for questions or link request: module admin

The Axioms of the Whitehead Russel Calculus

name: proptheo2, module version: 1.00.00, rule version: 1.00.00, orignal: proptheo2, author of this module: Michael Meyling

Description

This module notates the original axioms of the Whitehead-Russel calculus, the so called "primitive propositions". These five primitive propositions could be deduced by our four axioms.

Uses the following modules:


At first we show a little proposition to demonstrate the basic proof methods of propositional calculus:

Proposition 1
      (P Þ (Q Ú P))     (lem1)

Proof:

1 ((P Þ Q) Þ ((A Ú P) Þ (A Ú Q))) add axiom axiom4
2 ((B Þ Q) Þ ((A Ú B) Þ (A Ú Q))) replace proposition variable P by B in 1
3 ((B Þ (Q Ú P)) Þ ((A Ú B) Þ (A Ú (Q Ú P)))) replace proposition variable Q by (Q Ú P) in 2
4 (((P Ú Q) Þ (Q Ú P)) Þ ((A Ú (P Ú Q)) Þ (A Ú (Q Ú P)))) replace proposition variable B by (P Ú Q) in 3
5 (((P Ú Q) Þ (Q Ú P)) Þ ((ØP Ú (P Ú Q)) Þ (ØP Ú (Q Ú P)))) replace proposition variable A by ØP in 4
6 (((P Ú Q) Þ (Q Ú P)) Þ ((P Þ (P Ú Q)) Þ (ØP Ú (Q Ú P)))) reverse abbreviation impl in 5 at occurence 1
7 (((P Ú Q) Þ (Q Ú P)) Þ ((P Þ (P Ú Q)) Þ (P Þ (Q Ú P)))) reverse abbreviation impl in 6 at occurence 1
8 ((P Ú Q) Þ (Q Ú P)) add axiom axiom3
9 ((P Þ (P Ú Q)) Þ (P Þ (Q Ú P))) modus ponens with 8, 7
10 (P Þ (P Ú Q)) add axiom axiom2
11 (P Þ (Q Ú P)) modus ponens with 10, 9
qed


This is the first primitive proposition, its equal to our first axiom:

Proposition 2
      ((P Ú P) Þ P)     (prin1)

Proof:

1 ((P Ú P) Þ P) add axiom axiom1
qed


Now comes the second primitive proposition. It looks simular to our second axiom, but we have to use our first proposition to prove it:

Proposition 3
      (Q Þ (P Ú Q))     (prin2)

Proof:

1 (P Þ (Q Ú P)) add proposition lem1
2 (P Þ (A Ú P)) replace proposition variable Q by A in 1
3 (Q Þ (A Ú Q)) replace proposition variable P by Q in 2
4 (Q Þ (P Ú Q)) replace proposition variable A by P in 3
qed


The third primitive proposition:

Proposition 4
      ((P Ú Q) Þ (Q Ú P))     (prin3)

Proof:

1 ((P Ú Q) Þ (Q Ú P)) add axiom axiom3
qed


The fourth primitive proposition was proved with the other primitve propositions by P. Bernays. Here comes the sledgehammer:

Proposition 5
      ((P Ú (Q Ú A)) Þ (Q Ú (P Ú A)))     (prin4)

Proof:

1 ((P Þ Q) Þ ((A Ú P) Þ (A Ú Q))) add axiom axiom4
2 ((P7 Þ Q) Þ ((A Ú P7) Þ (A Ú Q))) replace proposition variable P by P7 in 1
3 ((P7 Þ P8) Þ ((A Ú P7) Þ (A Ú P8))) replace proposition variable Q by P8 in 2
4 ((P7 Þ P8) Þ ((P9 Ú P7) Þ (P9 Ú P8))) replace proposition variable A by P9 in 3
5 (P Þ (Q Ú P)) add proposition lem1
6 ((P Þ (Q Ú P)) Þ ((A Ú P) Þ (A Ú (Q Ú P)))) replace proposition variable Q by (Q Ú P) in 1
7 ((A Ú P) Þ (A Ú (Q Ú P))) modus ponens with 5, 6
8 (((A Ú P) Þ P8) Þ ((P9 Ú (A Ú P)) Þ (P9 Ú P8))) replace proposition variable P7 by (A Ú P) in 4
9 (((A Ú P) Þ (A Ú (Q Ú P))) Þ ((P9 Ú (A Ú P)) Þ (P9 Ú (A Ú (Q Ú P))))) replace proposition variable P8 by (A Ú (Q Ú P)) in 8
10 ((P9 Ú (A Ú P)) Þ (P9 Ú (A Ú (Q Ú P)))) modus ponens with 7, 9
11 ((Q Ú (A Ú P)) Þ (Q Ú (A Ú (Q Ú P)))) replace proposition variable P9 by Q in 10
12 ((P Ú Q) Þ (Q Ú P)) add axiom axiom3
13 ((D Ú Q) Þ (Q Ú D)) replace proposition variable P by D in 12
14 ((D Ú (A Ú (Q Ú P))) Þ ((A Ú (Q Ú P)) Ú D)) replace proposition variable Q by (A Ú (Q Ú P)) in 13
15 ((Q Ú (A Ú (Q Ú P))) Þ ((A Ú (Q Ú P)) Ú Q)) replace proposition variable D by Q in 14
16 (((Q Ú (A Ú (Q Ú P))) Þ P8) Þ ((P9 Ú (Q Ú (A Ú (Q Ú P)))) Þ (P9 Ú P8))) replace proposition variable P7 by (Q Ú (A Ú (Q Ú P))) in 4
17 (((Q Ú (A Ú (Q Ú P))) Þ ((A Ú (Q Ú P)) Ú Q)) Þ ((P9 Ú (Q Ú (A Ú (Q Ú P)))) Þ (P9 Ú ((A Ú (Q Ú P)) Ú Q)))) replace proposition variable P8 by ((A Ú (Q Ú P)) Ú Q) in 16
18 ((P9 Ú (Q Ú (A Ú (Q Ú P)))) Þ (P9 Ú ((A Ú (Q Ú P)) Ú Q))) modus ponens with 15, 17
19 ((Ø(Q Ú (A Ú P)) Ú (Q Ú (A Ú (Q Ú P)))) Þ (Ø(Q Ú (A Ú P)) Ú ((A Ú (Q Ú P)) Ú Q))) replace proposition variable P9 by Ø(Q Ú (A Ú P)) in 18
20 (((Q Ú (A Ú P)) Þ (Q Ú (A Ú (Q Ú P)))) Þ (Ø(Q Ú (A Ú P)) Ú ((A Ú (Q Ú P)) Ú Q))) reverse abbreviation impl in 19 at occurence 1
21 (((Q Ú (A Ú P)) Þ (Q Ú (A Ú (Q Ú P)))) Þ ((Q Ú (A Ú P)) Þ ((A Ú (Q Ú P)) Ú Q))) reverse abbreviation impl in 20 at occurence 1
22 ((Q Ú (A Ú P)) Þ ((A Ú (Q Ú P)) Ú Q)) modus ponens with 11, 21
23 (P Þ (P Ú Q)) add axiom axiom2
24 (A Þ (A Ú Q)) replace proposition variable P by A in 23
25 (A Þ (A Ú P)) replace proposition variable Q by P in 24
26 (Q Þ (Q Ú P)) replace proposition variable A by Q in 25
27 (P Þ (Q Ú P)) add proposition lem1
28 (P Þ (A Ú P)) replace proposition variable Q by A in 27
29 ((Q Ú P) Þ (A Ú (Q Ú P))) replace proposition variable P by (Q Ú P) in 28
30 (((Q Ú P) Þ P8) Þ ((P9 Ú (Q Ú P)) Þ (P9 Ú P8))) replace proposition variable P7 by (Q Ú P) in 4
31 (((Q Ú P) Þ (A Ú (Q Ú P))) Þ ((P9 Ú (Q Ú P)) Þ (P9 Ú (A Ú (Q Ú P))))) replace proposition variable P8 by (A Ú (Q Ú P)) in 30
32 ((P9 Ú (Q Ú P)) Þ (P9 Ú (A Ú (Q Ú P)))) modus ponens with 29, 31
33 ((ØQ Ú (Q Ú P)) Þ (ØQ Ú (A Ú (Q Ú P)))) replace proposition variable P9 by ØQ in 32
34 ((Q Þ (Q Ú P)) Þ (ØQ Ú (A Ú (Q Ú P)))) reverse abbreviation impl in 33 at occurence 1
35 ((Q Þ (Q Ú P)) Þ (Q Þ (A Ú (Q Ú P)))) reverse abbreviation impl in 34 at occurence 1
36 (Q Þ (A Ú (Q Ú P))) modus ponens with 26, 35
37 ((Q Þ P8) Þ ((P9 Ú Q) Þ (P9 Ú P8))) replace proposition variable P7 by Q in 4
38 ((Q Þ (A Ú (Q Ú P))) Þ ((P9 Ú Q) Þ (P9 Ú (A Ú (Q Ú P))))) replace proposition variable P8 by (A Ú (Q Ú P)) in 37
39 ((P9 Ú Q) Þ (P9 Ú (A Ú (Q Ú P)))) modus ponens with 36, 38
40 (((A Ú (Q Ú P)) Ú Q) Þ ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) replace proposition variable P9 by (A Ú (Q Ú P)) in 39
41 ((P Ú P) Þ P) add axiom axiom1
42 (((A Ú (Q Ú P)) Ú (A Ú (Q Ú P))) Þ (A Ú (Q Ú P))) replace proposition variable P by (A Ú (Q Ú P)) in 41
43 ((((A Ú (Q Ú P)) Ú (A Ú (Q Ú P))) Þ P8) Þ ((P9 Ú ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) Þ (P9 Ú P8))) replace proposition variable P7 by ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P))) in 4
44 ((((A Ú (Q Ú P)) Ú (A Ú (Q Ú P))) Þ (A Ú (Q Ú P))) Þ ((P9 Ú ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) Þ (P9 Ú (A Ú (Q Ú P))))) replace proposition variable P8 by (A Ú (Q Ú P)) in 43
45 ((P9 Ú ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) Þ (P9 Ú (A Ú (Q Ú P)))) modus ponens with 42, 44
46 ((Ø((A Ú (Q Ú P)) Ú Q) Ú ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) Þ (Ø((A Ú (Q Ú P)) Ú Q) Ú (A Ú (Q Ú P)))) replace proposition variable P9 by Ø((A Ú (Q Ú P)) Ú Q) in 45
47 ((((A Ú (Q Ú P)) Ú Q) Þ ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) Þ (Ø((A Ú (Q Ú P)) Ú Q) Ú (A Ú (Q Ú P)))) reverse abbreviation impl in 46 at occurence 1
48 ((((A Ú (Q Ú P)) Ú Q) Þ ((A Ú (Q Ú P)) Ú (A Ú (Q Ú P)))) Þ (((A Ú (Q Ú P)) Ú Q) Þ (A Ú (Q Ú P)))) reverse abbreviation impl in 47 at occurence 1
49 (((A Ú (Q Ú P)) Ú Q) Þ (A Ú (Q Ú P))) modus ponens with 40, 48
50 ((((A Ú (Q Ú P)) Ú Q) Þ P8) Þ ((P9 Ú ((A Ú (Q Ú P)) Ú Q)) Þ (P9 Ú P8))) replace proposition variable P7 by ((A Ú (Q Ú P)) Ú Q) in 4
51 ((((A Ú (Q Ú P)) Ú Q) Þ (A Ú (Q Ú P))) Þ ((P9 Ú ((A Ú (Q Ú P)) Ú Q)) Þ (P9 Ú (A Ú (Q Ú P))))) replace proposition variable P8 by (A Ú (Q Ú P)) in 50
52 ((P9 Ú ((A Ú (Q Ú P)) Ú Q)) Þ (P9 Ú (A Ú (Q Ú P)))) modus ponens with 49, 51
53 ((Ø(Q Ú (A Ú P)) Ú ((A Ú (Q Ú P)) Ú Q)) Þ (Ø(Q Ú (A Ú P)) Ú (A Ú (Q Ú P)))) replace proposition variable P9 by Ø(Q Ú (A Ú P)) in 52
54 (((Q Ú (A Ú P)) Þ ((A Ú (Q Ú P)) Ú Q)) Þ (Ø(Q Ú (A Ú P)) Ú (A Ú (Q Ú P)))) reverse abbreviation impl in 53 at occurence 1
55 (((Q Ú (A Ú P)) Þ ((A Ú (Q Ú P)) Ú Q)) Þ ((Q Ú (A Ú P)) Þ (A Ú (Q Ú P)))) reverse abbreviation impl in 54 at occurence 1
56 ((Q Ú (A Ú P)) Þ (A Ú (Q Ú P))) modus ponens with 22, 55
57 ((P7 Ú (A Ú P)) Þ (A Ú (P7 Ú P))) replace proposition variable Q by P7 in 56
58 ((P7 Ú (Q Ú P)) Þ (Q Ú (P7 Ú P))) replace proposition variable A by Q in 57
59 ((P7 Ú (Q Ú A)) Þ (Q Ú (P7 Ú A))) replace proposition variable P by A in 58
60 ((P Ú (Q Ú A)) Þ (Q Ú (P Ú A))) replace proposition variable P7 by P in 59
qed


The fifth primitive proposition is our fourth axiom:

Proposition 6
      ((P Þ Q) Þ ((A Ú P) Þ (A Ú Q)))     (prin5)

Proof:

1 ((P Þ Q) Þ ((A Ú P) Þ (A Ú Q))) add axiom axiom4
qed