Logic

Every argument tries to convince us of something on the strength of something else: from certain premises, a conclusion is drawn. But what makes that move legitimate? Why does “all men are mortal; Socrates is a man; therefore Socrates is mortal” strike us as impeccable, while “some swans are white; this animal is a swan; therefore this animal is white” does not? The answer to that question — what makes a piece of reasoning hold — is the business of logic, one of philosophy’s oldest and most technically demanding disciplines. Born as a reflection on the art of demonstration and debate, logic has traveled twenty-five centuries to become, in the twentieth, a formal science capable of clarifying — and also of exposing the limits of — mathematical reason itself. ...

When a mathematician states that 2 + 2 = 4, the claim seems indubitably true. But what exactly makes it true? Does the number 2 exist somewhere — in a Platonic realm, on a sheet of paper as a symbol, or as a construction of the mind? And how do we come to know mathematical truths without observing any empirical fact? These questions, deceptively simple at first glance, lie at the heart of one of philosophy’s most demanding fields: the philosophy of mathematics. In the nineteenth and twentieth centuries they ceased to be peripheral speculation and became an intellectual emergency, triggering one of the most fertile crises in the history of thought — one whose reverberations still shape logic, computation, and contemporary philosophy of mind. ...

Picture a universe made up of an infinity of points of view, each of them a minuscule soul that, without ever looking out of a window, mirrors the entire cosmos within itself — and which is nonetheless in perfect agreement with all the others, as though every clock in a vast hall had been set once and for all to strike the same hour for eternity. This is the world of Gottfried Wilhelm Leibniz (1646–1716): the last great universal scholar of the modern age — mathematician, logician, jurist, diplomat, historian and theologian — who attempted the most ambitious of syntheses, reconciling the mechanical science of his time with a metaphysics of substances and with Christian theology. His philosophy is at once one of the most coherent systems ever built and one of the bridges that connect Aristotle to the mathematical logic of the twentieth century. ...

Mário Ferreira dos Santos (1907–1968) is, in sheer scale of output, the largest systematic philosophical project ever attempted in Brazil. His Encyclopaedia of Philosophical and Social Sciences, planned in more than fifty volumes, is an undertaking without parallel in Brazilian intellectual history — and rare even in the international context of the 20th century, in which academic philosophy progressively abandoned systematic form. Whatever limitations one may identify in the work — and they exist —, Mário Ferreira’s project deserves serious academic examination: for its scope, its theoretical ambition and the singular position it occupies between Thomist scholasticism, Pythagoreanism, phenomenology and the German dialectical tradition. This article presents the life, the work and the central doctrine — Concrete Philosophy — with critical rigour and without hagiography. ...

In 1996, the Brazilian philosopher Olavo de Carvalho published a short, dense book titled Aristóteles em Nova Perspectiva: Introdução à Teoria dos Quatro Discursos (Aristotle in a New Perspective: Introduction to the Theory of the Four Discourses). Regardless of the controversies surrounding the author as a public figure, this book represents his most philosophically articulate contribution and deserves to be examined on its own terms. Its central thesis — that Aristotle’s Organon, together with the Rhetoric and the Poetics, forms a unified hierarchy of four types of discourse — engages real problems in the history of philosophy and shares ground with contemporary neo-Aristotelian readings by Pierre Aubenque, Enrico Berti, Alasdair MacIntyre and Martha Nussbaum. ...

Abelardo (Peter Abelard) The most brilliant and controversial intellectual of the 12th century. Also famous for his tragic love affair with Heloise (castrated by order of her uncle). He proposed conceptualism in the dispute over universals and introduced dialectical reason into theology. “I understand in order to believe” — reason must examine before accepting by faith (an inversion of Anselm of Canterbury). Born in Brittany into a family of the minor nobility, Abelard renounced his military inheritance to devote himself to the liberal arts, becoming an itinerant master who drew crowds of students to the schools of Paris. His career spans the decisive moment when teaching ceased to be a monopoly of the monasteries and migrated to the urban schools, foreshadowing the medieval university. A tireless polemicist, he challenged his own teacher William of Champeaux over the universals and later found himself in open conflict with the monastic authority embodied by Bernard of Clairvaux, who saw in his confidence in reason a danger to faith. This tension between dialectic and the contemplative tradition marks his entire body of work and helps explain the condemnations he suffered at ecclesiastical councils. ...

Al-Farabi Abū Naṣr Muḥammad ibn Muḥammad al-Fārābī (c. 872–c. 950) was one of the greatest philosophers of the medieval Islamic world, called by the tradition “the Second Teacher” (al-Muʿallim al-Thānī) — the first being Aristotle. Born in the region of Farab (present-day Kazakhstan/Uzbekistan), he worked principally in Baghdad and Aleppo under the patronage of the Hamdanid court. His work spans logic, political philosophy, metaphysics, philosophy of music, and theory of the sciences. ...

Aristotle Born in Stagira, in Macedonia, around 384 BCE, the son of Nicomachus — physician to the Macedonian court — Aristotle entered Plato’s Academy at the age of seventeen, where he remained for some twenty years, until his master’s death. He later tutored Alexander the Great and, in 335 BCE, founded the Lyceum in Athens, a school whose members became known as Peripatetics (from their habit of discussing while walking). With Alexander’s death and the wave of anti-Macedonian feeling, he left Athens in 323 BCE — so that the city, according to ancient tradition, “might not sin twice against philosophy” — and died the following year in Chalcis. The body of texts we have inherited from him is the largest and most systematic of antiquity, spanning logic, physics, biology, psychology, metaphysics, ethics, politics, rhetoric, and poetics. ...

Bertrand Russell Born in 1872 into an influential British aristocratic family — he was the grandson of a prime minister — Bertrand Russell had one of the longest and most varied careers in philosophy: he was a logician, mathematician, essayist, educator, and political activist, spanning nearly a century of history. He studied at Cambridge, received the Nobel Prize in Literature in 1950, and, faithful to his pacifism, was imprisoned during the First World War and led, in his nineties, the campaign against nuclear weapons (the Russell-Einstein Manifesto, 1955). He is, with Frege and Wittgenstein, one of the founders of analytic philosophy. ...

Boethius “The last Roman and the first Scholastic.” He translated and commented on the Logic of Aristotle into Latin, transmitting it to the Middle Ages. The Isagoge of Porphyry — which Boethius translated and commented on — (introduction to the Categories) launched the debate of universals that will dominate Scholasticism. Imprisoned and condemned to death by Theodoric, he wrote The Consolation of Philosophy in prison — one of the most widely read works of the Middle Ages. ...

Clarence Irving Lewis (1883–1964) was born in Stoneham, Massachusetts, and graduated from Harvard (1906), where his teachers included William James and Josiah Royce — the former’s pragmatism and the latter’s absolute idealism would permanently mark his philosophy. He taught at the University of California, Berkeley from 1911 to 1920, before returning to Harvard, where he remained until his retirement in 1953 as Edgar Pierce Professor of Philosophy. His work joins two fields that at first glance seem distant — formal logic and epistemology — around a single concern: how concepts and deductive systems, though conventional in origin, can still organize experience objectively. ...

Charles Sanders Peirce American logician, mathematician, philosopher, and scientist. Founder of pragmatism and modern semiotics. One of the most original minds in the history of philosophy, though neglected in his lifetime. His work profoundly influenced William James, John Dewey, and the entire analytic and continental tradition of the 20th century. Trained in chemistry at Harvard University and employed for decades by the U.S. Coast and Geodetic Survey, Peirce never secured a permanent academic post — due in part to personal difficulties and to the hostility of influential academic figures such as the astronomer Simon Newcomb; it was William James who, paradoxically, supported him in his years of hardship and became his greatest publicist. This institutional marginality did not prevent him from developing, across hundreds of unpublished manuscripts, a philosophical architecture of extraordinary depth: an original phenomenology, advanced formal logic, a comprehensive theory of signs, and an evolutionary metaphysics. ...

Chrysippus of Soli (c. 280–c. 207 BCE) was the third scholarch (head) of the Stoic school in Athens and is considered, alongside Zeno of Citium, the second founder of Stoicism. While Zeno founded the school and established its fundamental theses, it was Chrysippus who systematically elaborated Stoic logic, physics, and ethics, producing a monumental body of work that ancient tradition estimated at over 700 titles — almost all now lost. What we know of his thought comes from secondary sources: Diogenes Laërtius, Plutarch, Cicero, Sextus Empiricus, and Epictetus. ...

Gottfried Wilhelm Leibniz Born in Leipzig in 1646, Gottfried Wilhelm Leibniz was the last great universal sage: at once philosopher, mathematician, logician, physicist, jurist, historian, and diplomat. He invented — independently of Newton — the infinitesimal calculus, whose notation we still use, conceived the binary arithmetic that underlies computing, and designed calculating machines. He spent much of his life in the service of the House of Hanover and died in 1716, leaving a body of work scattered across thousands of letters and few published books. Philosophically, he sought the great reconciliation of modern science, the metaphysical tradition, and faith. ...

Gottlob Frege A German mathematician born in 1848, Gottlob Frege spent almost his entire career as a professor at Jena, in relative obscurity — his genius would be fully recognized only after his death, in 1925, above all thanks to Russell, Wittgenstein, and the Vienna Circle. Today he is regarded as the founder of modern logic and one of the fathers of analytic philosophy. His first great achievement was the creation, in the Begriffsschrift (1879), of a predicate logic that retired the old Aristotelian syllogistic: with quantifiers (“for all,” “there exists”), variables, and functions, Frege gave logic the precision and expressive power it needed to analyze mathematics. His larger aim was logicism — to demonstrate that arithmetic reduces to pure logic. The project, set out in the Basic Laws of Arithmetic, was struck head-on by a letter from Russell (1902) revealing a contradiction in the system; Frege never fully recovered from the blow. ...

Kurt Godel Kurt Gödel is widely regarded as the greatest logician since Aristotle. Born in Brünn (now Brno, Czech Republic) on 28 April 1906, and died in Princeton on 14 January 1978, his work fundamentally transformed mathematical logic, the foundations of mathematics, and the philosophy of mind. His two Incompleteness Theorems (1931) partially dismantled David Hilbert’s formalist programme and established intrinsic limits to any sufficiently powerful axiomatic system. Key Concepts First Incompleteness Theorem (Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, 1931): Gödel proved that any consistent formal system capable of expressing elementary arithmetic necessarily contains propositions that can neither be proved nor refuted within the system itself. The proof strategy is remarkably ingenious: Gödel encoded formal syntax by means of numbers (the so-called Gödel numbering), thereby constructing a sentence that, in its own arithmetic numbering, asserts “I am not provable in this system.” If the system could prove this sentence, it would be inconsistent; since it cannot, the sentence is true yet undecidable. The theorem showed that completeness — the capacity of a formal system to decide every well-formed proposition — is incompatible with consistency, for sufficiently expressive systems. ...

Ludwig Wittgenstein Born in Vienna in 1889, into one of the wealthiest and most cultured families of the Austro-Hungarian Empire, Ludwig Wittgenstein began by studying aeronautical engineering, but reflection on the foundations of mathematics led him to logic and from there to Cambridge, to study with Bertrand Russell. His biography is as singular as his thought: he fought in the First World War, gave away his inherited fortune, worked as a village schoolteacher, a gardener, and even an architect, before returning to academic philosophy. He is the most influential figure in twentieth-century analytic philosophy — and, very rarely, the author of two distinct and equally decisive philosophies, both centered on language. ...

Porphyry Porphyry of Tyre — in Greek Porphýrios, a name adopted in place of his Semitic birth name Malchus (“king” in Phoenician) — was born around 234 CE in Tyre (on the coast of present-day Lebanon) and died around 305 CE. A caveat: he should not be confused with Porphyry of Gaza (5th century), a Christian bishop with the same name, nor with other late-antique figures bearing it. After studying in Athens with Cassius Longinus, in around 263 CE he joined the circle of Plotinus in Rome, of whom he became the most celebrated pupil. He served as editor of the Enneads (posthumously published c. 301), arranging the master’s writings into six groups of nine treatises and prefacing them with the Vita Plotini (Life of Plotinus) — a fundamental biographical source on Plotinus. His influence, however, reaches far beyond his editorial work: through the Isagoge, Porphyry shaped the entry of Aristotelian logic into the medieval world. ...

Rudolf Carnap Rudolf Carnap was the most systematic and influential figure of the Vienna Circle and of logical empiricism, a movement that sought to refound philosophy on the rigor of modern logic and fidelity to experience. Trained in physics, logic, and philosophy in Germany, he studied with Gottlob Frege at Jena before joining the Vienna Circle in the 1920s. With the rise of Nazism he emigrated to the United States, where he taught at Chicago and at the University of California, Los Angeles. His work spans the logical construction of knowledge, the syntax and semantics of language, the theory of confirmation, and inductive logic, always guided by the ideal of conceptual clarity. ...

Saul Kripke Saul Aaron Kripke was arguably the most technically gifted analytic philosopher of his generation. Born in Omaha, Nebraska, he showed extraordinary aptitude from adolescence: he published his first mature logical article — a completeness proof for modal logic — at eighteen, and corresponded with professional logicians while still in high school. A professor at Princeton and later at the CUNY Graduate Center, Kripke transformed metaphysics, philosophy of language, and modal logic to such a degree that virtually all analytic philosophy after the debates surrounding Naming and Necessity is, in some sense, a response to him. ...