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Provability with Minimal Type Theory

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Abstract

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 substantially greater expressiveness than the 1D space of FOPL syntax. The increase in expressiveness over other formal systems of logic shows the Pathological Self-Reference Error of expressions previously considered to be sentences of formal systems. MTT shows that these expressions were never truth bearers, thus never sentences of any formal logic system.
1
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
root
Numbers 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
ResearchGate has not been able to resolve any citations for this publication.
Q → R // axiom Proof (using finite string rewrite rules) Logical_Inference
  • Provable
// 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")