The Significance of the New Logic is a textbook of modern mathematical logic, written by Willard Van Orman Quine in Portuguese and published in São Paulo in 1944, after he had spent the southern winter of 1942 teaching at the Free School of Sociology and Politics. It sets out, in four chapters, the theory of composition, the theory of quantification, identity and existence, and class, relation, and number—the “new logic” of Frege, Russell, and Whitehead, offered to Brazilian philosophers and scientists as a working instrument. The present volume gives the book in English for the first time, alongside Quine’s 1942 São Paulo lecture on the American resurgence of logic, an early essay of his on logic, and a long editorial study of his philosophical development.
“Quine would go on to develop revolutionary ideas about semantic holism and ontology, and this book provides a snapshot of his views on logic and language at a pivotal stage of his intellectual development.”
Quine’s preface names the three ideals the systematization tries to reconcile: rigor in the theoretical details, convenience in practical applications, and simplicity of presentation, with simplicity the chief aim. The book is a manual, not a treatise, but it carries philosophical freight. Two doctrines make their first appearance in its pages: the distinction between pure and impure designation, with the example of Giorgione, and the virtual theory of classes, which lets logic in the strict sense do much of the work of class theory without admitting classes into its ontology. The editors argue that the book is driven as much by the philosophy of logic and ontology as by pedagogy, and that its treatment of meaning, synonymy, and necessity—later extracted into “Notes on Existence and Necessity”—grew out of Quine’s exchanges with Carnap and Tarski.
From Truth Tables to the Virtual Theory of Classes
The Brazilian Lectures and Their Book
The volume opens with the editorial apparatus surrounding the first full English translation of Quine’s O Sentido da Nova Lógica, a book he wrote in Portuguese from lectures delivered in Brazil in 1942. The editors note that Quine, who modestly called his Portuguese faulty, had portions mimeographed for students and wanted his Brazilian lectures preserved as a book for greater reach. Logic in Brazil had arrived through Jesuit scholastic courses and gained strength from positivism—the national flag adopted in 1889 still carries the motto Ordem e Progresso. Manuel Amoroso Costa’s 1929 work was the first Brazilian book to reference mathematical logic, and Vicente Ferreira da Silva’s 1940 booklet the first devoted to symbolic logic alone; Ferreira da Silva, a lawyer drawn to existentialism, assisted Quine and helped edit the book. Quine had reviewed that 116-page booklet, faulting its proposal to divide meaningful from meaningless propositions. The real inauguration of contemporary Brazilian logic came with Quine’s teaching at the Free School of Sociology and Politics of São Paulo from June to September 1942, yielding the book two years later.
The Four Chapters and Quine’s Preface
The book’s four chapters run from the theory of composition through quantification to identity and existence, ending with class, relation, and number. Quine’s preface says the systematization reconciles three ideals: rigor in theoretical detail, convenience in practice, and simplicity of presentation, the last being the main objective. Chapter I splits logic into theory of deduction and theory of classes, with deduction divided into composition and quantification. Chapter II credits Frege with most of modern logic’s content, quantification theory included. Chapter III stresses the contrast between object and symbol, treats identity as simple and fundamental, accepts only material existence for physical objects, lays out the principles of identity, argues for eliminating every substantive but pronouns, and takes up sense, synonymy, and necessity. Chapter IV shows how mathematics depends on abstract objects reducible to classes and attributes.
Quine at Harvard and the Resurgence of Logic
Quine taught mathematical logic at Harvard from 1936 to 1939, producing Mathematical Logic in 1940, still under the sway of Principia Mathematica; he called Frege’s Begriffsschrift of 1879 the true beginning of mathematical logic. Paradoxes and the possibility of defining notions on more fundamental concepts, he argued, fed the resurgence of logic, and he presented Gödel’s 1931 theorem in plain terms: given any systematization of logic, there will be logical and even arithmetical truths provably unprovable. The Brazilian turn in logic proper came in the 1950s with Farah, da Costa, Zimbarg, Castrucci, Teixeira, Hegenberg, and Barbosa. Aristotelian logic’s limits justified a new logic, urgent given mathematics’ higher infinity of infinite numbers and classes; paradoxes threatened but stimulated the field, and only a more sophisticated logic could secure mathematics’ foundations. The editors compare the book to Tarski’s Introduction to Logic.
The São Paulo Lecture and the Book’s Reception
On July 30, 1942, Quine gave a well-attended public lecture at the United States–Brazil Cultural Union, Os Estados Unidos e o Ressurgimento da Lógica, defending modern logic as revolutionary and promising; it appears here in translation. Bennett’s 1945 review warned that readers might think all important problems settled, noted the absence of exercises and classroom drill, and regretted the book reached only Portuguese readers. N. C. da Costa’s review recognized its importance for Brazilian logic, observing that several Brazilian logicians began their studies with it. Quine’s account of interpreting modal logic, and of the failure of intersubstitutivity for co-referential names in modal contexts, also stems from this Brazil work; he dictated a translation titled “Notes on Existence and Necessity,” published in 1943. The translation follows the 1944 first edition, with the 1996 second edition a reprint in modernized orthography whose preface Quine supplied and which is translated here; the São Paulo lecture uses the 2004 reprint. Three footnote types mark Quine’s original notes, his second-edition notes, and editorial notes. The editors add that his Portuguese was far from faulty: he had taken a second-year class at Harvard after a stay in the Azores and listened to Brazilian short-wave radio before flying to Rio.
The New Logic in Its Historical Moment
The 1930s and 1940s were years of upheaval in logic and its philosophy, when the new logic—modern mathematical logic rather than the old Aristotelian paradigm—gathered force across Europe and the United States, taken up by analytic philosophers, mathematical logicians, pragmatists, logical positivists, and the Polish School.
The Turn to Instrumentalism
The book closes by locating the theory of classes outside logic proper, in mathematics. Quine’s virtual theory of classes lets strict logic do much of the work usually assigned to class theory without positing any special logical ontology, though it cannot legitimate all of classical mathematics. The final pages describe Gödel’s proof and Quine’s estimate of its philosophical significance: neither logical nor mathematical truth can consist in provability, since no formulation of “theorem” can cover all truths formulable even in elementary arithmetic without also covering falsehoods—and all the more so for logic, which can express arithmetic and much else. Even the potential infinite is affected; Carnap’s notes from the previous year record that Quine and Tarski both had qualms about it and about Carnap’s reliance on possibility. Quine calls the upshot a victory for those who obstinately refuse to recognize abstract objects: mathematics is a useful tool “without content in the full sense,” a body of ideograms that behave as though meaningful because we derive theorems from assumed axioms by the methods of quantification theory. Greg Frost-Arnold’s footnote notes the close resemblance to a view Tarski expressed in 1941. So the book ends in instrumentalism about classical mathematics—justified by fruitful application to natural science, its statements not taken to describe anything abstract. Quine later swore off the instrumentalist treatment as he outgrew his finitistic and nominalistic leanings, yet points of contact with his mature philosophy remain: his indispensability arguments admit abstract mathematical posits but justify them by usefulness to physics. He held firm, though, that higher set theory, having no applications, is merely recreational. Acknowledgments and a references list follow, then the front matter and the start of the Introduction.
Escaping Cantorian Theory and Russell’s Crisis
The Introduction proposes breaking free of set theory’s problematic reaches by formulating mathematics’ fundamental principles so that they suffice for familiar, useful theories but not for “the more outlandish parts of Cantorian theory.” Deduction methods must be made explicit and studied intensively regardless. A pressing motivation arose at the century’s beginning, when Bertrand Russell discovered that reasoning principles tacitly accepted in mathematics could involve contradiction. This precipitated a crisis: deductive principles had to be made explicit, carefully formulated, and even revised.
Grelling’s Paradox and the Need for Foundations
Paradoxes entertain as well as trouble. Grelling’s concerns denotation: an adjective is heterological if it does not denote itself—’long’ and ‘monosyllabic’ qualify, ‘short’ and ‘pentasyllabic’ do not—so ‘heterological’ is heterological if and only if it is not. This depends on the non-logical notions of adjective and denotation; Russell’s original paradox, concerning the class of all classes not members of themselves, is purely logical. Such paradoxes created the need for logic’s current revival. A further motivation is the search for perspicuous foundations: Cantor’s transfinite numbers prompted attempts to define number and infinity from more fundamental notions, while the infinitesimal (absurd: positive yet infinitely small) was eliminated by Weierstrass’s theory of limits, and the imaginary number needed clarification too. The reduction program seemed independent of the new logic, yet analysis came to depend on exact understanding of class, relation, and words like ‘if,’ ‘not,’ ‘all,’ ‘some.’ Surprisingly, all mathematical notions proved reducible to auxiliary logical notions, and these to just three—corresponding to ‘neither…nor,’ ‘is,’ and ‘all,’ plus pronouns. That ‘2 + 2 = 4’ would be meters long in primitive notation does not diminish the result: pure mathematics is reducible to logic. The new logic is also a practical research instrument, as Gödel’s partial solution of the Continuum Problem shows.
What Logic Is, and What It Is Not
Logic applies to any objects whatsoever—not a universal science from whose laws the special sciences’ laws are inferred, but a general science included in all sciences as their common part, the common denominator Tarski speaks of. Logical truths are marked by logical vocabulary, reducible to a few words; a statement is logically true if only logical vocabulary occurs essentially, so that substituting for other components cannot make it false while substituting for logical words can. The author proposes no philosophical thesis about logical truth, particularly no conventionalism holding such truths established by arbitrary linguistic conventions: an analogous description would fit chemical truth, yet nobody concludes chemistry rests on convention. The demarcation depends on “occurring essentially,” whose definition depends on the general truth/falsity distinction, so it does not rest on notational features alone—we cannot know whether a word occurs essentially by inspecting only the statement and not the world. A purely notational criterion would practically guide discovery of logical truths, and no such criterion is hoped for in chemistry, but logical truth seems susceptible, since in practice we recognize it by inspecting the statement itself.
Quantifiers and the Ambiguities of Ordinary Speech
Quine begins by showing that a quantifier, like the tilde, governs the shortest phrase its parentheses allow. “Everything is not combustible” splits into two readings: the false claim that nothing burns, and the true denial that everything burns. The difference between these truths and falsehoods is purely a matter of grouping, and quantifier notation makes explicit what ordinary language conceals. “Everything” does not behave like an ordinary substantive; “something” and “nothing” are pseudo-substantives translatable by quantifiers plus denial. “No bird has fur” becomes a quantified conditional, and its denial, “Some birds have fur,” follows the same pattern.
Logical Pronouns and the Priority of Variables
The letters x and y are logical pronouns, auxiliaries to the quantifier: the quantifier answers to “everything is such that,” the pronoun to “it.” The common view that pronouns stand in for names is mistaken—pronouns are logically more fundamental than names. Distinct letters handle nested claims like “Every person understands some language,” and “Everything is identical to everything” reduces to (x)(y) x = y, unlike the trivial (x) x = x. Quine models these pronouns on algebraic variables but avoids the word “variables” to stress that no varying objects are involved, crediting Peano with noting the pronominal character of the variable.
Matrices, Schemata, and Complex Translation
Matrices such as “x = y” have the form of statements but contain free pronouns and are neither true nor false; schemata are mere diagrams and cannot occur inside quantified statements. Quantifiers contain only pronouns, never schematic letters, and we cannot define them by substitution of names, since some objects have no names. A tangled sentence about a radio salesman is then translated centripetally, introducing quantifiers step by step; how far to analyze depends on one’s aims, though some choices are simply wrong.
Vacuous Quantification and Quantificational Truth
A matrix lacking free “x” contains “x” vacuously, and such quantifiers may be dropped, now treated as ungrammatical. Quantificational truth is broader than compositional truth: a statement true under any change of components, such as “If something affects everything, then it affects itself.” A matrix is quantificationally valid if quantificationally true or convertible to such by prefixing quantifiers. Deriving valid matrices generally proves easier than deriving only those without free pronouns.
Rules of Proof and the Limits of Mechanism
Further rules allow replacing parts of a valid matrix with compositional equivalents, interchanging “(x)(fx * gx)” with “(x)fx * (x)gx,” and quantifying any valid matrix with respect to any pronoun. From these, forms are proved valid and reused to shorten later proofs. The method determines a class of valid matrices that exhausts all valid matrices—a result essentially due to Gödel. Yet finding a derivation is non-systematic; Church proved no mechanical criterion is possible, and for broader mathematical truths even mechanically authenticable proof becomes impossible, as Gödel’s earlier result showed.
Implication, Equivalence, and the Syllogism
Quantificational implication relates statements whose conditional is quantificationally true; equivalence is reciprocal implication. Quine illustrates proof-writing with abbreviations and shows that order matters when quantifiers are separated by a tilde—”(x) ~ (y)fxy” is not equivalent to “(y) ~ (x)fxy,” illustrated by divisibility among whole numbers. Turning to the syllogism, known since Aristotle, he formalizes “No β is γ and every α is β; therefore, no α is γ” and proves it, then examines variants. Traditional theory distinguished fifteen forms; another nine require a supplemental existence hypothesis, illustrated with “horned,” “unicorn,” and “hypomorph.”
Names, Premises, and the Monadic Theory
Traditional logic treated “Socrates is human” artificially, as a universal or particular. Quine argues it resembles the clause “x is human,” and that statements containing names can be proved without introducing names into the theory: use a free pronoun, establish the matrix’s validity, then derive the singular case—a principle of application lying outside pure quantification theory. He then works a deduction from premises about a guard and Fiorecchio’s subordinates, noting two restrictions: premises admit no substitution, and quantification must not capture a pronoun free in a premise. For monadic schemata a mechanical criterion exists, due to Löwenheim, resting on a quadruple transformation into canonical form; the general theory must rely on non-systematic search. Practical uses in insurance are noted, though the greatest value remains theoretical.
Identity and Its Confusions
Identity means sameness: every object is identical to itself and nothing else. Heraclitus’s river paradox confuses identity with time; Wittgenstein’s remark fails to distinguish the empty “Cicero = Cicero,” the false “Cicero = Catiline,” and the informative “
Possible Objects and the Meaning of ‘Pegasus’
Some philosophers widen the universe to include concrete objects that are merely possible, so that both Bucephalus and Pegasus fall within it; Bucephalus is actualized, Pegasus is not, and ‘exist’ is reserved for the actualized while ‘be’ covers everything, concrete or abstract. The motive is to make ‘Pegasus does not exist’ say something about an object. Quine finds the remedy useless: the trouble returns with ‘the spinster wife of Pegasus does not exist,’ since she belongs to no domain of possibles. There is no paradox in the original sentence at all; ‘Pegasus’ means something whether or not it designates, and the puzzle arises only from confusing meaning with designation.
Ontology and the Principle of Application
‘What is there?’ is a question about quantification, since the quantifier says ‘whatever x may be.’ Quantificational truths survive any choice of objects, provided there is at least one, yet ontology is not wholly indifferent: a quantification denying that x is the product of nine multiplications of 1 by 9 comes out false under classical arithmetic and true under other views, and ‘(x)~(x whinnies · x flies)’ fails under an ontology that counts Pegasus as a possible concrete object. Vocabulary alone does not betray ontology, since substantives can be used non-designatively; pronouns reveal the designative use. The principle of application licenses inference from quantification to instances built with names, and it lapses with a non-name like ‘Pegasus’: from ~(y) x = y we may not infer ~(y)~ Pegasus = y, nor generalize from (y)~ Pegasus = y to (∃x)~ x = y. The principle is one only by courtesy, valid when the substantive is a genuine name, and whether ‘Pegasus’ names is a question for natural science. Names, in any case, are theoretically dispensable by paraphrase.
Descriptions as Eliminable Abbreviations
Descriptions are singular substantives beginning with ‘the’—’the author of Os Sertões’ is (ιx) x wrote Os Sertões—and a description names when exactly one object satisfies its matrix. Since that book has one author, ‘Euclides da Cunha = (ιx) x wrote Os Sertões.’ Any context g(ιx)fx can be paraphrased into ~(y)~(gy · fy · (x)~(fx · ~x = y)), so descriptions are eliminable abbreviations; the paraphrase’s intuitive rightness depends on the description designating, but since we use descriptions only when we believe they do, every use can be so treated. Two ambiguities—the order of eliminating two descriptions, and which matrix plays the role—are settled by stipulations about shorter matrices and the alphabetical priority of the pronoun. Quine credits the elimination to Russell (1905), with Frege perhaps first. Proofs establish the equivalence of ‘y = (ιx)fx’ with ‘fy · (x)~(fx · ~x = y)’; for a description to exist is for it to designate, and the principle of application is then proved for descriptions.
Europize, and the Pronoun as Sole Vehicle of Reference
Any name can be turned artificially into a description: coining ‘europize’ for ‘to be Europe’ makes ‘Europe’ into ‘(ιx) x europizes.’ This rids the language of non-descriptive names but neither avoids presupposing objects nor analyzes their nature; it reduces all substantives to descriptions, after which the principle of application suffices generally. Since descriptions are eliminable, every substantive except the pronoun can go, leaving the pronoun as the only vehicle of reference. Ontological commitment is then simply the objects in the range of the quantifier.
Classes, Attributes, and the Claims of Mathematics
Mathematics rests on abstract objects—numbers, functions, relations, classes, attributes—reducible to classes or attributes; the nominalist must reject classical mathematics or treat it as a useful machine with a fictitious ontology. Classes are not spatial fusions: the class of the thirteen South American continental countries differs from the class of Brazil and the rest, since only the first contains Paraguay, though both yield the same continent. Classes and attributes differ only in identity: attributes may differ though holding of the same things, while classes with the same members are identical. The sole criterion of attribute identity is logical equivalence, which gives awkward results: ‘x > the number of planets’ and ‘x > 9’ are not logically equivalent, so a certain statement is false, yet substituting on the true identity ‘the number of planets = 9’ yields it from a truth. Occurrences of names inside attribute-names are therefore not purely designative—a restriction that makes attributes inadequate for mathematics. Sense attributes, colors and sounds, may be more intuitive, but the difficulty stands.
The Psychological and Logical Study of Reasoning
The book closes with an appendix on the nature and history of logic. Psychology, the author insists, is a science but no part of logic; epistemology, by contrast, is often treated as a subdivision of logic. Reasoning therefore has a psychological side and a logical side. Logic in the broadest sense takes in epistemology, but the current and correct sense covers only inductive and deductive logic, leaving epistemology to stand apart. Its practical aim is to establish rules that guide and ease reasoning—hence the old phrase “art of reasoning.” Inductive logic studies how natural science proposes general laws from a handful of observed cases. Any collection of facts can be fitted by infinitely many laws, just as infinitely many continuous curves can touch a few discrete points while wandering freely between and beyond them; choosing the law that fits what has not yet been observed depends on intelligent conjecture, which inductive logic advances largely through pure theory, above all statistics, a branch of mathematics. Deductive logic deals with reasoning that will not settle for conjecture: from given sentences, known truths or assumed hypotheses, it draws inexorable consequences. Mathematics lives on this, and natural science uses it to test conjectural laws, since a hypothesis clashing with established fact must be given up. The great recent progress has been in deductive logic, to which the word “logic” is increasingly confined; statistics belongs to mathematics, and the practical side of inductive logic usually goes by the name scientific methodology, much of which really belongs to epistemology.
Aristotle, Procrustes, and the Primordial Light of Reason
To guide reasoning, deductive logic must work largely in pure theory, with general and abstract structures, so “art of reasoning” sounds odd for something as theoretical as the new logic, today the theoretical basis of pure mathematics. Even Aristotle’s formal logic, chiefly the theory of the syllogism, was mainly theoretical and survived the Middle Ages intact; Kant could call formal logic perfected two thousand years earlier. Yet people habitually made fruitful deductions that existing logic could not touch. Suppose you know that everyone entering a building unaccompanied by a member of a certain firm is stopped by the guard, that some of Fiorecchio’s subordinates entered unaccompanied, and that the guard never stopped any of Fiorecchio’s subordinates; you conclude that one or more of those subordinates belong to the firm. Reducing such reasoning to traditional logic takes Aristotle’s methods plus those of Procrustes. Ancient logic illustrated only reasoning that needed no guide—all men are mortal, Socrates is a man, therefore Socrates is mortal. Complex reasoning was done in the primordial light of reason, as in mathematics, without any reform of formal logic.
Cantor and the Higher Infinities
The need for a new examination of deduction came chiefly from mathematics, with the arrival of the higher infinities. At the end of the nineteenth century Georg Cantor discovered multiple degrees of infinity: two classes can both be infinite though one is larger, and transfinite numbers measure such differences, the collection of transfinite numbers being itself infinite at a higher level. Cantor showed there are just as many even numbers as integers, as many integers as integers and fractions combined, but more real numbers than integers. Ordinary reasoning that leans on intuition or good sense fails here; beyond finite numbers and classes, or beyond the first degree of infinity, imagination is useless, and we must steer blindly by rigorous valid rules of deduction. Modern techniques proved their worth in the hands of Löwenheim, Skolem, and Gödel. Mathematicians who turn their backs on transfinite numbers resemble the March Hare in Alice in Wonderland, who drank tea because it was always six o’clock, his watch broken, shifting places around the table and complaining when Alice asked what they did after finishing the round. The theory deserves attention for coherence: Cantor built it on already accepted principles and entrenched deductive methods, and repudiating it would sacrifice familiar useful mathematics sharing the same foundations. Perhaps fundamental principles could be framed to suffice for familiar theories without the outlandish Cantorian parts, but even setting transfinite arithmetic aside, deduction’s methods must be made explicit and studied intensively.
Russell’s Paradox and the Crisis of Principles
An even more pressing motive arose at the start of the century, when Bertrand Russell discovered that principles of reasoning tacitly accepted in mathematics, and perhaps outside it, could involve contradiction—precipitating a crisis. Deductive logic’s principles had to be made explicit, carefully formulated, even revised. The paradoxes are curious and entertaining. One characteristic case turns on denotation: ‘human’ denotes every human being, ‘green’ every green thing, ‘long’ every long thing.
What the Book Argues About Logic and Ontology
The book’s most consequential claim is that logic proper has no ontology. Quine credits Tarski with the insight that whatever logic asserts is what can be asserted about the objects of any of the sciences; the theory of classes, by contrast, implies an ontology. The virtual theory of classes is the technical device that makes the contrast stick. Class-membership notation, with variables on both sides of ‘∈’, prima facie affirms the existence of classes; the virtual theory drops the primitive membership predicate and translates “y is a member of the class of f things” into the plain ‘fy’, so that no variable ranges over classes at all. Set-theoretic principles are then stated schematically, through predicate letters. The gain is ontological economy; the cost is strength, since the virtual theory is weaker than full arithmetic. Quine may have hoped it could supply the finitist, physicalistic mathematics Tarski wanted, one that would not require Russell and Whitehead’s implausible claim that infinitely many objects exist a priori. The war interrupted the project, and by the time Quine returned to set theory his nominalist commitment had faded.
Running alongside this is a stubborn refusal of positivist anti-metaphysics. Chapter III opens by rebutting the early Wittgenstein’s treatment of identity: the Tractatus view that identity statements are empty or nonsensical, endorsed by the Vienna Circle, is dismissed on the ground that it fails to distinguish objects from their names. Concatenating two distinct names of the same thing with the identity sign is perfectly sensible. Quine had already declined, in 1934, to treat the question “What manner of things are these, whose names are sentences?” as meaningless; he answered it instead, denying that there are such entities as propositions. Five years later he put the point bluntly: to dismiss the issue between nominalism and realism as a metaphysical pseudoproblem is to cut the Gordian knot too deep. In Brazil he made it part of his mission to show that the new logic makes ontology scientifically respectable again, capable of answering “What is there?” and “What is real?”—aiming, as one reviewer put it, to overturn the conclusions of the Vienna Circle.
Sense, Quotation, and the Giorgione Problem
The book’s treatment of meaning and modality is where its philosophical threads knot together. Two names can designate the same object without being synonymous: ‘Evening Star’ and ‘Morning Star’ pick out one planet by different criteria, so “the Evening Star is the Morning Star” is an astronomical truth, not a verbal one. Synonymy is needed for indirect quotation, which demands a synonymous sentence and not merely a logically equivalent one, and for the Kantian notion of the analytic: a statement is analytic if it is logically true or reducible to a logical truth by substituting synonyms, as “No bachelor is married” reduces to “No unmarried man is married.” Necessity is then defined off analyticity—’necessarily’ applied to a statement is true just when the statement is analytic—and ‘possibly’ and ‘impossible’ are defined in turn.
The famous example carries the argument. “Giorgione is so-called because of his size” says something true of the painter Giorgio Barbarelli, but the name occurs there in a dual role: something is said by means of it, and something is said of it, since the Italian suffix ‘-one’ connotes largeness. Substitute ‘Barbarelli’ and the sentence turns false; existentially generalize and it turns meaningless. The occurrence is designative but not purely designative, and quotation makes the point explicit: “Giorgione was called ‘Giorgione’ because of his size” survives both substitution and generalization. The same opacity infects ‘believes that’, ‘knows that’, ‘says that’, and ‘necessarily’. Pronouns cannot be sent into such positions: “There is something necessarily greater than 7” is meaningless, while “Necessarily something is greater than 7” is true. Quine’s extensionalist instinct—to state science wholly in extensional terms, modeled on extensional mathematics—runs up against these contexts, and he explains them with only physical objects and human language, tentatively, rather than positing Fregean senses.
A Textbook That Changed a Country’s Logic
The book’s afterlife is part of its significance. Brazilian logic before Quine ran through Jesuit scholastic courses and the positivist current that gave the national flag its motto, Ordem e Progresso; the first Brazilian book to reference mathematical logic was Amoroso Costa’s in 1929, and the first devoted to symbolic logic was Vicente Ferreira da Silva’s 1940 booklet—the one Quine reviewed, criticizing its proposal to divide meaningful from meaningless propositions. Ferreira da Silva assisted Quine in São Paulo and helped edit O Sentido. The true inauguration of contemporary Brazilian logic came with Quine’s visit, and N. C. da Costa’s review noted that several Brazilian logicians began their studies with the book. Bennett’s 1945 review, by contrast, warned that readers might think all important problems disposed of, and regretted that the book’s appeal was confined to Portuguese readers—a limitation this translation removes.
The book ends with Gödel. Quine describes the proof and draws out its philosophical upshot: neither logical nor mathematical truth can consist in provability, since no formulation of theoremhood can cover all the true statements formulable even in elementary arithmetic without also covering false ones. He calls the result a victory for those who obstinately refuse to recognize abstract objects, and treats mathematics as a useful tool “without content in the full sense”—ideograms that behave as though they have meaning because of the methods of quantification used to derive theorems from assumed axioms. That instrumentalism about classical mathematics is a position the mature Quine abandoned, though his later indispensability arguments, which admit abstract mathematical posits and justify them by their usefulness to physics, keep a point of contact with it. He never stopped holding that higher set theory, having no applications, is merely recreational.
