PEIRCE EXISTENTIAL GRAPH SYSTEM

One axiom

P0 VOID : VOID,

Five rules of inference

P1 even deletion: (gx)->(x) when (..) is even number of nested [..]

P2 odd insertion: <x>-><gx> when <..> is odd number of nested [..]

P3 iteration: g[x]->g[gx] (half of GSB generation)

P4 deiteration: g[gx]->g[x] (half of GSB generation)

P5 double cut: [[x]]=x (GSB reflection)

SIMPLE EXISTENTIAL GRAPH SYSTEM

One axiom: Consistency

M0 : [x[ x ]]

One rule of inference: Iteration

M1: g[x]->g[gx] .

THEOREM P3 ITERATION G[X]->G[GX]

P3 is equal to N1

THEOREM P4 DEITERATION

Proof

[g[x][g[ x]]] N0 iconsistency of g[x]

[g[gx][g[x]]] N1 iteration of g

g[gx]->g[x] D1 definition of ->

THEOREM P5 DOUBLE CUT [[X]]=X

P5a. [[x]]->x

Proof

[[x] [ ]] N0 consistency of [x]

D1 definition of

P5b. x->[[x]]

proof:

[x [ x ]] N0 indifference of x

[x [ x[x[x]]]] N0 indifference of x

[x [ [ [x]]]] P4 deiteration of x

x->[[x]] D1 definition of ->

LEMMA: INVERSION

(X->Y)->([Y]->[X])

Proof

[[x[y]] [y] x ] N0 consistency of x[y]

[[x[y]] [y][[x]] ] P5 double cut

[[x[y]][[[y][[x]]]]] P5 double cut

(x->y)->([y]->[x]) definition

LEMMA 2: ADDITION

[[gx[gy]] gx[gy] ] N0 consistency of gx[gx]

[[x[y]] gx[gy] ] P4 deiteration of g

[[x[y]][[gx[gy]]]] P5 double cut

[x[y]]->[gx[gy]] definition

0-DEPTH DELETION

(GX->Y)->(X->Y)

proof

[gx[y][gx[y]]] N0 consistency of gx[y]

[gx[y][ x[y]]] deiteration of g

(gx->y)->(x->y) definition

1-DEPTH INSERTION

[X]->[GX]

proof

gx->x 0-depth deletion

[x]->[gx]] inversion

2-DEPTH DELETION

[X[GY]]->[X[Y]]

proof

[y]->[gy] 1-depth insertion

x[y]->x[gy] addition

[x[gy]]->[x[y]] inversion

3-DEPTH INSERTION

[X[Y[Z]]->[X[Y[GZ]]]

proof

[y[gz]]-> [y[z]] 2-depth deletion

x[y[gz]]->x[y[z]] addition

[x[y[z]]]->[x[y[gz]]] inversion

P1 EVEN-DEPTH DELETION

P2 ODD-DEPTH INSERTION

In general

(2n+1)-depth insertion can be proved by inverting of added 2n-depth deletion

2n-depth deletion can be proved by inverting of added (2n-1)-depth insertion

THEOREM P0

VOID

Proof

[x[]]=

=[[]] substitution

= VOID double cut