The ultimate foundation of
[a priori] Truth

Simple English:
For any natural (human) or formal (mathematical) language L we know that an expression X of language L is true if and only if there are expressions Γ of language L that connect X to known facts.

The only way that you know that a {dog} is not a type of {cat} is that the terms: {dog} and {cat} are defined to have a set of {BaseFact} properties, and some of these properties are defined to be mutually exclusive. A BaseFact is the ultimate ground of the notion of True.

BaseFact is an expression X of (natural or formal) language L that has been assigned the semantic property of True by making it a member of the collection named: BaseFacts.   (Similar to a math Axiom).
(1) BaseFacts that contradict other BaseFacts are prohibited.
(2) BaseFacts must specify Relations between Things.
The above is the complete specification for a BaseFact.

To verify that an expression X of language L is True or False only requires a syntactic logical consequence inference chain (formal proof) from one or more BaseFacts to X or ~X.    (Backward chaining reverses this order).

True(L, X) ↔ ∃Γ ⊆ BaseFacts(L) (Γ  ⊢ X)
False(L, X) ↔ ∃Γ ⊆ BaseFacts(L) (Γ  ⊢ ~X)

Sentence (mathematical logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: As the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.

Defining a Generic Decidability Decider:
∀L ∈ Formal_Systems
∀X ∈ Closed-WFF(L)
~True(L, X) ∧ ~False(L, X) → Incorrect(L, X)

A language L is a set of finite strings of characters from a
defined alphabet specifying relations to other finite strings.
These finite strings could be tokenized as single integer values.

A Relation is identical to the common notion of a Predicate
from Predicate Logic, essentially a Boolean valued function.

Finite string Expression X expresses relation R of language L.

True(L, X) ↔ ∃Γ ⊆ BaseFacts(L) (Γ ⊢ X)
01 There_Exists (2)(6)
02 Subset_Of      (3)(4)
03 Γ
04 BaseFacts        (5)
05 L
06 Provable          (3)(7)
07 X

Copyright 2018 (and many other years since 1997) Pete Olcott