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The simplest existential graph system

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Armahedi Mahzar
Armahedi Mahzar

This document summarizes the Peirce Existential Graph system and the Simple Existential Graph system. It defines the axioms and rules of inference for each system. It then proves several theorems about the rules of inference for the Peirce system, including theorems showing rules P3, P4, and P5 are valid. It introduces some lemmas about insertion, deletion, and inversion and uses these to prove rules P1 and P2 are also valid.

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The simplest existential graph system
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

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The simplest existential graph system

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