314 lines
12 KiB
Markdown
314 lines
12 KiB
Markdown
---
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title: Propositional Logic
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TARGET DECK: Obsidian::STEM
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FILE TAGS: formal-system::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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**Propositional logic** is a logical system derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and biconditionals ($\Leftrightarrow$). A **proposition** is a sentence that can be assigned a truth 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 that can be assigned 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: 1708199272076-->
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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 $\Rightarrow$. In classical logic, it has truth table $$\begin{array}{c|c|c} p & q & p \Rightarrow q \\ \hline T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array}$$
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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 do you write "$P$ if $Q$" in propositional logic?
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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 do you write "$P$ if $Q$" using "necessary"?
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Back: $P$ is necessary for $Q$.
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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: 1717853966420-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ if $Q$" using "sufficient"?
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Back: $Q$ is sufficient for $P$.
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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: 1717853966425-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ only if $Q$" in propositional logic?
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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 do you write "$P$ only if $Q$" using "necessary"?
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Back: $Q$ is necessary for $P$.
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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: 1717853966429-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ only if $Q$" using "sufficient"?
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Back: $P$ is sufficient for $Q$.
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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: 1717853966432-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ is necessary for $Q$" in propositional logic?
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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 do you write "$P$ is necessary for $Q$" using "if"?
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Back: $P$ if $Q$.
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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: 1717853966435-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ is necessary for $Q$" using "only if"?
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Back: $Q$ only if $P$.
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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: 1717853966438-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ is sufficient for $Q$" in propositional logic?
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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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How do you write "$P$ is sufficient for $Q$" using "if"?
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Back: $Q$ if $P$.
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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: 1717853966441-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ is sufficient for $Q$" using "only if"?
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Back: $P$ only if $Q$.
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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: 1717853966444-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ if $Q$" using "only if"?
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Back: $Q$ only if $P$.
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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: 1717853966449-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ is sufficient for $Q$" using "necessary"?
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Back: $Q$ is necessary for $P$.
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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: 1717853966454-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ only if $Q$" using "if"?
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Back: $Q$ if $P$.
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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: 1717853966458-->
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END%%
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%%ANKI
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Basic
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How do you write "$P$ is necessary for $Q$" using "sufficient"?
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Back: $Q$ is sufficient for $P$.
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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: 1717853966462-->
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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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<!--ID: 1715969047070-->
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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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* “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759).
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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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* “Principle of Explosion,” in _Wikipedia_, July 3, 2024, [https://en.wikipedia.org/w/index.php?title=Principle_of_explosion](https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=1232334233). |