the close relationship between proof- and valuation-theory.

Z Logic and Its Consequences

by Jonathan Bowen

Martin C. Henson, Steve Reeves and Jonathan P. Bowen. CAI: Computing and Informatics, 22(4):381-415, 2003. In Dines Bjørner (ed.), special issue on The Logics of Formal Specification Languages. doi:10289/157

This paper provides an introduction to the specification language Z from a logical perspective. The possibility of... more This paper provides an introduction to the specification language Z from a logical perspective. The possibility of presenting Z in this way is a consequence of a number of joint publications on Z logic that Henson and Reeves have co-written since 1997. We provide an information as well as formal introduction to Z logic and show how it may be used, and extended, to investigate issues such as equational logic, the logic of preconditions, the issue of monotonicity and both operation and data refinement.

Equivalence Relations & Topological Automorphism Groups In Simple Theories

by Aviv Keren

M.Sc. Thesis

On interpretations of bounded arithmetic and bounded set theory

by Richard Pettigrew

(2009) in Notre Dame Journal of Formal Logic 50(2):141-152

In their paper 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which... more In their paper 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

Theorem 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic I
Delta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong’s interpretation of the arithmetic in the set theory. Instead, I am forced to produce a different interpretation.

Contradicciones y paradigmas: Un enfoque paraconsistente

by Lorenzo Peña

Publ. in:
Relativismo cultural y filosofía: Perspectivas norteamericana y latinoamericana
ed. by Marcelo Dascal
México: UNAM, 1992, pp. 43-81. ISBN 968-36-2779-X

Propone el presente trabajo las cuatro siguientes tesis:
1. Es verdadero el relativismo justificacional o de... more Propone el presente trabajo las cuatro siguientes tesis:
1. Es verdadero el relativismo justificacional o de avales: no hay aval a favor de una creencia que no sea relativo.
2. Es falso el relativismo de la verdad: no es verdad que ninguna creencia tenga verdad más que con relación a algún ente o punto de referencia.
3. Los relativistas han contribuido a proyectar luz sobre algunas cuestiones, como la búsqueda de algún género de convergencia.
4. Una convenientísima estrategia en pos de una convergencia puede articularse aplicando una lógica paraconsistente gradualística (infinivalente), e.d. una lógica que, dando cabida a grados de verdad y de falsedad, haga aceptable la existencia de creencias que sean [hasta cierto punto] verdaderas y [hasta cierto punto] falsas.

The present paper argues that:
1. warrant relativism is true -- any belief warrant is relative;
2. [truth] relativism is false (not every belief can be true only as regards some particular entity or reference-point);
3. there are valuable insights relativists have provided us with, one of them being the search for some kind of convergence;
4. a most convenient convergence policy can be articulated by applying a paraconsistent gradualistic (infinite-valued) logic, i.e. a logic which, by allowing degrees of truth and falseness, makes room for some beliefs being both [up to a point] true and yet [to some extent] false.

PALABRAS CLAVE.- contradicciones, paradigmas, relativismo, lógica paraconsistente

KEYWORDS.- contradictions, paradigms, relativism, paraconsistent logic

READING BETWEEN THE LINES IN CONSTRUCTIVE SET THEORY

by Raymond Turner

J Logic Computation (1997) 7 (2): 229-250.
doi: 10.1093/logcom/7.2.229

We formulate and investigate various ways of (conservatively) extending Martin-Löf's type theories with separation... more We formulate and investigate various ways of (conservatively) extending Martin-Löf's type theories with separation types and choice principles and demonstrate how these extenstions can be employed to formalize Bishop's mathematical practice of hiding and recovering witnessing information.

LOGICS OF TRUTH

by Raymond Turner

Notre Dame Journal of Formal Logic
Volume 31, Number 2, Spring 1990
Logics

This paper surveys three recent semantic theories of truth and
compares them from the perspective of their... more This paper surveys three recent semantic theories of truth and
compares them from the perspective of their underlying logics. In particular, the underlying logic of the Gupta-Herzberger theory is investigated, and an
analysis of modal logics of truth arising from this semantic theory is given.

A THEORY OF PROPERTIES

by Raymond Turner

A Theory of Properties
Ray Turner
The Journal of Symbolic Logic
Vol. 52, No. 2 (Jun., 1987), pp. 455-472
(article consists of 18 pages)
Published by: Association for Symbolic Logic
Stable URL: http://www.jstor.org/stable/2274394

Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all... more Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory
permitted the representation of the property which holds of everything which does
not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads
immediately to a contradiction. The subsequent history of foundational studies was
dominated by attempts to formulate theories of properties and sets which would not
succumb to the Russell argument. Among such are Russell's simple theory of types
and the development of various iterative conceptions of set. All of these theories
ban, in one way or another, the self-reference responsible for the paradoxes; in this
sense they are all "typed" theories. The semantical paradoxes, involving the concept
of truth, induced similar nightmares among philosophers and logicians involved in
semantic theory. The early work of Tarski demonstrated that no language that
contained enough formal machinery to respresent the various versions of the Liar
could contain a truth-predicate satisfying all the Tarski biconditionals. However,
recent work in both disciplines has led to a re-evaluation of the limitations imposed
by the paradoxes.
In the foundations of set theory, the work of Gilmore [1974], Feferman [1975],
[1979], [1984], and Aczel [I9801 has clearly demonstrated that elegant and useful
type-free theories of classes are feasible. Work on the semantic paradoxes was given
new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of
Gupta [I9821 and Herzberger [1982]. These papers demonstrate that much room is
available for the development of theories of truth which meet almost all of Tarski's
desiderata.