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| title | TARGET DECK | FILE TAGS | tags | |||
|---|---|---|---|---|---|---|
| Natural Deduction | Obsidian::STEM | formal-system::natural-deduction |
|
Overview
Natural deduction is a proof system typically used alongside classical truth-functional prop-logic and pred-logic 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.
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.
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 and pred-logic#Universals 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.
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.
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.
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.
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
END%%
Implication
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}
and {\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E_1 \Rightarrow E_2, E_1 \ \hline E_2 \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.
END%%
%%ANKI
Basic
How is {\Rightarrow}{\text{-}}I expressed in schematic notation?
Back: {\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}
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
END%%
Existential Quantification
TODO
Universal Quantification
TODO
Bibliography
- Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.