Absolute Logic – logic without axioms

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In first-order predicate calculus “logic” (formal system) depends on the choice of an axiomatic system. However equally expressible first-order predicate logic can be done without axioms.

I construct an expressive first-order predicate logic without axioms in which only implications can be proved. Then I consider its philosophical and theological implications.

We have the same rules for constructing valid logical statements as in the usual first-order predicate logic (we can optionally have notations for equality, sets, etc.)

We have a “sequence” (tuple) of proved statements consisting initially of one statement 1 => 1 (the axiom).

New statements are added by inference rules:

• Modus ponens.
• Substitution (as usual).
• A and B |- A
• A and B |- B

Theorem: If a theorem is provable in the usual first-order predicate logic, its an implication in our logic from conjunction of the axioms of the “usual” logic in our logic (provided that we have only 0, =>, “and”) logical symbols

Proof: We need to prove that implication of every axiom from conjuntion of the axioms is provable and that modus ponens and substitution work. Modus ponens and substitution are direct. It remains to prove that axioms are “implied”.

Obviously, it’s enough to be able to prove “(A and B) => A” and “(A and B) => B”, but that’s inference rules. End of proof.

So, every statement of any usual logic maps to a statement of our logic in a constant (dependent only on the “usual” logic) size complexity, every proof maps to a proof in our logic in a linear size complexity (dependent only on the “usual” logic). The backward mapping is even more trivial: it’s the identity function.

Put short: Our logic is equivalent to every traditional logic.

So the essence of logic is absolute (not dependent on axioms), except of the axiom 1 => 1.

It’s interesting to note that (not checked yet!) truth (“1”) is not provable (it recalls Cantor’s theorems).

1 => 1 can be considered either as the sole axiom or as an inference rule from an empty set of premises, because when axiom is one and always the same, the concept of axioms isn’t relevant (interesting to consider).

What are analogues of Cantor’s theorems in this system?

Funny enough, the above maps to the informal language of philosophiers both as “Truth is absolute.” and “Truth is relative.”

“The truth is absolute.” recalls the word “absolute” from various philosophies and religions. “Truth is relative.” makes irrational the belief that we need a fixed religious system like Torah, Gospel, or Koran, because it now makes no sense to believe in any other axiomatic system than that truth implies truth. So, now the Gospel “which brings down ideas” is now itself fallen (as Jesus prayed “let my will not be”).

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Published

By Victor Porton

I am the chief editor of this journal and creator of this site.

1. Can in my logic be proved falseness of anything (A => 0)?

If it cannot, we get “logic without false”, “a truth-only logic”, “nothing can be falsified” (speaking like a propagandist) what looks like to be related with my Not Science project.

2. The above article can be easily rewritten for any lambda calculi of the “lambda calculi cube” and again make the conclusion that we need no axioms.

But in lamda calculus intuitionistic logic is also true, so also making the axiom “truth implies truth” not necessary (just a theorem).

So, the most relevant logic is the intuitionistic one, isn’t it?

Intuitinists invented intuitionism to get rid of infinity (and God). How could we define infinity in my logic?

3. [“Truth is relative.” makes irrational the belief that we need a fixed religious system like Torah, Gospel, or Koran, because it now makes no sense to believe in any other axiomatic system than that truth implies truth.] makes somehow rational the funny statement preached for example by Russian Orthodoxes that your religion should depend on the place where you live or where you were born or even of your nationality.

4. I’d say Cantor theorem extends to this system as in my system it’s impossible to formulate “I’m not contradictory.” (at least, directly), because (apparently, need to check), there are no way to define self-descriotion of the system in it.

5. The above in some sense obsoletes axiomatic method: It makes no sense to choose a fixed system of axioms, because we now choose them individually for every theorem as implication premises.

6. The above logic can be generalized to infinite formulas (see my “Axiomatic theory of formulas” on this site), but we need to decide if the end term of an infinite chain of implications is true or not.

1. This obsoletes my own formerly great discovery of Algebraic General Topology (such as funcoids and reloids). In the case of my theory of actions of partially ordered semigroups, the theory could be easily discovered by generalizing Kuratowski variant of topological spaces or theory of operator semigroups.

However axiomatic theories still make sense to describe sets of objects. In funcoids theory, it’s relevant in my theory of mapping between funcoids. How could we describe and discover such things as these mappings in a routine way rather than through my feat?

1. So, mappings between funcoids that I discovered are mappings between sets defined by conjunctions of axioms. We could exclude axioms one-by-one and obtain bigger sets and wider mappings. However, the direct generalization of my theorems to wider mappings may be irrelevant (because false) generalizations. What is a good heuristic strategy in this case?

2. This in turn in some sense at least partly obsoletes my revolt because of my poverty and consequentially no degree:

My revolt was caused by non-publication of my algebraic general topology (and consequently of discontinuous analysis) and therefore stagnation of science.

But if we follow this new logic in “everyday” academic research, discovering and publishing theorems covering concepts and properties of funcoids, reloids, and even actions of partially ordered semigroups become inevitable (except of the case of ceased civilization) by the academic science even if the present irrational broken legal system, and broken traditions of academies and job hunting, that gives rights to receive salaries and grants paid, remain.

Now I speak as a scientific economist, not as a religious person fighting for justice. However the fact that injustice (that includes me having no degree and my pleas being not considered by courts) remain a serious economical trouble, also accordingly theories of many (most?) scientific economists.

1. Also using my logic partly solves the “wall” between fundamental and applied science, an important case of common goods tragedy (that works in fundamental science are important but workers oftentimes cannot obtain financing of research work, equipment, editing, subsidiaries, communication, and wide prestigious publication):

Removing superfluous axioms is expected to often make applied mathematical research much more fundamental and therefore much more useful without hindering (in short term) it’s monetary compensation.

3. There is however a trouble (that could be classified as a trouble of too much freedom) that for example discovery of funcoids could be “eliminated” by using as presumptions predicates like X is a proximity (or even metrjc) space instead of an implicit conjunction of proximity axioms, making us unable (not willing) to eliminate superfluous axioms.

To solve this trouble, we could always trace a theorem to its primary implicants (“new axioms” of my system), but that would be indetermined by having multiple proofs (and even cycles of equivalent – implying each other – theorems) of an axiom, multiple definitions of a predicate.

How to eliminate (or make less harmful) this indeterminancy?

1. Obviously, this trouble could be solved by never defining any symbols (or defining some symbols but never defining predicate symbols).

However, this would lead to up to exponential on the number of defined symbols growth of size of theorem statements.

Which variant could be a reasonable middle between these two extremes?

1. A middle is allowing defining predicates only which have no conjuntion at top level. Is this middle reasonable/effective?

1. If it is, then we have the algorithm of generalization. It remains to write it in details and enter it into a computer.

But generalization was counted the greatest feature of human mind. I discovered the main part of the algorithm of technological singularity?!!!