Semantic WFF(x) specified syntactically According to Wikipedia: x ⊨ y is a semantic rather than syntactic relationship. I specify this relationship as syntactic because I can see how this relationship can be formalized using Rudolf Carnap (1952) Meaning Postulates. Hypothesis: WFF(x) can be applied to the semantics of formalized declarative sentences such that: WFF(x) ↔ ( ~True(x) ↔ False(x) ) // (see proof sketch below) For clarity we focus on atomic propositions expressing a single relation between two Things. Alfred Tarski: // metalanguage M defines expressions in object language L ∀x True(x) ↔ φ(x) // Tarski's Formal correctness of True(x) formula Sketch of a proof of the hypothesis: Thing : Relation : Binary-Relation // inheritance hierarchy ∀a ∈ Binary-Relation ∃b ∈ types & ∃c ∈ types | Compatible-Types(a, b, c) Get-Binary-Relation(x) ↦ ( binary-relation ∈ Binary-Relation ∨ Ø ) ∀x True(x) ↔ φ(x) // Tarski's Formal correctness of True(x) formula φ(x) ↔ WFF(x) & binary-relation(arg1, arg2) WFF(x) ↔ ( Get-Binary-Relation(x) & Compatible-Types( binary-relation, arg1, arg2 ) ) Truth Teller Paradox: "This sentence is true" ↔ x ⊨ True(x) To evaluate True(x) we begin with WFF(x) corresponding to: (a) Binary-Relation(x) == true // Logical-Entailment is a binary relation (b) Compatible-Types( Logical-Entailment, x, True(x) ) The second argument to Logical-Entailment specifies infinite recursion, thus ~WFF(x). Copyright Pete Olcott 1997, 1999, 2004, 2016,