**In orthodox economic analysis – also called ‘economic logic’ – the concepts ‘economic’, ‘rational’ and ‘logical’ are not clearly distinguished. This is a major source of confusion.in the text of the book the terms economic and rational are extensively explained. In thus appendix the term ‘logical’ is discussed.**

When discussing the problem of logic, pretty soon it becomes clear that the term logic is a vague and fuzzy concept. In our daily language we use it when we mean to say that we understand the mechanism(s) behind a particular phenomenon. But when we study logic as a field of science, most of the time we refer to classical or orthodox logic, which is based on a series of laws or axioms that are always assumed when making a statement. However, also within logic as a science there are different approaches – some are close to the classical approach, while others are close to the more popular use of the term.

Aristotle (384 – 322 BC) was one of the first philosophers who dealt with logic extensively. For him logic is a tool to argue convincingly. He analysed sentences by making a difference between subject and predicate. For instance, in the sentence ‘all human beings are mortal’ the part ‘human beings’ is the subject, while the part ‘are mortal’ is the predicate. Syntactically the sentence is correct: there is a subject and there is a predicate that says something about the subject. Whether the sentence is true and has meaning are different questions. Aristotle developed the syllogism, which is a tool to derive conclusions from a series of assumptions. The truth of the conclusion is a logical implication of the truth of the assumptions. Later Chrysippus (280 – 206 BC) introduced so-called logical connectives, such as ‘and’, ‘or’, ‘if’ and ‘then’. This made him able to analyse complex statements and the (logical) structure of large texts (propositional calculus).

For centuries Aristotle’s analyses dominated the study of logic. Then Leibniz (1646 – 1716) came with the idea to abstract from concrete sentences and to use algebraic equations in his search for the logical structure in our language. He formulated a law, stating: a = b. According to him this identity is the basis of all logic. He derived four axioms (or laws) from this identity, which were sufficient to analyse an infinitely large number of sentences. These axioms are:

(1) a = a; this is an expression of the law of identity;

(2) if a = b and b = c, then a = c; this is the law of substitution;

(3) a = not (not a); this is the law of non-contradiction; [¬ (a & ¬ a)]

(4) a is b = not –b is not –a; this is the law of the excluded middle. [a v ¬ a]

While the first two laws are about the identity of a particular element, are the third and fourth laws basically saying that something is ‘true OR false’ – not ‘true AND false’ and not something ‘in between true and false’.

This set of laws describe what is typical logical. It is the basis for a system of rules of thought. Leibniz appeared to be able to isolate these rules from the practical grammar, which is embedded in interpretation and judgment. This is the logical world, describing the rules that are necessary conditions for understandable thinking. These rules are not sufficient conditions, since we cannot check whether our logical statements are true in the empirical sense. Many philosophers followed Leibniz, of which Kant is the most important one. In his “Kritik der Reinen Vernunft” (1781) Kant interpreted the axioms of logic as describing the structure of thinking: the human mind is not able to understand texts that do not meet these logical conditions. Actually Kant was saying that our mind frames information in such a way that we will never be able to directly observe ‘reality’ – our observation always runs via the framework that is set by the mind.

Mathematicians tried to use the logical axioms to see whether or not it could function as a foundation for mathematics. Frege (1848 – 1925) introduced so-called quantifiers, such as ‘all’, ‘some’, ‘most’ to get a link between logic and mathematics. But others, especially Hilbert (1862 – 1943), tried to proof the independence of typical mathematical axioms, such as statements about the existence of numbers (1 + 1 = 2) and lines (two lines are parallel if these intersect each other in infinity), from the typical logical axioms. However, Gödel (1906 – 1978) showed that any system of axioms is incomplete or inconsistent. Take, for instance, the sentence: ‘the law of identity is unprovable’. If this statement is true, then logic is an incomplete system, namely based on an axiom that cannot be related to other axioms; it can just be assumed. If this statement is false, then the system is inconsistent.

The result found by Gödel can be interpreted as follows: logicians have discovered that human thinking has a logical aspect. This can be presented as an aspect-system. This system is based on the assumption that every thing has an identity. We can never logically proof the truth of this identity axiom. It is saying that the world is knowable – it identifies itself to us. If someone is denying it, we cannot proof he is wrong. Mathematicians have discovered that in our imagination not only logical axioms but also typical mathematical symbols like lines and numbers play a role. We cannot prove their existence (logically); we can only assume their existence on the basis of our experience of understanding what we mean when we talk about one, two and three. A simple acceptance leads to adoption of a huge number of other statements when applying logic to the typical mathematical axioms.

Human thinking is subject to restriction, like all other aspects of human life. An important function of logic is the discovery of contradictions or paradoxes. The Zeno or movement paradox shows our difficulty when dealing with the problem of finity versus infinity. If we lag behind someone else in a particular race, we can never pass the other: the time we need to bridge the gap, is used by the other to stay ahead. If we are faster, we will approach but never reach the other. This paradox shows our difficulty to imagine finity; we can only imagine the end of something, not the end of everything; we always ask ourselves what is next. We cannot imagine ‘nothing’, only ‘some thing’. It is a contradiction to imagine ‘no thing’. In many races we see athletes lagging behind but running faster, at a particular moment they pass the other. This matter of experience can only be explained, when avoiding the finity/infinity–trap. Another paradox is the so-called heap-paradox. It is about a heap of sand. When we take one grain of sand from the heap, it is still a heap. When we continue taking away grains of sand from the heap, we end up with one grain, which is definitely not a heap. But when does the set of grains end with being a heap? We can solve this problem only by making an agreement on the definition of a heap of sand. This paradox makes clear that an identity is not fixed. Everything is in constant change and in our real world, in contrast to our logical world, it is not necessarily be true that a = a. A third paradox is called a self-referential paradox. If a statement states something about itself, it can be wrong or self-evident. For example: ‘this sentence is false’. This is paradoxical. But the sentence ‘this axiom is true’ does not add something relevant; it just stresses the character of the axiom, to which the sentence refers, namely it can not be proven and is assumed because of its self-evident character. So is the axiom ‘a = a’ self-evident, while the statement (a) = (not –a) is a contradiction, which cannot be accepted as part of our system of logic.

So far we can draw the conclusion that the logical world is an isolated world that describes just one aspect of the real world. If we want to develop a ‘logical’ system that can be applied to real world problems, we might adjust this classical or orthodox world, so as to make it more practical.

This is exactly what happened in the world of logicians and mathematicians. Two important experiences were playing a role in the development of logical and mathematical systems. In the first place, we must acknowledge that our information and knowledge of reality is very limited. In the second place, our brain capacity is not only limited, but also is framed in a particular way. In terms of physical and chemical processes (the hard ware) we have just a limited control over it. In terms of thinking and feeling (the operating system and other soft ware) we all are framed in ways, which are also affected by the cultural context that surrounds us. One important implication is the discovery that in practice our thinking does not take place like a logical machine. The search for relevant information takes place within particular frames of interpretation and analysis. These frames (or world views, see Weber) organise information available in an efficient and directive way. Within these frames humans are rational, in the sense of logically arranging the positive and negative effects of particular strategies. But the “choice” of frame is not a rational-logical affair.

This is the reason why logicians and mathematicians have tried to develop alternative approaches to the problem of modelling human thinking. We will discuss two alternative logics, namely ‘fuzzy logic’ and ‘intuitionistic logic’. Fuzzy logic rejects the law of the excluded middle and allows as truth values any real number between 0 and 1, where 0 stands for completely false and 1 stands for completely true. So, something is true or false to a certain extent only. Related to this problem is the problem of fuzzy concepts. Basically every concept is just an attempt to approach the indicated reality; it is never reality itself. When we discuss important phenomena such as unemployment or inflation it is not exactly clear what we mean by this. If experts try to define and measure it, there appear big differences. Intuitionistic logic, as developed by Brouwer (1881-1966) and others, not only rejects the law of the excluded middle but also the double negative elimination (p = ¬ ¬ p). This type of logic accepts a position of ‘I don’t know’, besides true or false.

When analysing the mind classical logic imagines the mind as a logical machine. New approaches picture the mind as a neural net, which can be in any of a variety of states. In a logical machine thinking is the same as formal deduction: deriving logical implications from premises. In a neural net, however, first a process of categorization takes place. How? We don’t know! Then we observe the situation and try to find relationships between different categories. When we feel pain when touching a very hot thing and we have this experiences a few times, we tend to draw a general conclusion that we will always feel pain when touching a very hot thing. This ‘inductive inference’ is just a psychological fact – we cannot justify it. This approach might lead to pattern recognition, which arouses positive sentiments connected to the idea that we understand something. Particular relationships are recognised as being part of a bigger whole, giving these particulars meaning. When getting to relationships it is (classical) statistics rather than (classical) mathematics that plays a role.

Since all these fields – not only logic, but also mathematics and statistics – are facing the problem of uncertainty and severe lack of information and knowledge, we see the emergence of fuzzy and intuitionistic statistics and fuzzy and intuitionistic mathematics. The common denominator is their attempt to make the classical approach more practical. Philosophers and mathematicians such as Wittgenstein and Keynes (John Maynard, 1883 – 1946, also a famous non-classical economist) tried to say something systematic about interpretation and judgment and about probability and belief in a non-classical way.

Now we can understand what is meant by the axiom of orthodox economics, saying that classical logic can be applied. Although there is no authority that can fix the meaning of the word ‘classical’ here – again a fuzzy concept – it is quite acceptable to define the hard core of classical logic as the four laws of Leibniz.

References

(1) Dan Cryan, Sharon Shatil, Bill Mablin, Logic, Icon Books, 2001.

(2) Sheila Dow, Economic Methodology, An Inquiry, OxfordUniversity Press, 2002.

(3) Timothy Gowers, Mathematics, OxfordUniversity Press, 2002.

(4) Chris Horner, Emrys Westacott, Thinking through Philosophy, CambridgeUniversity Press, 2000.

(5) Roger Scruton, Kant, OxfordUniversity Press, 1996.

Dr.Piet Keizer

Utrecht University School of Economics

30-12-2006.