Sunday, April 11, 2010

Logic Lecture April 12: Modern Logic, Russell and Mill

BCC Logic
Eric Gerlach
4/12/2010

MODERN EUROPEAN LOGIC: RUSSELL AND MILL

From here on out in the course, we will be studying modern European logic. Today we will examine the positivist position of Bertrand Russell and the skeptical position of John Stuart Mill, putting the positivist 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.


Bertrand Russell (1872-1970) is a big rationalist philosopher of the Anglo-American positivist tradition, a tradition that believes in doing the groundwork for facts. Thus, Russell often argues against Hegel, skepticism and instrumentalism (or Utilitarianism, the position of Mill next) by name as he does in the text.


First Text: Russell’s 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 similarly certain of a good and just God whereas Russell is a staunch atheist who believes in the positive truth of science, not religion.

Russell argues that if we want to know anything other than the 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.

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.
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 become 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. 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.

Unfortunately for Russell, he was writing the piece before the work of Einstein, when Newton’s laws of nature was the dominant view of science. 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 our conceptions of the regularities of nature. Another philosopher of science, Thomas Kuhn, would use the term “paradigm” to describe scientific conceptions in a similarly relativistic and non-positivistic light. As we will see, Mill would be quite happy by this change of language.


Second Text: Russell’s 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 a metaphor of whittling).

Russell has faith in a ‘minimal theory’, what people are trying to find with STRING theory. He gives the Hegelians and Skeptics that there is always SOME uncertainty in truth, some relations of views, institutions, room for error, but 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.

Some things must be more than opinion, as opinions can contradict each other but the true fact never contradicts itself or other things. We should give the greatest weight (note the use of metaphor, again) to that which is most regular and most certain, thus 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 (different times, parts, purple, etc).

As a positivist, 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 Bivalence (a statement must be either true or false, but not neither).



John Stuart Mill (1806-1873), skeptic and Utilitarian, comes from the opposing side of the debate, very much against Russell’s arguments.

Utilitarianism, which Russell attacks as ‘Instrumentalism’ says that there is no truth to things other than how they are used. This is a very similar position to the “existentialism” of Avicenna and Ockham. Both sides agree that the world has regular patterns, and that we use our minds to come up with concepts of the 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.

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’s ‘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.


Third Text: Mill’s Logic & 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 Referentiality 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.