Defining a Decidability Decider By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that have the semantic error of Pathological self-reference(Olcott 2004). The foundation of this system requires the notion of a BaseFact that anchors the semantic notions of True and False. When-soever a formal proof from BaseFacts of language L to a closed WFF X or ~X of language L does not exist X is decided to be semantically incorrect. 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 the same as a Predicate from Predicate Logic, essentially a Boolean valued function. A BaseFact is an expression X of (formal or formalized natural) 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. Finite string Expression X expresses relation R of language L. 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) Provable(Γ, X) False(L, X) ↔ ∃Γ ⊆ BaseFacts(L) Provable(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) Copyright 2018 (and many other years since 1997) Pete Olcott