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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An accessible resource to help you puzzle out logic concepts
Logic For Dummies, 2nd Edition explains all the varied ways we use logic in philosophy, science, and everyday life. College students taking a logic course and lifelong learners alike can benefit from this accessible guide to logic concepts—such as syllogisms, constructing proofs and refutations, propositional and predicate logic, symbolic logic, modal and fuzzy logic, deductive and inductive reasoning, and beyond. With real-world examples, fun logic problems, and fully worked out proofs, you have plenty of opportunities to follow along and apply what you've learned. Logic For Dummies, 2nd Edition helps you connect the logical dots!
Inside:
- Grasp formal and informal logic with clear explanations and practice problems
- See how logic shows up in everyday life and discover how to spot logical fallacies
- Work through logical proofs and refutations, with answer explanations to strengthen your understanding
- Sharpen your ability to reason through complex issues with truth tables
Logic For Dummies, 2nd Edition is the logical choice for anyone who wants or needs to learn the concepts, theories, and formulas of logical reasoning.
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Logic For Dummies®
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Introduction
About This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: Getting Started with Logic
Chapter 1: Taking a Logical Perspective
Getting an Overview of Practical Logic
Building and Evaluating Logical Arguments
Making Logical Conclusions Simple with the Laws of Thought
Combining Logic and Math
Chapter 2: Following Logical Developments from Aristotle to AI
Classical Logic — from Aristotle to the Enlightenment
Modern Logic — the 17th, 18th, and 19th Centuries
Logic in the 20th Century and Beyond
Chapter 3: Just for the Sake of Argument
Defining Logic as it Relates to Arguments
Studying Examples of Arguments
Thinking Again: Logical Fallacies
Understanding What Logic Is and Isn’t
Whose Logic Is It, Anyway?
Part 2: Flourishing with Formal Sentential Logic
Chapter 4: Engaging with Formal Affairs
Observing the Formalities of Sentential Logic
Deploying the Five SL Operators
Seeing How SL Is Like Simple Arithmetic
(Not) Getting Lost in Translation
Chapter 5: Embracing the Value of Evaluation
Evaluating Is the Bottom Line
Making a Statement
Recognizing the Eight Forms of SL Statements
Revisiting Evaluation
Chapter 6: Turning the Tables: Evaluating Statements with Truth Tables
Putting It All on the Table: The Joy of Brute Force
Constructing Baby’s First Truth Table
Putting Truth Tables to Work
Building New Tautologies and Contradictions
Chapter 7: Taking the Easy Way Out: Creating Quick Tables
Dumping the Truth Table for a New Friend: the Quick Table
Outlining the Quick Table Process
Planning Your Strategy
Working Smarter (Not Harder) with Quick Tables
Chapter 8: Knowing How Truth Grows on Trees
Understanding How Truth Trees Work
Showing Consistency or Inconsistency
Testing for Validity or Invalidity
Separating Tautologies, Contradictions, and Contingent Statements
Checking for Semantic Equivalence or Inequivalence
Part 3: Exploring Proofs, Syntax, and Semantics in SL
Chapter 9: Providing Proof for What You Propose
Bridging the Premises-Conclusion Divide
Engaging the Eight Implication Rules in SL
The If-Rules: Modus Ponens and Modus Tollens
The And-Rules: Conjunction and Simplification
The Or-Rules: Addition and Disjunctive Syllogism
The Double-If-Rules: Hypothetical Syllogism and Constructive Dilemma
Chapter 10: Applying Equivalence Rules
Distinguishing Implications from Equivalences
Discovering the Ten Valid Equivalences
Chapter 11: Making Big Assumptions with Conditional and Indirect Proof
Conditioning Your Premises with Conditional Proof
Thinking Indirectly: Verifying Arguments with Indirect Proof
Combining Conditional and Indirect Proof
Chapter 12: Polishing Off Your Proofs
Easy Proofs: Taking a Gut Approach
Moderate Proofs: Recognizing When to Use Conditional Proof
Difficult Proofs: Figuring Out What to Do When the Going Gets Tough
Chapter 13: Putting Operators in Their Place
Making Do with the Five SL Operators
Downsizing the Operators — a Semi-true Story
Chapter 14: Syntactical Maneuvers and Semantic Considerations
Working WFF Us or Against Us
Comparing SL to Boolean Algebra
Part 4: Questing After Quantification Logic
Chapter 15: Expressing Quantity with Quality: Quantification Logic
Taking a Quick Look at Quantification Logic
Expressing Quantity with Two New Operators
Picking Out Statements and Statement Forms
Chapter 16: Embracing QL Translations
Translating the Four Basic Forms of Categorical Statements
Discovering Alternative Translations of the Basic Forms
Identifying Statements in Disguise
Chapter 17: Proving Arguments with QL
Applying SL Rules in QL
Transforming Statements with Quantifier Negation
Exploring the Four Quantifier Rules
Chapter 18: Expanding Logical Relations
Relating to QL Relations
Identifying Constants with Identities
Chapter 19: Planting a Quantity of Truth Trees
Applying Your Truth Tree Knowledge to QL
Reaching for the Sky: Non-Terminating Trees
Part 5: Meditating on Modern Developments in Logic
Chapter 20: Computer Logic
Remembering the Early Versions of Computers
Decoding the Modern Age of Computers
Ushering in Artificial Intelligence
Chapter 21: Sporting Propositions: Non-classical Logic
Opening Up to Possibility
Getting into a New Modality
Taking Logic to a Higher Order
Moving Beyond Consistency
Making Logical Sense of Quantum Logic
Chapter 22: Paradox and Axiomatic Systems
Grounding Logic in Set Theory
Discovering the Axiomatic System for SL
Proving Consistency and Completeness
Examining Gödel’s Incompleteness Theorem
Pondering the Meaning of It All
Part 6: The Part of Tens
Chapter 23: Ten (or So) Quotes about Logic
Chapter 24: Ten Big Names in Logic
Aristotle (384–322 BCE)
Peter Abelard (1079–1142)
Gottfried Leibniz (1646–1716)
George Boole (1815–1864)
Georg Cantor (1845–1918)
Gottlob Frege (1848–1925)
David Hilbert (1862–1943)
Bertrand Russell (1872–1970)
Kurt Gödel (1906–1978)
Alan Turing (1912–1954)
Chapter 25: Ten Tips for Passing a Logic Exam
Breathe
Start by Glancing over the Whole Exam
Warm Up with an Easy Problem First
Fill In Truth Tables Column by Column
If You Get Stuck on a Proof, Jot Down Everything
If You REALLY Get Stuck on a Proof, Move On
If Time Is Short, Finish the Tedious Stuff
Check Your Work
Admit Your Mistakes
Stay Focused Until the Bitter End
Index
About the Author
Connect with Dummies
End User License Agreement
List of Tables
Chapter 2
TABLE 2-1 The Square of Oppositions
Chapter 3
TABLE 3-1 The Cans and Cannots of Logic
Chapter 4
TABLE 4-1 The Five Logical Operators
Chapter 5
TABLE 5-1 The Eight Forms of SL Statements
Chapter 6
TABLE 6-1 Number of Constants and Rows in a Truth Table
TABLE 6-2 Truth Table Tests for a Variety of Logical Conditions
TABLE 6-3 Building New Tautologies and Contradictions
Chapter 14
TABLE 14-1 Corresponding Symbols in SL and Boolean Algebra
TABLE 14-2 Comparing SL and Boolean Algebra
TABLE 14-3 Properties Common to Boolean Algebra and Arithmetic (and All Other Se...
Chapter 16
TABLE 16-1 Translations of the Four Basic Forms of Categorical Statements
TABLE 16-2 Alternative Translations of the Four Basic Forms of Categorical State...
Chapter 17
TABLE 17-1 The Four QN Rules
TABLE 17-2 Four Equivalent Ways to Write
All
and
Not All
Statements
TABLE 17-3 Four Equivalent Ways to Write
Some
and
No
Statements
TABLE 17-4 The Four Quantifier Rules in QL and Their Limitations
List of Illustrations
Chapter 3
FIGURE 3-1: A look at a logic tree can help you tell where your arguments fall.
Chapter 7
FIGURE 7-1: The six easiest types of SL statements.
FIGURE 7-2: Four not-so- easy types of SL statements.
FIGURE 7-3: The six difficult types of SL statements.
Chapter 8
FIGURE 8-1: The eight types of SL statements with decompositions.
Chapter 21
FIGURE 21-1: Compromising on the cost of a television.
FIGURE 21-2: A mind-bending shell game.
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
About the Author
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Logic For Dummies®, 2nd Edition
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Introduction
You use logic every day, and I bet you don’t even realize it. For instance, consider these examples of common situations in which you may use logic:
Planning an evening out with a friend
Asking your boss for a day off or for a raise
Picking out a shirt to buy among several that you like
Explaining to your kids why homework comes before TV
All these scenarios require you to use logic to clarify your thinking and get other people to see things from your perspective.
Even if you don’t always act on it, logic is natural, at least to humans. And logic is one of the big reasons humans have lasted so long on a planet filled with lots of other creatures that are bigger, faster, more numerous, and more ferocious. Because logic is already a part of your life, after you notice it, you’ll see it working (or not working) everywhere you look.
In this book, I show you how logic arises naturally in daily life. When you recognize this, you can figure out how to refine certain types of thinking down to their logical essence. Logic gives you the tools for working with what you already know (the premises) to get you to the next step (the conclusion). It’s also great for helping you spot flaws in arguments, such as unsoundness, hidden assumptions, or just plain unclear thinking.
About This Book
Logic has been around a long time — almost 2,400 years and counting! The ancient Greek philosopher Aristotle was the first to systematically study logic and attempt to formalize related ideas, and many later philosophers followed his lead. With so many people (past and present) thinking and writing about logic, you may find it difficult to know where to begin. But never fear, I wrote this book with beginners in mind.
If you’re taking an introductory course in logic, you can supplement your knowledge with the information in this book. I explain just about everything you’re studying in class simply, with lots of step-by-step examples. At the same time, if you’re just interested in understanding what logic is all about, this book is a great place to start.
Logic is one of the few areas of study taught in two different college departments: math and philosophy. The reason it can fit into two seemingly different categories has a historical basis:
Its philosophical origin:
Logic began with Aristotle and has been developed by philosophers for centuries.
Its transition to computation:
About 150 years ago, mathematicians found that logic was an essential tool for grounding their work as foundational math became more and more abstract.
One of the most important results of this overlap is formal logic, which takes ideas from philosophical logic and applies them in a mathematical framework. Formal logic is usually taught in philosophy departments as a purely computational (that is, mathematical) exercise.
When writing this book, I tried to balance both of these aspects of logic. Generally speaking, the book begins where logic began — with philosophy — and ends where it transitioned — in mathematics.
To help you navigate throughout this book, I use the following conventions:
Italics
are used for emphasis and to highlight new words and terms defined in the text. They’re also used for constants and variables in equations.
Boldface
text indicates keywords in bulleted lists and also true (
T
) and false (
F
) values in equations and tables. It’s also used for the 18 rules of inference in sentential logic (SL) and the 5 rules of inference in quantification logic (QL).
Sidebars are shaded gray boxes that contain text that’s interesting but not critical to your understanding of the chapter or topic.
Twelve-point boldface text (
T
and
F
) is used in running examples of truth tables and quick tables to indicate information that’s just been added. It’s used in completed truth tables and quick tables to indicate the truth value of the entire statement.
Parentheses are used throughout statements, instead of a combination of parentheses, brackets, and braces. Here’s an example:
Foolish Assumptions
Logic For Dummies is for anybody who wants to know about logic: what it is, where it came from, why it was invented, and even where it may be going. In this book, I give you an overview of logic in its many forms and provide you with a solid base of knowledge to build upon.
Here are a few things I assume about you because of your interest in this book:
You want to find out more about logic, whether you’re taking a course or just curious.
You can distinguish between true and false statements about commonly known facts, such as “George Washington was the first U.S. president,” and “The Statue of Liberty is in Tallahassee.”
You understand simple arithmetic.
You can grasp simple algebra, such as solving for
x
in the equation .
Icons Used in This Book
Throughout this book, you find four icons that highlight different types of information:
I use this icon to point out the key ideas you need to know. Make sure you understand the information in these paragraphs before reading on.
This icon highlights helpful hints that show you the easy way to get things done. Try them out, especially if you’re enrolled in a logic course.
Don’t skip these icons! They draw your attention to common errors you want to avoid and help you recognize where logic traps are hiding so you don’t take a wrong step and get caught.
This icon alerts you to interesting, but unnecessary, trivia that you can read or skip over as you like.
Beyond the Book
This book comes with a free access-anywhere Cheat Sheet that includes a quick summary of logic basics, including logical operators, truth tables, implication rules, equivalence rules, and more. To get this Cheat Sheet, simply go to www.dummies.com and type Logic For Dummies Cheat Sheet in the Search box.
Where to Go from Here
Feel free to skip around in this book as you like. When I discuss a new topic that depends on more basic ideas, I refer to the chapter where I introduced those basics. If you only need info on a certain topic, check out the Index or the Table of Contents — they’ll point you in the right direction.
If you have some background in logic and already have a handle on the Part 1 stuff, you can jump forward to where the action is. Each part builds on the previous parts, so if you can understand the material in Part 3, you probably don’t need to concentrate on Parts 1 and 2 (unless, of course, you just want a little review).
If you’re taking a logic course, you may want to read Parts 3 and 4 carefully. You may even try to reproduce the proofs in those chapters with the book closed. Better to find out what you don’t know while you’re studying than while you’re sweating out an exam!
If you forge ahead to Parts 4 and 5, you’re probably ready to tackle some fairly advanced ideas. If you’re itching to get to some meaty logic, check out Chapter 22. This chapter on logical paradoxes has some really cool stuff to take your thinking to warp speed. Bon voyage!
Part 1
Getting Started with Logic
IN THIS PART …
See how you constantly use logic to turn the facts at hand into a better understanding of the world.
Get a perspective on the history of logic, with a look at various theories that have developed over the centuries.
Understand the basic structure of a logical argument, focus on key concepts such as the premises and the conclusion, and discover how to identify and counter a variety of logical fallacies.
Chapter 1
Taking a Logical Perspective
IN THIS CHAPTER
Seeing the world from a logical point of view
Using logic to build valid arguments
Applying the laws of thought
Understanding the connections between math and logic
You and I live in an illogical world. If you doubt this fact, just watch a few YouTube videos. Or really listen to the person sitting at the next barstool. Or, better yet, spend the weekend with your in-laws.
With so many people thinking and acting illogically, why should you be any different? Wouldn’t it be more sensible just to be as illogical as the rest of the human race?
Well, okay, being illogical on purpose is probably not the best idea. For one thing, how can trying to be illogical possibly be sensible? For another, if you’ve picked up this book in the first place, you probably aren’t built to be illogical. Let’s face it — some folks thrive on chaos (or claim to), and others don’t.
In this chapter, I introduce you to the basics of logic and how it applies to life. I tell you about a few words and ideas that are key to logic. And I touch briefly on the connections between logic and math.
Getting an Overview of Practical Logic
Whether you know it or not, you already understand a lot about logic. In fact, you already have a built-in logic detector. Don’t believe me? Take this quick test to see whether you’re logical:
Q:
How many pancakes does it take to shingle a doghouse?
A:
23, because bananas don’t have bones.
If the answer seems illogical to you, that’s a good sign that you’re at least on your way to being logical. Why? Simply put, if you can spot something that’s illogical, you must have a decent sense of what’s actually logical.
In this section, I start with what you already understand about logic (though you may not be aware of that knowledge) and build toward a foundation that can help you in your study of logic.
Bridging the gap from here to there
Most children are naturally curious. They constantly want to know why everything is the way it is. And for every because answer they receive, they have one more why question. For example, consider these common kid questions:
Why does the sun rise in the morning?
Why do I have to go to school?
Why does the car start when you turn the key?
Why do people break the law when they know they could go to jail?
When you think about it, every great mystery seeks a solution via these types of questions: Even when the world doesn’t make sense on its own, why does it feel like it should?
Kids sense from an early age that even though they don’t understand something — like why the sun rises in the morning — the answer must be somewhere. And they think, “If I’m here and the answer is somewhere else, what do I have to do to get there?” (Often, their determination to get there leads them to bug their parents with more questions.)
Human desire to get from here to there — from ignorance to understanding — is one of the main reasons logic came into existence as philosophy and science. Logic grew out of an essential human need to make sense of the world and, as much as possible, gain some control over it.
Understanding cause and effect
One way to understand the world is to notice the connection between cause and effect.
As you grow from a child to an adult, you begin to piece together how one event causes another. Typically, you can express the connections between cause and effect by using an if-statement (which is a basic conditional structure in logic). For example, consider these if-statements:
If
I let my favorite ball roll under the couch,
then
I can’t reach it.
If
I do all my homework before Dad comes home,
then
he’ll play catch with me before dinner.
If
I practice on my own this summer,
then
the coach will pick me for the team in the fall.
If
I keep showing up at job interviews prepared and with confidence,
then
I’ll eventually get a job.
Understanding how if-statements work is an important aspect of logic.
Breaking down if-statements
Every if-statement is made up of two smaller statements called sub-statements: The antecedent, which follows the word if, and the consequent, which follows the word then. For example, consider this if-statement:
If it is 5 p.m., then it’s time to go home.
In this if-statement, the antecedent is the sub-statement it is 5 p.m. The consequent is the sub-statement it’s time to go home.
Every sub-statement can stand as a complete statement in its own right.
Stringing if-statements together
In many cases, the consequent of one if-statement becomes the antecedent of another. When this happens, you get a string of consequences, which the Greeks called a sorites (suh-rye-tease). For example:
In this case, you can link these if-statements together to form a new if-statement:
If it’s 5 p.m., then I need to call my husband to make reservations at the restaurant.
Thickening the plot
As you gain more life experience, you may find that the connections between cause and effect become more and more sophisticated:
If
I let my favorite ball roll under the couch,
then
I can’t reach it,
unless
I scream so loud that Grandma gets it for me,
though if
I do that more than once,
then
she gets annoyed and puts me back in my highchair.
If
I practice on my own this summer
but
not so hard that I blow my knees out,
then
the coach will pick me for the team in the fall
only if
there’s a position open,
but if
I do
not
practice,
then
the coach will
not
pick me.
Knowing everything and more
As you begin to understand the world and what you find in it, you begin to make more general statements about it. For example:
All horses are friendly.
All 5-year-olds are silly.
Every teacher at that school is out to get me.
Every time I hear a phone ring in our house, it’s my sister’s phone.
Words like all and every allow you to categorize things into sets (groups of objects) and subsets (groups within groups). For example, when you say, “All horses are friendly,” you mean that the set of all horses is contained within the set of all friendly things — that is, horses are a subset of friendly things.
Explaining existence itself
You also discover connections within the world by figuring out what exists and doesn’t exist. For example:
Some of my teachers are nice.
There is at least one student in school who likes me.
No one in the chess club can beat me.
There is no such thing as a Martian.
Existence statements like these generally use wording that points to connections (or not):
An intersection of sets:
Words like
some
,
there is
, and
there exists
show an overlapping of sets called an
intersection.
For example, when you say, “Some of my teachers are nice,” you mean that there’s an intersection between the set of your teachers and the set of nice things.
No intersection between sets:
Words like
no
,
there is no
, and
none
show that there’s no intersection between sets. For example, when you say, “No one in the chess club can beat me,” you mean that there’s no intersection between the set of all the chess club members and the set of all the chess players who can beat you.
Identifying a few logical words
As you can see, certain words show up a lot as you begin to make logical connections. Some of these common words are:
if … then
and
but
or
not
unless
though
every
all
only if
each
there is
there exists
some
there is no
none
Taking a closer look at words like these is an important step in the development of logic. When you do, you begin to see how these words enable you to divide and categorize the world in different ways (and therefore understand it better).
Building and Evaluating Logical Arguments
When people say “Let’s be logical” about a given situation or problem, they often mean “Let’s follow steps like these”:
Figure out what we know to be true.
Spend some time thinking about it.
Find the best course of action.
In logical terms, this three-step process involves building a logical argument — an explanation or rationale that contains a set of premises at the beginning and a conclusion at the end. In many cases, the premises and the conclusion will be linked by a series of intermediate steps. In the following sections, I discuss the steps in the order you’re likely to encounter them.
Generating premises
The premises are the facts of the matter: the statements that you know or believe to be true. In many situations, writing down a set of premises is a great first step to problem-solving.
For example, suppose you’re a school board member trying to decide whether to support the construction of a new school that would open for the school term that begins in September. Everyone is excited about the project, but you make some phone calls and piece together your facts to construct the following premises:
The funds for the project won’t be available until March.
The construction company won’t begin work until they receive payment.
The entire project will take at least eight months to complete.
So far, you have only a set of three premises. But when you put these premises together, you get closer to the conclusion, which will complete your logical argument. In the next section, I show you how to combine the premises.
Bridging the gap with intermediate steps
Sometimes a logical argument is just a set of premises followed by a conclusion. In many cases, however, an argument also includes intermediate steps that show how the premises gradually lead to that conclusion.
Using the school construction example from the previous section, you may want to spell things out like this:
According to the information we have, we won’t be able to pay the construction company until March, so they won’t be done until at least eight months later, which is November. But the school term begins in September. Therefore …
The word therefore indicates a conclusion and is the beginning of the final step, which I discuss in the next section.
Forming a conclusion
The conclusion is the outcome of your logical argument. If you’ve worked through the intermediate steps in a clear progression, the conclusion should be fairly obvious. For the school construction example I’ve been using, here it is:
The building won’t be complete before the school term begins in September.
If the conclusion doesn’t make sense, something may be wrong with your argument. In some cases, an argument may not be valid. In others, you may have missing premises that you’ll need to add.
Deciding whether the argument is valid
After you’ve built a logical argument, you need to be able to decide whether it’s valid, which means that it’s a good argument.
To test a logical argument’s validity, you assume that all the premises are true and then see whether the conclusion follows automatically from them. If the conclusion follows automatically, you know it’s a valid argument. If not, the argument is invalid.
Understanding enthymemes (assumptions)
The school construction argument (from the example in the previous sections) may seem valid, but you also may have a few doubts. For example, if another source of funding became available, the construction company could start earlier than March and perhaps finish by the September deadline. And so, the argument has a hidden premise, or assumption, called an enthymeme (en-thi-meem), as follows:
There is no other source of funds for the project.
Logical arguments about real-world situations (in contrast to mathematical or scientific arguments) almost always have enthymemes — that is, hidden assumptions that may go unacknowledged. So, the clearer you understand the enthymemes hidden in an argument, the better chance you have of making sure your argument is valid.
Uncovering hidden premises in real-world arguments is more closely related to rhetoric — the study of how to make clear and convincing arguments — than it is to logic. I touch upon rhetoric, logical fallacies, and other details about the structure of logical arguments in Chapter 3.
Making Logical Conclusions Simple with the Laws of Thought
As a basis for understanding logic, philosopher Bertrand Russell set down three laws of thought. These laws are all grounded in ideas dating back to Aristotle, who founded classical logic more than 2,300 years ago. (See Chapter 2 for more on the history of logic.)
All three laws of thought are basic and easy to understand. The important thing to note is that these laws enable you to draw logical conclusions about statements even if you aren’t familiar with the real-world circumstances they’re discussing.
The law of identity
The law of identity states that every individual thing is identical to itself.
For example:
Jason Sudeikis is Jason Sudeikis.
My cat, Ian, is my cat, Ian.
The Washington Monument is the Washington Monument.
Without any information about the world, you can see from logic alone that all these statements are true. The law of identity tells you that any statement in the form of X isX must be true. In other words, any individual thing in the universe is the same as itself.
The law of the excluded middle
The law of the excluded middle states that every statement is either true or false.
For example, consider these two statements:
My first name is Mark.
My first name is Algernon.
Again, without any information about the world, you know logically that each of these statements is either true or false. According to the law of the excluded middle, no third option is possible — in other words, statements can’t be partially true or false. Rather, in logic, every statement is either completely true or completely false.
As it happens, I’m content that the first statement is true and relieved that the second is false.
The law of non-contradiction
The law of non-contradiction states that given a statement and its opposite, one is true, and the other is false.
For example:
My first name is Algernon.
My first name is not Algernon.
Even if you don’t know my name, you can be sure from logic alone that one of these statements is true and the other is false. In other words, because of the law of non-contradiction, my first name can’t both be and not be Algernon.
Combining Logic and Math
Throughout this book, I often prove my points with examples that use math. (Don’t worry — none of my examples is more complicated than what you learned in fifth grade or before.) Math and logic go great together for two reasons, which I explain in the following sections.
Math is good for understanding logic
Throughout this book, as I explain logic to you, I sometimes need examples that are clearly true or false to prove my points. As it turns out, math examples are great for this purpose because, in math, a statement is always either true or false, with no gray area between.
On the other hand, sometimes random facts about the world may be more subjective, or up for debate. For example, consider these two statements:
Rihanna is an amazing singer.
Romeo and Juliet
is a lousy play.
Many people may well agree in this case that the first statement is true and the second is false, but both statements are up for debate. Now look at these two statements:
The number 7 is less than the number 8.
Five is an even number.
Clearly, there’s no disputing that the first statement is true and the second is false.
Logic is good for understanding math
As I discuss in the section “Making Logical Conclusions Simple with the Laws of Thought” earlier in this chapter, the laws of thought on which logic is based — such as the law of the excluded middle — depend on black-and-white thinking. And, well, not many subjects are more black-and-white than math. Even though you may find studying history, literature, politics, or the arts to be more fun, they contain many more shades of gray.
Math is built on logic like a house is built on a foundation. If you’re interested in the connection between math and logic, check out Chapter 22, which focuses on how math starts with obvious facts called axioms and then uses logic to form interesting and complex conclusions called theorems.
Chapter 2
Following Logical Developments from Aristotle to AI
IN THIS CHAPTER
Understanding the roots of logic
Examining classical and modern logic
Looking at 20th and 21st century developments in logic
When you think about how illogical humans can be, you may be surprised to discover how much they’ve developed the concepts of logic over the years. Here’s just a partial list of some varieties of logic that are floating around in the big world of premises and conclusions:
Boolean logic
Modern logic
Quantification logic
Classical logic
Multi-valued logic
Quantum logic
Formal logic
Non-classical logic
Sentential logic
Fuzzy logic
Predicate logic
Syllogistic logic
Informal logic
Propositional logic
Symbolic logic
As your eyes scan all these varieties of logic, you may feel a sudden urge to embrace your humanity fully and leave logic to the Vulcans. The good news, as you can soon discover, is that many types of logic are quite similar. After you’re familiar with a few of them, the rest become much easier to understand.
So, where did all these types of logic come from? Well, that’s a long story — in fact, it’s a story that spans more than 2,000 years. I know 2,000 years seems like quite a lot to cram into one chapter, but I guide you through only the most important details. So, get ready for your short history lesson.
Classical Logic — from Aristotle to the Enlightenment
The ancient Greeks had a hand in discovering just about everything, and logic is no exception. For example, Thales and Pythagoras applied logical argument to mathematics. Socrates and Plato applied similar types of reasoning to philosophical questions. But the true founder of classical logic was Aristotle.
When I talk about classical logic in this section, I’m referring to the historical period in which logic was developed, in contrast with modern logic, which I discuss later in the chapter. Classical logic, however, can also mean the most standard type of logic (which most of this book is about) in contrast with non-classical logic (which I discuss in Chapter 21). I try to keep the distinction clear as I go along.
Aristotle invents syllogistic logic
Before Aristotle (384–322 BCE), ancient scholars applied logical argument intuitively where appropriate in math, science, and philosophy. For example, given that all numbers are either even or odd, if you could show that a certain number wasn’t even, you knew, then, that it must be odd. The Greeks excelled at this divide-and-conquer approach. They regularly used logic as a tool to examine the world.
Aristotle, however, was the first to recognize that the tool itself could be examined and developed. In six writings on logic — later assembled as a single work called Organon, which means “tool” — he analyzed how a logical argument functions. Aristotle hoped that logic, under his new formulation, would serve as a tool of thought that would help philosophers understand the world better. The system he developed, based on step-by-step logical arguments called syllogisms, is also known as Aristotelian logic.
Aristotle considered the goal of philosophy to be scientific knowledge, and saw the structure of scientific knowledge as fundamentally logical. Using geometry as his model, he noted that science consisted of proofs, proofs of syllogisms (which I discuss later in this section), syllogisms of statements, and statements of terms. So, in Organon, he worked from the bottom upward: The first book, the Categories, deals with terms; the second, On Interpretation, with statements; the third, Prior Analytics, with syllogisms; and the fourth, Posterior Analytics, with proofs.
Prior Analytics, the third book in Organon series, delves directly into what Aristotle called syllogisms, which are argument structures that, by their very design, appear to be indisputably valid.
The idea behind the syllogism was simple — so simple, in fact, that it had been taken for granted by philosophers and mathematicians until Aristotle noticed it. In a syllogism, the premises and conclusions fit together in such a way that, once you accept the premises as true, you must accept that the conclusion is true as well — regardless of the content of the actual argument being made.
For example, here’s Aristotle’s most famous syllogism:
Premises:
All men are mortal.
Socrates is a man.
Conclusion:
Socrates is mortal.
The following argument is similar in form to the first. And it’s the form of the argument, not the content, that makes it indisputable. Once you accept the premises as true, the conclusion follows as equally true.
Premises:
All clowns are scary.
Bobo is a clown.
Conclusion:
Bobo is scary.
Categorizing categorical statements
Aristotle focused much of his attention on understanding what he called categorical statements, which are simply statements that talk about whole categories of objects or people. Furniture, chairs, birds, trees, red things, Avengers movies, and cities that begin with the letter T are all examples of categories.
In keeping with the law of the excluded middle (which I discuss in Chapter 1), everything is either in a particular category or not in it. For example, a red chair is in the category of furniture, chairs, and red things, but not in the category of birds, trees, Avengers movies, or cities that begin with the letter T.
Aristotle broke categorical statements down into the following two types:
Universal statements, which usually start with a word like all or every — words that tell you something about an entire category. Here’s an example of a universal statement:
All dogs are loyal.
This statement relates to two categories and tells you that everything in the category of dogs is also in the category of loyal things. You can consider this a universal statement because it tells you that loyalty is a universal quality of dogs.
Particular statements, which usually start with some or at least one — words or phrases that tell you about the existence of at least one example within a category. Here’s an example of a particular statement:
Some bears are dangerous.
This statement tells you that at least one item in the category of bears is also in the category of dangerous things. This statement is considered a particular statement because it tells you that at least one particular bear is dangerous.
Understanding the square of oppositions
The square of oppositions — a tool Aristotle developed for studying categorical statements — organizes the four basic forms of categorical statements that appear frequently in syllogisms. These four forms are based on the positive and negative forms of universal and particular statements.
Aristotle organized these four types of statements into a simple chart similar to Table 2-1. His most famous example was based on the observation that all humans are mortal. However, the example in the table is inspired by my sleeping cat.
TABLE 2-1 The Square of Oppositions
Positive Forms
Negative Forms
Universal Forms
A:All cats are sleeping.
There doesn’t exist a cat that isn’t sleeping.
No cats are not sleeping.
Every cat is sleeping.
E:No cats are sleeping.
All cats are not sleeping.
There isn’t a cat that is sleeping.
There doesn’t exist a sleeping cat.
Particular Forms
I: Some cats are sleeping.
Not all cats are not sleeping.
At least one cat is sleeping.
There exists a sleeping cat.
O: Not all cats are sleeping.
Some cats are not sleeping.
There is at least one cat that isn’t sleeping.
Not every cat is sleeping.
As you can see from the table, each type of statement expresses a different relationship between the category of cats and the category of sleeping things. In English, you can express each type of statement in a variety of ways. For example, under the positive forms for universal statements in Table 2-1, you find instances of the English wording that use a double negative to create a positive statement. I list a few of the many ways to state the same idea in the table, but many more are possible in each case.
Aristotle noticed relationships among all these types of statements. The most important of these relationships is the contradictory relationship between the statements that are diagonal from each other in the table. With contradictory pairs, one statement is true and the other false.
For example, look at the A and O statements in Table 2-1. Clearly, if every cat in the world is sleeping at the moment, then A is true and O is false; otherwise, the situation is reversed. Similarly, look at the E and I statements. If every cat in the world is awake, then E is true and I is false; otherwise, the situation is reversed.
If you’re wondering, the letters for the positive forms A and I supposedly come from the Latin word AffIrmo, which means “I affirm.” Similarly, the letters for the negative forms E and O are said to come from the Latin word nEgO, which means “I deny.” The source of these designations is unclear, but you can rule out Aristotle, who spoke Greek, not Latin.
Euclid’s axioms and theorems
Although Euclid (c. 325–265 BCE) wasn’t strictly a logician, his contributions to logic were undeniable.
Euclid is best known for his work in geometry, which is still called Euclidean geometry in his honor. His greatest contribution to this field was his logical organization of geometric principles into axioms and theorems.
Euclid began with five axioms (also called postulates) — true statements that he believed were simple and self-evident. From these axioms, he used logic to prove theorems — true statements that were more complex and not immediately obvious. In this way, he succeeded in proving the vast body of geometry that logically followed from the five axioms alone. Mathematicians still use his logical organization of statements into axioms and theorems. (For more on this topic, see Chapter 22.)
Euclid also used a logical method called indirect proof. In this method, you assume the opposite of what you want to prove and then show that this assumption leads to a conclusion that’s obviously incorrect.
For example, a detective in a murder mystery may reason in the following indirect way:
If the butler committed the murder, then he must have been in the house between 7 p.m. and 8 p.m. (which is when the murder occurred).
But witnesses saw him in the city 20 miles away during that hour, so he couldn’t have also been in the house.
Therefore, the butler didn’t commit the murder.
Indirect proof is also called proof by contradiction and reductio ad absurdum, which is Latin for “reduced to an absurdity.” (Flip to Chapter 11 for more about how to use indirect proof.)
Chrysippus and the Stoics
While Aristotle’s successors developed his work on the syllogistic logic of categorical statements, another Greek school of philosophy, the Stoics, took a different approach. They focused on conditional statements, which are statements that take the form of an if-statement. For example:
If clouds are gathering in the west, then it will rain.
Most notable among these logicians was Chrysippus (279–206 BCE). He examined arguments using statements that were in this if-statement. For example:
Premises:
If clouds are gathering in the west, then it will rain.
Clouds are gathering in the west.
Conclusion:
It will rain.
Certainly, you can find connections between the Aristotelian and the Stoic approaches. Both approaches focused on sets of premises containing statements that, when true, tended to fit together in a way that forced the conclusion to be true as well. But friction between the two schools of thought caused logic to develop in two separate paths for more than a century — though over time they merged into a unified discipline.
Medieval logic and the extension of Aristotelian logic
After Aristotle, medieval logic — which flourished between the 5th and 15th centuries — extended and refined Aristotelian logic. Thinkers such as Boethius, Avicenna, Thomas Aquinas, and William of Ockham played pivotal roles in shaping logical theories that influenced philosophy and religion. They adapted Aristotle’s ideas about reasoning and developed new ways to of ordering arguments, which helped lay the foundation for later advances in science and philosophy. Their work bridged classical and modern thought, keeping logical thinking alive in medieval education and beyond.
Boethius: Affirming Aristotelian logic
The Roman philosopher Boethius (c. 477–524) helped preserve and spread Aristotle’s ideas on logic to the medieval world. He translated and explained Aristotle’s works, such as Categories and On Interpretation, making them easier to understand. He also advanced on theories about how statements fit together in logical arguments. His book The Consolation of Philosophy, while not strictly about logic, used logical reasoning to explore deep philosophical questions (see the sidebar “Boethius and The Consolation of Philosophy”).
His contributions ensured that Aristotelian logic remained a cornerstone of education in the Middle Ages.
BOETHIUS AND THE CONSOLATION OF PHILOSOPHY
In The Consolation of Philosophy, Boethius applied logical reasoning to profound philosophical questions, such as
The nature of fortune and fate: He questioned why good people suffer while bad people prosper. Through logical argumentation, he reasoned that true happiness does not come from wealth, power, or fame but from inner virtue.The problem of evil: He examined how evil can exist in a world governed by a just and all-powerful God. He argued that evil people are actually powerless because they lack true goodness, which is the highest form of being.Divine foreknowledge and free will: He addressed whether God's knowledge of the future means that human actions are predetermined. He concluded that God exists outside of time and sees all events at once, which allows for free will while maintaining divine omniscience (universal knowledge).Avicenna: Expanding Aristotelian logic
A prominent Persian philosopher, Avicenna (c. 980–1037) expanded Aristotle’s logic by introducing ideas about necessity, possibility, and impossibility, which anticipated later developments in modal reasoning (see Chapter 21). He also worked on the relationship between subjects and predicates in statements, improving Aristotle’s system of logic. His major work, The Book of Healing, deeply influenced both Islamic and European medieval philosophy. Avicenna’s ideas helped shift logic toward a more systematic and scientific approach, and they influenced many later thinkers.
Peter Abelard: Going modal
Peter Abelard (1079–1142) was a French philosopher and pioneering medieval logician who helped move logic beyond traditional patterns and explore how statements and ideas really work. He studied how language creates meaning and how varying words and phrases can affect whether a statement is true or false. Abelard’s focus on semantics — how statements express truth and meaning (see Chapter 14) — made him a forerunner of modern ideas in logic and philosophy of language.
Abelard was also one of the first to explore modal logic — the logic of possibility and necessity (see Chapter 21), using words such as can and must — which laid the groundwork for later advances in that area. In his writings, especially Sic et Non, he compared conflicting views from earlier thinkers to show how clear thinking could solve problems.
Thomas Aquinas: Connecting faith and reason
Thomas Aquinas (1225–1274), an Italian philosopher and priest, combined Aristotle’s logic with Christian theology, forming a comprehensive scholastic method that became a major part of later medieval philosophy. His most important work, Summa Theologica, used logical arguments to explain religious beliefs, showing that faith and reason could work together. Aquinas also deepened the understanding of analogy and the interpretation of religious language, helping theologians use logical reasoning more effectively in discussing divine matters. His influence was long-lasting, shaping the way scholars used logic in religious and philosophical discussions.
William of Ockham: Relating language and thought
English philosopher and theologian William of Ockham (c. 1287–1347) is best known for Ockham’s (or Occam’s) razor, the idea that the simplest explanation is usually the best. He challenged unnecessary assumptions in logic and supported nominalism, the idea that general concepts (like tree or redness) are just names, not real things. In his book Summa Logicae, he redefined how logical statements connect and developed ideas about how language and thought relate. Ockham’s work influenced later philosophy, including the development of scientific reasoning.
Jean Buridan: Consequences and clear thinking
Jean Buridan (c. 1300 – 1358) was a French philosopher and central figure in late medieval logic. He is best known for formalizing the theory of consequences (how one statement can follow logically from another) and advancing Abelard’s work on modal logic (statements of possibility and necessity that use words such as can and must).
Building on earlier ideas from medieval thinkers, Buridan went beyond simple patterns like syllogisms to improve how people could understand what makes an argument valid. He explained how words can stand for different things depending on the situation that they describe; this explanation was an early version of ideas we now use in modern logic. (See Chapter 3 for a look at how the meaning of words can vary, resulting in a variety of logical fallacies.) In books such as Summulae de Dialectica, he stressed how logical language must keep arguments clear and make sure that the conclusions truly follow from the starting points.
Instructors used Buridan’s books for hundreds of years to teach students how to reason well. His clear thinking helped connect the old style of logic with the new ways of reasoning that developed in the modern era.
Modern Logic — the 17th, 18th, and 19th Centuries
In Europe, as the Age of Faith gradually gave way to the Age of Reason in the 16th and 17th centuries, thinkers became optimistic about finding answers to questions about the nature of the universe.
Even though scientists (such as Isaac Newton) and philosophers (such as René Descartes) continued to believe in God, they looked beyond church teachings to provide answers about how God’s universe operated. When they did, they found that many of the mysteries of the world — such as the fall of an apple or the motion of the moon in space — could be explained and predicted by using mathematics. With this surge in scientific thought, logic became a fundamental tool of reason.
Leibniz and the Renaissance
Gottfried Leibniz (1646–1716) was arguably the greatest logician of the Renaissance in Europe. Like Aristotle, Leibniz saw the potential for logic to become an indispensable tool for understanding the world. He was the first logician to take Aristotle’s work a significant step further by turning logical statements into symbols that could then be manipulated like numbers and equations. The result was the first crude attempt at symbolic logic.
In this way, Leibniz hoped logic would transform philosophy, politics, and even religion into pure calculation, providing a reliable method to answer all of life’s mysteries with objectivity. In a famous quote from The Art of Discovery (1685), he says:
The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: “Let us calculate, without further ado, to see who is right.”
Unfortunately, his dream of transforming all areas of life into calculation wasn’t pursued by the generation of thinkers that followed him. His ideas were so far ahead of their time that they weren’t recognized as important. After his death,
