314 lines
12 KiB
Markdown
314 lines
12 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 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.
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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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With respect to predicate logic, how many inference rules does natural deduction include?
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Back: $14$
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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: 1721655978496-->
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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 operators.
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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 proposition $E1$, $$\neg{\text{-}}I{:} \quad \text{TODO}$$
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and $$\neg{\text{-}}E{:} \quad \text{TODO}$$
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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? $$\begin{array}{c} P, Q, R \\ \hline P \land R \end{array}$$
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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? $$\begin{array}{c} P \land Q \\ \hline P \end{array}$$
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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: $$\lor{\text{-}}I{:} \quad \begin{array}{c} E_i \\ \hline E_1 \lor \cdots \lor E_n \end{array}$$
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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: $$\lor{\text{-}}E{:} \quad \begin{array}{c} E_1 \lor \cdots \lor E_n, E_1 \Rightarrow E, \ldots, E_n \Rightarrow E \\ \hline E \end{array}$$
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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? $$\begin{array}{c} P, Q \\ \hline R \lor P \end{array}$$
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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? $$\begin{array}{c} P \lor Q, P \Rightarrow R, Q \Rightarrow R \\ \hline P \end{array}$$
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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 $E1$ and $E2$, $${\Rightarrow}{\text{-}}I: \quad \text{TODO}$$
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and $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$
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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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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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Cloze
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Natural deduction rule {1:$\Rightarrow$}-{1:$E$} is also known as {modus ponens}.
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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: 1721665541953-->
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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: $$\begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$
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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: $$\begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$
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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? $$\begin{array}{c} P \Rightarrow Q, P \\ \hline R \end{array}$$
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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? $$\begin{array}{c} P \Rightarrow Q, P \\ \hline Q \end{array}$$
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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 $E1$ and $E2$, $${\Leftrightarrow}{\text{-}}I: \quad \begin{array}{c} E1 \Rightarrow E2, E2 \Rightarrow E1 \\ \hline E1 \Leftrightarrow E2 \end{array}$$
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and $${\Leftrightarrow}{\text{-}}E: \quad \begin{array}{c} E1 \Leftrightarrow E2 \\ \hline E1 \Rightarrow E2, E2 \Rightarrow E1 \end{array}$$
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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: $$\begin{array}{c} E1 \Rightarrow E2, E2 \Rightarrow E1 \\ \hline E1 \Leftrightarrow E2 \end{array}$$
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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? $$\begin{array}{c} P \Rightarrow Q, Q \Rightarrow P \\ \hline Q \Leftrightarrow P \end{array}$$
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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: $$\begin{array}{c} E1 \Leftrightarrow E2 \\ \hline E1 \Rightarrow E2, E2 \Rightarrow E1 \end{array}$$
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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? $$\begin{array}{c} P \Leftrightarrow Q \\ \hline Q \Rightarrow P \end{array}$$
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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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