This is my 200th post: Problematic self reference?

Consider the statement (call it p):

This statement is false.

Is p true or false? If it is true, it contradicts itself. So it must be false. But that is what p states, so it must be true. Liar’s Paradox.

On the face of it, it seems that the problem with this paradox is either self-reference or bi-valued logic itself. If the problem is bi-valued logic, it seems that a lot of math is suspect, in particular proof by contradiction. Proof by contradiction is a staple of math as I know it, and giving it up would be difficult.

For a long time, I thought that the problem here was inappropriate self-reference. But not all forms of self-reference lead to contradictions. Self-referencing statements are even used in proofs of theorems, the Godel’s incompleteness theorem for example. The title of this post “This is my 200th post” is a self-referencing statement (incidentally, it is true). It does not lead to any contradiction. If the problem is inappropriate self-reference, what forms of self-reference are problematic? Change the liar’s paradox to “This statement is true.” and there is no contradiction. The form of self-reference is clearly the same. What then?

q: This statement is true.

If q is false, it contradicts itself. So q must be true. This does not lead to any contradiction. Assuming bi-valued logic is valid, q is true. But what does q mean? What is true? As I see it now, the problem with the liar’s paradox is that just like q, it is empty, devoid of any content. There is nothing in it that can be either true or false. In other words, it is not a statement at all because it does not state anything about the world. Mere adherence to rules of grammar does not produce statements.

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Rationality

In common usage, people sometimes tend to use the words rational and logical somewhat interchangeably. The purpose of this post is to distinguish between these.

Logic:

Logic is the set of rules that allows me to evaluate an arguement independent of its content, purely from its structure. Just as I use grammar to parse a sentence and determine the relationships between the words in the sentence, I use logic to parse an arguement and determine the relationships between the statements in the arguement. Just as a grammatical sentence may be meaningless (Colorless green ideas sleep furiously), a logical arguement may be meaningless or irrelevant. However, the analogy with grammar only goes so far. There are many different grammars and all of them are equally valid within the context of their application – a given language. Any consistently applied way of meaningfully combining words in a sentence forms a grammar. Grammar is a matter of convention. The same is not true of logic. The word itself has no plural. This is a striking fact. Think about it. It indicates that man cannot even conceive of a plural for logic. There can be no such thing as my logic vs your logic. Logic is the structure of coherant thought. It is a part of the mental apparatus that man is born with. It is implicit in the capacity to think. By implicit, I mean that I cannot choose to think illogically (though I may make mistakes). To identify mistakes in thinking, the implicit rules of logic need to be made explicit by identifying them. This is a science. Like all sciences, the science of logic also presupposes several things. In particular, it presupposes man’s ability to use logic (implicitly). Whether the word logic refers to the implicit set of rules or to the science which deals with identifying them depends on context. In this post, I am going to use the word logic to refer to the implicit set of rules.

Reason:

Reason is the faculty of understanding and integrating sensory material into knowledge. Reason does not work automatically. To reason, man has to consciously choose to think and to direct his thoughts to achieve understanding. By directing thoughts, I mean preventing thoughts from wandering by staying focussed. Reasoning involves the use of logic. It also involves several other techniques. “Reason employs methods. Reason can use sense-perception, integration, differentiation, reduction, induction, deduction, philosophical detection, and so forth in any combination as a chosen method in solving a particular problem.” [Burgess Laughlin in a comment on an old post] Deduction obviously uses logic. I believe induction does too but the science on inductive logic is nowhere as well developed as it is on deductive logic. Sense-perception, integration and differentiation don’t use logic (Note: integration and differentiation refer to grasping the similarities and differences between various things). Reason then is not simply the faculty of using logic.

Rationality:

“Rationality is man’s basic virtue, the source of all his other virtues… The virtue of rationality means the recognition and acceptance of reason as one’s only source of knowledge, one’s only judge of values and one’s only guide to action.” [Ayn Rand, in The Virtue of Selfishness]

In the discussion that motivated this post, a colleague argued that if the use of reason does not guarantee correct decisions, it cannot be one’s only guide to action. A gut feeling or intuition might sometimes be a better guide to action. There are two separate issues here – the fact that the use of reason cannot guarantee correct decisions and the claim that intuition can be an alternative guide to action.

Consider intuition first. Intuition is an involuntary automatic evaluation of the available choices. There is no conscious awareness of the reasons for the evaluation. Intuition is a learnt response from previous experience. As such intuition is extremely helpful in any decision making process. However, the fact remains that in every voluntary decision – the sort of decision where there is enough time to reason – intuition is only one of the inputs to the use of reason. As long as I make a decision consciously and deliberately, reason remains the only guide to action. The only alternative is to evade the responsibility of a choice. Relying upon intuition is not irrational in itself. I might decide that I do not have sufficient knowledge to reach a decision and choose to rely upon intuition instead. As long as I identify the lack of knowledge, my decision is fully rational. Identifying the lack of knowledge (and hopefully doing something about it) will actually allow me to learn from the new experience and improve my intuitions for future use. Blindly relying on intuition – by default instead of by choice – will actually weaken my intuition in the long run. Intuition is one of the most valuable tools for decision making but it needs to be carefully cultivated by the use of reason for it to be good or useful.

It is important to stress that rationality (in the context of making a decision) involves the use of all my knowledge to the best of my ability. In particular, this includes knowledge of the time available, the relevance of prior experience and any known gaps in knowledge. It is this last aspect of rationality – the use of known gaps in knowledge – that is the motivation for the field of probability. Probability is about quantifying uncertainty by making use of all known information and postulating equal likelihood where no information is available. The consistent use of the equal likelihood postulate is at the heart of probability theory and it is what gives probability its precise mathematical characteristics. In modeling an outcome for an uncertain event, I start with a uniform distribution (every outcome is equally likely) and use available information to transform it into a more appropriate distribution. The parameters of the transformation represent a quantitative use of known information. The shape of the final distribution represent a qualitative use of the known information.

With this brief treatment of probability, I can now address the obvious fact that the use of reason cannot guarantee correct decisions. Consider an example. I have historical data for the exchange rate between a pair of currencies. I also have market quoted prices for various financial instruments involving the currency pair. To model the exchange rate at some future time with a probability distribution, I can use the historical data to establish the shape of the distribution and the market quoted prices to obtain the parameters of the distribution. If I had more information (say a model for other parameters that affect the exchange rate), I could incorporate that too. A decision based on such a model would be a rational decision. On the other hand, I could say that since the model does not guarantee success, I will simply use a uniform distribution (Ouch!! That is not even possible since the range for the exchange rate is unbounded. Let me simply restrict the range to an intuitive upper bound) with the arguement that the uniform distribution might actually turn out to be better. Yes, it might turn out to be better, but the arguement that it should be used is still invalid (Consequentialism is invalid and I am not going to argue this). Not all decisions can be formulated with precise mathematics like this, but the principle is the same. It is always better to use all my knowledge to the best of my ability.

Another aspect of the original discussion remains unaddressed – the claim that rationality is subjective. Since this post has already got long enough, I will just stress here that there is a difference between context-dependent and subjective.

Intuitions and a-priori knowledge

In a comment on my post on hypotheticals, Krishnamurthy asked:

When you say “Rationality means that man must instead find principles on which to base his actions “, the question arises about how to arrive at those principles. If he cannot use his intuition, and if he cannot do the complicated expected utility maximization, then he can only arrive at the principles by evaluating the outcomes of his previous actions. But to evaluate he would need some principles to begin with. (on second thought, even to do expected utility maximization, he would need to make some evaluations). how does a human being find the principle to base his actions on ?

I hold that knowledge can never be a-priori. To see why, consider these questions

Does a digital balance know how to measure weight?
Does a computer know how to add numbers?
Does my heart know how to pump blood?
Do my eyes and brain know how to distinguish objects from each other?
Does a parrot that recites 2 + 2 = 4 know that 2 + 2 = 4?

My answer to all these questions would be no. There is no knowledge involved here. Knowledge, in the sense applicable to a human mind, involves the exercise of free will. An entity that does not have free will cannot have any knowledge. It is like a machine that does certain things because that is its nature. Since no exercise of free will can occur before one exists, knowledge cannot be a-priori.

Now consider the human mind. I believe that the mind is built with the capacity to use logic, but not with the knowledge of the laws of logic. This is a subtle point. What I am saying is that the mind has an inbuilt ability to determine whether something makes sense. But active effort is required to use this ability. And further effort is required to identify why it makes sense. Men obviously have been using logic for millenia. But it took Aristotle to identify the laws of logic. The operation of the laws of logic is part of the nature of the mind but the knowledge of the laws of logic is not. It takes active effort to grasp the laws of logic – to realize that when something “makes sense”, it is because that something is consistent with the laws of logic. The faculty that is capable of doing this grasping is reason. Man is born with the faculty of reason. But it is the use of reason that results in knowledge.

Recollect the time when you learnt the truth table for “p AND q” where “p” and “q” are propositions. How did you grasp that the truth table was correct? I did so by substituting actual propositions for “p” and “q” and verifying the values in the truth table. This indicates that knowledge of the truth table was not a-priori but the ability to verify particular propositions was. The truth-table was <i>induced</i> from the ability to verify particular propositions. More importantly, this also indicates that in the absence of any particular propositions, I could not have induced the truth table for “p AND q”. This is another reason that knowledge cannot be a-priori.

The ability to understand and evaluate propositions and to induce principles is inbuilt. If you want to call this ability intuition (I call it reason), I have no problem accepting the validity of intuitions, provided effort is made to express the result of this “intuition” in terms of the laws of logic, observations and any other principles one has already validated. But I don’t think this is what anybody means by intuition. For example, the Merriam Webster dictionary defines intuition as
1: quick and ready insight
2 a: immediate apprehension or cognition b: knowledge or conviction gained by intuition c: the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
Note the parts I have emphasized. They all indicate that intuition is knowledge achieved without active effort and without the use of reason.

Does this answer your question?

Government and education

A while back I came across this infuriating story (via A Little Lower than The Angels) of a man who did not send his children to a public school against the law of his state and so was shot dead by the agents of the state. Since I have written a bit lately on the moral and political implications of public education, this is a good time to relate this story to that debate. The legal murder of John Singer is the logical conclusion to any arguement that advocates public education. Here’s how.

a) The state has the power to tax me to provide public education.

b) Therefore I have a legal responsibility to the state for the welfare of others.

c) Therefore the state may decide that my children’s education is essential to the welfare of others (free and compulsory education)

d) Therefore  the state may decide what this education must consist of.

e) Therefore the state may punish me (ultimately by death if I resist) if I refuse to accept the state’s requirements.

Do you agree with (a) but not with (e)? Examine your premises. Logic has a way of catching up with people even if they do not choose to be logical.

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