History Of Logic by Marcos Benevides

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Notes from the book “History of Logic: from Aristotle to Tarski”.

Intro

Roughly speaking, the development of logic in Europe has had three “golden ages”:

The Greek period

From the beginning of the Greek golden period we get many anticipations of later developments. From Aristotle we get a wonderfully worked out system of “term-logic”, known as the “syllogism”, which was a precursor to the first-order logic, or “predicate calculus”, and which became easily the most influential contribution in logic for two thousand years. From the Stoics we get the first formulations of elements of propositional logic, with definitions of connectives given via truth-values, but we also see many other interesting formulations of logical problems, such as proposals for the nature of implication that are very similar to some contemporary theories on subjunctive and indicative conditionals. Much of the best work form the Stoic period, notably the work of Chrysippus, has been lost to history, but we know indirectly that he wrote much on logic generally, and specifi cally about paradoxes, including the Liar and the Sorites. It was for a very long time thought that the Aristotelian and Stoic conceptions of logic were at odds with one another, before their reconciliation in the modern era under the umbrella of first-order logic.

The fact that Aristotle’s achievement was so great actually had the unexpected consequence of holding back logical progress.

The medieval period

While learning in the West declined, the study of logic was kept alive in the thriving Islamic world under the Abbasid Caliphate, reaching great heights of sophistication with notable contributions from:

From the start of the twelfth century onwards, Europe became increasingly exposed to the Arabic world, and with this exposure came a rediscovery of the ancient Greek texts on logic, especially those of Aristotle, along with the benefit of the sophisticated Arabic commentaries. As these texts circulated around Europe we find the beginning of our second golden age of logic, from about 1100 to 1400, the high medieval or “scholastic” period, with notable contributors including Peter Abelard, William of Sherwood, Peter of Spain, William of Ockham and Paul of Venice. Curiously, during this period there was a combination both innovative progress and reverence for ancient authority, most particularly that of Aristotle. The progress came in the form of greater understanding of quantification, reference and logical consequence, a more intricate treatment of semantics, and developments in modal and temporal logic.

In the early decades of the sixteenth century, across northern Europe, we find the introductory arts course being adapted to meet the requirements of an influx of students from the professional classes. Within this arts course, with its humanist predilection for Greek and Latin eloquence, and legal and ethical instruction, there was an acknowledged need for some rigorous underpinning of instruction in ‘clear thinking’. But the meticulous introduction to formal logic and semantic theory provided by the scholastic programme came to look increasingly unsuitable for this purpose. Lisa Jardine 1982 : 805

The birth of Calculus

While the study of pure logic may have declined in the early scientific period, there were a great many advances in mathematics, in particular in algebra and geometry. By the end of the sixteenth century European mathematicians had recovered many of the works from antiquity that had been lost, and largely improved upon the Arabic algebraic developments. This period of mathematical progress arguably culminated with the creation of the infinitesimal calculus of Leibniz (1646–1716) and Newton (1642–1727). Leibniz himself was an outstanding mind who worked in many fields and, against the intellectual tide of his day, he conducted a solitary programme of innovative research in formal logic. Although the main body of Leibniz’s works on logic remained relatively unknown until the turn of the twentieth century, in retrospect he is hugely significant for articulating a completely new vision of the subject – one that stressed the use of the recently developed mathematical and algebraic methods.

Leibniz’s motivations for pursuing an algebraic treatment of logic were deeply rooted in his philosophical views. He conceived of a mathematically reconstructed logic as a medium that would refl ect the nature of thought more faithfully than the comparatively clumsy linguistic approach of the medieval logicians. The idea was that this new logic, being symbolic rather than linguistic, would actually improve the way that we think, by laying out the real elements of thought with perfect clarity, and would therefore be a crucial key in advancing science generally. In his ambition, Leibniz saw the possibility for a universal language of thought based on logic; a lingua philosophica (or ‘philosophical language’), which is something that would be recognizable to any student of analytic philosophy today. Despite the details of Leibniz’s logical innovations remaining largely out of sight in the Royal Library of Hanover until the turn of the twentieth century, the general character of his vision managed to exert an infl uence on a number of early pioneers in the modern age of logic. For instance, Leibniz’s ideal of a perfect language of thought heavily infl uenced two of the most important logicians of the early modern period; George Boole (1815–1864) and Gottlob Frege. This influence is reflected in the titles of their best known works on logic – Boole’s 1854 work “The Laws of Thought” and Frege’s 1879 work “Begriffsschrift” (which means ‘concept- script’). Although Boole and Frege diff ered signifi cantly in their approaches, both were trying to realize the mathematical vision of logic that Leibniz had envisaged.

The modern era

Aristotelian quantifier phrases take two predicates as arguments: they have the form ‘All A are B’, ‘Some A is B’, ‘No A is B’ or ‘Not all A are B’. Fregean or Peircian quantifi ers are unary and introduce a bound variable : a variable standing

One of the fundamental insights of the algebraic approach to logic was that logical symbols are subject to interpretation, as symbols in algebra are (for example, the geometric interpretation of complex numbers). Th is leads naturally to the idea of a model : an interpretation of the vocabulary of a logical system which satisfies its axioms. We therefore come to the notion of a Boolean algebra: any model of Boole’s axioms of logic. Such algebras consist of a set of atoms, the elements of the algebra; distinguished elements 0 and 1; and conjunction (intersection or multiplication), disjunction (union or addition) and complementation (negation or subtraction) operations on the set of elements. The study of Boolean algebras was begun by Marshall Stone, who proved a number of important results, including the representation theorem that bears his name: every Boolean algebra is isomorphic to (that is, has the same structure as) a Boolean algebra of sets.

We can trace the birth of model theory to Löwenheim’s paper “Über Möglichkeiten im Relativkalkül”. Working within Schröder’s version of the calculus of relations, Löwenheim showed that if a first-order sentence has a model, then it has a countable model. Th e Norwegian mathematician Thoralf Skolem improved Löwenheim’s proof in a paper of 1920, and in doing so generalized the result from single sentences to (possibly infi nite) sets of sentences. The result is important because it shows that first-order logic is not, in general, able to fix the cardinalities of its models: there can, for example, be models of the theory of real numbers which are countable, or models of set theory where sets that the theory holds to be uncountable are actually (from the external, model-theoretic perspective) countable. The completeness theorem for first-order logic, later proved by Gödel in his 1929 PhD thesis, is an easy corollary of Skolem’s work from the early 1920s, but this was not understood until later.