413 lines
14 KiB
Markdown
413 lines
14 KiB
Markdown
---
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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 ($\Leftrightarrow$). A **proposition** is a sentence that can be assigned a truth or false value.
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%%ANKI
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Cloze
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{Propositional} logic is also known as {zeroth}-order logic.
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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: 1715897257085-->
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END%%
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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 proposition?
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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 propositions fall within?
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Back: Atomic and molecular propositions.
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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 proposition?
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Back: One that cannot be broken up into smaller propositions.
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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 proposition?
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Back: One that can be broken up into smaller propositions.
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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} proposition can be broken up into {atomic} propositions.
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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 proposition?
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Back: The latter has an associated 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 are constant propositions?
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Back: Propositions that contain only constants as operands.
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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: 1707422675517-->
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END%%
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%%ANKI
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Basic
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How does Lean define propositional equality?
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Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
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Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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Tags: lean
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<!--ID: 1706994861298-->
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END%%
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%%ANKI
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Basic
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How does Lean define `propext`?
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Back:
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```lean
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axiom propext {a b : Prop} : (a ↔ b) → (a = b)
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```
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Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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Tags: lean
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<!--ID: 1706994861300-->
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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* or *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* or *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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%%ANKI
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Basic
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Given propositions $p$ and $q$, $p \Leftrightarrow q$ is equivalent to the conjunction of what two expressions?
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Back: $p \Rightarrow q$ and $q \Rightarrow p$.
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Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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## Sets
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A **state** is a function that maps identifiers to $T$ or $F$. A proposition 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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What is a state?
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Back: A function mapping identifiers to values.
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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: 1706994861314-->
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END%%
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%%ANKI
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Basic
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Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
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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: 1706994861318-->
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END%%
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%%ANKI
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Basic
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Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
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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: 1706994861320-->
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END%%
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%%ANKI
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Basic
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A proposition is well-defined with respect to what?
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Back: A state to evaluate against.
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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: 1706994861316-->
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END%%
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%%ANKI
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Basic
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What proposition represents states $\{(b, T), (c, T)\}$ and $\{(b, F), (c, F)\}$?
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Back: $(b \land c) \lor (\neg b \land \neg c)$
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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: 1706994861337-->
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END%%
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%%ANKI
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Basic
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What set of states does proposition $a \land b$ represent?
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Back: $\{\{(a, T), (b, T)\}\}$
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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: 1706994861339-->
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END%%
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%%ANKI
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Basic
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What set of states does proposition $a \lor b$ represent?
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Back: $\{\{(a, T), (b, T)\}, \{(a, T), (b, F)\}, \{(a, F), (b, T)\}\}$
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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: 1715895996324-->
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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 $b \lor \neg c$"?
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Back: $b \lor \neg c$ 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: 1706994861341-->
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END%%
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%%ANKI
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Basic
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What is the weakest proposition?
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Back: $T$
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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: 1706994861348-->
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END%%
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%%ANKI
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Basic
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What set of states does $T$ represent?
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Back: The set of all 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: 1706994861350-->
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END%%
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%%ANKI
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Basic
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What is the strongest proposition?
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Back: $F$
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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: 1706994861352-->
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END%%
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%%ANKI
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Basic
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What set of states does $F$ represent?
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Back: The set of no 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: 1706994861354-->
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END%%
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%%ANKI
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Basic
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What does a proposition *represent*?
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Back: The set of states in which it is true.
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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: 1706994861335-->
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END%%
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%%ANKI
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Basic
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When is $p$ stronger than $q$?
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Back: When $p \Rightarrow q$.
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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: 1706994861343-->
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END%%
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%%ANKI
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Basic
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If $p \Rightarrow q$, which of $p$ or $q$ is considered stronger?
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Back: $p$
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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: 1715631869202-->
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END%%
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%%ANKI
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Basic
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When is $p$ weaker than $q$?
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Back: When $q \Rightarrow p$.
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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: 1706994861346-->
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END%%
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%%ANKI
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Basic
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If $p \Rightarrow q$, which of $p$ or $q$ is considered weaker?
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Back: $q$
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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: 1715631869207-->
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END%%
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%%ANKI
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Basic
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Why is $b \land c$ stronger than $b \lor c$?
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Back: The former represents a subset of the states the latter represents.
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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: 1706994861356-->
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END%%
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%%ANKI
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Basic
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Given sets $a$ and $b$, $a = b$ is equivalent to the conjunction of what two expressions?
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Back: $a \subseteq b$ and $b \subseteq a$.
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Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Cloze
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{$a \Rightarrow b$} is to propositional logic as {$a \subseteq b$} is to sets.
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Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Cloze
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{$a \Leftrightarrow b$} is to propositional logic as {$a = b$} is to sets.
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Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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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). |