Definition:Hilbert Proof System/Instance 1
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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
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 3.3$: Definition $3.9$
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. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. In particular: This still needs to be inserted into the flow in the correct place. Work on Hamilton has not gone far yet. This is here so that we do not lose track of the fact that this is cited, as it is not named as such in Hamilton itself. If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code. |
- 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.): $\S 2$: Formal statement calculus: $2.1$ The formal system $L$