“The history of logic should be of interest to anyone with aspirations to thinking that is correct, or at least reasonable.”
“Reflecting on the history of logic forces us to reflect on what it means to be a reasonable cognitive agent, to think properly.”
“In the Critique of Pure Reason (1781), Immanuel Kant stated that no progress in logic had been made since Aristotle.”
“He therefore concludes that the logic of his time had reached the point of completion. There was no more work to be done.”
“Two hundred years later, after the astonishing developments in the 19th and 20th centuries, with the mathematisation of logic at the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong.”
“Why did Kant disregard the scholastic tradition? And, more generally, what explains the ‘decline’ of logic after the scholastic period?”
“scholastic logic became less and less prominent after the end of the Middle Ages, except for educational purposes at universities (but again, in watered-down versions).”
“Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous.”
“Thomas Aquinas, for example, held that logic is about ‘second intentions’, roughly what we call second-order concepts, or concepts of concepts. But as late as in the 16th century, the Spanish theologian Domingo de Soto could write with confidence that ‘dialectic is the art or science of disputing’.”
“In these dialogues, Socrates regularly engages in the practice of refutation (elenchus), which consists in an exchange of questions and answers in which interlocutors would be led to grant the opposite of what they stated at the beginning, on the basis of their answers along the way.”
“And two of Aristotle’s logical texts, the Topics and the Sophistical Refutations, are explicitly about dialectical practices and contain a regimentation of these practices by way of an abstract description of their structural features.”
“Disputations were chiefly tied to the university culture of the later Middle Ages. Tellingly, in the passage above, Descartes speaks of the ‘logic of the Schools’, therefore criticising a whole approach to education based on disputations and their underlying logic.”
“The excessive formalism of scholastic disputations came to be vilified and mocked, for example in Molière’s play Le Malade imaginaire (1673) or The Imaginary Invalid, where the pedantic and rather foolish Thomas Diafoirus resorts to disputational vocabulary to make a point about love”
“Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences.”
“The point, though, is that there was a dramatic break in philosophical style in the early modern period: compare a great text of medieval thought, Summa Theologica (1265-1274) by Thomas Aquinas, which is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation.”
“In the modern period, a number of philosophers came to see the nature of logic in terms of the faculties of mind.”
“This leads us back to Kant, for whom logic pertained above all to the structure of thought as such and the operations of the mind, such as in his interpretation of Aristotelian categories.”
“Boole developed a method to ‘calculate’ whether an argument was correct or not, in stark contrast with the catalogue-based approach of traditional logic. Boole’s work launched what is known as the algebra of logic tradition, and pushed forward the idea of using mathematical symbolism in connection with logic.”
“Boole was responding to a growing interest in logic among mathematicians, and he mostly remained Kantian in his thought, but his work undoubtedly represents a turning point in the history of logic: it kicks off the mathematical period singled out by Bocheński as one of the most prolific periods in the history of logic.”
“The other towering figure in 19th-century logic is Frege.”
“While Boole (whom Frege harshly criticised) used mathematics to analyse logic (syllogisms), Frege’s project was to use logic to analyse mathematics.”
“More specifically, Frege wanted to provide mathematics with purely logical foundations, an effort that became known as the logicist programme.”
“He intended to derive all truths of arithmetic from purely logical principles (axioms), using only logical rules.”
“To this end, however, Frege had to ‘mathematise’ logic so as to make it suitable for the logicist programme, in particular by taking the mathematical notion of function as the main conceptual building block of his system.”
“Explicitly inspired by the 17th-century tradition of artificial languages (Leibniz’s work in particular), Frege devised an entirely new notation for his system, which he called ‘concept-script’ (or Begriffsschrift).”
“In the preface to his book Begriffsschrift (1879), he offers an illuminating description of the motivations for introducing this new language:”
“My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed, I had to bend every effort to keep the chain of inferences free of gaps. In attempting to comply with this requirement in the strictest possible way, I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present concept-script. Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed, so that its origin can be investigated.”
“The idea that ordinary language is expressively inadequate to account for mathematical (or even logical) reasoning became a recurring theme in the ensuing tradition of mathematical logic, so much so that the term ‘symbolic logic’ became synonymous with this tradition.”
“Doing logic came to mean simply working with special symbols, not with ordinary words. In this respect, it is worth noting that the humanist authors had criticised the Latin of scholastic logicians precisely as ‘too artificial’, and even the Greek language that Aristotle relies on for syllogistic logic is regimented and removed from ordinary ways of speaking at the time. In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage.”
“Frege was the first to realise that not only the axioms could be expressed in logical terms – the rules of inference themselves leading from axioms to further truths also required a rigorous treatment.”
“Unfortunately, Frege’s impressive theoretical cathedral lay on shaky grounds, as revealed by Russell with the discovery of the paradox that bears his name. Frege’s logicist system allows for the existence of a collection that both does and does not belong to itself – contradiction!”
“(An intuitive rendition of the key idea in Russell’s paradox is the so-called barber paradox: imagine a barber who shaves all those, and those only, who do not shave themselves. The question is, does the barber shave himself? If he does, then he doesn’t; if he doesn’t, then he does.)”
“It was with the goal of recovering and further developing the logicist programme that Russell and his collaborator Alfred North Whitehead went on to develop the monumental (and somewhat messy) system presented in their Principia Mathematica (1910).”
“Essentially, all major developments in logic in the 20th century are premised, directly or indirectly, on the existence of Principia Mathematica.”
“To return to Bocheński’s characterisation of the three grand periods in the history of logic, two of them, the ancient period and the medieval scholastic period, were closely connected to the idea that the primary application of logic is for practices of debating such as dialectical disputations.”
“The third of them, in contrast, exemplifies an entirely different rationale for logic, namely as a foundational branch of mathematics, not in any way connected to the ordinary languages in which debates are typically conducted.”
“However, traces of logic’s dialogical origins persist in recent developments, which means that taking the dialogical or dialectical perspective into account is essential to come to a thorough understanding of the nature of logic even in its more recent, mathematical instantiations – also because mathematics itself is very much a dialogical affair.”
“The history of logic also leads us to question the overly individualistic conception of knowledge and of our cognitive lives that we inherited from Descartes and others, and perhaps to move towards a greater appreciation for the essentially social nature of human cognition.”