Minimal Type Theory (MTT) shows exactly how all of the constituent parts of an expression relate to each other (in 2D space) when this expression is formalized using a directed acyclic graph (DAG). This provides greater expressiveness than the 1D space of FOPL syntax. X @ ~True(X) // assign alias operator "@" explained "@" means the LHS is assigned as an alias for the RHS . This extension to FOPL syntax provides the means for: (1) Meaningful names to be assigned to expressions. (2) Predicates to have other Predicates as terms. // enabling HOL of an unlimited finite order (3) An Expression to refer directly to itself. https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence A formula A is a syntactic consequence within some formal system FS of a set Γ of formulas if there is a formal proof in FS of A from the set Γ: Γ ⊢FS A Translation to MTT notational conventions: Γ ⊢FS A ≡ ( ∃Γ ⊂ FS (Γ ⊢ A) ) First Order Predicate Logic Syntax used the the basis for the Minimal Type Theory Language: sentence : atomic_sentence | sentence IMPLIES sentence | sentence IFF sentence | sentence AND sentence | sentence OR sentence | sentence PROVES sentence // enhancement | quantifier IDENTIFIER sentence // MTT syntax is different | '~' sentence %prec NOT | '(' sentence ')' ; atomic_sentence : IDENTIFIER '(' term_list ')' // ATOMIC PREDICATE | IDENTIFIER // SENTENTIAL VARIABLE (enhancement) ; term : IDENTIFIER '(' term_list ')' // FUNCTION | IDENTIFIER // CONSTANT or VARIABLE ; term_list : term_list ',' term | term ; quantifier : THERE_EXISTS | FOR_ALL ; Minimal Type Theory augments the above syntax in two key ways: (a) Adding the Assign Alias Operator: "@" (b) Requiring every variable to be associated with a specific type. 2 Provable(L, X) @ L ∈ Formal_Systems, X ∈ Finite_Strings, ∃Γ ⊂ L (Γ ⊢ X) 00 root (1)(4)(7)(10) 01 ∈ (2)(3) 02 L 03 Formal_Systems 04 ∈ (5)(6) 05 X 06 Finite_Strings 07 ∃ (8) 08 ⊂ (9)(2) 09 Γ 10 ⊢ (9)(5) ⊂ Γ ⊢ FS L ∈ X root ∈ S ∃ Numbers on Directed Graph Edges indicate Order of Evaluation (1) (1)(2)(1)(2) (2) (1) (4) (2) (1) (2) (3) 3 Refutable(L, X) @ L ∈ Formal_Systems, X ∈ Finite_Strings, ∃Γ ⊂ L (Γ ⊢ ~X) 00 root (1)(4)(7)(10) 01 ∈ (2)(3) 02 L 03 Formal_Systems 04 ∈ (5)(6) 05 X 06 Finite_Strings 07 ∃ (9) 08 ⊂ (9)(2) 09 Γ 10 ⊢ (9)(11) 11 ~ (5) ⊂ Γ ⊢ FS L ∈ X root ∈ S ∃ Numbers on Directed Graph Edges indicate Order of Evaluation ~ (1) (1)(2)(1)(2) (1) (4) (2) (1) (2) (3) (2) 4 ~Provable(L, X) @ L ∈ Formal_Systems, X ∈ Finite_Strings, ~∃Γ ⊂ L (Γ ⊢ X) 00 root (1)(4)(7)(11) 01 ∈ (2)(3) 02 L 03 Formal_Systems 04 ∈ (5)(6) 05 X 06 Finite_Strings 07 ~ (8) 08 ∃ (9) 09 ⊂ (10)(2) 10 Γ 11 ⊢ (10)(5) ⊂ Γ ⊢ FS L ∈ X root ∈ S ∃ Numbers on Directed Graph Edges indicate Order of Evaluation ~ (1) (1)(2)(1)(2) (2) (1) (4) (2) (1) (2) (3) 5 G @ ∀L ∈ Formal_Systems, ~∃Γ ⊂ L (Γ ⊢ G) "@" means the LHS is assigned as an alias for the RHS . There is no referencing / dereferencing needed, G is one and the same thing as the expression that refers to G. (Unlike Tarksi naming) G is not referring to its name, G is referring to itself. 00 root (1)(5)(9) // G is an alias for this node 01 ∀ (2) 02 ∈ (3)(4) 03 L 04 Formal Systems 05 ~ (6) 06 ∃ (7) 07 ⊂ (8)(3) 08 Γ 09 ⊢ (8)(0) // cycle indicates infinite evaluation loop error FS ⊂ ∃ ⊢ Γ ~ ∈ ∀ L rootNumbers on Directed Graph Edges indicate Order of Evaluation (1) (1) (1) (1) (1) (1) (2) (2) Cycle Indicates Error (2) (2) (1) (3) In the case of Pathological Self-Reference (PSR) the second argument to the ⊢ predicate forms and infinite loop instead of ever reaching its expected sentential variable. This prevents the evaluation of the expression from ever completing. 6 Gödel's Proof (Revised Edition) 2001 Nagel, Newman, and Hofstadter page 97 (G) ~(∃x) Dem (x, Sub(n, 17, n) ) completing the substitution (G) ~(∃x) Dem (x, G) converting to common notation (G) ~(∃x)(x ⊢ G) Example of Provable(L, R) WFF of L (1) P // premise (2) P → Q // axiom (3) Q → R // axiom Proof (using finite string rewrite rules) Logical_Inference("P", "P → Q") ∴ "Q" Logical_Inference("Q", "Q → R") ∴ "R" ∴ Provable("R") All of the above copyright 2017 Pete Olcott