---
title: Proof Systems
TARGET DECK: Obsidian::STEM
FILE TAGS: formal-system
tags:
  - formal-system
  - logic
---

## Overview

A deductive system found in certain formal systems. Proof systems are syntactic in nature. Proofs are sequences of WFFs, each of which are entailed from previous ones in the sequence. A theorem is then the last WFF in any valid sequence of WFFs.

%%ANKI
Basic
*Why* do we say formal systems using proof systems are purely syntactic?
Back: No meaning is attached to the WFFs of the formal language.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%ANKI
Cloze
Proof systems are {syntactic}, not {semantic}.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%ANKI
Basic
How is it purely syntactic proof systems can produce theorems?
Back: Inference rules define how to produce theorems from sequences of WFFs.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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%%ANKI
Basic
What kind of deductive system is natural deduction usually categorized as?
Back: A proof system.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%ANKI
Basic
In a proof system, a proof is used to produce what?
Back: A theorem.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%ANKI
Basic
In a proof system, a proof is a sequence of what?
Back: Well-formed formulas.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%ANKI
Basic
In a proof system, how are WFFs produced from one another?
Back: Via inference rules.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

## Bibliography

* “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).