Modern logic

Logic was established as a branch of philosophy by Greek philosophers and systematically described by Aristotle. In the form of syllogism, it survived virtually unaltered until the late 19th century. Change finally came when mathematicians, notably Gottlob Frege, recognised that logic and mathematics are inseparable, and that mathematics consists of arguments and demonstrations based on logic. This inspired a very British school of thought, analytic philosophy, based on the idea that philosophy, too, is derived from logic. Its leading figure, Bertrand Russell, was the first to analyse the language of philosophical statements as logical propositions.

The analytic approach influenced philosophers in the German-speaking world, where it evolved into logical positivism. Russell’s influence, however, was seen most importantly in the linguistic philosophy of Ludwig Wittgenstein. Alongside the philosophical analysis of language, there was also a growing interest in linguistics as a science, which had its own philosophical implications in turn.

Mathematics and logic

In his book Begriffsschrift [1879], German mathematician Gottlob Frege overturned views of logic that had held firm for more than 2,000 years. At the time, it was regarded as a set of “rules” derived from the way that we think. But Frege proposed, for the first time, that logic is objective and has nothing to do with the way we use it. Logical propositions are objective truths, and either follow one from another, or do not, irrespective of human psychology.

Frege also pointed out that mathematics consists of a series of arguments and demonstrations to show how one thing follows from another, and therefore has the same principles as logic.

Until this point, the prevailing view had been that mathematics, too, was a human creation, like language, but Frege’s arguments showed that mathematics was objective and universal. We do not create it, but discover it in the same way that we discover physical laws. Frege’s linking of mathematics and logic also showed that there was far more to logic than the syllogism, making it a much more powerful tool for philosophy.

Logic and the syllogism

Like all philosophers, Aristotle sought to justify his theories with rational arguments. Yet, he was not content with the simple reasoning used by early philosophers, nor even the dialectical method developed by Socrates. Instead, he proposed a system of logic whereby the information in two statements or “premises” can be used to reach a conclusion. For example, from the premises “All men are mortal” and “Socrates is a man”, we can infer the conclusion “Socrates is mortal”.

In the first formal study of logic, Aristotle broke down this form of logical argument, known as syllogism, into three parts: a major premise, a minor premise and a conclusion. Each of these parts contains two terms, presented in various forms such as “All A are B”, “Some A are B”, “No A are B” or “Some A are not B”. Using his talent for analysis and classification, he then categorised the possible combinations of the different forms of premise and conclusion, identifying those that presented valid arguments, and those that were invalid.  

Logic vs epistemology

Frege’s insight that logic is not a creation of the human mind, but consists of universal objective truths, had profound implications for philosophy. Mathematics, he said, followed the same principles as logic, and so was equally objective. But the process of reasoning that underlies philosophy is also a series of arguments and demonstrations, governed by logical principles and is similarly not a human creation. Philosophical truths should therefore be objective, and discoverable in the same way that we can discover the truths of mathematics, independent of the way our minds work.

Much of modern philosophy, however, has been based on epistemology, the study of how we know what we know, and what we can know, all of which concerns what is going on in our minds and is effectively irrelevant to philosophical truth. For philosophy to discover objective truths about our knowledge, it now became clear, it must have its foundations in logic, not epistemology.

Principia Mathematica

Frege’s work went virtually unnoticed outside a small circle of mathematicians, and was ignored by philosophers until it was discovered by Bertrand Russell. Russell had studied both philosophy and mathematics at Cambridge University and had independently concluded that arithmetic, and possibly all of mathematics, could be derived from logic, stating the case in Principles of Mathematics [1903]. He and his colleague Alfred North Whitehead then attempted to prove this in the massive three-volume Principia Mathematica, completed ten years later.

Following this, he turned his attention fully to philosophy, unsurprisingly concentrating on logic [and probably influenced in this choice by his godfather, John Stuart Mill, who had written the definitive 19th century book on the subject]. Like Frege, Russell recognised the implications of breakthroughs in logic and mathematics for philosophy. They had shown that logic is an objective set of universal truths, and Russell believed that with science and philosophy both based on logic, we should be able to discover an objective knowledge of the world.

Analytic philosophy

Russell came from the tradition of empiricism and, along with his colleague G.E. Moore, led the British movement in philosophy away from the idealism that had dominated the 19th century.

Having demonstrated that mathematics and logic were inseparably connected in the Principia, Russell set out to show that logic should be the basis for philosophical enquiry: because it consists of objective, universal truths, logic is a firmer foundation for our knowledge of the world than the speculation offered by epistemology.

Applying techniques of logical analysis to statements about our knowledge enables an objective evaluation of them, an approach to philosophical enquiry that became known as analytic philosophy.

In order to analyse philosophical statements logically, they first have to be presented in a logical form, similar to the propositions of mathematics, using the formal grammar and signs and symbols of logic. A philosophical argument can then be shown as valid or invalid using the rules of logic, in much the same way as a mathematical proof.

Truth and logic

Analytic philosophy offered the prospect of a more scientific method of examining philosophical arguments. Translating them into a series of logical propositions, however, can be problematic. Before their validity can be tested, the sometimes flowery and convoluted language of philosophers must be presented as simply as possible. There is no doubt, that the concepts of philosophy are frequently not as clear-cut as those in a mathematical or logical proposition and, unlike numbers and symbols, words and phrases present problems of meaning. So, it is sometimes impossible to establish the truth of a statement.

Seemingly similar statements in the same logical form, may have very different logical implications. For example, if we say, “The King of Spain is bald”, the meaning is clear, and we can check the facts to establish empirically if it is true or false. However, the statement “The King of France is bald” has exactly the same logical form, and an obvious meaning, but is it true or false? There is no King of France, so how can we check? Despite its valid logical form, such a statement may have no meaning at all.

Tractatus Logico-Philosophicus

As a student of Bertrand Russell, Ludwig Wittgenstein adopted the principles of analytic philosophy, then set out to establish the limits of our knowledge using logic in the same way that Kant and Schopenhauer had previously used epistemology. In Tractatus Logico-Philosophicus [1921], he showed that in trying to understand the world, we describe it using language that consists of propositions with logical structures: “The totality of propositions is language.” The world, “the totality of facts, not things”, is also structured, and language pictures the world by making a representation, or “map”, in the same logical form as the world it describes. All that can be meaningfully said is “the totality of true propositions”, statements that can be empirically verified about the phenomenal world of experience.

Philosophy should confine itself to this world, since language limits the things it can meaningfully discuss, in effect, to statements about natural science. Subjects such as ethics and religion are still important, Wittgenstein said, but are “mystical” – we can make no meaningful propositions about them.

The project had a broad goal: to identify the relationship between language and reality and to define the limits of science.

The Tractatus is written in an austere and succinct literary style, containing almost no arguments as such, but consists of altogether 525 declarative statements, which are hierarchically numbered and meant to be self-evident. The work is recognised by philosophers as one of the most significant philosophical works of the twentieth century.

A prominent view set out in the Tractatus is the “picture theory”, sometimes called the picture theory of language. The picture theory is a proposed explanation of the capacity of language and thought to represent the world. Although something need not be a proposition to represent something in the world, Wittgenstein was largely concerned with the way propositions function as representations.

According to the theory, propositions can “picture” the world as being a certain way, and thus accurately represent it either truly or falsely.

Logical positivism

Thanks to Russell, Frege’s work on logic reached a much wider audience and readership. By the 1920s, it was not only British thinkers who incorporated the new logic into their philosophy: a group of scientists and mathematicians known as the Vienna Circle formed with the aim of establishing a philosophical basis for science. They believed that it is the job of science, not philosophy, to provide truths about the world, but the job of philosophy is to provide a logical framework in which science can work.

The view they developed, known as logical positivism, applied the techniques of logic to the statements of science in the same way that analytic philosophy approached philosophical statements. To talk clearly and objectively about scientific ideas and theories, we must first analyse the language of scientific statements as logical propositions to ascertain their meaning, ruling out any that are meaningless.

For the logical positivist, a statement can be accepted as true only if it meets strict logical criteria and can be empirically verified; anything else cannot be proved and so is effectively meaningless.

Language as a tool

By the mid-20th century, analytic philosophy had become the predominant philosophy of the English-speaking world. Logical positivism was also influential, as members of the Vienna Circle fled from Nazism to Britain and the United States, and its strict criteria of meaningfulness were applied to all forms of language, not just science. Analytic philosophy had become largely an analysis of language, rather than of philosophical questions. But not all British philosophers accepted this trend, and some, such as G.E. Moore, argued for a “common sense” approach to the logical analysis of language.

Wittgenstein also returned to the argument, realising that his Tractatus was self-contradictory: it was full of propositions that do not picture the world, and so are meaningless. In its place, he developed an entirely different linguistic philosophy, dropping the metaphor of language “picturing” reality and replacing it with language as a tool. Each word or concept does not mean a specific thing, but derives its meaning from the intention of its user and the context in which it is used. Today’s mobile phone text and language used between users is a reflective example of how meaning between other users might differ.

Linguistics

Both British and continental philosophy became increasingly concerned with language as the 20th century progressed. Analytic philosophy called for the analysis of philosophical statements to render them in a logical form, prompting an interest in the philosophy of language itself. In France, the tradition of literary philosophy spawned structuralism, a philosophy defined and based on linguistic structure. This trend was mirrored in the growth of linguistics itself.

The aim of this young and developing science was to study language using scientific methods, but there was also some cross-fertilisation with linguistic philosophy. Among other things, linguistics is concerned with the structure of language, in terms of its grammar, semantics and so on, and this influenced the structuralist movement. It also studies the way in which languages have changed and evolved, how they differ from one another, and whether there is an underlying structure common to all languages. This, too, has philosophical implications, in the study of the way that we acquire language and use it to express our ideas.

At the core of linguistics is the applied scientific study of language. It entails a comprehensive, systematic, objective, and precise analysis of all aspects of language – whether cognitive, social, environmental, biological, or structural.

Universal grammar

One of the questions that occupied linguistics was whether there is a common grammatical structure to all human languages. There are families of related languages that share similar grammars, but the many different families seem to have little or nothing in common. An answer from this problem came from Noam Chomsky, later better known for his critical analysis of political power. He noticed that children became proficient in their mother tongue far more quickly than one might expect from the amount of stimulus they are given, and regardless of the particular language a child is learning. Chomsky concluded that we must have some innate knowledge of the structure of language, and that this structure must be common to all languages – a universal or “generative” grammar.

The notion of language having a universal formal structure was analogous to the link between logic and mathematics, but the idea of an a priori understanding of it harked back to Cartesian rationalism and was difficult to square with linguistics’ claim to be a scientific discipline in the empiricist tradition.

Artificial intelligence

Advances in science and technology have produced machines capable of performing tasks that previously could only be done by human thought. Sophisticated computer programming has taken them beyond the realm of simple calculation, and given them “artificial intelligence”, including the ability to use language. So machines can mimic human behaviour – but can they actually think? As recorded within Science Book: Computer Science, mathematician and computer pioneer Alan Turing proposed a simple test of a machine’s ability to show intelligence. A machine and a human are both asked questions in normal language, to which they give replies in the same language. If an impartial judge cannot tell which answers are which, the machine has passed the test and is considered to be showing intelligence. Modern computers are increasingly able to do this successfully, especially as we programme them to process information in the same way as human brains and even introduce such concepts as “fuzzy logic”. But can we therefore say that they are capable of “thinking”? Will they ever be capable of what we understand to be consciousness?

PHILOSOPHY AND SCIENCE IN THE 20TH CENTURY

Most of the sciences evolved from branches of philosophy, complementing them with scientific theories describing the physical world. But as the pace of progress accelerated with the scientific revolution of the Enlightenment, the natural sciences largely replaced metaphysics, and by the end of the 19th century psychology and neuroscience began to provide a scientific alternative to the philosophy of the mind. Neuroscience, for instance, can tell us more about the way our brains work than introspective philosophy.

In the 20th century, Albert Einstein’s theories seemed to provide a comprehensive explanation of the physical universe, but many aspects of the new physics threw up almost as many questions as answers – problems that science alone could not explain. And just as science appeared to be replacing aspects of philosophy, some philosophers turned their attention to science itself. Karl Popper proposed a practical answer to the problem of induction, the basis of scientific methods, while Paul Feyerabend questioned the notion of a single reliable scientific method, based on Thomas Kuhn’s idea that science makes advances not in a smooth progression, but in radical jumps.  

Scientific answers to metaphysical questions

When Einstein formulated his theories of relativity at the beginning of the 20th century, he ushered in a completely new way of understanding the universe. This overturned the old Newtonian view, replacing it with a comprehensive description of the physical universe according to a new set of physical laws. For the first time, time and space themselves were explained scientifically, and the substance of the universe was defined in terms of energy. Many of the metaphysical questions that had concerned philosophy from its earliest beginnings seemed finally to be answered.

Yet, as Einstein would have happily admitted, the new physics was not a definitive answer, nor did it negate the importance of Newton’s contribution. It was not “right” or “true”, but simply a more accurate explanation than Newton’s, which was perfectly good for its time – as a pragmatist would say, it was a valid explanation. And just as Newtonian physics marked a stage in the history of science, so too Einstein’s theories might one day be replaced with something that fits the facts even better.

New philosophical questions

While Einstein’s theories appeared to bring an end to a lot of metaphysical speculation, the physics that he pioneered also posed fresh questions. And rather than disproving old philosophical theories, it appeared to confirm or at least complement many. The definition of matter as energy, for example, has more than a superficial resemblance to the idealism of Schopenhauer or even Hindu and Buddhist philosophy. Many of the concepts of quantum mechanics, meanwhile, are hard for even physicists to understand, and seem almost mystical – such as Heisenberg’s uncertainty principle and the “observer effect” [the necessity of observation in “fixing” the properties of a microscopic quantum system], which seem to defy common sense, but have an uncanny resemblance to Berkeley’s philosophy.

The Big Bang theory of the origin of the universe, meanwhile, also reopened the philosophical debate about the nature of reality and causality – and the possibility that there is more than one universe presents even more food for philosophical thought.

Falsifiability

The method science uses to establish theories is based on induction – inferring a general rule from individual instances. Even after Hume pointed that induction cannot logically be used to show anything with certainty, scientists continued to use the methods of observation and experiment with remarkable success.

The problem of induction, however, still nagged philosophers until the 1930s, when Karl Popper proposed a different way of viewing scientific methodology. He argued that while vast numbers of positive instances of an event cannot prove a scientific theory conclusively, a single negative instance can decisively show it to be false. As an example, he gave the hypothesis: “All swans are white”; this cannot be proved true by reference to any number of observations of white swans, but can be proved false or “falsified” by the appearance of just one black swan.

The true criterion of a scientific theory is not that it is inferred by induction, but that it should be falsifiable, capable of being shown as false by observation or experiment.  

Paradigm shifts

The philosophy of science developed in the 20th century as an important branch in its own right, especially after Karl Popper’s ground-breaking work.

As well as examining the philosophical and logical basis of scientific methodology, philosophers turned their attention to the nature of scientific progress. It had been assumed that advances in science were a process of continuous evolution, until in 1962 Thomas Kuhn introduced the notion of paradigm shifts.

In his analysis of the history of science, Kuhn suggested that periods of “normal” science are interrupted by periods of “crisis”. During the periods of normality, scientists work within an agreed framework, or paradigm, and any anomalies in their work are overlooked or dismissed. If, however, the anomalies become significant, this provokes a crisis, in which they must be accounted for by new theories and a shift to a new paradigm. Normal science now resumes under the new framework, until the next crisis. Historical examples of such paradigm shifts include the revolutionary theories of Copernicus, Newton and Einstein. Newton’s ideas of gravity, for instance, served as a framework for more than 200 years, but were overturned by Einstein’s general theory of relativity, which is almost universally accepted by modern science.

Against Method

Kuhn’s depiction of the history of scientific progress as a series of periodic revolutions was given an anarchistic spin by his colleague Paul Feyerabend. When a paradigm shift occurs, all the accepted concepts are affected and so, Feyerabend argued in 1975, there is no single, permanent framework for establishing scientific truth. New approaches and methods are adopted and become the consensus of scientists, but they would not have been considered valid before. Scientific progress, therefore, is not achieved by following strict rules – on the contrary, advances are made when these are broken.

Feyerabend rejected attempts to justify the scientific method as Popper had done, proposing that there are no universal methodological rules and attempts to find such rules hinder scientific progress. In particular, he attacked the criterion of consistency with known facts, pointing out that no major theory was consistent with an older one covering the same ground. If science is to continue to advance, he argued, it would be better to ignore any prescriptive theories of methodology.

---

_{This completes the page on Modern Logic. Amendments to the above entries may be made in future.}