**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