notebook/notes/formal-system/proof-system/natural-deduction.md

---
title: Natural Deduction
TARGET DECK: Obsidian::STEM
FILE TAGS: formal-system::natural-deduction
tags:
  - logic
  - natural-deduction
  - programming
---

## Overview

Natural deduction is a proof system typically used alongside classical truth-functional [[prop-logic|propositional]] and [[pred-logic|predicate]] logic. It is meant to mimic the patterns of reasoning that one might "naturally" make when forming arguments in plain English.

%%ANKI
Basic
Why is natural deduction named the way it is?
Back: It is mean to mimic the patterns of reasoning one might "naturally" make when forming arguments in plain English.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978485-->
END%%

## Axioms

Natural deduction is interesting in that it has no axioms.

%%ANKI
Basic
How many axioms does natural deduction include?
Back: $0$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978490-->
END%%

## Inference Rules

Scoped to just propositional logic, there are 10 inference rules corresponding to an "introduction" and "elimination" of each propositional logic operator. When extending to predicate logic, we also include an introduction and elimination rule for both the [[pred-logic#Existentials|existential]] and [[pred-logic#Universals|universal]] quantifers.

%%ANKI
Basic
With respect to propositional logic, how many inference rules does natural deduction include?
Back: $10$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978493-->
END%%

%%ANKI
Basic
With respect to predicate logic, how many inference rules does natural deduction include?
Back: $14$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978496-->
END%%

%%ANKI
Basic
How are natural deduction's inference rules categorized into two?
Back: As introduction and elimination rules.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978499-->
END%%

%%ANKI
Basic
With respect to propositional logic, how are natural deduction's inference rules categorized into five?
Back: As an introduction and elimination rule per propositional logic operators.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978506-->
END%%

### Negation

For proposition $E1$, $$\neg{\text{-}}I{:} \quad \text{TODO}$$
and $$\neg{\text{-}}E{:} \quad \text{TODO}$$

### Conjunction

For propositions $E_1, \ldots, E_n$, $$\land{\text{-}}I{:} \quad \begin{array}{c} E_1, \ldots, E_n \\ \hline E_1 \land \cdots \land E_n \end{array}$$
and $$\land{\text{-}}E{:} \quad \begin{array}{c} E_1 \land \cdots \land E_n \\ \hline E_i \end{array}$$

%%ANKI
Basic
In natural deduction, how is conjunction introduction denoted?
Back: As $\land{\text{-}}I$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656449679-->
END%%

%%ANKI
Basic
In natural deduction, how is conjunction elimination denoted?
Back: As $\land{\text{-}}E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656449704-->
END%%

%%ANKI
Basic
How is $\land{\text{-}}I$ expressed in schematic notation?
Back: $$\begin{array}{c} E_1, \ldots, E_n \\ \hline E_1 \land \cdots \land E_n \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978513-->
END%%

%%ANKI
Basic
How is $\land{\text{-}}E$ expressed in schematic notation?
Back: $$\begin{array}{c} E_1 \land \cdots \land E_n \\ \hline E_i \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721655978517-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P, Q, R \\ \hline P \land R \end{array}$$
Back: $\land{\text{-}}I$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656730330-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P \land Q \\ \hline P \end{array}$$
Back: $\land{\text{-}}E$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656601607-->
END%%

### Disjunction

For propositions $E_1, \ldots, E_n$, $$\lor{\text{-}}I{:} \quad \begin{array}{c} E_i \\ \hline E_1 \lor \cdots \lor E_n \end{array}$$
and $$\lor{\text{-}}E{:} \quad \begin{array}{c} E_1 \lor \cdots \lor E_n, E_1 \Rightarrow E, \ldots, E_n \Rightarrow E \\ \hline E \end{array}$$

%%ANKI
Basic
In natural deduction, how is disjunction introduction denoted?
Back: As $\lor{\text{-}}I$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656416280-->
END%%

%%ANKI
Basic
In natural deduction, how is disjunction elimination denoted??
Back: As $\lor{\text{-}}E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656416284-->
END%%

%%ANKI
Basic
How is $\lor{\text{-}}I$ expressed in schematic notation?
Back: $$\lor{\text{-}}I{:} \quad \begin{array}{c} E_i \\ \hline E_1 \lor \cdots \lor E_n \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656416288-->
END%%

%%ANKI
Basic
How is $\lor{\text{-}}E$ expressed in schematic notation?
Back: $$\lor{\text{-}}E{:} \quad \begin{array}{c} E_1 \lor \cdots \lor E_n, E_1 \Rightarrow E, \ldots, E_n \Rightarrow E \\ \hline E \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656416291-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P, Q \\ \hline R \lor P \end{array}$$
Back: $\lor{\text{-}}I$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656730337-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P \lor Q, P \Rightarrow R, Q \Rightarrow R \\ \hline P \end{array}$$
Back: $\lor{\text{-}}E$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721656601613-->
END%%

### Implication

For propositions $E1$ and $E2$, $${\Rightarrow}{\text{-}}I: \quad \text{TODO}$$
and $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$

%%ANKI
Basic
In natural deduction, how is implication introduction denoted?
Back: As ${\Rightarrow}{\text{-}}I$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665510225-->
END%%

%%ANKI
Basic
In natural deduction, how is implication elimination denoted?
Back: As ${\Rightarrow}{\text{-}}E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665541946-->
END%%

%%ANKI
Basic
*Modus ponens* is associated with which propositional logic operator?
Back: $\Rightarrow$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665541949-->
END%%

%%ANKI
Basic
Does *modus ponens* correspond to an introduction or elimination rule?
Back: Elimination.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665541951-->
END%%

%%ANKI
Cloze
Natural deduction rule {1:$\Rightarrow$}-{1:$E$} is also known as {modus ponens}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665541953-->
END%%

%%ANKI
Basic
How is ${\Rightarrow}{\text{-}}E$ expressed in schematic notation?
Back: $$\begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665510228-->
END%%

%%ANKI
Basic
How is *modus ponens* expressed in schematic notation?
Back: $$\begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721665541955-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P \Rightarrow Q, P \\ \hline R \end{array}$$
Back: N/A.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244354-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P \Rightarrow Q, P \\ \hline Q \end{array}$$
Back: ${\Rightarrow}{\text{-}}E$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244357-->
END%%

### Biconditional

For propositions $E1$ and $E2$, $${\Leftrightarrow}{\text{-}}I: \quad \begin{array}{c} E1 \Rightarrow E2, E2 \Rightarrow E1 \\ \hline E1 \Leftrightarrow E2 \end{array}$$
and $${\Leftrightarrow}{\text{-}}E: \quad \begin{array}{c} E1 \Leftrightarrow E2 \\ \hline E1 \Rightarrow E2, E2 \Rightarrow E1 \end{array}$$

%%ANKI
Basic
In natural deduction, how is biconditional introduction denoted?
Back: As ${\Leftrightarrow}{\text{-}}I$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244359-->
END%%

%%ANKI
Basic
In natural deduction, how is biconditional elimination denoted?
Back: As ${\Leftrightarrow}{\text{-}}E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244361-->
END%%

%%ANKI
Basic
How is ${\Leftrightarrow}{\text{-}}I$ expressed in schematic notation?
Back: $$\begin{array}{c} E1 \Rightarrow E2, E2 \Rightarrow E1 \\ \hline E1 \Leftrightarrow E2 \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244362-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P \Rightarrow Q, Q \Rightarrow P \\ \hline Q \Leftrightarrow P \end{array}$$
Back: ${\Leftrightarrow}{\text{-}}I$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244367-->
END%%

%%ANKI
Basic
How is ${\Leftrightarrow}{\text{-}}E$ expressed in schematic notation?
Back: $$\begin{array}{c} E1 \Leftrightarrow E2 \\ \hline E1 \Rightarrow E2, E2 \Rightarrow E1 \end{array}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244366-->
END%%

%%ANKI
Basic
Which natural deduction inference rule is used in the following? $$\begin{array}{c} P \Leftrightarrow Q \\ \hline Q \Rightarrow P \end{array}$$
Back: ${\Leftrightarrow}{\text{-}}E$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721666244364-->
END%%

## Bibliography

* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.