Self-reference in Paradoxes

Simple Instruction for Generating Paradoxes

The trick with which classical logical systems can be broken consists of two instructions:

1: A statement refers to itself.
2: In the reference or in the statement there is a negation.

This constellation always results in a paradox.

A famous example of a paradox is the barber who shaves all the men in the village, except of course those who shave themselves (they don’t need the barber it). The formal paradox arises from the question of whether the barber shaves himself. If he does, he’s one of those men who shave themselves and, as the statement about the barber says, he doesn’t shave those men. So he doesn’t shave himself. Therefore, he is one of the men who don’t shave themselves – and those men he shaves.

In this way, the truth of the statement, whether or not he shaves himself, constantly changes back and forth between TRUE and FALSE. This oscillation is typical of all genuine paradoxes, such as the lying Cretan or the formal proof in Gödel’s incompleteness theorem, where the truth of a statement oscillates continuously between true and false and thus cannot be determined. In addition to the typical oscillation, the barber example also clearly shows the two conditions for the true paradox mentioned above:

1. Self-reference: Does he shave HIMSELF?
2. Negation: He does NOT shave men who shave themselves..

At this point, reference can be made to Spencer-Brown, who developed a calculation that clearly demonstrates these relationships. The calculation is presented in his famous book ‘Laws of Form’. Felix Lau commented the ideas to us laymen in his book ‘Die Form der Paradoxie’.

False Paradoxes (Zenon: Achille and the Turtoise)

These ‘classical’ and true paradoxes can be juxtaposed with ‘false’ paradoxes. A good example is the ‘paradox’ of Achilles and the tortoise. This false paradox does not contain a genuine logical problem, as in the case of the barber, but is built around an inadequately chosen model which leads to Zenon’s surprising, but wrong conclusion. The times and distances that the two competitors are getting shorter and shorter and are thus approaching a value that cannot be exceeded within the selected (wrong) model. This means that Achilles cannot overtake the tortoise in the model. In reality, however, there is no reason for the times and distances to be distorted in such a way.

The impossibility of overtaking the turtoise exists only in the model, which is incorrectly selected in a subtle way. A measurement system that fudges in this way is of course not appropriate. In reality, it is only a perfidious choice of model, and not a real paradox. Accordingly, the two criteria for genuine paradoxes are not present.

The Crucial Role of Model Selection

The example of Achilles and the tortoise shows the importance of a correct choice of model. The choice of model always takes place outside the representation of the solution and is not the subject of a logical proof. Rather, the choice of model has to do with the relationship of logic to reality. It takes place on a superordinate meta-level.

I postulate that the field of logic should necessarily include the choice of model and not just the calculation within the model. How do we choose a model? If logic is the study of correct thinking, then this question must also be addressed by logic.

The Meta-level is Necessary for Choosing the Model

The interaction of two levels, namely an observed level and a superordinate, observing meta-level, plays a necessary role in the choice of model, which always takes place on the meta-level, regarding the observed level.

The play of the two levels can be ovserved, too. This is exactely the case in the true pradoxes, e.g. in the barber paradox. The self-reference in the true paradox inevitably introduces the two levels. This happens because an observed statement refers to itself and exists thus twice, once on the level under consideration, on which it is quasi the ‘object’, and secondly on the meta-level, on which it is referring to itself. The oscillation in the paradox arises through a ‘loop’, i.e. through a circular process between the two levels from which the logical system cannot escape – and it is oscillating because the negation makes it twist by each turn.

Selfreferential Loops and Paradoxes

Incidentally, there are two types of selfreferential logical loops, as Spencer-Brown and Lau point out:
– a negative one (with negation), which leads to a paradox
– a positive one (with confirmation), which leads to a tautology. In other words: self-reference in logical systems is always dangerous!

In order to correctly handle paradoxes in logical systems, it is worth introducing a ‘meta-leap’ – a relationship between the ovserved level and the meta-level.

The acceptance of the two levels and their relation is crucial for understanding the relationship between reality and formal logic.

Self-reference and First-Order-Logic (FOL)

Self-reference causes classical logical systems such as FOL to break down.

See also: Paradoxes and Logic
More on the topic of logic -> Logic overview page

Translation: Juan Utzinger

Logic in Practice and Theory

Computer programs consist of algorithms. Algorithms are instructions on how and in what order an input is to be processed. Algorithms are nothing more than applied logic and a programmer is a practising logician.

But logic is a broad field. In a very narrow sense, logic is a part of mathematics; in a broad sense, logic is everything that has to do with thinking. These two poles show a clear contrast: The logic of mathematics is closed and well-defined, whereas the logic of thought tends to elude precise observation: How do I come to a certain thought? How do I construct my thoughts when I think? And what do I think just in this moment, when I think about my thinking? While mathematical logic works with clear concepts and rules, which are explicit and objectively describable, the logic of thinking is more difficult to grasp. Are there any rules for correct thinking, just as there are rules in mathematical logic for drawing conclusions in the right way?

When I look at the differences between mathematical logic and the logic of thought, something definitely strikes me: Thinking about my thinking defies objectivity. This is not the case in mathematics. Mathematicians try to safeguard every tiny step of thought in a way that is clear and objective and comprehensible to everyone as soon as they understand the mathematical language, regardless of who they are: the subject of the mathematician remains outside.

This is completely different with thinking. When I try to describe a thought that I have in my head, it is my personal thought, a subjective event that primarily only shows itself in my own mind and can only be expressed to a limited extent by words or mathematical formulae.

But it is precisely this resistance that I find appealing. After all, I wish to think ‘correctly’, and it is tempting to figure out how correct thinking works in the first place.

I could now take regress to mathematical logic. But the brain doesn’t work that way. In what way then? I have been working on this for many decades, in practice, concretely in the attempt to teach the computer NLP (Natural Language Processing). The aim has been to find explicit, machine-comprehensible rules for understanding texts, an understanding that is a subjective process, and – being subjective – cannot be easily brought to outside objectivity.

My computer programmes were successful, but the really interesting thing is the insights I was able to gain about thinking, or more precisely, about the logic with which we think.

My work has given me insights into the semantic space in which we think, the concepts that reside in this space and the way in which concepts move. But the most important finding concerned time in logic. I would like to go into that closer and for this target we first look at paradoxes.

Real Paradoxes

Anyone who seriously engages with logic, whether professionally or out of personal interest, will sooner or later come across paradoxes. A classic paradox, for example, is the barber’s paradox:

The Barber Paradox

The barber of a village is defined by the fact that he shaves all the men who do not shave themselves. Does the barber shave himself? If he does, he is one of the men who shave themselves and whom he therefore does not shave. But if he does not shave himself, he is one of the men he shaves, so he also shaves himself. As a result, he is one of the men he does not have to shave. So he doesn’t shave – and so on. That’s the paradox: if he shaves, he doesn’t shave. If he doesn’t shave, he shaves.

The same pattern can be found in other paradoxes, such as the liar paradox and many others. You might think that these kinds of paradoxes are far-fetched and don’t really play a role. But paradoxes do play a role, at least in two places: in maths and in the thought process.

Russell’s Paradox and Kurt Gödel’s Incompleteness Theorems

Russel’s paradox has revealed the gap in set theory. Its ‘set of all sets that does not contain itself as an element’ follows the same pattern as the barber of the barber paradox and leads to the same kind of unsolvable paradox. Kurt Gödel’s two incompleteness theorems are somewhat more complex, but are ultimately based on the same pattern. Both Russel’s and Gödel’s paradoxes have far-reaching consequences in mathematics. Russel’s paradox has led to the fact that set theory can no longer be formed using sets alone, because this leads to untenable contradictions. Zermelo had therefore supplemented the sets with classes and thus gave up the perfectly closed nature of set theory.

Gödel’s incompleteness theorems, too, are ultimately based on the same pattern as the Barber paradox. Gödel had shown that every formal system (formal in the sense of the mathematicians) must contain statements that can neither be formally proven nor disproven. A hard strike for mathematics and its formal logic.

Spencer-Brown and the “Laws of Form”

Russel’s refutation of the simple set concept and Gödel’s proof of the incompleteness of formal logic suggest that we should think more closely about paradoxes. What exactly is the logical pattern behind Russel’s and Gödel’s problems? What makes set theory and formal logic incomplete?

The question kept me occupied for a long time. Surprisingly, it turned out that paradoxes are not just annoying evils, but that it is worth using them as meaningful elements in a new formal logic. This step was exemplarily demonstrated by the mathematician Georg Spencer-Brown in his 1969 book ‘Laws of Form’, including a maximally simple formalism for logic.

I would now like to take a closer look at the structure of paradoxes, as Spencer-Brown has pointed them out, and the consequences this has for logic, physics, biology and more.

continue: Paradoxes and Logic (part2)

Translation: Juan Utzinger

Hans Rudolf Straub

Tag Archives: Paradox

Self-reference 2 (Paradoxes)