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Logic: consistent and complete
1. Logic: consistent and complete
In logic (containing arithmetic), one would like the following to hold.
Consistency means that only true propositions can be proven true.
Complete means that everything that is true can be proven true.
However, such logic systems cannot be both consistent and complete. Therefore, a complete system allows everything to be proven true and everything to be proven false.
Every system that is complete will have inconsistencies.
Every system that is consistent will not be complete.
Some interesting ideas come out of these ideas of what is "
true".
2. Truth types: logic, reality, opinion
The word "
truth" has a wide range of meanings. For the present purposes, the following are considered levels of truth.
Logical truth: abstract symbol manipulation
Reality truth: traditional science from facts and models
Human truth: opinions and everything else
These areas can overlap but the distinction is, nonetheless, useful. Human opinion truth may or may not be connected to reality truth and/or logical truth.
3. Related ideas
The following ideas of logical truth are related.
Russell logical paradox as in "This statement is false.".
Mathematical incompleteness (Formal predicate logic systems, Gödel, 1924, 1931).
Computational decidability (Turing Machine and the Halting Problem, Turing, 1936)
Randomness determination (Algorithmic Information Theory, Kolmogorov, Chaitin)
Connections can be made of logical truth to reality truth and human truth.
4. Bertrand Russell
Bertrand Russell (1872-1970) was a famous mathematician and humanist/socialist. The Russell Paradox (1901), a fundamental paradox in logic. is named after him. The Russell Paradox appears in reality in many forms.
This statement is false.
You must hate all forms of hate.
You must not tolerate intolerance.
It is interesting when people are pressed to reconcile, say, the toleration paradox, with ideas such as, say, inclusiveness, they fall back on the type system of Russell that did not resolve the logical issue. The only solution appears to be to give up on actual logic but pretend to be using logic.
5. Short forms
Some short forms of the Russell Paradox are the following.
This statement is false.
I am lying.
Is no your answer to this question?
Pinocchio: My nose will grow now.
There is no resolution of the Russell paradox in logic.
6. David Hilbert
David Hilbert (German mathematician) was an influential German mathematician. In 1928, he proposed finding a consistent mathematical system that will allow all possible truths to be decided. This would allow the automatic, or mechanical, proving of all possible mathematical truths. In 1931, Gödel proved that this could not be done.
Hilbert started the separation of mathematics from philosophers (opinion truth) and reality truth. Today, (pure) mathematics is a logical truth consisting of symbol manipulation with no direct connection with reality truth or opinion truth.
7. Gödel and incompleteness
Kurt Gödel stunned the mathematical world in 1931 by proving that it is impossible to find a consistent mathematical system that will allow all possible truths to be decided. This is called the
incompleteness Theorem.
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
The incompleteness theorem of
Kurt Gödel (Austrian logician, mathematician, philosopher) says that, for any formal proof system that includes arithmetic, there exist true statements that cannot be proven within the system. Part of Gödel's motivation was the belief that
there is no finite description of truth. That is, truth is infinite.
8. Uncertainty
It is easy to confuse "incompleteness" for logical systems with "uncertainty" from reality systems.
In simple terms, Heisenberg's Uncertainty Principle states that one cannot measure both the momentum and position of particles exactly at the same time.
9. Lack of information
It is easy to confuse "incompleteness" for logical systems with a lack of information from reality systems.
In "incompleteness" and "computability" (below), there is no lack of information no impreciseness in measurement. The "incompleteness" is an inherent part of logical systems (that contain arithmetic).
In the "halting problem" (below), the program will have access to every bit of code and data that is to be analyzed, without ambiguity, but the (general) problem cannot be solved.
10. Book: Gödel Escher and Bach
The book
Gödel, Escher, Bach: An eternal golden braid, from 1979 by Douglas Hofstadter, contains many recursive and self-referential themes.
Examples of Russell's paradox are covered.
A detailed explanation of Gödel's incompleteness theorem is covered at a high level for intuitive understanding purposes.
11. Implications
The incompleteness theorem has manifestations in both computing and information theory.
Halting Problem (Turing, 1936)
Information and randomness as part of AIT (Algorithmic Information Theory) (Chaitin, Kolmogorov)
12. Alan Turing: halting problem
Alan Turing (1912-1954) developed the ideas that proved the limits of computing before the first programmable digital computer was built.
Claude Shannon (1939) showed that one could built such a computer.
The
halting problem (Turing, Turing machine, 1936) result: It is impossible to write a computer program that looks at another computer program (and its data) and determines whether that other computer program eventually halts.
The possible answers for a computation of an undecidable problem are yes (true), no (false), or maybe (wait forever). One may be able to go "
outside the system" to determine a better answer.
An abstract (or physical) computer can be called a
Turing Machine. A
Turing complete programming language can compute any computable function.
[waiting for a program to stop, secure form submission, virus detection]
13. Gregory Chaitin
Information theorist
Gregory Chaitin (determining randomness) showed that there is no algorithm for determining whether a sequence of symbols is random. One must be informed of this. That is, one must go outside the system.
This forms the basis of
AIT as in the smallest program that can output a given output.
14. Formal systems
Here we are interested in the work by Gödel on consistency and completeness. Formal systems (with arithmetic) have the following properties.
Every system that is complete will have inconsistencies.
Every system that is consistent will not be complete.
We are only interested in systems that include arithmetic and will try to make an interesting connection to reality truth and human (opinion) truth. Let us model these ideas.
15. Models and reality
A model is an abstraction of reality.
Essentially, all models are wrong, but some are useful. George Box, Statistician.
The best material model of a cat is another, or preferably the same, cat. Norbert Wiener (and A. Rosenblueth).
A model is a useful fiction.
16. Model
A model will now be constructed based on the ideas of completeness and consistency.
Note that the model is intended to model relevant characteristics at the extremes.
There are always exceptions to any model. Here, the general patterns are of interest.
17. People
For modeling purposes, we will group people into four categories.
People who desire and only use complete systems. They are logically happy.
People who try to be both complete and consistent. They are logically crazy since that cannot be done.
People who do not try to be consistent nor complete. They will be logically ignored.
People who desire and only use consistent systems. They are logically happy.
Thus, there are only two groups of concern. What might be the properties of each group?
18. Complete
People who desire and only use complete systems are logically happy but not consistent.
19. Complete and consistent
People who try to be both complete and consistent are logically crazy since both cannot be logically achieved. This group will be logically ignored.
20. Neither complete nor consistent
People who do not try to be consistent nor complete will be logically ignored.
21. Consistent
People who desire and only use consistent systems are logically happy but not complete.
22. Groups
We are left with two groups.
Complete group - with many subgroups. These are the "green people".
Consistent group - with many subgroups. These are the "purple people".
23. Colors
To avoid offending any (known) human group, the colors used are for the "
green people", the "
purple people", the "
gray people" and the "
blue people".
Human skin color tends to be variations of "
red", "
yellow", "
black" and "
white". Related (composite) colors include "
orange".
The remaining basic colors are "
green", "
blue", "
magenta" and "
cyan". Related (composite) colors include "
purple".
Note: The color "
gray" was used as neither "
cyan" nor "
magenta" looked sufficiently different from the other colors used.
Note: For those who may not be able to see those colors, there is text to differentiate the groups.
24. Complete groups
Each group of people in the
complete set of groups will not be bothered by inconsistencies (by definition) and will pretty much agree with each other, as long as no details are discussed or encountered. They revise what it means to be complete. In general, they do not realize that this is not possible.
25. Consistent groups
Each group of people in the
consistent set of groups will have decided on a different set of consistent rules to use. Everyone in these groups is always trying to add, remove or refine rules to get a complete set. In general, they do not realize that this is not possible.
26. Relations between groups
Each group in the
complete set will agree with each other unless details are discussed but will all agree that the consistent set are too rigid and inflexible to be part of their set groups.
Each group in the
consistent set will disagree with each other on which set of consistent rules is the best but will all agree that the complete set of groups are not consistent and cannot be part of their set groups.
Thus, there is a natural and logical division of groups into what is called equivalence sets.
27. Trees and forests
The
complete group tends to look at the big picture and sees the forest but not the trees. Trees, as details, just bring out inconsistencies which are better ignored.
The
consistent group tends to look at the trees, as details, and tends to miss the big picture. The details need to be consistent even if the big picture is not clear.
28. Accusations
The
complete group tells the
consistent group that "
you are incorrect". What they mean is that "
you are incomplete".
The
consistent group tells the
complete group that "
you are wrong". What they mean is that "
you are inconsistent".
29. Semantics
The words "right" and "correct" are usually synonyms.
The words "wrong" and "incorrect" are usually synonyms.
It may happen that members of each group attribute slightly different meanings to these words.
30. Self-assessment
The complete group says "we are right" but mean that "we are complete".
The complete group says "we were wrong" but mean that "we were complete" but changed what it means to be "complete".
The consistent group says that "we are right" but mean that "we are consistent".
The consistent group says that "we were right" but mean that "we were consistent" but changed what it means to be "consistent".
31. Put it together
Let us put together the above concepts into a diagram.
Philosophers are people who know less and less about more and more, until they know nothing about everything. Scientists are people who know more and more about less and less, until they know everything about nothing. Konrad Lorenz (Austrian zoologist)
32. Knowledge
The
complete group (
philosophers)
are people who know less and less about more and more, until they know nothing about everything. They see the forest but tend to ignore the trees.
The
consistent group (
scientists)
are people who know more and more about less and less, until they know everything about nothing. The see the trees but tend to ignore the forest.
33. Discussion
The above discussion has been in abstract logical terms.
Discuss: Are there groups of people who tend to fit the above descriptions? If not, why not? If so, discuss the following.
What additional characteristics might be important?
What of the above characteristics may not fit well?
34. End of page
