website/content/blog/proofdef.md

---
title: "Theorem Proving Definitions"
date: 2019-12-29T11:21:07-05:00
draft: false
tags:
    - Formal Methods 
images: []
math: true
medium_enabled: true
---

When I look into a new field, sometimes I get confused by the whole new set of vocab terms I need to encounter. This post will serve to keep me straight with the terms involved in theorem proving. 

| Word              | Definition                                                   |
| ----------------- | ------------------------------------------------------------ |
| Modus Ponens      | If $P$ implies $Q$ and $P$ is asserted to be true, then $Q$ must be true. |
| Complete          | If every formula having the property can be derived using the system.  (i.e The system does not miss a result) |
| Negation-Complete | Either $\phi$ or $\neg \phi$ can be proved in the system.    |
| Consistent        | For any provable formula $\phi$, the negation ($\neg \phi$) cannot be provable. (Cannot derive a contradiction) |
| Decidable         | An effective method exists for deriving the correct answer in a finite time. |
| Sound             | Every formula that can be proved in the system is logically valid with respect to the semantics of the system. (i.e The system does not produce a wrong result) |

Hopefully, I'll come back and add more terms as I get confused.