The conceptual foundations of decision-making in a democracy

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Paradoxes: A Sure Sign of an Error of Thought.

All paradoxes have the same basic structure: a deduction from assumptions which we consider to be true leads to a conclusion which we know to be false. The rules of deductive thinking, logic and mathematics are based on the assumption that a fact cannot at the same time exist and not exist, that there are no miracles in nature. If we accept these rules, then paradoxes then must result from an error of thought, either in the truth of the assumptions or in the rules or process of deduction. The deduction-process of the well-known paradoxes has been checked over and over again by the best minds of history, so we can safely assume that it was correctly performed. The rules of deduction have never been proven wrong in their application to actual situations in the myriad of times they were applied and thus are anyway not a prime suspect. The error must then lie in the assumptions. The explicit assumptions also have never been questioned: they are axioms. The most promising approach then is to search for implicit assumptions, and one wonders why that approach has hardly ever been taken. Apparently some assumptions seem so evident that they have become part of our unconscious thinking. If the error indeed lies in assumptions, then it must be possible to express in common language the misconception of reality which is at the root of the paradox. It will turn out that the major causes of paradoxes are either a lack of understanding of the information process or the - implicit - belief in an autonomous existence of abstract concepts. We will show both at work, taking as an example the most famous paradoxes: ‘the liar’ and ‘Achilles versus the tortoise’.

Note that part of the ‘de-construction’ of paradoxes presented in this chapter is already implied by Kant and Tarski. Kant resolves his antinomies by noting that we cannot from purely transcendental concepts ever draw conclusions as to our real world of experience, and thus cannot on the basis of totally abstract concepts deduce the truth of any statement about that real world; we can only decide about the formal correctness of that statement. Tarski developed the philosophically acceptable notion of truth as correspondence of statement to fact and noted that it is applicable exclusively to statements about relations between objects and that these statements must be unambiguously formulated, must conform to a formal language such as logic and mathematics. My contribution is to show that their reasoning finds empirical support in our knowledge about information, especially its functional nature.

THE LIAR: MISUNDERSTANDING THE INFORMATION PROCESS. First formulated by Zeno, there are two modern versions of this paradox:

1)	‘I am lying now’
2)	‘The sentence on page 123 is false’. On page 123 there is just one statement: ‘This sentence is false.’

An information process always consists of a value-giving subject which assigns a value to stimuli received from the world outside this value-giving subject, attempts to construct a representation about some object outside the current information process. Outside refers to everything except that which is engaged in the specific process we are talking about. The current process itself can never be an object in this process. It can become an object only in a sub-

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sequent process. Concerning the paradoxes, the very first thing to do is to determine who is the value-giving subject, and what is the purpose it has in mind when engaging in the information process concerned. In these paradoxes, the value giving subject is obviously whoever decides that the statement involved is a lie. Whenever we use the qualification of true or false, the purpose we have is to assess whether a statement purporting to represent some element of our reality does indeed correspond to that element, which obviously cannot be the statement itself. If the statement refers only and entirely to itself, there is no possibility of a difference between the statement and the reality about which it purports to talk. Even a tautology is not in first instance written as a = a, but as a = b. The identity of a and b is not a foregone conclusion, it is the contribution of the tautology.

In both versions of the paradox there is no element of the world outside the statement to which the statement could correspond: the information is about nothing, so there is simply no information process, as shown below.

ad 1, ‘I am lying now’. In that statement, the value giving subject is ‘I’ and the value is ‘lying, i.e. ‘not true’. It says: -a. That is all there is. For ‘now’ is not an object, but only the chronological situation of the process itself. There is no paradox, but only an incomplete, meaningless, sentence referring to nothing outside itself, and thus not amenable to an evaluation as to its truth. The statement ‘I am beautiful’ is of a totally different nature, for the object is the appearance of the ‘surface’-elements of the living being making the statement, while the value-giving entity is a set of the neurons in the brain of that being which are instrumental in making the assessment which gives value to the image of the being. So the word ‘I’ appears in the statement in two guises: as object (its ‘durface) and as value-giving subject (neurons in the brain), it refers to two different parts of the same individual; it illustrates the appropriateness of distinguishing between the notion of I and of individual, person.

ad2, ‘The sentence on page 123 is a lie’. First note that through all paradoxes of this type runs one implicit assumption: ‘if an expression is not true, it must be false’ and vice versa. The notion that it may simply not be intended to convey anything about the real world is not even considered, which is surprising. For a ‘normal’ person confronted with a sentence which sounds a bit odd will ask: ‘do you really want to say something, and if you do, what is it?’ Even in the logic of propositions - which ‘knows’ only two meanings, true and false - there is a third alternative for evaluation, namely whether the sentence is well-formed or not, whether or not it is a sentence, a ‘formula’, of the language concerned. Let us look at the sentence on page 123 (‘This sentence is false’) and attempt to translate it into a sentence of the logic of propositions. ‘Is false’ obviously is the negation.

The negation is only a connective. If the statement is to be a formula of logic, ‘this sentence’ must be a propositional variable, must be whatever the statement is about, say A. The statement then reads (-A). A propositional variable is a reference to an unspecified aspect of reality about which we want to talk in the language of logic. A propositional variable can stand for any word or sentence, say ‘my nephew is a bore’, provided this variable refers to some element of reality. We must now decide on the propositional variable of the paradox. If we chose for A, then it means ‘anything except A’. Using the above example, it means ‘my nephew is not a bore’.

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Until you have specified to which element of reality A refers, the sentence does not refer anything, has no meaning. By no kind of philosophically acceptable concept of truth can it be branded true or false. If it refers to the whole sentence, then it states B = -A. As long as -A refers to nothing, B also refers to nothing and cannot be evaluated in terms of true and false. The whole sentence on page 123 then is neither true nor false, it is not a formula of the logic of propositions, only a potential element of one. It is an incomplete sentence even in common language, it is ‘non-sense’. The first sentence, saying ‘the sentence on page 123 is false’, is indeed false because the statement referred to is not a complete sentence and thus cannot be qualified as either true or false.

This explanation of the Paradox is of the same nature as that of Russell's, who says that you have to choose whether you are talking in the object language, i.e. use the sentence to talk about something other than the sentence itself, or whether you want to talk in a meta-language which can also talk about the sentence itself. To that effect Russell has to establish a rule and then prohibit the sentence as not respecting that rule. The well-established tenet that whenever we can do without extra conditions we should do so pleads for the above explanation which relies only on the established rules of logic and of common language.

Zeno's form of the paradox was: ‘All Cretans are liars’ plus: ‘I am a Cretan’. Those sentences are legitimate sentences both in logic and common language. It is a paradox because he himself is a Cretan. So he will have lied. But if he did, all Cretans tell the truth, also Zeno. But he just told a lie ... etc. The solution to his ‘paradox’ is so simple that I have never understood why it has fascinated so many. Applying the axiomatisation I advocate, the most elementary logic shows that underlying Zeno's paradox there is an assumption, namely that the Cretan population is either composed of persons who never lie or who always lie; it excludes the possibility that Cretans sometimes tell the truth and sometimes lie according to what they consider appropriate under the circumstances. That assumption is clearly unwarranted: one might assume that there can exist people who always tell the truth, whatever the consequences, but there will never be people who always tell a lie, for that would serve no purpose whatsoever as they will soon become known as such. Again the paradox is created by an ad hoc use of language and is not intended to communicate something about our world, but to create a paradox.

ACHILLES AND THE TORTOISE: ASSUMING THE AUTONOMOUS EXISTENCE OF ABSTRACT CONCEPTS. A tortoise challenges Achilles to a race, betting that he will not overtake it if Achilles gives him a head start of 100 m. Achilles accepts. Running one hundred times faster than the tortoise, Achilles has covered the 100 m head start by the time the tortoise has covered his first meter. But by the time Achilles has covered this meter, the tortoise has covered another cm. When Achilles has covered this cm, the tortoise has advanced one tenth of a mm, and so on... Conclusion: Achilles will never overtake the tortoise. Yet we know that in reality Achilles will overtake the tortoise. This apparent paradox has even spawned the assertion that we cannot rely on our senses.

It is difficult for me to understand why this ‘paradox’ has not been relegated to the realm of entertaining riddles right at its inception. For if taken seriously, if we really want to register if

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and when Achilles will overtake the tortoise, we immediately see that the above description of the race never was designed to register in any practical way the location at which Achilles overtakes the tortoise. In fact, by establishing the rule that we never let Achilles run beyond the distance which separated him from the tortoise at the previous measurement, we have effectively precluded the eventuality that he could overtake it.

The perniciousness of that paradox lies in the choice of an iterative procedure without acknowledging its inherent limitations. Had we chosen algebra, we would have found that the tortoise will be overtaken at 100 divided by point nine metres, an irrational number which we can only approximate: 111,111... Had we chosen other figures for the head start and difference in speed which would not result in an irrational number, no paradox would have arisen.

The only objective, other than creating a paradox, which we can think of for establishing such an observation procedure is to approximate - to whatever precision we want - the geometrical point at which Achilles overtakes the tortoise. A geometrical point is a totally abstract construction defined as having an infinitely small size. Any size, however small, which we would attribute to a point ipso facto disqualifies that point as a geometrical one. As such a point has no ‘real’ existence, it can never be measured in practice. It is an illustration for the assertion that abstract concepts have no observer- independent existence. We do not establish iterative procedures for measuring abstract concepts, but to decide in a practical way about properties of concrete items. The precision required from a measurement is dictated by its use. The procedure of the ‘paradox’ is perfectly adequate to establish at which nanometre Achilles will overtake the tortoise.

Both this paradox and the one about the liar arise from the failure to acknowledge the instrumental character of any information process, including language and the abstract concepts we create. We must expect to run into problems if we use them for purposes for which they have not been developed. The paradoxes also underline how dependent the adequacy of communication is on a shared objective and on the respect of the rules of conduct which we can deduce from this shared objective.