notebook/notes/logic/prop-logic.md

417 lines
14 KiB
Markdown
Raw Blame History

This file contains invisible Unicode characters!

This file contains invisible Unicode characters that may be processed differently from what appears below. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to reveal hidden characters.

This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.

---
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 ($\Leftrightarrow$). A **proposition** is a sentence that can be assigned a truth or false value.
%%ANKI
Cloze
{Propositional} logic is also known as {zeroth}-order logic.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715897257085-->
END%%
%%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 proposition?
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 propositions fall within?
Back: Atomic and molecular propositions.
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 proposition?
Back: One that cannot be broken up into smaller propositions.
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 proposition?
Back: One that can be broken up into smaller propositions.
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} proposition can be broken up into {atomic} propositions.
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 proposition?
Back: The latter has an associated 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 are constant propositions?
Back: Propositions that contain only constants as operands.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707422675517-->
END%%
%%ANKI
Basic
How does Lean define propositional equality?
Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
Reference: Avigad, Jeremy. Theorem Proving in Lean, n.d.
Tags: lean
<!--ID: 1706994861298-->
END%%
%%ANKI
Basic
How does Lean define `propext`?
Back:
```lean
axiom propext {a b : Prop} : (a ↔ b) → (a = b)
```
Reference: Avigad, Jeremy. Theorem Proving in Lean, n.d.
Tags: lean
<!--ID: 1706994861300-->
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%%
%%ANKI
Basic
Given propositions $p$ and $q$, $p \Leftrightarrow q$ is equivalent to the conjunction of what two expressions?
Back: $p \Rightarrow q$ and $q \Rightarrow p$.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047070-->
END%%
## Sets
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.
%%ANKI
Basic
What is a state?
Back: A function mapping identifiers to values.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861314-->
END%%
%%ANKI
Basic
Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861318-->
END%%
%%ANKI
Basic
Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
Back: No.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861320-->
END%%
%%ANKI
Basic
A proposition is well-defined with respect to what?
Back: A state to evaluate against.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861316-->
END%%
%%ANKI
Basic
What proposition represents states $\{(b, T), (c, T)\}$ and $\{(b, F), (c, F)\}$?
Back: $(b \land c) \lor (\neg b \land \neg c)$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861337-->
END%%
%%ANKI
Basic
What set of states does proposition $a \land b$ represent?
Back: $\{\{(a, T), (b, T)\}\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861339-->
END%%
%%ANKI
Basic
What set of states does proposition $a \lor b$ represent?
Back: $\{\{(a, T), (b, T)\}, \{(a, T), (b, F)\}, \{(a, F), (b, T)\}\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715895996324-->
END%%
%%ANKI
Basic
What is sloppy about phrase "the states in $b \lor \neg c$"?
Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861341-->
END%%
%%ANKI
Basic
What is the weakest proposition?
Back: $T$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861348-->
END%%
%%ANKI
Basic
What set of states does $T$ represent?
Back: The set of all states.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861350-->
END%%
%%ANKI
Basic
What is the strongest proposition?
Back: $F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861352-->
END%%
%%ANKI
Basic
What set of states does $F$ represent?
Back: The set of no states.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861354-->
END%%
%%ANKI
Basic
What does a proposition *represent*?
Back: The set of states in which it is true.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861335-->
END%%
%%ANKI
Basic
When is $p$ stronger than $q$?
Back: When $p \Rightarrow q$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861343-->
END%%
%%ANKI
Basic
If $p \Rightarrow q$, which of $p$ or $q$ is considered stronger?
Back: $p$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869202-->
END%%
%%ANKI
Basic
When is $p$ weaker than $q$?
Back: When $q \Rightarrow p$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861346-->
END%%
%%ANKI
Basic
If $p \Rightarrow q$, which of $p$ or $q$ is considered weaker?
Back: $q$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869207-->
END%%
%%ANKI
Basic
Why is $b \land c$ stronger than $b \lor c$?
Back: The former represents a subset of the states the latter represents.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861356-->
END%%
%%ANKI
Basic
Given sets $a$ and $b$, $a = b$ is equivalent to the conjunction of what two expressions?
Back: $a \subseteq b$ and $b \subseteq a$.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047071-->
END%%
%%ANKI
Cloze
{$a \Rightarrow b$} is to propositional logic as {$a \subseteq b$} is to sets.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047073-->
END%%
%%ANKI
Cloze
{$a \Leftrightarrow b$} is to propositional logic as {$a = b$} is to sets.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047074-->
END%%
## Bibliography
* 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).