71 lines
2.9 KiB
Markdown
71 lines
2.9 KiB
Markdown
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---
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title: Proof Systems
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TARGET DECK: Obsidian::STEM
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FILE TAGS: formal-system
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tags:
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- formal-system
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- logic
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---
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## Overview
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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.
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%%ANKI
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Basic
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*Why* do we say formal systems using proof systems are purely syntactic?
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Back: No meaning is attached to the WFFs of the formal language.
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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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<!--ID: 1721561534205-->
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END%%
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%%ANKI
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Cloze
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Proof systems are {syntactic}, not {semantic}.
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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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<!--ID: 1721561534211-->
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END%%
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%%ANKI
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Basic
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How is it purely syntactic proof systems can produce theorems?
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Back: Inference rules define how to produce theorems from sequences of WFFs.
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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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<!--ID: 1721561534217-->
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END%%
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%%ANKI
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Basic
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What kind of deductive system is natural deduction usually categorized as?
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Back: A proof system.
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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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<!--ID: 1721561534223-->
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END%%
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%%ANKI
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Basic
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In a proof system, a proof is used to produce what?
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Back: A theorem.
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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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<!--ID: 1721561534230-->
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END%%
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%%ANKI
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Basic
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In a proof system, a proof is a sequence of what?
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Back: Well-formed formulas.
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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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<!--ID: 1721561534235-->
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END%%
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%%ANKI
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Basic
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In a proof system, how are WFFs produced from one another?
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Back: Via inference rules.
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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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<!--ID: 1721561534241-->
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END%%
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## Bibliography
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* “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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