Impossible logic

A)    The impossibility of logic

1. Suppose you build a computing machine, and you give the order: “You will never say if this sentence is true.” If the sentence is true, then the machine should say that the sentence is false. If it is false, the machine can tell the truth that the sentence is false. So we will never know the correct answer. This is a problem that Godel introduced, showing that logic is not immune to inconsistencies. Logic is not a ‘perfect machine of truth.’ Godel even quantified the problem with a theorem, which simply says that for each theory Τ there is a sentence G which states that “G cannot be answered by Τ.” If G could be proved by the axioms of Τ, then Τ would have a theorem G, which is contradictory, so Τ would be inconsistent. But if Τ is consistent then G cannot be proved by T, thus T is incomplete.

2. This a description of Kurt Gödel’s incompleteness theorem, by Solomon Feferman: “Actually there are two incompleteness theorems, and what people have in mind when they speak of Gödel’s theorem is mainly the first of these. Like Heisenberg’s uncertainty principle, it has captured the public imagination with the idea that there are absolute limits to what can be known. More specifically, it’s said that Gödel’s theorem tells us there are mathematical truths that can never be proved. Among postmodernists it’s used to support skepticism about objective truth; nothing can be known for sure. And in the bibliography of Christianity and mathematics it’s asserted that theologians can be comforted in their failure to systematize revealed truth because mathematicians cannot grasp all mathematical truths in their systems either.” [1]

B)    Reduction to infinity

3. One way to prove that a logical argument is true, is to prove that the opposite of this argument is false. This is the process of reduction to absurdity. An example is the following one:

- The Earth cannot be flat; otherwise (supposing the opposite), we would find people falling off the edge. [2]

Thus the original argument is true (reduction to the absurd).

4. This is in fact a very week argument. A similar argument is the following one:

- The Earth cannot be round; otherwise (supposing the opposite), we would again find people falling off the opposite side.

Thus the original argument is true (the Earth cannot be round either).

5. This example showed us that logic is absurd on its own. The deeper problem however with the method of reduction is that it may take an infinite number of steps to prove that an argument is true, or to contradict a false argument. For example we can suppose that the elementary particles of matter can be divided into smaller particles. The purpose of this assumption is to find in the end a particle which is really fundamental. However, as observation shows, there will always be particles smaller and smaller (electrons and quarks are considered at the moment fundamental, but sooner or later their constituents will be found).

6. This is another example: If we call P₀ the sentence: ‘There is a fundamental indivisible particle of matter,’ then we have the following steps:

P₀: ‘There is a fundamental indivisible particle of matter (presumably the atom).’

P₁: ‘The atom can be divided into electrons and nucleons (protons and neutrons).’

P₂: ‘The nucleons can be divided into quarks.’

P₃: ‘Probably quarks can be divided into smaller fundamental particles (yet unknown).’

…

P_n: ‘Those yet unknown particles would in turn be divided into even smaller particles.’

P_n+1: ‘And so on…’

…

Thus it can never be proven if there is a fundamental particle of matter.

C)    The moral basis of logic

7. The previous process can also refer to logic itself. If we suppose that logic consists of a fundamental argument (an axiom) from which all other arguments are produced then we may never find an axiom which cannot be divided into smaller parts. Thus we can never prove if logic is based on a fundamental principle.

8. Leaving the microcosm, such a basic principle on a large scale can be God. On one hand we might say:

- God cannot exist. If God existed then there wouldn’t be so much injustice in the world (assuming that God would be a moral being who would have never let injustice rule the world).

9. On the other hand, assuming that God can never be immoral we have to conclude that God cannot help us, thus God may be great but not omnipotent:

- God exists but He is not omnipotent. If He were omnipotent, He wouldn’t let people suffer.

10. In the same sense, since God is presumed as an axiom of logic, we may simply conclude that logic is not omnipotent, even if it exists on its own:

- Logic exists but it is not omnipotent. It can cannot answer all questions.

11. Thus logic cannot perceive absolute Good. Although logic is based on a moral axiom (the existence of God in the form of absolute Good), it cannot prove it. This does not necessarily make logic immoral, but it shows that Truth is something which is neither ‘right’ or ‘wrong,’ thus it cannot be reached by using a packet of ‘yes’ and ‘no’ sentences.  

D)    The facts of perception

12. We cannot find everything with logic. This however raises the paradox that by assuming things which cannot be understood we have already included those things in logic (even if they are incomprehensible).

13. Hermann von Helmholtz, who is often credited with the first study of visual perception in modern times, examined the human eye and concluded that it was optically rather poor. The poor-quality information gathered via the eye seemed to him to make vision impossible. He therefore concluded that vision could only be the result of some form of unconscious inferences: a matter of making assumptions and conclusions from incomplete data, based on previous experiences. [3]

14. In his own words, “The problems which that earlier period considered fundamental to all science were those of the theory of knowledge: What is true in our sense perceptions and thought? And in what way do our ideas correspond to reality? Philosophy and the natural sciences attack these questions from opposite directions, but they are the common problems of both. Philosophy, which is concerned with the mental aspect, endeavors to separate out whatever in our knowledge and ideas is due to the effects of the material world, in order to determine the nature of pure mental activity. The natural sciences, on the other hand, seek to separate out definitions, systems of symbols, patterns of representation, and hypotheses, in order to study the remainder, which pertains to the world of reality whose laws they seek, in a pure form. Both try to achieve the same separation, though each is interested in a different part of the divided field...” [4]

E)     Diving into the irrational

15. Deep beneath our rational thought lie processes of the mind which are unconscious. Such processes or actions rise from the unconscious and produce effects on our mind, involuntary and ambiguous thoughts, which we then rationalize. Therefore logic can be nothing else than the orderly way we conceive and treat those processes. However by challenging logic we face the danger of becoming incomprehensible or even paranoid. But perhaps paranoia is the first stage, after which we may discover a new dimension of thinking.

16. According to its formal definition, paranoia is an instinct or thought process believed to be heavily influenced by anxiety or fear, often to the point of delusion and irrationality. Paranoid thinking typically includes persecutory delusions, or beliefs of conspiracy concerning a perceived threat towards oneself. Making false accusations and the general distrust of others also frequently accompany paranoia. For example, an incident most people would view as an accident or coincidence, a paranoid person might believe was intentional. [5]

17. Psychoanalysis, which was founded by Sigmund Freud, can be defined as a therapeutic method related to the study of the unconscious mind. During psychoanalytic sessions the patient expresses his or her thoughts, including free associations, fantasies and dreams, from which the analyst infers the unconscious conflicts causing the patient’s problems. [6]

18. Surrealism used the method of free association in order to achieve artistic results. Andre Breton defined surrealism as “psychic automatism in its pure state, by which one proposes to express…the actual functioning of thought…in the absence of any control exercised by reason, exempt from any aesthetic or moral concern.” [7]

19. In addition, Salvador Dali invented his ‘paranoiac- critical method,’ which he described as, “spontaneous method of irrational knowledge based on the critical and systematic objectivity of the associations and interpretations of delirious phenomena.” [8]

20. Dali used his method to relate objects that were otherwise unrelated. He did this through the use of optical illusions and juxtaposing images. He also said that, “The subconscious has a symbolic language that is truly a universal language, for it speaks with the vocabulary of the great vital constants, sexual instinct, feeling of death, physical notion of the enigma of space- these vital constants are universally echoed in every human being.” [9]

21. If reality comes about as a certain combination of some of its constituent things then the causal and moral method which logic uses to bring things together in a certain way could be just one possible way (and apparently not the best one). What if there is an infinite number of ways (no matter how irrelevant or extreme they might seem) to combine everything which is found in the unconscious? What if the unconscious contains a deeper and wider reality, waiting for us to be explored?

22. In that sense what we call paranoia may be the result of the human mind coming in contact with a higher transcendental and universal intelligence. This kind of intelligence is related to Carl Jung’s notion of the collective unconscious. At a first stage such a contact may create fear and withdrawal from that deeper reality. But should we let fear or aversion deprive us from the opportunity of discovering and exploring a greater truth?

F)     Quantum logic

23. Although logic cannot perceive infinity (logic infinitely approaches the problem of infinity without ever reaching infinity), it has an idea about the infinity it is trying to approach. Can we overpass logic by using logic, or do we need another way of thinking, and if yes then of what kind is that new thinking?

24. An example is the failure of the distributive law according to quantum logic:

- Let’s suppose that we have a particle which has a momentum p, and can be found either at position q or at position r. The distributive law is stated as follows:

p and (q or r) = (p and q) or (p and r)

The problem however is that in quantum mechanics, according to the uncertainty principle, it is impossible to know both the momentum (p) and the position (q or r) of a particle at the same time. Thus both results (p and q) or (p and r) are false in quantum logic, and the distributive law of classical logic fails. [10]

G)    The little black box

25. Quantum logic offers examples of how an answer can be neither ‘yes’ nor ‘no’ but a combination somewhere in between. Here is another example about how we may treat our own thought:

26. Imagine that there exists in the universe a little black box which contains all possible information about everything. This information can be represented by bits, things, letters or words. Logic would be a certain way by which we combine those things or words. Irrational thought could be a combination which does not make sense, or which produces contradictory results.

27. Perhaps in the beginning we are afraid to open this box because some of its contents could be scary or unbearable (for example the information about the date when we will die). But sooner or later we have to open the box because it is the same box where our own thought stems from. Thus instead of avoiding to open the box, or pretending that it doesn’t exist, we had better open it and learn how to control its contents, as well as our thoughts which are the products of these contents.

28. Thus the secret cannot be to abolish logic or its moral codes but, using instead our imagination, to find different ways to combine the things of which logic consists, reconsidering consequently the answers about what is true or false, thus also good or bad, purposeful or coincidental. Even if there are questions which might never be answered, it seems that the problems are created by our own mind in the first place. Therefore our own thought seems to be both the problem and the solution. It is the little black box waiting to be opened and to be explored.

[1]: [http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf]

[2]: [https://en.wikipedia.org/wiki/Reductio_ad_absurdum]

[3]: [http://en.wikipedia.org/wiki/Visual_perception]

[4]: [http://www.marxists.org/reference/subject/philosophy/works/ge/helmholt.htm]

[5]: [https://en.wikipedia.org/wiki/Paranoia]

[6]: [https://en.wikipedia.org/wiki/Psychoanalysis]

[7]: [https://www.moma.org/learn/moma_learning/themes/surrealism]

[8]: [https://en.wikipedia.org/wiki/Paranoiac-critical_method]

[9]: [http://emerald.tufts.edu/programs/mma/fah188/clifford/Subsections/Paranoid%20Critical/paranoidcriticalmethod.html]

[10]: [https://en.wikipedia.org/wiki/Quantum_logic]

8/24/2018

Picture: [http://subbody.net/image/calabi2odorustring.jpg]