289 lines
13 KiB
Markdown
289 lines
13 KiB
Markdown
---
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title: Logic
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TARGET DECK: Obsidian::STEM
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FILE TAGS: formal-system
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tags:
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- formal-system
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- logic
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---
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## Overview
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A **formal system** is an abstract structure and formalization of an axiomatic system used for inferring theorems from axioms by a set of inference rules. It consists of two components: a **formal language** and a **deductive system**.
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%%ANKI
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Cloze
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{Deductive} reasoning begins with {a set of premises} and draws conclusions by {applying valid inferences}.
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Reference: “Deductive Reasoning,” in _Wikipedia_, June 16, 2024, [https://en.wikipedia.org/w/index.php?title=Deductive_reasoning](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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<!--ID: 1721991104381-->
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END%%
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%%ANKI
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Cloze
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{Inductive} reasoning begins with {a body of observations} and draws conclusions by {generalizing the observations}.
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Reference: “Inductive Reasoning,” in _Wikipedia_, May 6, 2024, [https://en.wikipedia.org/w/index.php?title=Inductive_reasoning](https://en.wikipedia.org/w/index.php?title=Inductive_reasoning&oldid=1222455892).
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<!--ID: 1721991104385-->
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END%%
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%%ANKI
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Cloze
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Deduction is to {premises} whereas induction is to {observations}.
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Reference: “Inductive Reasoning,” in _Wikipedia_, May 6, 2024, [https://en.wikipedia.org/w/index.php?title=Inductive_reasoning](https://en.wikipedia.org/w/index.php?title=Inductive_reasoning&oldid=1222455892).
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<!--ID: 1721991207202-->
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END%%
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%%ANKI
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Cloze
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Deduction is to {valid inferences} whereas induction is to {generalizations}.
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Reference: “Inductive Reasoning,” in _Wikipedia_, May 6, 2024, [https://en.wikipedia.org/w/index.php?title=Inductive_reasoning](https://en.wikipedia.org/w/index.php?title=Inductive_reasoning&oldid=1222455892).
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<!--ID: 1721991207211-->
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END%%
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%%ANKI
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Basic
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Do formal systems employ inductive or deductive reasoning?
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Back: Deductive reasoning.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534091-->
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END%%
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%%ANKI
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Basic
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Is the study of logic concerned with deductive or inductive reasoning?
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Back: Deductive reasoning.
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Reference: “Deductive Reasoning,” in _Wikipedia_, June 16, 2024, [https://en.wikipedia.org/w/index.php?title=Deductive_reasoning](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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<!--ID: 1721561534094-->
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END%%
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%%ANKI
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Basic
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Which of deductive or inductive reasoning can provide "genuinely new information"?
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Back: Inductive reasoning.
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Reference: “Deductive Reasoning,” in _Wikipedia_, June 16, 2024, [https://en.wikipedia.org/w/index.php?title=Deductive_reasoning](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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<!--ID: 1721561534099-->
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END%%
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%%ANKI
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Basic
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What does it mean for inductive reasoning to provide "genuinely new information"?
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Back: Information not found in the premises can be produced.
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Reference: “Deductive Reasoning,” in _Wikipedia_, June 16, 2024, [https://en.wikipedia.org/w/index.php?title=Deductive_reasoning](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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<!--ID: 1721561534103-->
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END%%
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%%ANKI
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Basic
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Which of deductive or inductive reasoning is considered more formal?
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Back: Deductive reasoning.
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Reference: “Deductive Reasoning,” in _Wikipedia_, June 16, 2024, [https://en.wikipedia.org/w/index.php?title=Deductive_reasoning](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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<!--ID: 1721561534107-->
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END%%
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%%ANKI
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Basic
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What two parts make up a formal system?
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Back: A formal language and a deductive system.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534111-->
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END%%
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%%ANKI
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Basic
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Which of a formal system or a deductive system is defined in terms of the other?
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Back: The formal system.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534115-->
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END%%
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%%ANKI
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Basic
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What does a formal system specify that a deductive system does not?
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Back: A formal language.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534119-->
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END%%
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%%ANKI
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Basic
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What does a formal system specify that a formal language does not?
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Back: A deductive system.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534124-->
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END%%
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%%ANKI
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Basic
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The term "formal system" can be seen as a portmanteau of what other terms?
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Back: A **formal** language and a deductive **system**.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534129-->
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END%%
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A formal language comprises of an **alphabet** and a **formal grammar**. The grammer has associated rules describing how symbols of the alphabet are manipulated to create **well-formed formulas** (WFFs). The **syntax** of a language describes the set of possible expressions that are valid utterances. The **semantics** of a language describe what these valid utterances actually mean.
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%%ANKI
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Basic
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What two parts make up a formal language?
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Back: An alphabet and a formal grammer.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534134-->
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END%%
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%%ANKI
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Basic
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What is the purpose of a formal language's alphabet?
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Back: It defines the set of valid symbols used in the language.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534138-->
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END%%
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%%ANKI
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Basic
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What is the purpose of a formal language's grammar?
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Back: It defines how to construct well-formed formulas from the alphabet.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534142-->
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END%%
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%%ANKI
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Basic
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What is WFF an acronym for?
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Back: **W**-ell-**f**ormed **f**ormula.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534147-->
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END%%
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%%ANKI
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Basic
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Is the following a WFF of propositional logic? $(a \Rightarrow b) \Leftrightarrow c$
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Back: Yes.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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Tags: formal-system::propositional
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<!--ID: 1721561534150-->
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END%%
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%%ANKI
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Basic
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Is the following a WFF of propositional logic? $(a \Rightarrow b( \Leftrightarrow c$
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Back: No.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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Tags: formal-system::propositional
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<!--ID: 1721561574169-->
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END%%
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%%ANKI
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Basic
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Is the following a WFF of propositional logic? $\forall x, x \in A \Rightarrow x = y$
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Back: No.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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Tags: logic::propositional logic::predicate
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<!--ID: 1722075143977-->
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END%%
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%%ANKI
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Basic
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With respect to formal languages, what is syntax?
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Back: The set of possible expressions that are valid in the language.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534154-->
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END%%
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%%ANKI
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Basic
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With respect to formal languages, what is semantics?
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Back: What valid expressions of the language mean.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534157-->
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END%%
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A deductive system can be further subcategorized as either a [[formal-system/proof-system/index|proof system]] or a [[formal-system/logical-system/index|logical system]]. In both cases, the general principle is to define a (possibly empty) set of **axioms** alongside a set of **inference rules** that together can be used to infer **theorems**.
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%%ANKI
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Basic
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How are deductive systems further subcategorized?
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Back: As proof systems and logical systems.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534160-->
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END%%
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%%ANKI
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Basic
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Axioms belong to which of the two parts of a formal system?
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Back: The deductive system.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534163-->
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END%%
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%%ANKI
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Basic
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Inference rules belong to which of the two parts of a formal system?
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Back: The deductive system.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534167-->
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END%%
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%%ANKI
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Basic
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An alphabet belongs to which of the two parts of a formal system?
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Back: The formal language.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534170-->
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END%%
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%%ANKI
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Basic
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A formal grammar belongs to which of the two parts of a formal system?
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Back: The formal language.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534173-->
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END%%
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%%ANKI
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Cloze
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A deductive system derives {1:theorems} using {1:inference rules} starting with {1:axioms}.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534176-->
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END%%
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%%ANKI
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Cloze
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{1:Proof} systems are {2:syntactic} whereas {2:logical} systems are {1:syntactic and semantic}.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534180-->
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END%%
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%%ANKI
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Cloze
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A {metalanguage} refers to the {language used to talk *about* a formal system}.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534185-->
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END%%
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%%ANKI
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Cloze
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An {object} language refers to the {formal language found in a formal system}.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534189-->
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END%%
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%%ANKI
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Basic
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In a discussion of propositional logic, what does the metalanguage refer to?
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Back: The natural language being used to describe propositional logic.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534194-->
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END%%
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%%ANKI
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Basic
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In a discussion of propositional logic, what does the object language refer to?
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Back: The formal language of propositional logic.
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Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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<!--ID: 1721561534200-->
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END%%
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## Bibliography
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* “Deductive Reasoning,” in _Wikipedia_, June 16, 2024, [https://en.wikipedia.org/w/index.php?title=Deductive_reasoning](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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* “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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* “Inductive Reasoning,” in _Wikipedia_, May 6, 2024, [https://en.wikipedia.org/w/index.php?title=Inductive_reasoning](https://en.wikipedia.org/w/index.php?title=Inductive_reasoning&oldid=1222455892). |