Definition:Hilbert Proof System/Instance 1

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This page has been identified as a candidate for refactoring of advanced complexity.
In particular: Note that in Definition:Classical Propositional Logic (which needs refactoring) this system is attributed to Jan Łukasiewicz. This is also documented and explored in 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.), which User:Prime.mover needs to get working on.
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Definition

This instance of a Hilbert proof system is used in:


Let $\LL$ be the language of propositional logic.


$\mathscr H$ has the following axioms and rules of inference:


Axioms

Let $\mathbf A, \mathbf B, \mathbf C$ be WFFs.

The following three WFFs are axioms of $\mathscr H$:

Axiom 1:    \(\ds \mathbf A \implies \paren {\mathbf B \implies \mathbf A} \)      
Axiom 2:    \(\ds \paren {\mathbf A \implies \paren {\mathbf B \implies \mathbf C} } \implies \paren {\paren {\mathbf A \implies \mathbf B} \implies \paren {\mathbf A \implies \mathbf C} } \)      
Axiom 3:    \(\ds \paren {\neg \mathbf B \implies \neg \mathbf A} \implies \paren {\mathbf A \implies \mathbf B} \)      


Rules of Inference

The sole rule of inference is Modus Ponendo Ponens:

From $\mathbf A$ and $\mathbf A \implies \mathbf B$, one may infer $\mathbf B$.


Source of Name

This entry was named for David Hilbert.


Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
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