---
title: Logic
TARGET DECK: Obsidian::STEM
FILE TAGS: formal-system
tags:
  - 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
Cloze
{Deductive} reasoning begins with {a set of premises} and draws conclusions by {applying valid inferences}.
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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%%ANKI
Cloze
{Inductive} reasoning begins with {a body of observations} and draws conclusions by {generalizing the observations}.
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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%%ANKI
Cloze
Deduction is to {premises} whereas induction is to {observations}.
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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%%ANKI
Cloze
Deduction is to {valid inferences} whereas induction is to {generalizations}.
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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%%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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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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](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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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](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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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](https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&oldid=1229329170).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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%%ANKI
Basic
What is WFF an acronym for?
Back: **W**-ell-**f**ormed **f**ormula.
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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%%ANKI
Basic
Is the following a WFF of propositional logic? $(a \Rightarrow b) \Leftrightarrow c$
Back: Yes.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
Tags: formal-system::propositional
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%%ANKI
Basic
Is the following a WFF of propositional logic? $(a \Rightarrow b( \Leftrightarrow c$
Back: No.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
Tags: formal-system::propositional
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%%ANKI
Basic
Is the following a WFF of propositional logic? $\forall x, x \in A \Rightarrow x = y$
Back: No.
Reference: “Formal System,” in _Wikipedia_, May 10, 2024, [https://en.wikipedia.org/w/index.php?title=Formal_system](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
Tags: logic::propositional logic::predicate
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%%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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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**.

%%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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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%%ANKI
Basic
An alphabet belongs 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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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END%%

%%ANKI
Basic
A formal grammar belongs 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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
<!--ID: 1721561534173-->
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
<!--ID: 1721561534185-->
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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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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%%ANKI
Basic
In a discussion of propositional logic, what does the metalanguage refer to?
Back: The natural language being used to describe propositional logic.
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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%%ANKI
Basic
In a discussion of propositional logic, what does the object language refer to?
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](https://en.wikipedia.org/w/index.php?title=Formal_system&oldid=1223254138).
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## Bibliography

* “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).
* “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).
* “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).