George Boole Research Papers - Academia.edu (original) (raw)

The following are the slides of my presentation at the meeting "GBMS Theme 2: From Boole's Algebra of Logic to Boolean Algebra, and Beyond" within the BOOLE CONFERENCES, Cork, Ireland, 27 - 28 Aug 2015. The aim of the presentation was to... more

The following are the slides of my presentation at the meeting "GBMS Theme 2: From Boole's Algebra of Logic to Boolean Algebra, and Beyond" within the BOOLE CONFERENCES, Cork, Ireland, 27 - 28 Aug 2015. The aim of the presentation was to trace the notion of symbolic knowledge that can be found implicitly in George Boole’s Algebra of Logic. G. W. Leibniz introduced around 1684 the idea of symbolic knowledge (cogitatio caeca or cogitatio symbolica) in order to draw a fundamental distinction between forms of cognitive representations. It can be described as knowledge obtained by means of a sign system. Even if in Boole’s work the notion of symbolic knowledge is not explicitly mentioned, it was presupposed in the methodology of British algebraists at the beginning of the 19th Century that constituted the base for Boole’s algebraic approach to logic. In the paper the presence of different features of symbolic knowledge in Boole´s work will be shown. Boole conceived two basic ways of producing symbolic knowledge: (a) by manipulation of signs according to rules (this way would be knowledge by formal calculus in the stricter sense), (b) by application of the sign system to a new domain, so that new properties of the domain can be known. In the last case knowledge of formal structures is involved and it could be possible to speak of ‘structural knowledge’ as a form of symbolic knowledge. The following conclusions will be also drawn:
(1) The algebraic formulation of logic played in Boole a more pragmatic than semantic role, namely the solution of logical problems.
(2) Boole solved logical problems through ‘computations’ in algebra. However, compared with the preceding attempts at logical calculi in the 18th Century, Boole devised an algebraic structure for logic, so that a whole new perspective for the analysis of logic opened up.

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... more

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.

This note explores an important problem of the history of science, namely the influence of Indian logic on George Boole's The Laws of Thought. The theories that have been proposed to explain the origins of Boole's algebra have ignored his... more

This note explores an important problem of the history of science, namely the influence of Indian logic on George Boole's The Laws of Thought. The theories that have been proposed to explain the origins of Boole's algebra have ignored his wife Mary's claim that he was deeply influenced by Indian logic. This note examines this claim and argues that Boole's focus was more than a framework for propositions and that he was trying to mathematize cognitions as is assumed in Indian logic and to achieve this, he believed an algebraic approach was the most reasonable. By exploring parallels between his work and Indian logic, we can explain several peculiarities of his algebraic system.

Boolean induction, Bulletin of Symbolic Logic. TBA XX (201X) XXX–YYY. ► JOHN CORCORAN, Boolean induction. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: corcoran@buffalo.edu George Boole (1815–1864), founder... more

"Hypnosis ranks amongst the most fundamental ideas that made the Victorian age. Together with progress, creativity, techno-science and industrialization, evolutionism and its by-product eugenism, and, last but not least, the emergent... more

"Hypnosis ranks amongst the most fundamental ideas that made the Victorian age. Together with progress, creativity, techno-science and industrialization, evolutionism and its by-product
eugenism, and, last but not least, the emergent feminist movement, it gave a peculiar flavor to its main trait: the faith in the superiority (if not the superior rationality) of Western civilization and in its colonial duties."

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. We give fresh new emphasis on the goal-directed nature of deduction and on evidence that Aristotle’s practice... more

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. We give fresh new emphasis on the goal-directed nature of deduction and on evidence that Aristotle’s practice exemplifies goal-directedness even if his explicit theory does not deal with it. There is also new emphasis on the fact that the categorical syllogistic was intended as a prototype, like first-order monadic logic, and not as a “universal logic”, like Principia Mathematica. This paper adds to our appreciation of Aristotle penetrating insights into the nature of demonstrative knowledge while at the same time bringing to light more of the beginner’s oversights and omissions inevitable in original logical investigations.

This book is alleged to be a comprehensive but largely elementary description of mathematical logic including its historical development, its most important achievements and its implications for philosophy. Although the intended audience... more

This book is alleged to be a comprehensive but largely elementary description of mathematical logic including its historical development, its most important achievements and its implications for philosophy. Although the intended audience includes both mathematicians
and philosophers, 'the book presupposes no knowledge of logic and only high school mathematics". The author avows an aim "to stimulate long-lived interest by communicating some of the excitement and beauty of the subject". Details and techniques are eschewed in favor of emphasis on the development of an overview.
The book is also intended by the author to serve as an introduction to logic. His opinion of the shortcomings of most introductions to logic is perceptive and insightful.

CORCORAN INTRODUCES BOOLE’S LAWS OF THOUGHT. Corcoran, J. 2003. Introduction. In Boole,G. 1854/2003, vii-xxxv. Laws of Thought. Amherst, NY: Prometheus Books. Amazon.com book search: BOOLE-CORCORAN. INTRODUCTION The publication of The... more

CORCORAN INTRODUCES BOOLE’S LAWS OF THOUGHT.
Corcoran, J. 2003. Introduction. In Boole,G. 1854/2003, vii-xxxv. Laws of Thought.
Amherst, NY: Prometheus Books.
Amazon.com book search: BOOLE-CORCORAN.
INTRODUCTION
The publication of The Laws of Thought in 1854 launched modern mathematical logic.
The author George Boole (1815–1864) was already a celebrated mathematician specializing in the branch of mathematics still known as analysis. If, as Aristotle (384–322 B.C.E.) tells us, we do not understand a thing until we see it growing from its beginning, then those who want to understand modern mathematical logic should begin with The Laws of Thought. There are many wonderful things about this book besides its historical importance. For one thing, the reader does not need to know any mathematical logic. There was none available to the audience for which it was written—even today a little basic algebra and a semester’s worth of beginning logic is all that is required. For another thing, the book is exciting reading. The reader comes to feel through Boole’s intense, serious, and sometimes labored writing that the birth of something very important is being witnessed.
Of all the foundational writings concerning mathematical logic, this one is the most accessible to the non-expert and it has the most to offer the non-expert. The secondary literature on Boole is lively and growing, as can be seen from an excellent recent anthology (Gasser 2000) and a complete bibliography that is now available (Nambiar 2003). Boole’s manuscripts on logic and philosophy, once nearly inaccessible, are now in print (Grattan-Guinness and Bornet 1997). This is a good time to start to study Boole.

This paper is more a series of notes than a scholarly treatise. It focuses on certain achievements of Aristotle, Boole and Tarski. The notes presented here using concepts introduced or formalized by Tarski contribute toward two main... more

This paper is more a series of notes than a scholarly treatise. It focuses on certain achievements
of Aristotle, Boole and Tarski. The notes presented here using concepts introduced or formalized
by Tarski contribute toward two main goals: comparing Aristotle’s system with one Boole constructed
intending to broaden and to justify Aristotle’s, and giving a modern perspective to both logics. Choice
of these three logicians has other advantages. In history of logic, Aristotle is the best representative
of the earliest period, Boole the best of the transitional period, and Tarski the best of the most recent
period. In philosophy of logic, all three were amazingly successful in having their ideas incorporated
into mainstream logical theory. This last fact makes them hard to describe to a modern logician who
must be continually reminded that many of the concepts, principles, and methods that are taken to be
“natural” or “intuitive” today were all at one time discoveries.
Keywords: Counterargument, countermodel, formal epistemology, formal ontology, many‑sorted,
metalogic, one‑sorted, proof, range‑indicator, reinterpretation.

Mark Wilson argues that the standard categorizations of "Theory T thinking"— logic-centered conceptions of scientific organization (canonized via logical empiricists in the mid-twentieth century)—dampens the understanding and appreciation... more

Mark Wilson argues that the standard categorizations of "Theory T thinking"— logic-centered conceptions of scientific organization (canonized via logical empiricists in the mid-twentieth century)—dampens the understanding and appreciation of those strategic subtleties working within science. By "Theory T thinking," we mean to describe the simplistic methodology in which mathematical science allegedly supplies ‘processes’ that parallel nature's own in a tidily isomorphic fashion, wherein "Theory T’s" feigned rigor and methodological dogmas advance inadequate discrimination that fails to distinguish between explanatory structures that are architecturally distinct. One of Wilson's main goals is to reverse such premature exclusions and, thus, early on Wilson returns to John Locke's original physical concerns regarding material science and the congeries of descriptive concern insofar as capturing varied phenomena (i.e., cohesion, elasticity, fracture, and the transmission of coherent work) encountered amongst ordinary solids like wood and steel are concerned. Of course, Wilson methodologically updates such a purview by appealing to multiscalar techniques of modern computing, drawing from Robert Batterman's work on the greediness of scales and Jim Woodward's insights on causation.

There is something distressing in the fact that this book, coauthored by a reputable logician, published by a reputable press and favorably reviewed by reputable reviewers, is nevertheless so marred that it cannot begin to serve its... more

There is something distressing in the fact that this book, coauthored
by a reputable logician, published by a reputable press and
favorably reviewed by reputable reviewers, is nevertheless so marred
that it cannot begin to serve its avowed purpose. The claims made by
the authors, publishers and reviewers amount to an elaborate and
cruel hoax. Where the reader is lead to expect clarity, insight and
rationality instead he finds himself in a largely indecipherable swamp
scattered with bizarre little islands of Kafkaesque puzzles and Alicein-
Wonderland meaning shifts.

Studies handling the lights and the colors as key components of computation attracted the scientists and engineers since these studies are potentially applicable to the signal processing through optical interconnections between electronic... more

Studies handling the lights and the colors as key components of computation attracted the scientists and engineers since these studies are potentially applicable to the signal processing through optical interconnections between electronic devices. Here, a novel optical computing model is proposed by modifying the known optical parallel logic gates, after employing the printed colors as the input/output data, instead of shadowgram images projected on a screen. The proposed approach allows spontaneous and parallel Boolean operations by simply overlaying the colors printed on films or duplicating the prints on papers. By setting a limited number of color prototypes with known CIELAB color coordinates, prediction of color changes due to duplicated color printing and/or film-on-paper overlaying of the printed colors was performed through Boolean “AND” operation and their experimental confirmation after actual color reading was also performed. Furthermore, possible applications of this CIELAB-coded printable logic gate system in natural computing and development of novel color barcodes were discussed.

Between Art and Truth with MachineLearning: A brief history of ArtificialIntelligence from the code of Ur-Nammu to the Generative NeuralNetworks. From autofiction to the open collective author. English translation of the French original... more

Between Art and Truth with MachineLearning: A brief history of ArtificialIntelligence from the code of Ur-Nammu to the Generative NeuralNetworks. From autofiction to the open collective author. English translation of the French original “Entre l’art et la vérité“, in La Cinquième Saison: Revue littéraire romande, volume 6, ‘Portrait des robots’. 2019.

Did Boole create a new paradigm to replace Aristotle’s? Or did he merely show the untenability of the Aristotelian paradigm thus inadvertently revealing a vacuum to be filled by Peano, Russell, or someone else? We raise these and related... more

Did Boole create a new paradigm to replace Aristotle’s? Or did he merely show the untenability of the Aristotelian paradigm thus inadvertently revealing a vacuum to be filled by Peano, Russell, or someone else? We raise these and related questions in the course of reviewing Peckhaus’s rich study:“The mathematical origins of nineteenth-century algebra of logic”. It is widely agreed that a revolutionary paradigm-shift took place in logic in the 1800s. Logical research before 1800 was dominated by a single paradigm whose core is traceable to ancient Greek logic, principally Aristotelian. Logical research after 1900 has been dominated by a radically different paradigm governing the work of logicians such as Frege, Peano, Russell, Hilbert, Gödel, Tarski, and Church.
Justifying the word ‘mathematical’ in its title, the paper under review observes, whereas “the old logic” had been pursued almost exclusively by people regarded as philosophers, “the new logic” involved many people regarded as mathematicians. The paper addresses the issue of identifying the paradigm-shift: when did “the new logic” begin, who were the leading actors, what were the motivations, how did the paradigm gain acceptance?

This 1969 study, Corcoran's first essay, concerns logical systems considered as theories. By searching for the problems which the post-Boole systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria... more

This 1969 study, Corcoran's first essay, concerns logical systems considered as theories. By searching for the problems which the post-Boole systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as well as with logical truth and logical consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, "strong soundness" and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of deductions. The first deals with the concept of rigor or "gaplessness". The second is a completeness condition for a system of deductions. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual, post-Boole historical development of this area of logic. The author's subsequent historical research reveals that Aristotle's logic fits the most sophisticated post-Boole paradigm and that Boole's fits the intermediate paradigm. The upshot is that from this viewpoint, logic declined from Aristotle to Frege and thereafter developed to regain Aristotle's sophistication, which was not appreciated until modern logicians--especially Jaskowski--found it for themselves. ---John Corcoran, 2014

John Corcoran is a logician, philosopher, mathematician, linguist, and historian of logic. His philosophical work stems from trying to understand proof and demonstrative knowledge. This has led to research on several interconnected... more

John Corcoran is a logician, philosopher, mathematician, linguist, and historian of logic. His philosophical work stems from trying to understand proof and demonstrative knowledge. This has led to research on several interconnected topics: the interrelations of objectual, operational, propositional, and expert knowledge; the nature of logic; the nature of mathematical logic; information-theoretic foundations of logic; conceptual structures of metalogic; relationships of logic to epistemology and ontology; and roles of proof theory and model theory in logic. Corcoran’s interests, hypotheses, and conclusions continue to evolve but many are foreshadowed in his earliest works, especially his 1973 paper “Gaps between logical theory and mathematical practice”.
Corcoran’s papers have been translated into Arabic, Czech, Dutch, Greek, Italian, Korean, Persian, Portuguese, Spanish, Russian, Turkish, and Ukrainian. His 1989 signature essay “Argumentations and logic” has been translated into five languages and his 1999 instructional essay “Critical thinking and pedagogical license” has been translated into six languages. Several of his papers have been reprinted. His 2015 article “Existential import today”, co-authored with the Iranian logician Hassan Masoud, is currently first on its journal’s most-read list.

J M Keynes was very clear about the importance of the relational ,propositional logic that he was going to deploy in the A Treatise on Probability: “This chapter has served briefly to indicate, though not to define, the subject... more

Pragmatism, according to Peirce, “is a sort of instinctive attraction for living facts.” Living facts, like living organisms, are cellular. Interaction in cellular ontologies is not causal, in which objects touch, attract or repel each... more

Pragmatism, according to Peirce, “is a sort of instinctive attraction for living facts.” Living facts, like living organisms, are cellular. Interaction in cellular ontologies is not causal, in which objects touch, attract or repel each other, but semiotic, in which subjects communicate with each other across borders defined by coded interfaces or protocols. Peirce-inspired research by Fernando Zalamea suggests that a new pragmatist logic is emerging with the addition of “horosis,” coined from the Greek word for “border,” to the Kantian dualism of analysis and synthesis.

CONTENTS:
I. Articles, II. Abstracts, III. Books, IV. Miscellaneous, V. Reviews, VI. Refereeing

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow. In other words, It... more

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow. In other words, It is one thing for a given premise-conclusion argument to be valid or to be invalid and it is quite another thing for it to be shown either that the given argument is valid or that it is invalid. Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and countermodels are studied in formal semantics (model theory).
The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole's first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole's symbolic derivations. Boole's 1847 work is shown to be deficient in both respects. It is of course to be expected that Boole's work would fall short of modern achievements. But, while acknowledging Boole's greatness in other respects, we establish conclusively that Boole's criteria for validity and invalidity of premise-conclusion arguments is even inferior to Aristotle.

In the last two decades, a number of researchers have been engaged in the study of natural computing systems that employ physical, chemical, and biological properties as direct media for manifesting computations. Among such attempts,... more

In the last two decades, a number of researchers have been engaged in the study of natural computing systems that employ physical, chemical, and biological properties as direct media for manifesting computations. Among such attempts, studies focusing on the use of lights as key computation components in particular have attracted the attention of researchers and engineers, since these studies are potentially applicable to signal processing through optical intercon-nections between electronic devices. Our research team has recently been engaged in the study of a novel color-based natural computing model. Our recent works included using CIELAB-coded colors on printed-paper to compute Boolean conjunctions (AND operations). In this study, we performed Boolean operations based on CIELAB-coded colors by placing color-printed films over aluminum-coated reflectors with and/or without color. The results of the operations were gathered by testing the color codes printed on the films for negation or highlighting. This type of CIELAB-based color computing has a wide range of potential applications, such as a method for security or access control to secured systems. Such applications could match paired color keys on which the arrays of color codes could be printed and optically computed.

In 1847 Boole described “laws of the mental processes” that “render logic possible” [1, pp. 4–6]. He considered “mental acts”—operations applying to classes and yielding respective subclasses: Boole wrote: "Now the several mental... more

In 1847 Boole described “laws of the mental processes” that “render logic possible” [1, pp. 4–6]. He considered “mental acts”—operations applying to classes and yielding respective subclasses:
Boole wrote: "Now the several mental operations […] are subject to peculiar laws. [Some concern] repetition of a given operation or the succession of different ones. […] These will […] be ranked as necessary truths […] probably they are noticed for the first time in this Essay."
Our Boolean operation-class calculus [BOC] represents—probably for the first time in this lecture—Boole’s 1847 laws, but in modernized form.
This rich system is what Boole actually presented, only a distant ancestor of Boolean Algebra

Is Nature logical? Is uncertainty hardwired into Nature? Where do the paradoxes or aporia lie? Might paradoxes exist in the object (& not in the thinking)? What is the ontological status of mathematical objects? Are laws of Nature really... more

Is Nature logical? Is uncertainty hardwired into Nature? Where do the paradoxes or aporia lie? Might paradoxes exist in the object (& not in the thinking)? What is the ontological status of mathematical objects? Are laws of Nature really written in the language of mathematics? Is mathematics a form of philosophical speculation? Did mechanization precede mathematization? Are axioms more pragmatic than theoretical? Can all mathematics be reduced to (but not derived from) logic? ‘How’ is mathematics able to explain reality? Can the quantum or the cosmological scales be visualized using Euclidean geometry? Does theoretical physics sometimes invent its own mathematics to make progress? Can (sub/un- conscious) thoughts or dreams be concretized? Is pure mathematics an extension of deductive logic, & always incomplete? Can mathematics be semi empirical (as: axioms + data)? Is a causal explanation always necessary? Is it possible to have a theory of everything for anything? What is the future of the limits (& uncertainties) of mathematics? Would large scale collaboration and computation help push the limits? SynTalk thinks about these & more questions using concepts from theoretical physics & mathematics (Prof. Rajesh Gopakumar, ICTS-TIFR, Bangalore), statistics (Prof. Shyama Prasad Mukherjee, ex-University of Calcutta, Kolkata), & philosophy (Prof. Babu Thaliath, JNU, New Delhi).

CORCORAN ON SCHOLAR CORCORAN 2017 BIO The Iranian Association for Logic will publish a translation of one of my articles. They asked for a bio to introduce me to the Iranian readers. This is what will be published. But I would still... more

CORCORAN ON SCHOLAR CORCORAN 2017 BIO
The Iranian Association for Logic will publish a translation of one of my articles. They asked for a bio to introduce me to the Iranian readers. This is what will be published. But I would still welcome your frank suggestion for improvements. Who knows? Some other publication might want my bio.

R.P. Loui's 1991 article contains a number of errors as regards (a) some supposed conflict between Keynes's work in inductive logic and Russell's work in deductive logic at "…the beginning of this century, to Keynes' dispute with Russell... more

R.P. Loui's 1991 article contains a number of errors as regards (a) some supposed conflict between Keynes's work in inductive logic and Russell's work in deductive logic at "…the beginning of this century, to Keynes' dispute with Russell over logic and probability."and (b) overlooks that Keynes built on Boole's foundations .These errors detract from his overall assessment in favor of Keynes's position. Keynes's and Russell's two approaches are not opposed to each other ,but are supplementary. Loui's limited reading of commentary written by Russell on Keynes's inductive approach(Russell regarded Keynes's contribution as being the best available from 1912 through 1959)accounts for his errors.

Les noms de mathématiciens parmi les plus grands sont aussi des noms de philosophes. Mais le rapport entre mathématiques et philosophie qui est ainsi attesté n'a pas toujours été le même. Il est passé par trois états principaux dont il... more

Les noms de mathématiciens parmi les plus grands sont aussi des noms de philosophes. Mais le rapport entre mathématiques et philosophie qui est ainsi attesté n'a pas toujours été le même. Il est passé par trois états principaux dont il fallait faire le récit et tirer toutes les conséquences.

This book could have an immediate impact on current debates in history of logic. Among other things it could help clarify two “priority” issues. The first concerns the origin of symbolic logic, the distinctively modern form of logic per... more

This book could have an immediate impact on current debates in history of logic. Among other things it could help clarify two “priority” issues. The first concerns the origin of symbolic logic, the distinctively modern form of logic per se: a field of study which has had more or less continuous development since it was founded by Aristotle. In Prior Analytics, Aristotle addressed the two central problems of logic per se: (1) how to show that a given conclusion follows from given premises that do formally imply it and (2) how to show that a given conclusion does not follow from given premises that do not formally imply it. In a way, Aristotle was moving toward seeking a decisive test or criterion for determining if the conclusion follows and also one for determining if the conclusion does not follow. In particular, the first issue is whether any person or persons deserve to be regarded as having initiated symbolic logic and if so who and how.
The second, an entirely separate issue, concerns the branch of mathematics whose theorems are metatheorems about logics (or parts of logics or applications of logics) and which is known today as mathematical logic—even though it might with equal justice be called mathematics of logic. This includes two principle subfields: proof theory, more properly called derivation theory, which investigates purely syntactic entities called derivations, and model theory, more properly interpretation theory, which studies set-theoretic interpretations of formalized languages. In short, the issue is whether any person or persons deserve to be regarded as having founded mathematical logic and if so who and how. At stake is identification of the origins of proof theory and model theory.

Boole's Solutions Fallacy is the mistake of using " being a solution to an equation " as a sufficient condition for " being a consequence of it ". Let us say that an x-equation is an equation having x as its only unknown: x(x-1) = 0, xx =... more

Boole's Solutions Fallacy is the mistake of using " being a solution to an equation " as a sufficient condition for " being a consequence of it ". Let us say that an x-equation is an equation having x as its only unknown: x(x-1) = 0, xx = x, x = 0 and x = 1 are x-equations. Let us say that an Lx-equation (or left-x-equation) is an equation having x as its left term: x = x, x = (1-x), x = 0 and x = 1 are Lx-equations, but x(x-1) = 0 and xx = x are not. In Boole's logical algebra or algebraic logic, every Lx-equation that is a consequence of a given x-equation is a solution to it: x = 0 is an Lx-equation that is a consequence of xx = 0 and is a solution, 00 = 0. However, not every Lx-equation that is a solution to a given x-equation is a consequence of it; x = (1-x), x = 0 and x=1 are Lx-equations that are all solutions to x(x-1) = 0, but none are consequences. Boole never committed the fallacy in cases as simple as this. Nevertheless, as this paper shows, this fallacy is involved in many of Boole's trains of thought. Boole's confusion of solutions of equations with deductions from equations was so compete that he even spoke of deducing solutions. This fallacy is an integral part of one of Boole's proudest " discoveries " – that logic is a form of analysis or algebra. Boole thought that he had finally determined the true nature of logical deduction; he thought he discovered that deducing a consequence from given premises is really solving a system of equations. On page 1, Velleman wrote the following sentence. When you solve an equation for x you are using the information given by the equation to deduce what the value of x must be. COMMENT: Velleman is saying that solving an equation is deducing from it. In other words, solving an equation is deducing conclusions from it until a solution is reached. He did not limit or qualify his remark in any way. He did not say 'often', 'sometimes', etc. At the outset, we must be clear that solving an equation involves two steps: one of discovery and one of justification. In the discovery step we find a tentative solution, " what the value of x might be " , a trial solution. In the justification step, we verify that the tentative solution is correct. For example, for the presented equation 4 = (x + 1) 2 we guess x = 1 as a solution, which we verify by " deducing the presented equation from it " —not by deducing the solution from the presented equation. Velleman has this backward.

Review-essay: Corcoran reviews a recent structuralism paper Review of: Hodesdon, K. “Mathematica representation: playing a role”. Philosophical Studies (2014) 168:769–782. Mathematical Reviews. MR 3176431. ABSTRACT: This 4-page... more