Mathematical Logic

In this thesis, we will study the * embedding of classical first-order logic in first-order S4, which is based on the * translation, originally introduced in Fitting (1970). The initial, main part is dedicated to a detailed modeltheoretic proof of the soundness of the embedding. This will follow the proof sketch in Fitting (1970). We will then outline a proof procedure for a proof-theoretic replication of the soundness result. Afterwards, a potential proof of faithfulness of the * embedding, read in terms of soundness and completeness, will be discussed. We will particularly highlight the many difficulties coming with it. In the final section, we will relate this discussion to the debate on notational variance in French (2019). We will do this by showing how a weaker version of French's notion of 'expressive equivalence' conforms to the model-theoretic soundness result. We will then conclude that the soundness result without completeness might contain rather little overall insight by relating it to the extensibility of classical logic to S4.

We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies on exactly which integers are involved (. Our interest was first drawn to the topic by the following surprising result—paraphrasing Russell’s page 66].

Russell 1919’s Fraction-Separator Theorem: the fraction whose numerator is the sum of the numerators of two unequal given fractions and whose denominator is the sum of the denominators is between the two given fractions—excluding cases when the sum of the denominators equals zero.

Although Russell gives none, proof is obtainable from page 270 of De Morgan 1831.
REFERENCE LINKS

Russell 1919’s fraction-separator theorem. Bulletin of Symbolic Logic. 24 (2018) 381.
https://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separator-theorem?source=link
https://www.academia.edu/34438558/2_Russell_1919_s_fraction-separator_theorem_090117.pdf
Semiotic confusions in Russell 1919. Bulletin of Symbolic Logic. 24 (2018) 382–3.
(Co-author: Kevin Tracy) https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC
https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC
HOW CAN THIS BE IMPROVED?

Abstract. The satis ability (SAT) problem is a core problem in mathemat-ical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computer-aided design, computer-aided manufacturing, machine vision, database, ...

In this work is we prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. In order to obtain this result we need to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasimodular form E2. We prove some auxiliary model completeness results with the same functions composed with appropriate change of variables. In the conclusion we make some remarks about the noneffectiveness of our proof and the difficulties to be overcome to obtain an effective model completeness result.

CITE AS: Corcoran, John. 1999.  “Laws of thought”.  Cambridge Dictionary of Philosophy.  R. Audi, Ed.  Cambridge: Cambridge UP. p. 489.

Laws of thought are laws by which or in accordance with which valid thought proceeds, or that justify valid inference, or to which all valid deduction is reducible. Laws of thought are rules that apply without exception to any subject matter of thought, etc.; sometimes they are said to be the object of logic. The expression, rarely used in exactly the same sense by different authors, has long been associated with three equally ambiguous expressions: the law of identity (ID), the law of contradiction (or non-contradiction; NC), and the law of excluded middle (EM).
The so-called “logical truths” of standard first-order logic as discussed in mainstream logic works such as Corcoran-Masoud 2014 and 2015 are uninterpreted sentences and thus not even true.
Nevertheless, they too have been called laws of thought even though they are not laws in any normal sense. Other people unacquainted with traditional usage are inclined to think that a law of thought is a law that might be arrived at by thought, which could make ‘law of thought’ coextensive with ‘law’.
Sometimes, these three expressions—ID, NC, and EM— are taken as propositions of formal ontology having the widest possible subject matter, propositions that apply to entities per se: (ID), every thing is (i.e., is identical to) itself; (NC) no thing having a given quality also has the negative of that quality (e.g., no even number is non-even); (EM) every thing either has a given quality or has the negative of that quality (e.g., every number is either even or non-even).

"Formal methods aim to apply mathematically-based techniques to the development of computer-based systems, especially at the specification level, but also down to the implementation level. This aids early detection and avoidance of errors through increased understanding. It is also beneficial for more rigorous testing coverage. This talk presents the use of formal methods on a real project. The Z notation has been used to specify a large-scale high integrity system to aid in air traffic control. The system has been implemented directly from the Z specification using SPARK Ada, an annotated subset of the Ada programming language that includes assertions and tool support for proofs. The Z specification has been used to direct the testing of the software through additional test design documents using tables and fragments of Z. In addition, Mathematica has been used as a test oracle for algorithmic aspects of the system. In summary, formal methods can be used successfully in all phases of the lifecycle for a large software project with suitably trained engineers, despite limited tool support.
"

Ristampa dell'omonimo Cap. 3 del Vol. III, Le discipline filosofiche, de La Filosofia, diretta da Paolo Rossi, UTET, Torino 1995, pp. 187-322; nuova ristampa in N. Abbagnano, G. Fornero, P. Rossi, Filosofia. Storie, Parole, Temi, Vol. 19, Corriere della sera, 2019, pp. 177-312.

Si Proust a su raconter les intermittences du coeur chez son Narrateur, Whitehead a observé sur Socrate les intermittences de l'intelligence. Les secondes ne parviennent à l'échelle des premières que dans la théorie de l'identité personnelle où devient pertinent chez Whitehead le concept d'ancestral d'un relation tel qu'il passe de Frege aux Principia Mathematica.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

1981. A note on categoricity and completeness, History and Philosophy of Logic 2, 113–19.

Current study of axiomatic method presupposes concepts and results of string theory and set theory. But axiomatic method was vigorously pursued in the last half of the nineteenth century without explicit reference to the theory of strings, which was not axiomatized until the 1930s (independently by Tarski 1934 and by Hermes
1938). For historical remarks on string theory see Corcoran, Frank and Maloney1974. Moreover, although it is natural to read modern set-theoretic ideas in early methodological work, it is clear that early postulate theorists such as Dedekind, Peano, Hilbert, Huntington and Veblen were not working in any formal set-theoretic framework. Indeed, close correspondence between modern precise explications of methodological concepts and imprecise formulations found in the writings of early postulate theorists suggests that the creators of modern mathematical logic, particularly Carnap, Tarski, and Church, may have been guided by the goal of putting the early ideas on a more mathematically precise basis.

"Science is simply common sense at its best; that is, rigidly accurate in observation and merciless to fallacy in logic."
~ Thomas Henry Huxley

Author Information: Professor, Department of Paediatrics, University of Florida Department of Pediatrics, University of Florida, Gainesville, Florida, U.S.A. 32610.

Truth-preservation, implication-preservation, and cognition-preservation.

This is one in a series of presentations designed to alert the philosophical community that claims made for the importance of truth-preservation are often overblown, frequently false, and—even when strictly speaking true--misleading and counter-productive. Truth-preservation per se is not nearly as important as implication-preservation; implication-preservation per se is not nearly as important as, cognition-preservation

First days of a logic course
This short paper sketches one logician’s opinion of some basic ideas that should be presented on the first days of any logic course. It treats the nature and goals of logic. It discusses what a student can hope to achieve through study of logic. And it warns of problems and obstacles a student will have to overcome or learn to live with. It introduces several key terms that a student will encounter in logic.

‘‘Complex’’ is a special attribute we can give to many kinds of systems. Although it is used often as a synonym of ‘‘difficult,’’ it has a specific epistemological meaning, which is going to be shared by the incoming science of complexity. ‘‘Difficult’’ is an object which, by means of an adequate computational power, can be deterministically or stochastically predictable. On the
contrary ‘‘complex’’ is an object which can not be predictable because of logical impossibility or because its predictability would require a computational power far beyond any physical feasibility, now and forever. For complexity refers to some observing system, it is always subjective, and thus it is defined as observed irreducible complexity. Human systems are affected by several sources of complexity, belonging to three classes, in order of descending restrictivity. Systems belonging to the first class are not predictable at all, those belonging to the second class are predictable only through an infinite computational capacity, and those belonging to the third class are predictable only through a trans-computational capacity. The first class has two sources of complexity: logical complexity, directly deriving from self-reference and Goedel’s incompleteness theorems, and relational complexity,
resulting in a sort of indeterminacy principle occurring in social systems. The second class has three sources of complexity: gnosiological complexity, which consists of the variety of possible perceptions; semiotic complexity, which represents the infinite possible interpretations of signs and facts; and chaotic complexity, which characterizes phenomena of nonlinear dynamic systems. The third class coincides with computational complexity, which basically coincides with the mathematical concept of intractability. Artificial, natural, biological and human systems are characterized by the influence of different sources of complexity, and the latter appear to be the most complex.

1972. Weak and Strong Completeness in Sentential Logic, Logique et Analyse 59/60, 429–34.  MR0337476 (49 #2245)

This is another study illustrating the fruitfulness of thinking of “logics” as three-part systems composed of a language, a semantics [or model theory] and a deductive system [or proof theory]. For simplicity we take a fixed language for sentential logic [or propositional logics] with a fixed standard two-valued semantics, but we “vary” deductive systems. In the 1960s when this paper was written, it was still widely assumed that a “logic” was “determined” by its theorems, by what it could “prove”. The vague assumption was that once the theorems were fixed, it was somehow automatic which conclusions were derivable from given premises.
The purpose of this paper is to give a method which weakens deriving power of a sentential logic while leaving proving power unchanged. In particular if the method is applied to a strongly complete logic the resultant will not have strong completeness but it will retain weak completeness. Moreover, because of the nature of the method two added facts obtain.
First, weak completeness of the resultant is always an obvious corollary to weak completeness of the operand, so that no new completeness proof need be constructed. Second, failure of strong completeness in the resultant is always obvious, so no multi-valued matrices need be constructed. The method applies to all sentential logics formulated using only axiom schemes and schematically statable rules. These results reaffirm, generalize, and simplify a result of Hiz [Journal of Symbolic Logic, 24 (1959), pp. 193-202].

Resumo: Neste artigo analisamos as críticas apresentadas por George Berkeley, em The analyst (1734), ao método das fluxões e à inconsistência intrínseca à noção de infinitésimo do cálculo diferencial e integral, introduzido por Isaac Newton. Procuramos mostrar que as críticas de Berkeley não eram de todo infundadas, uma vez que foram necessários quase duzentos anos para que viesse a ser introduzida por Karl Weierstrass a definição rigorosa de limite, que propiciou uma solução para o problema dos infinitésimos. São mencionadas ainda duas outras teorias contemporâneas, com abordagens distintas para a solução da questão do infinitésimo: a análise não-standard de Abraham Robinson e o cálculo diferencial paraconsistente proposto por Newton da Costa. Apesar de serem citados alguns autores importantes para o desenvolvimento do cálculo, este artigo não se propõe a analisar suas obras e não pretende apresentar uma história do cálculo diferencial e integral. Palavras-chave: George Berkeley, método das fluxões, infinitésimo, cálculo diferencial paraconsistente.

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

The purpose of this meta-analysis was to synthesize findings from 42 interventions (randomized control trials and quasi-experimental studies) on instructional approaches that enhance the mathematics proficiency of students with learning disabilities. We examined the impact of four categories of instructional components: (a) approaches to instruction and/or curriculum design, (b) formative assessment data and feedback to teachers on students' mathematics performance, (c) formative data and feedback to students with LD on their performance, and (d) peer-assisted mathematics instruction. All instructional components except for student feedback with goal-setting and peer-assisted learning within a class resulted in significant mean effects ranging from 0.21 to 1.56. We also examined the effectiveness of these components conditionally, using hierarchical multiple regressions. Two instructional components provided practically and statistically important increases in effect size–teaching students to use heuristics and explicit instruction. Limitations of the study, suggestions for future research, and applications for improvement of current practice are discussed.

Tarski’s proof of the law of identity

  Tarski’s LEIBNIZ’S LAW [Introduction to Logic, Sect. 17] is the second-order sentence in variable-enhanced English:

For everything x, for everything y: x = y iff
x has every property y has and y has every property x has.

  The Law of Identity, Tarski’s LAW OF REFLEXIVITY FOR IDENTITY, is:

Everything is equal to itself. For everything x: x = x.

  Tarski says it can be “derived” from Leibniz’s Law. Indeed, the paragraph following its statement begins with the word “PROOF” and ends with the words “which was to be proved”. This paper analyzes Tarski’s argumentation. We found several errors, four noted below.

Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come delle pietre miliari e come tali richiamati ancora oggi nella letteratura specialistica, investono aspetti fondamentali per la ricerca in campo matematico, quali la relazione tra verità e dimostrabilità o questioni relative alla natura e ai limiti della conoscenza matematica. Questo profilo nasce dal tentativo di correggere l’impressione che la grandezza del personaggio possa essere ricondotta a uno dei suoi molteplici contributi in particolare, come ad esempio il teorema di incompletezza per le teorie assiomatiche al quale egli deve gran parte della sua fama. Al contrario, il volume cerca di offrire una panoramica per quanto possibile esaustiva dell’opera logica e matematica di Gödel, così da valutarne meglio la portata.

Na busca de uma matemática anterior as formalizações, apresentamos uma pesquisa na qual enfocamos os aspectos ontológicos da lógica matemática existente e perceptíveis nos jogos digitais, particularmente no game paradigmático Myst. Demonstramos que por meio da resolução dos puzzles do jogo estamos desenvolvendo as estruturas cognitivas necessárias para a construção do conhecimento.

This is my completed PhD-thesis I never had chance to defend. It studies the famous problem in epistemic logic (logical omniscience) and proposes a method to solve it using some elements of the intuitionistic logic. It is written in Ukrainian but contains an English short summary.

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds.  Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001.

This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols.
It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes.
To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic.
Rejection of second-order logic is viewed as an incipient paradigm shift involving radical repudiation of a part of our scientific tradition that is defended by traditionalists. But it is also viewed as analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished.
Second-order logic may someday be gone, but it will never be forgotten.
Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor.
People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic.  And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject.  As Aristotle first said, we do not understand a discipline until we have seen its development.  It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been.

The focus of this study was to investigate mental computation conceptual frameworks that Heirdsfield (2001c) formulated to explain the difference between proficient (accurate and flexible) mental computers and accurate (but not flexible) mental computers. A further aim was to explore the potential for students’ developing efficient mental strategies.

The well-known necessary-consequence/necessary-consequent ambiguity—a species of segmentation, scope, and structural ambiguities—is found in conditional sentences such as:

(A) If zero is oblong, then necessarily some square is oblong.

A can read as expressing a proposition to the effect that “some square is oblong” is a necessary consequence of “zero is oblong”. This necessary-consequence segmentation parses A with the adverb as part of the discontinuous modal connective ‘if … then necessarily’. The consequent ‘some square is oblong’ is non-modal. Alternatively, the adverb can be taken instead as part of the consequent yielding the necessary-consequent segmentation: A can read as expressing roughly that the modal proposition “necessarily some square is oblong” is a material consequence of “zero is oblong”. Brackets indicate segmentation:

Necessary-consequence segmentation
[If] zero is oblong, [then necessarily] some square is oblong.

Necessary-consequent segmentation
If zero is oblong, then [necessarily some square is oblong].

Similar confusing ambiguities arise in logic texts where ‘valid’, ‘logical’, ‘necessary’, ‘sound’, ‘derivable’, and the like occur before ‘conclusion’, ‘inference’, ‘deduction’ and the like.

This paper, supplementing [1], compares corresponding segmentations of various conditional sentences. E.g., taking both “necessities” as logical, the following have different truth-values—first true, second false. The second implies the first.

[If] Twain is Clemens, [then necessarily] Clemens is Twain.
If Twain is Clemens, then [necessarily Clemens is Twain].

The first says that “Clemens is Twain” follows logically from “Twain is Clemens”; the second that “Clemens is Twain” is logically necessary if Twain is Clemens.