Liar Paradox refuted with Olcott universal Truth predicate

∀x True(x) ↔ ⊢x                // Olcott universal Truth predicate
∀x False(x) ↔ ⊢~x             // universal False predicate
∀x ~True(x) ↔ ~⊢x           // universal ~True predicate
∀x ~False(x) ↔ ~⊢~x       // universal ~False predicate
∀x ((~True(x) ∧ ~False(x)) ↔ ~Logic_Sentence(x))

The proof that the Olcott universal Truth predicate is correct is twofold:
(1) It always corresponds to the intuitive notation of True(x)
(2) Applied to logic and math it makes them both consistent and complete.

We don’t need to specify any particular formal system, we are examining this at the level of the formal language, which only requires the conventional semantics associated with the logical operators.

“This sentence is not true.” — Formalized as: LP ↔ ~⊢LP
English: A sentence of formal language is materially equivalent to a statement of its own unprovability.
LP ↔ ~⊢LP // This sentence is not provable
If ⊢LP this contradicts its assertion: (~⊢LP) ∴ ~True(LP)
If ⊢~LP this contradicts its assertion: ~(~⊢LP) ∴ ~False(LP)
∀x ((~True(x) ∧ ~False(x)) ↔ ~Logic_Sentence(x))
∴ ~Logic_Sentence(LP)

“This sentence is false.” — Formalized as: LP ↔ ⊢~LP
English: A sentence of formal language is materially equivalent to a statement of its own refutation.
LP ↔ ⊢~LP // This sentence is false
If ⊢LP this contradicts its assertion: (⊢~LP) ∴ ~True(LP)
If ⊢~LP this contradicts its assertion: ~(⊢~LP) ∴ ~False(LP)
∀x ((~True(x) ∧ ~False(x)) ↔ ~Logic_Sentence(x))
∴ ~Logic_Sentence(LP)

Gödel’s 1931 Incompleteness sentence: G ↔ ~Provable(x)
English: A sentence of formal language is materially equivalent to a statement of its own unprovability.
G ↔ ~⊢G // This sentence is not provable
If ⊢G this contradicts its assertion: (~⊢G) ∴ ~True(G)
If ⊢~G this contradicts its assertion: ~(~⊢G) ∴ ~False(G)
∀x ((~True(x) ∧ ~False(x)) ↔ ~Logic_Sentence(x))
∴ ~Logic_Sentence(G)

Tarski’s 1936 Undefinability sentence: ~Provable(x) ↔ True(x)
English: The unprovability of a sentence of formal language is materially equivalent to a statement of its own Truth.
Assuming that the Olcott universal Truth predicate True(x) ↔ Provable(x)
is correct the Tarski sentence ~Provable(x) ↔ True(x) is rejected.

<Kurt Gödel>
     ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA
     MATHEMATICA AND RELATED SYSTEMS I by Kurt Gödel Vienna

     (pages 40-41)

     The analogy between this result and Richard’s antinomy leaps to the eye;
     there is also a close relationship with the “liar” antinomy,14 since
     the undecidable proposition [R(q); q] states precisely that q belongs
     to K, i.e. according to (1), that [R(q); q] is not provable. We are
     therefore confronted with a proposition which asserts its own
     unprovability.

     14 Every epistemological antinomy can likewise be used for a similar
     undecidability proof.
</Kurt Gödel>

Reference to (14) Thus showing the error that causes the undecidability of the Liar Paradox would equally apply this same error to the 1931 Incompleteness Theorem.
∴ Logic_Sentence(G ↔ ~Provable(G)) == False

Decomposition of a proposition into its two distinct Semantic properties:
(1) Its Assertion. // What it is claiming to be true.
(2) The Satisfiability or Falsifiability of its Assertion.

     Theories that characterize or define a relativized concept of truth
     (truth in a model, truth in an interpretation, valuation, or possible
     world) set out from the start in a direction different from that proposed
     by Convention T. Because they substitute a relational concept for the
     single-place truth predicate of T-sentences, such theories cannot carry
     through the last step of the recursion on truth or satisfaction which is
     essential to the quotation-lifting feature of T-sentences.
     (Davidson, Donald 1973: 68)

Davidson, Donald 2001
Inquiries into Truth and Interpretation (Second Edition)
CLARENDON PRESS · OXFORD
Essay 5. In Defence of Convention T (1973)

Copyright 2018 Pete Olcott