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”).