92 lines
3.5 KiB
Markdown
92 lines
3.5 KiB
Markdown
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---
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title: Predicate Logic
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TARGET DECK: Obsidian::STEM
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FILE TAGS: logic::predicate
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tags:
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- logic
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- predicate
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---
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## Overview
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A branch of logic that uses quantified variables over non-logical objects. A **predicate** is a sentence with some number of free variables. A predicate with free variables "plugged in" is a [[prop-logic|proposition]].
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%%ANKI
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Cloze
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{Predicate} logic is also known as {first}-order logic.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1715897257076-->
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END%%
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%%ANKI
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Basic
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What is a predicate?
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Back: A sentence with some number of free variables.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1715897257082-->
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END%%
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%%ANKI
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Basic
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What distinguishes a predicate from a proposition?
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Back: A proposition does not contain free variables.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708199272110-->
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END%%
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%%ANKI
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Basic
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How are propositions defined in terms of predicates?
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Back: A proposition is a predicate with $0$ free variables.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708199272115-->
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END%%
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%%ANKI
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Basic
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Why is "$3 + x = 12$" *not* a proposition?
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Back: Because $x$ is a variable.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708199272121-->
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END%%
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## Sets
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A **state** is a function that maps identifiers to values. A predicate can be equivalently seen as a representation of the set of states in which it is true.
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%%ANKI
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Basic
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Is $(i \geq 0)$ well-defined in $\{(i, -10)\}$?
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Back: Yes.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1715898219881-->
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END%%
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%%ANKI
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Basic
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Is $(i \geq 0)$ well-defined in $\{(j, -10)\}$?
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Back: No.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1715898219890-->
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END%%
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%%ANKI
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Basic
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What predicate represents states $\{(i, 0), (i, 2), (i, 4), \ldots\}$?
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Back: $i \geq 0$ is even.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1715898219895-->
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END%%
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%%ANKI
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Basic
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What is sloppy about phrase "the states in $i + j = 0$"?
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Back: $i + j = 0$ is not a set but a representation of a set (of states).
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1715898219903-->
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END%%
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## Bibliography
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* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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