Logic: Russell, Mill & Modern European Logic

From here on out in the course, we will be studying modern European logic.  Today we will examine the logical positivist position of Bertrand Russell and the utilitarian position of John Stuart Mill, putting the dogmatic vs. skeptic duel into modern European times.  I gave you several texts from Russell and Mill in your reader to give you the debate in the author’s words.  Next class, we will begin studying Ludwig Wittgenstein and his truth table method.

As we have noted, in the ancient world logic was about the foundations of debate.  In the modern world, particularly fueled by the success of algebraic code-breaking of nature (cryptography from Islamic civilization, discussed last time) logic became a mathematical language that many hoped would provide and prove the foundations of algebraic mathematics.  As physicists used mathematics to uncover the mechanics of the universe, mathematicians used logic to uncover the mechanics of mathematics.  Unfortunately for Newton’s laws and Wittgenstein’s truth tables, both of these searches uncovered much and produced many useful systems and tools, but neither succeeded in uncovering the full and final mechanisms.

Bertrand Russell (1872-1970) was born into one of the most prominent aristocratic families of Britain, and retained the title ‘Lord’ even when arrested for demonstrating as a pacifist against World War One and dabbling in socialism.  With Frege and Wittgenstein, Russell is one of the founders of the anti-idealist Analytic school of Philosophy, the school which broke from the European Continental tradition and became the philosophical tradition of Britain and America.  As one author put it, on the European continent, philosophy wants to be art, and in the island of Britain and in America, philosophy wants to be science.  Just as Russell was very influential in Britain and America, on the continent, in Germany and France particularly, the German philosophers Hegel, who argued that reality is contradiction, and Nietzsche, who was deeply influenced by Heraclitus, took philosophy in a romantic and skeptical direction.

Russell is a highly influential rationalist philosopher of the Anglo-American Analytic tradition, a tradition that believes in doing the groundwork for scientific facts.  Russell often argues against skepticism and instrumentalism (or Utilitarianism, the position of Mill next) by name as he does in the text I gave you.  He also argues against Hegel and the Hegelians, who we will study in the coming weeks, who argue that reality is composed of contradictions and oppositional tensions.  Analytical positivism assumes that there are underlying facts that are absolutely and universally true, much like the atomic truths of the Vaisheshika and Nyaya schools of ancient India and Aristotle of ancient Greece.

In the first of Russell’s texts that I gave you, The Problem of Induction, Russell says that we know the self and sense data exist.  Interestingly, Descartes started modern European philosophy with a very similar position, though Descartes was certain of a good and just God whereas Russell is a staunch atheist who believes in the positive truth of science, not religion.  On a side note, philosophy in the middle ages and up through the European Enlightenment of the 1600s and 1700s was almost entirely supportive and uncritical of Christianity, and with the Enlightenment and through modern times in the English and American Analytic tradition been almost entirely supportive and uncritical of science, in spite of the fact that philosophy is the deep questioning of assumptions and traditions.  Apparently, one can be uncritical of assumptions and traditions while being critical of assumptions and traditions in the name of religion or science equally.  But let us return to Russell, my skepticism aside.

Russell argues that if we want to know anything other than the obvious and immediate, we need to gather things in our conceptions into general principles.  We can perceive lightning following thunder again and again, but we must draw a general principle from this if we want to know the relationship between the two.  Notice this is exactly what the Nyaya sutra said about the relationship between induction and deduction, and what the Charvakas called illusion.  Two questions can be asked here.  First, how do we know even the obvious or immediate without drawing generalities?  If I see an apple, don’t I have to draw generalities about the parts to see it as a whole, and don’t I have to draw generalities about apples to see it as an apple?  Second, why do we need to gather things into concepts or general principles?  Russell wants to argue against Mill and ‘Instrumentalism’ which argues that our concepts are not absolute truth but merely truth as it is useful.  Is Russell’s necessity different from utility?

Russell admits that this creates a problem (the central issue between positivism and skepticism, in fact): if we only know general principles by induction, then how can we know anything for certain?  Russell takes the same position as Aristotle, and argues that we cannot be satisfied with mere opinion, but need knowledge of certain principles that are always true in order to have knowledge.  It is true that we seek certainty, but if the skeptic is right and there is only more or less certain opinion, the relatively more certain being what we often call objective knowledge, then we are satisfied with mere opinion, even if it is only the relatively certain opinions that satisfy us.  Is any situation of knowledge/opinion entirely satisfactory?  Russell argues that we need a founding principle, a starting point of certainty, or otherwise all of our sciences, including mathematics, are mere opinions.  We must believe that there are certainties out there, and that our induction sometimes leads us to certain truth.

Russell says that science finds the laws of uniformity in nature, and thus gives us knowledge of certainties.  Remember, science in Russell’s time has made major advances since algebraic code-breaking, and is not simply the observation of nature as it was in Gotama and Aristotle’s time, but Russell’s position is identical to that of Gotama in ancient India and Aristotle in ancient Greece.  While in ancient times science was largely observation of nature, with Islamic algebra and the increasing levels of mechanics infused into daily life science became not just passive observation but active experimentation.

Unfortunately for Russell, he was writing the piece before the work of Einstein, when Newton’s laws of nature were the dominant model of science and physics.  Einstein’s work did much to swing scientists away from the use of the word “law” and toward use of the word “theory” to describe proposed conceptions of the regularities of nature, humanity and the universe.  Another philosopher of science, Thomas Kuhn, would use the term “paradigm” to describe scientific conceptions in a similarly relativistic and hypothetical light.  As we will see, Mill would be quite happy by this change of language.  It is also similar to the Jain principle of sayadvada, that all human truth is put forward as a best guess, as a hypothetical proposition or theory.

In the second text I gave you, The Empiricist Answer to Skepticism, Russell calls out Hegelians and skeptics by name and says we cannot get the sort of independent certainties we want to call facts or laws by these methods.  He says we can indeed whittle away at our theories to get to the pure data, the facts without interpretation but as they must be.  Wittgenstein will later use the metaphor of bedrock (like Russell uses the metaphor or analogy of whittling).

Russell has faith in a ‘minimal theory’, what people are trying to find with string theory in physics today.  He concedes to the Hegelians and skeptics that there is always SOME uncertainty in truth, some relations of views institutions, room for error, but Russell argues that some data has “independent credibility”, meaning we know its ‘just true’, ‘right there’, and we can assume correctly that this is true data or fact.

Russell argues that some things must be more than opinion, as opinions can contradict each other but the true fact never contradicts itself or other things.  This is diametrically opposite Hegel, who argues that contradictions are what compose all things and their situations.  Russell argues we should give the greatest weight (note the use of metaphor and analogy, again) to that which is most regular and most certain, and in this way form our principles and propositions.  This is the method which Russell believes answers the skeptic.  Interestingly, the examples Russell uses suffer from the same problems that Aristotle had before, arguing that what is red cannot be blue.  The Jains, Buddhists, and Sextus Empiricus would argue that things can be red and blue at different times, parts, together as the color purple, etc.

Russell is a big believer in the Principle of Non-Contradiction (a statement must be true or false, but not both) and the Principle of the Excluded Middle or alternatively titled Principle of Bivalence (a statement must be either true or false, but not neither).  These two principles are, as we have seen in the ancient world and we will continue to see in modern thought, the central issue dividing dogmatic/positivist and skeptical/relativist thought.


John Stuart Mill (1806-1873 CE) was born in London, influenced by Ancient Greek, French, and liberal thought.  He is considered the major founder of Utilitarianism and one of the first champions of individual freedom, women’s rights, and the abolition of slavery.  His father wrote a history of India, and Mill was for a time involved with his father in the British East India company, the corporation that helped Britain maintain their economic hold over India.  

Mill’s family was friends with the Bentham family, from whom Mill took up his consequentialist, ‘happy principle’ thought.  Mill’s father had Bentham tutor Mill as a child with the idea that Mill would become a champion of consequentialist thought.  However, it was Mill who found the name ‘utilitarian’ in a Christian text that used the term to describe an evil way to strive for good without principles.  Mill picked up the name and developed the thinking in line with Bentham, his mentor, becoming its famous spokesman.  Mill is a central thinker in Logic, Economics and Ethics.  In all of these subjects, he advocated rethinking basic principles and assumptions based on the ongoing experiences of their usefulness.  For Mill and Utilitarianism, the true is not true in itself but true because it is useful for creating happiness and avoiding pain.  

Political laws, ethical morals, mathematical rules, and scientific understandings are to be continuously examined and developed such that they are best for A) the greatest number of people, and B) over the longest period of time.  Apart from this practical value, Mill argues there is no objective truth to things.  Rather, the objective of truth is its beneficial use.  If a certain logic or form of mathematics is useful, proving itself a valuable tool that can be put to good work, then this is the only proof that it is solid and sure.

Utilitarianism, which Russell attacks as ‘Instrumentalism’ says that there is no truth to things other than how they are used and for whom they are used.  This is a very similar position to the “existentialism” of Avicenna and Ockham, with a social bent.  Both Russell and Mill agree that the world has regular patterns, and that we use our minds to form concepts of these regularities.  However, Mill and utilitarians argue that we are often fooled by our use and concepts of things into thinking that we grasp the essence of the thing when in fact all we are doing is using and grasping.  What is useful to some may not be useful to others, and what is useful for many years may not be useful years later.

As algebraic science made many new discoveries, it is interesting that human beings again split along positivist and skeptic lines.  Does science show us that the truth is out there and can be discovered, or that we should always be suspicious of our certain truths because new truths can overturn the old truths?

Utilitarianism was a powerful force in Britain against more traditional thinkers like Russell, who is fighting back against Hegelian process theory and Utilitarian ‘its whatever we want to do with it or make of it’ theory.  Russell believes that things are factually as they are, and we should be objective to have true knowledge.  A Utilitarian would say we can arrange situations such that we have knowledge, but there is no ‘true view’, purpose or nature of things.  For instance, Utilitarianism was very useful for feminism, that there is no ‘true position’ of women but it is however we position things such that they are most useful.

In the text Logic and Mathematics, Mill asks: if we admit that all is induction (British Empiricism, which Russell embraces) then why do we say there are “exact sciences”?  We also similarly say that there are ‘hard sciences’.  He argues that this is an illusion due to the fact that objects of math are conceptions and thus imaginary, hence they have perfect straight edges like an ideally straight line.  A perfectly straight line, the example he uses, with no width, like a point, cannot exist outside of the imagination.  It is a real being of the world, but imaginary and not physical.  Some (Russell) say without perfection of a sort there is no math, science or knowledge possible (eternal facts as well), but Mill argues this is silly as we have these things yet do not have an instance of a perfectly straight line in the real world.

Russell argued that we can strip down or “whittle” to the pure straight edged truth, but Mill argues that this helps us to focus our observation and thinking but it does nothing to guarantee that our knowledge is certain at all.  We can ignore aspects of a thing to focus on particular aspects or parts, but this does not successfully or completely take these factors out of the picture, even as far as relevance to the parts that are in focus.  This is a very important ground for the positivist/skeptic debate today.  If we take a banana and put it in a lab, are we more or less capable of seeing its truths there?  If we create abstractions about bananas with our minds, are these getting into the thing or away from it?

Russell wanted a first certain principle of non-contradiction to give us certainty.  Mill argues that there are no first principles of geometry, mathematics or anything else.  Mill argues that the ‘first principles’ are in fact simply generalized observations of real world situations.  Mill turns specifically to the two principles of Non-Contradiction and Excluded Middle with this skepticism.  These principles are in fact observation in practice.  We can see that belief and disbelief oppose one another, that they are “oppositional mental states” just as we can see that opposing stories tend to mean that someone is somewhat mistaken.  This is the simple contrast between any two opposites.  Mill argues that the two principles are merely useful generalizations, as are all “true” concepts used by human beings whether scientists, philosophers or common folk.

The Problem of Self Referentially is one of the most famous and interesting problems with the principles of Non-Contradiction and Excluded Middle.  This problem was very important for Russell’s career, and it brought him both great success and frustration.

The classic example is the liar’s paradox.  Consider the sentence, “This sentence is false”.  According to non-contradiction, this must be either true or false but not both, but if it is true then it is false and if it is false then it is true.  It cannot be fully resolved either way.

Russell pointed out a similar paradox in the work of Frege, the German analytic philosopher who was central in trying to turn Logic towards founding arithmetic, who Russell and then Wittgenstein followed and overturned (as Wittgenstein later did to his teacher Russell).  Frege thought he could prove that mathematics is simply a system of non-contradictory sets, which he called Set Theory.  Russell realized, however, that there are still contradictions possible in this system of mere sets.  He asked, “Is the set of all sets which are not members of themselves a member of itself?”  Just like the liar paradox, if it is, then it isn’t, and vice-versa.

Russell’s point is that there is a paradox that results from self-reference (‘this sentence’ or ‘this set’).  Russell tried to use Frege’s work to finish a foundation of math, which he never completed to his own satisfaction.  He thought that he could come up with a ‘theory of types’ that would prevent the paradox (so a ‘set’ is of objects but a ‘type’ is a group of sets, not objects), but it lead to further paradoxes.  At this time, he discovered a young Austrian genius who showed up at his office hours one day simply to argue endlessly against everything Russell said, who Russell believed would finish his work but in the end came to stand completely against proof theory: Ludwig Wittgenstein.

Recently, a different version of the liar’s paradox has been proposed that supposedly does not suffer from self-reference: consider two people standing next to each other, one with a sign that reads “Her sign is false” and another with a sign that reads, “His sign is true”.  Like Russell’s theory of types, this version of the paradox tries to eliminate self-reference to tidy up the problem, but the paradox remains.