3: Deductive Logic I - Aristotelian Logic
- Page ID
- 24333
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
In this chapter and the next we will study two deductive logics—two approaches to evaluating deductive arguments. The first, which is the subject of the present chapter, was developed by Aristotle nearly 2,500 years ago, and we’ll refer to it simply as Aristotelian Logic; the second, the subject of the next chapter, has roots nearly as ancient as Aristotle’s but wasn’t fully developed until the 19th century, and is called Sentential logic.
- 3.1: Deductive Logics
- In this chapter and the next we will study two deductive logics—two approaches to evaluating deductive arguments. The first, which is the subject of the present chapter, was developed by Aristotle nearly 2,500 years ago, and we’ll refer to it simply as Aristotelian Logic; the second, the subject of the next chapter, has roots nearly as ancient as Aristotle’s but wasn’t fully developed until the 19th century, and is called Sentential logic.
- 3.2: Classes and Categorical Propositions
- For Aristotle, the fundamental logical unit is the class. Classes are just sets of things—sets that we can pick out using language. The simplest way to identify a class is by using a plural noun: trees, clouds, asteroids, people—these are all classes. Names for classes can be grammatically more complex, too. We can modify the plural noun with an adjective: ‘rich people’ picks out a class.
- 3.3: The Square of Opposition
- The four types of categoricals are related to one another in systematic ways; we will look at those relationships. The relationships are inferential: we can often infer, for example, from the truth of one of the four categoricals, whether the other three are true or false. These inferential relationships among the four categorical propositions are summarized graphically in the Square of Opposition.
- 3.4: Operations on Categorical Sentences
- We continue our exploration of the portion of natural language to which Aristotle’s logic restricts itself—the standard form sentences expressing categorical propositions. To familiarize ourselves more intimately with these, we will look at how they respond when we perform various operations on them, when we manipulate them in various ways. We will examine three operations: conversion, obversion, and contraposition. Each of these alters the standard form sentences in some way.
- 3.5: Problems with the Square of Opposition
- The Square of Opposition is an extremely useful tool: it neatly summarizes, in graphical form, everything we know about the relationships among the four types of categorical proposition.
- 3.6: Categorical Syllogisms
- As we’ve said, Aristotelian Logic limits itself to evaluating arguments all of whose propositions—premises and conclusion—are categorical. There is a further restriction: Aristotelian Logic only evaluates categorical syllogisms. These are a special kind of argument, meeting the following conditions: A categorical syllogism is a deductive argument consisting of three categorical propositions (two premises and a conclusion).
Thumbnail: Bust of Aristotle. Marble, Roman copy after a Greek bronze original by Lysippos from 330 BC; the alabaster mantle is a modern addition. (Public domain via Wikipedia)