Axioms of Propositional Calculus

name: propaxiom, module version: 1.00.00, rule version: 1.00.00, original: propaxiom, author of this module: Michael Meyling

Description

This module includes the axioms of propositional calculus. These axioms (together with some rules) allow the deduction of all theorems of propositional calculus. To learn about possible conclusions click through the following `.. used by..' list.
This file is part of the project `Hilbert II' see project homepage.

Content

We only need the logical disjunction (`or') and the negation (`not'). The other operands will be defined by them. (It would be possible to use only one operator: `sheffer's or' or `sheffer's and'.) At first we note an abbreviation that defines the implication:

1. Abbreviation
@F0 → @F1 ≡ ¬@F0 ∨ @F1     (impl)

This means that we could transform a part formula that matches the pattern on the left side into the right side and we could transform a part formula that matches the pattern on the right side into the formula on the left side.

The logical conjunction (`and') could be defined with de Morgan:

2. Abbreviation
@F0 ∧ @F1 ≡ ¬(¬@F0 ∨ ¬@F1)     (and)

This is still an abbreviation.

The logical equivalence is defined as usual:

3. Abbreviation
@F1 ↔ @F2 ≡ (@F1 → @F2) ∧ (@F2 → @F1)     (equi)

Now we come to the first axiom of propositional calculus. This axiom enables us to get rid of an unnecessary disjunction:

4. Axiom
(P ∨ P) → P     (axiom1)

This expresses the `idempotence' of the disjunction.

Axiom of weakening:

5. Axiom
P → (P ∨ Q)     (axiom2)

If a proposition is true, any alternative may be added without making it false.

The commutativity of the disjunction is expressed with the third axiom:

6. Axiom
(P ∨ Q) → (Q ∨ P)     (axiom3)

The last axiom is:

7. Axiom
(P → Q) → ((A ∨ P) → (A ∨ Q))     (axiom4)

8. Rule Declaration
modus ponens     (rule1)

For an explanation see modus ponens or modus ponens (o)

9. Rule Declaration

10. Rule Declaration

11. Rule Declaration
replace proposition variable     (rule4)

For an explanation see replace proposition Variable or replace proposition variable (o)

12. Rule Declaration
use abbreviation     (rule5)

For an explanation see use abbreviation or use abbreviation (o)

13. Rule Declaration
reverse abbreviation     (rule6)

For an explanation see reverse abbreviation or reverse abbreviation (o)

Cross References

This document is used by the following documents: