# Simple formal refutation of the Liar Paradox

The body of conceptual knowledge is entirely comprised of stipulated relations between expressions of language.  The fact that G is provably unsatisfiable eliminates the misconception of Gödel’s 1931 Incompleteness Theorem.

Within Arithmetic Truth and Provability are ONE-AND-THE-SAME
Provable(“2 + 3 = 5”) ↔ True(“2 + 3 = 5”)
Refutable(“2 + 3 = 7”) ↔ False(“2 + 3 = 7”)

There exists logic sentence LP such that LP is materially
equivalent to a proof of its own negation:  ∃LP (LP ↔ ⊢ ¬LP)
This truth table proves there is no Liar Paradox logic sentence
∃LP (LP    ↔     ⊢   ¬LP)
……….F…..F…..T……….// When LP ≡ “2 + 3 = 7” LP is False and Refutable
……….T…..F…..F……….// When LP ≡ “2 + 3 = 5” LP is True and ¬Refutable

Wittgenstein’s minimal essence of Gödel’s 1931  Incompleteness sentence
I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says ‘P is not provable in Russell’s system’.  ∃P (P ↔ RS ⊬ P)

Adapted for the minimal essence of the 1931 Incompleteness Theorem
There exists logic sentence G such that G is materially
equivalent to its own unprovability:  ∃G (G ↔ ⊬ G)
This truth table proves that logic sentence G does not exist
∃G (G   ↔     ⊬   G)
…….F….F….T……….// When G ≡ “2 + 3 = 7” G is False and ¬Provable
…….T….F….F……….// When G ≡ “2 + 3 = 5” G is True and ¬¬Provable