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title TARGET DECK FILE TAGS tags
Logic Obsidian::STEM formal-system
formal-system
logic

Overview

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.

%%ANKI Basic How are conclusions drawn in deductive reasoning? Back: From valid inferences on a set of premises. Reference: “Deductive Reasoning,” in Wikipedia, June 16, 2024, https://en.wikipedia.org/w/index.php?title=Deductive_reasoning.

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%%ANKI Basic How are conclusions drawn in inductive reasoning? Back: By generalizing from some body of observations. Reference: “Inductive Reasoning,” in Wikipedia, May 6, 2024, https://en.wikipedia.org/w/index.php?title=Inductive_reasoning.

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%%ANKI Basic Do formal systems employ inductive or deductive reasoning? Back: Deductive reasoning. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic Is the study of logic concerned with deductive or inductive reasoning? Back: Deductive reasoning. Reference: “Deductive Reasoning,” in Wikipedia, June 16, 2024, https://en.wikipedia.org/w/index.php?title=Deductive_reasoning.

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%%ANKI Basic Which of deductive or inductive reasoning can provide "genuinely new information"? Back: Inductive reasoning. Reference: “Deductive Reasoning,” in Wikipedia, June 16, 2024, https://en.wikipedia.org/w/index.php?title=Deductive_reasoning.

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%%ANKI Basic What does it mean for inductive reasoning to provide "genuinely new information"? Back: Information not found in the premises can be produced. Reference: “Deductive Reasoning,” in Wikipedia, June 16, 2024, https://en.wikipedia.org/w/index.php?title=Deductive_reasoning.

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%%ANKI Basic Which of deductive or inductive reasoning is considered more formal? Back: Deductive reasoning. Reference: “Deductive Reasoning,” in Wikipedia, June 16, 2024, https://en.wikipedia.org/w/index.php?title=Deductive_reasoning.

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%%ANKI Basic What two parts make up a formal system? Back: A formal language and a deductive system. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic Which of a formal system or a deductive system is defined in terms of the other? Back: The formal system. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic What does a formal system specify that a deductive system does not? Back: A formal language. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic What does a formal system specify that a formal language does not? Back: A deductive system. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic The term "formal system" can be seen as a portmanteau of what other terms? Back: A formal language and a deductive system. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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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.

%%ANKI Basic What two parts make up a formal language? Back: An alphabet and a formal grammer. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic What is the purpose of a formal language's alphabet? Back: It defines the set of valid symbols used in the language. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic What is the purpose of a formal language's grammar? Back: It defines how to construct well-formed formulas from the alphabet. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic What is WFF an acronym for? Back: W-ell-formed formula. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic Is the following a WFF of propositional logic? Back: (a \Rightarrow b) \Leftrightarrow c Back: Yes. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system. Tags: formal-system::propositional

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%%ANKI Basic Is the following a WFF of propositional logic? Back: (a \Rightarrow b( \Leftrightarrow c Back: No. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system. Tags: formal-system::propositional

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%%ANKI Basic Is the following a WFF of propositional logic? Back: \forall x, x \in A \Rightarrow x = y Back: No. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system. Tags: logic::propositional logic::predicate END%%

%%ANKI Basic With respect to formal languages, what is syntax? Back: The set of possible expressions that are valid in the language. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic With respect to formal languages, what is semantics? Back: What valid expressions of the language mean. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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A deductive system can be further subcategorized as either a formal-system/proof-system/index or a formal-system/logical-system/index. 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.

%%ANKI Basic How are deductive systems further subcategorized? Back: As proof systems and logical systems. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic Axioms belong to which of the two parts of a formal system? Back: The deductive system. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic Inference rules belong to which of the two parts of a formal system? Back: The deductive system. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic An alphabet belong to which of the two parts of a formal system? Back: The formal language. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic A formal grammar belong to which of the two parts of a formal system? Back: The formal language. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Cloze A deductive system derives {1:theorems} using {1:inference rules} starting with {1:axioms}. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Cloze {1:Proof} systems are {2:syntactic} whereas {2:logical} systems are {1:syntactic and semantic}. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Cloze A {metalanguage} refers to the {language used to talk about a formal system}. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Cloze An {object language} refers to the {formal language found in a formal system}. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic In a discussion of propositional logic, what is the metalanguage? Back: English (or whatever natural language is being used). Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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%%ANKI Basic In a discussion of propositional logic, what is the object language? Back: The formal language of propositional logic. Reference: “Formal System,” in Wikipedia, May 10, 2024, https://en.wikipedia.org/w/index.php?title=Formal_system.

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Bibliography