241 lines
9.3 KiB
Markdown
241 lines
9.3 KiB
Markdown
---
|
||
title: Propositional Logic
|
||
TARGET DECK: Obsidian::STEM
|
||
FILE TAGS: logic::propositional
|
||
tags:
|
||
- logic
|
||
- propositional
|
||
---
|
||
|
||
## Overview
|
||
|
||
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:
|
||
|
||
* A **sentence** is any grammatical string of words.
|
||
* A **predicate** is a sentence with free variables.
|
||
* A **statement** is a sentence that can be assigned a truth or false value.
|
||
* A predicate with free variables "plugged in" is a statement.
|
||
|
||
%%ANKI
|
||
Basic
|
||
What are the basic propositional logical operators?
|
||
Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861291-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is a propositional statement?
|
||
Back: A declarative sentence which is either true or false.
|
||
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).
|
||
<!--ID: 1708199272076-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What two categories do propositional statements fall within?
|
||
Back: Atomic and molecular statements.
|
||
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).
|
||
<!--ID: 1708199272083-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is an atomic statement?
|
||
Back: It cannot be broken up into smaller statements.
|
||
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).
|
||
<!--ID: 1708199272087-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is a molecular statement?
|
||
Back: It can be broken up into smaller statements.
|
||
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).
|
||
<!--ID: 1708199272091-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
A {molecular} statement can be broken up into {atomic} statements.
|
||
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).
|
||
<!--ID: 1708199272095-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What distinguishes a sentence from a statement?
|
||
Back: The latter is a sentence that can be derived a truth value.
|
||
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).
|
||
<!--ID: 1708199272099-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What distinguishes a sentence from a predicate?
|
||
Back: The latter is a sentence that contains free variables.
|
||
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).
|
||
<!--ID: 1708199272103-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What distinguishes a predicate from a statement?
|
||
Back: A statement does not contain free variables.
|
||
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).
|
||
<!--ID: 1708199272110-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How are statements defined in terms of predicates?
|
||
Back: A statement is a predicate with $0$ free variables.
|
||
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).
|
||
<!--ID: 1708199272115-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Why is "$3 + x = 12$" *not* a statement?
|
||
Back: Because $x$ is a variable.
|
||
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).
|
||
<!--ID: 1708199272121-->
|
||
END%%
|
||
|
||
## Implication
|
||
|
||
Implication is denoted as $\Rightarrow$. It has truth table
|
||
|
||
$p$ | $q$ | $p \Rightarrow q$
|
||
--- | --- | -----------------
|
||
$T$ | $T$ | $T$
|
||
$T$ | $F$ | $F$
|
||
$F$ | $T$ | $T$
|
||
$F$ | $F$ | $T$
|
||
|
||
Implication has a few "equivalent" English expressions that are commonly used.
|
||
Given propositions $P$ and $Q$, we have the following equivalences:
|
||
|
||
* $P$ if $Q$
|
||
* $P$ only if $Q$
|
||
* $P$ is necessary for $Q$
|
||
* $P$ is sufficient for $Q$
|
||
|
||
%%ANKI
|
||
Basic
|
||
What name is given to operand $a$ in $a \Rightarrow b$?
|
||
Back: The antecedent
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861308-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What name is given to operand $b$ in $a \Rightarrow b$?
|
||
Back: The consequent
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861310-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How does "$P$ if $Q$" translate with $\Rightarrow$?
|
||
Back: $Q \Rightarrow P$
|
||
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).
|
||
<!--ID: 1708199272127-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How does "$P$ only if $Q$" translate with $\Rightarrow$?
|
||
Back: $P \Rightarrow Q$
|
||
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).
|
||
<!--ID: 1708199272134-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How does "$P$ is necessary for $Q$" translate with $\Rightarrow$?
|
||
Back: $Q \Rightarrow P$
|
||
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).
|
||
<!--ID: 1708199272140-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How does "$P$ is sufficient for $Q$" translate with $\Rightarrow$?
|
||
Back: $P \Rightarrow Q$
|
||
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).
|
||
<!--ID: 1708199272145-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of *if* or *only if* map to *necessary*?
|
||
Back: *if*
|
||
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).
|
||
<!--ID: 1708199272151-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of *if* or *only if* map to *sufficient*?
|
||
Back: *only if*
|
||
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).
|
||
<!--ID: 1708199272157-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which logical operator maps to "if and only if"?
|
||
Back: $\Leftrightarrow$
|
||
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).
|
||
<!--ID: 1708199272163-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which logical operator maps to "necessary and sufficient"?
|
||
Back: $\Leftrightarrow$
|
||
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).
|
||
<!--ID: 1708199272168-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the converse of $P \Rightarrow Q$?
|
||
Back: $Q \Rightarrow P$
|
||
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).
|
||
<!--ID: 1708199272173-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
When is implication equivalent to its converse?
|
||
Back: It's indeterminate.
|
||
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).
|
||
<!--ID: 1708199272178-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the contrapositive of $P \Rightarrow Q$?
|
||
Back: $\neg Q \Rightarrow \neg P$
|
||
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).
|
||
<!--ID: 1708199272184-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
When is implication equivalent to its contrapositive?
|
||
Back: They are always equivalent.
|
||
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).
|
||
<!--ID: 1708199272189-->
|
||
END%%
|
||
|
||
## References
|
||
|
||
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
* 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).
|