ASBAL, A way to intepret any string as an axiom

A few years ago(2024, specifically), i created the language HAHAHA(which was very poorly defined), in which all its axioms where just random characters, and those where assumed to be "true". However, this does, to some extent, make sense? like, we just assign combinations of symbols truth values, so thats practically what HAHAHA. what makes it "underspecified", is that theres no way to derive theorems from these axioms(which is a crucial thing to have). So, i introduce Arbitrary String Based Axiomatic Logic, or, as in the title, just ASBAL, which is a way to intepret any strings as axoims, and ALSO find out which other strings are true in on specific ASBAL system. Aswell as this, i will introduce ASBAL_V, which introduces abitrary variables in ASBAL(in another blog post propably).

First, we must define some functions on strings over some alphabet Σ:


      1. S / y:
      S / y = LR
      S = LyR
      L ∈ (Σ∖y)*
      S, R ∈ Σ*
      y ∈ Σ
      2. |S|:
      |S| = 1 + |cdr(S)|
      |Λ| = 0
      S ∈ Σ*
      3. inter(S1,S2):
      inter(S1,S2) = car(S1) · car(S2) · inter(cdr(S1),cdr(S2))
      inter(S,Λ) = inter(Λ,S) = S

note that here, car(S) and cdr(S), are used to get the first character and everything but the first character of S respectively, and · is concatenation. Now, say our set of axiom strings is A, the set of true strings is T, and the false ones are F. Now, we have the rules for deriving truth values for strings!:


      1. S ∈ F iff S = x · (a ∈ A) · y  & S z · (b ∈ A) · w where a ≠ b & |S| > |a| & |S| > |b|
      2. S ∉ A iff S ∈ F
      3. S ∈ A iff S = x · a · y where |S| > |a| & a ∈ T
      4. S ∈ A iff inter(S,f) ∈ T where |f| ≤ |S| & f ∈ F

So, this is(propably) very confusing, so to put it in human words:

  1. S is false if it is a substring of two different true strings
  2. S is not true if its false(yes, this is needed). In other words, something is false unless proven otherwise(and if its proven true, it is excluded from. the false set even if it can be proven false)
  3. S is true if it is a superstring of a true string
  4. S is true, if, when interleaved with a false string, is true.

Now, you may want an example, but i think a seperate page would be better for that, so if i do that, i will link it from here.

Thanks for reading this post. -- Yayimhere.

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