241 lines
9.3 KiB
Markdown
241 lines
9.3 KiB
Markdown
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---
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title: Propositional Logic
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TARGET DECK: Obsidian::STEM
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FILE TAGS: logic::propositional
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tags:
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- logic
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- propositional
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---
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## Overview
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A branch of logic derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and biconditionals ($\Leftrightarrow$). There exists a hierarchy of terms used to describe a string of English:
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* A **sentence** is any grammatical string of words.
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* A **predicate** is a sentence with free variables.
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* A **statement** is a sentence that can be assigned a truth or false value.
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* A predicate with free variables "plugged in" is a statement.
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%%ANKI
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Basic
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What are the basic propositional logical operators?
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Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$
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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: 1706994861291-->
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END%%
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%%ANKI
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Basic
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What is a propositional statement?
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Back: A declarative sentence which is either true or false.
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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: 1708199272076-->
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END%%
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%%ANKI
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Basic
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What two categories do propositional statements fall within?
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Back: Atomic and molecular statements.
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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: 1708199272083-->
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END%%
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%%ANKI
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Basic
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What is an atomic statement?
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Back: It cannot be broken up into smaller statements.
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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: 1708199272087-->
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END%%
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%%ANKI
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Basic
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What is a molecular statement?
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Back: It can be broken up into smaller statements.
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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: 1708199272091-->
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END%%
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%%ANKI
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Cloze
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A {molecular} statement can be broken up into {atomic} statements.
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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: 1708199272095-->
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END%%
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%%ANKI
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Basic
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What distinguishes a sentence from a statement?
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Back: The latter is a sentence that can be derived a truth value.
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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: 1708199272099-->
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END%%
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%%ANKI
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Basic
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What distinguishes a sentence from a predicate?
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Back: The latter is a sentence that contains 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: 1708199272103-->
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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 statement?
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Back: A statement 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 statements defined in terms of predicates?
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Back: A statement 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 statement?
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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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## Implication
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Implication is denoted as $\Rightarrow$. It has truth table
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$p$ | $q$ | $p \Rightarrow q$
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--- | --- | -----------------
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$T$ | $T$ | $T$
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$T$ | $F$ | $F$
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$F$ | $T$ | $T$
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$F$ | $F$ | $T$
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Implication has a few "equivalent" English expressions that are commonly used.
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Given propositions $P$ and $Q$, we have the following equivalences:
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* $P$ if $Q$
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* $P$ only if $Q$
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* $P$ is necessary for $Q$
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* $P$ is sufficient for $Q$
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%%ANKI
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Basic
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What name is given to operand $a$ in $a \Rightarrow b$?
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Back: The antecedent
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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: 1706994861308-->
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END%%
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%%ANKI
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Basic
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What name is given to operand $b$ in $a \Rightarrow b$?
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Back: The consequent
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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: 1706994861310-->
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END%%
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%%ANKI
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Basic
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How does "$P$ if $Q$" translate with $\Rightarrow$?
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Back: $Q \Rightarrow P$
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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: 1708199272127-->
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END%%
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%%ANKI
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Basic
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How does "$P$ only if $Q$" translate with $\Rightarrow$?
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Back: $P \Rightarrow Q$
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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: 1708199272134-->
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END%%
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%%ANKI
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Basic
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How does "$P$ is necessary for $Q$" translate with $\Rightarrow$?
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Back: $Q \Rightarrow P$
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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: 1708199272140-->
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END%%
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%%ANKI
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Basic
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How does "$P$ is sufficient for $Q$" translate with $\Rightarrow$?
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Back: $P \Rightarrow Q$
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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: 1708199272145-->
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END%%
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%%ANKI
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Basic
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Which of *if*/*only if* map to *necessary*?
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Back: *if*
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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: 1708199272151-->
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END%%
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%%ANKI
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Basic
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Which of *if*/*only if* map to *sufficient*?
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Back: *only if*
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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: 1708199272157-->
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END%%
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%%ANKI
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Basic
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Which logical operator maps to "if and only if"?
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Back: $\Leftrightarrow$
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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: 1708199272163-->
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END%%
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%%ANKI
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Basic
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Which logical operator maps to "necessary and sufficient"?
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Back: $\Leftrightarrow$
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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: 1708199272168-->
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END%%
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%%ANKI
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Basic
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What is the converse of $P \Rightarrow Q$?
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Back: $Q \Rightarrow P$
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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: 1708199272173-->
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END%%
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%%ANKI
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Basic
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When is implication equivalent to its converse?
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Back: It's indeterminate.
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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: 1708199272178-->
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END%%
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%%ANKI
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Basic
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What is the contrapositive of $P \Rightarrow Q$?
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Back: $\neg Q \Rightarrow \neg P$
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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: 1708199272184-->
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END%%
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%%ANKI
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Basic
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When is implication equivalent to its contrapositive?
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Back: They are always equivalent.
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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: 1708199272189-->
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END%%
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## References
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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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