339 lines
13 KiB
Markdown
339 lines
13 KiB
Markdown
---
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title: Natural Deduction
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TARGET DECK: Obsidian::STEM
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FILE TAGS: formal-system::natural-deduction
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tags:
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- logic
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- natural-deduction
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- programming
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---
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## Overview
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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.
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%%ANKI
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Basic
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Why is natural deduction named the way it is?
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Back: It is mean to mimic the patterns of reasoning one might "naturally" make when forming arguments in plain English.
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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: 1721655978485-->
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END%%
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## Axioms
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Natural deduction is interesting in that it has no axioms.
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%%ANKI
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Basic
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How many axioms does natural deduction include?
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Back: $0$
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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: 1721655978490-->
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END%%
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## Inference Rules
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Scoped to propositional logic, there are 10 inference rules corresponding to an "introduction" and "elimination" of each propositional logic operator.
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%%ANKI
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Basic
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With respect to propositional logic, how many inference rules does natural deduction include?
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Back: $10$
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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: 1721655978493-->
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END%%
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%%ANKI
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Basic
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How are natural deduction's inference rules categorized into two?
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Back: As introduction and elimination rules.
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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: 1721655978499-->
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END%%
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%%ANKI
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Basic
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With respect to propositional logic, how are natural deduction's inference rules categorized into five?
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Back: As an introduction and elimination rule per propositional logic operator.
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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: 1721655978506-->
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END%%
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### Negation
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For propositions $E_1$ and $E_2$, $$\neg{\text{-}}I{:} \quad \begin{array}{c} \text{from } E_1 \text{ infer } E_2 \land \neg E_2 \\ \hline \neg E_1 \end{array}$$
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and $$\neg{\text{-}}E{:} \quad \begin{array}{c} \text{from } \neg E_1 \text{ infer } E_2 \land \neg E_2 \\ \hline E_1 \end{array}$$
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%%ANKI
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Basic
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In natural deduction, how is negation introduction denoted?
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Back: As $\neg{\text{-}}I$.
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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: 1721825479315-->
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END%%
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%%ANKI
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Basic
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In natural deduction, how is negation elimination denoted?
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Back: As $\neg{\text{-}}E$.
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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: 1721825479325-->
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END%%
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%%ANKI
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Basic
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How is $\neg{\text{-}}I$ expressed in schematic notation?
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Back: $$\begin{array}{c} \text{from } E_1 \text{ infer } E_2 \land \neg E_2 \\ \hline \neg E_1 \end{array}$$
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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: 1721825479330-->
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END%%
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%%ANKI
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Basic
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How is $\neg{\text{-}}E$ expressed in schematic notation?
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Back: $$\begin{array}{c} \text{from } \neg E_1 \text{ infer } E_2 \land \neg E_2 \\ \hline E_1 \end{array}$$
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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: 1721825479336-->
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END%%
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### Conjunction
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For propositions $E_1, \ldots, E_n$, $$\land{\text{-}}I{:} \quad \begin{array}{c} E_1, \ldots, E_n \\ \hline E_1 \land \cdots \land E_n \end{array}$$
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and $$\land{\text{-}}E{:} \quad \begin{array}{c} E_1 \land \cdots \land E_n \\ \hline E_i \end{array}$$
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%%ANKI
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Basic
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In natural deduction, how is conjunction introduction denoted?
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Back: As $\land{\text{-}}I$.
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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: 1721656449679-->
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END%%
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%%ANKI
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Basic
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In natural deduction, how is conjunction elimination denoted?
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Back: As $\land{\text{-}}E$.
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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: 1721656449704-->
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END%%
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%%ANKI
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Basic
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How is $\land{\text{-}}I$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_1, \ldots, E_n \\ \hline E_1 \land \cdots \land E_n \end{array}$$
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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: 1721655978513-->
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END%%
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%%ANKI
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Basic
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How is $\land{\text{-}}E$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_1 \land \cdots \land E_n \\ \hline E_i \end{array}$$
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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: 1721655978517-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \\ 2. & Q \\ 3. & R \\ \hline & P \land R \end{array}$$
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Back: $\land{\text{-}}I$
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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: 1721656730330-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \land Q \\ \hline & P \end{array}$$
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Back: $\land{\text{-}}E$
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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: 1721656601607-->
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END%%
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### Disjunction
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For propositions $E_1, \ldots, E_n$, $$\lor{\text{-}}I{:} \quad \begin{array}{c} E_i \\ \hline E_1 \lor \cdots \lor E_n \end{array}$$
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and $$\lor{\text{-}}E{:} \quad \begin{array}{c} E_1 \lor \cdots \lor E_n, E_1 \Rightarrow E, \ldots, E_n \Rightarrow E \\ \hline E \end{array}$$
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%%ANKI
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Basic
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In natural deduction, how is disjunction introduction denoted?
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Back: As $\lor{\text{-}}I$.
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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: 1721656416280-->
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END%%
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%%ANKI
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Basic
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In natural deduction, how is disjunction elimination denoted?
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Back: As $\lor{\text{-}}E$.
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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: 1721656416284-->
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END%%
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%%ANKI
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Basic
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How is $\lor{\text{-}}I$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_i \\ \hline E_1 \lor \cdots \lor E_n \end{array}$$
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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: 1721656416288-->
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END%%
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%%ANKI
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Basic
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How is $\lor{\text{-}}E$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_1 \lor \cdots \lor E_n, E_1 \Rightarrow E, \ldots, E_n \Rightarrow E \\ \hline E \end{array}$$
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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: 1721656416291-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \\ 2. & Q \\ \hline & R \lor P \end{array}$$
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Back: $\lor{\text{-}}I$
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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: 1721656730337-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \lor Q \\ 2. & P \Rightarrow R \\ 3. & Q \Rightarrow R \\ \hline & R \end{array}$$
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Back: $\lor{\text{-}}E$
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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: 1721656601613-->
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END%%
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### Implication
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For propositions $E_1, \ldots, E_n$, $${\Rightarrow}{\text{-}}I: \quad \begin{array}{c} \text{from } E_1, \cdots, E_n \text{ infer } E \\ \hline (E_1 \land \cdots \land E_n) \Rightarrow E \end{array}$$
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and $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E_1 \Rightarrow E_2, E_1 \\ \hline E_2 \end{array}$$
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%%ANKI
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Basic
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In natural deduction, how is implication introduction denoted?
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Back: As ${\Rightarrow}{\text{-}}I$.
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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: 1721665510225-->
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END%%
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%%ANKI
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Basic
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How is ${\Rightarrow}{\text{-}}I$ expressed in schematic notation?
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Back: $$\begin{array}{c} \text{from } E_1, \cdots, E_n \text{ infer } E \\ \hline (E_1 \land \cdots \land E_n) \Rightarrow E \end{array}$$
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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: 1721785548092-->
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END%%
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%%ANKI
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Basic
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In natural deduction, how is implication elimination denoted?
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Back: As ${\Rightarrow}{\text{-}}E$.
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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: 1721665541946-->
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END%%
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%%ANKI
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Basic
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*Modus ponens* is associated with which propositional logic operator?
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Back: $\Rightarrow$
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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: 1721665541949-->
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END%%
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%%ANKI
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Basic
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Does *modus ponens* correspond to an introduction or elimination rule?
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Back: Elimination.
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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: 1721665541951-->
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END%%
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%%ANKI
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Basic
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How is ${\Rightarrow}{\text{-}}E$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_1 \Rightarrow E_2, E_1 \\ \hline E_2 \end{array}$$
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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: 1721665510228-->
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END%%
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%%ANKI
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Basic
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How is *modus ponens* expressed in schematic notation?
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Back: $$\begin{array}{c} E_1 \Rightarrow E_2, E_1 \\ \hline E_2 \end{array}$$
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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: 1721665541955-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \Rightarrow Q \\ 2. & P \\ \hline & R \end{array}$$
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Back: N/A.
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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: 1721666244354-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \Rightarrow Q \\ 2. & P \\ \hline & Q \end{array}$$
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Back: ${\Rightarrow}{\text{-}}E$
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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: 1721666244357-->
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END%%
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### Biconditional
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For propositions $E_1$ and $E_2$, $${\Leftrightarrow}{\text{-}}I: \quad \begin{array}{c} E_1 \Rightarrow E_2, E_2 \Rightarrow E_1 \\ \hline E_1 \Leftrightarrow E_2 \end{array}$$
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and $${\Leftrightarrow}{\text{-}}E: \quad \begin{array}{c} E_1 \Leftrightarrow E_2 \\ \hline E_1 \Rightarrow E_2, E_2 \Rightarrow E_1 \end{array}$$
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%%ANKI
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Basic
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In natural deduction, how is biconditional introduction denoted?
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Back: As ${\Leftrightarrow}{\text{-}}I$.
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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: 1721666244359-->
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END%%
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%%ANKI
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Basic
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In natural deduction, how is biconditional elimination denoted?
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Back: As ${\Leftrightarrow}{\text{-}}E$.
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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: 1721666244361-->
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END%%
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%%ANKI
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Basic
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How is ${\Leftrightarrow}{\text{-}}I$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_1 \Rightarrow E_2, E_2 \Rightarrow E_1 \\ \hline E_1 \Leftrightarrow E_2 \end{array}$$
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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: 1721666244362-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \Rightarrow Q \\ 2. & Q \Rightarrow P \\ \hline & Q \Leftrightarrow P \end{array}$$
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Back: ${\Leftrightarrow}{\text{-}}I$
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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: 1721666244367-->
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END%%
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%%ANKI
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Basic
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How is ${\Leftrightarrow}{\text{-}}E$ expressed in schematic notation?
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Back: $$\begin{array}{c} E_1 \Leftrightarrow E_2 \\ \hline E_1 \Rightarrow E_2, E_2 \Rightarrow E_1 \end{array}$$
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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: 1721666244366-->
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END%%
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%%ANKI
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Basic
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Which natural deduction inference rule is used in the following? $$\begin{array}{rc} 1. & P \Leftrightarrow Q \\ \hline & Q \Rightarrow P \end{array}$$
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Back: ${\Leftrightarrow}{\text{-}}E$
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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: 1721666244364-->
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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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