Introduction toLogic is clear and concise, uses interesting examples (many philosophical in nature), and has easy-to-use proof methods. Its key features, retained in this Third Edition, include:
• simpler ways to test arguments, including an innovative proof method and the star test for syllogisms;
• a wide scope of materials, suiting it for introductory or intermediate courses;
• engaging examples, from philosophy and everyday life;
• useful for self-study and preparation for standardized tests, like the LSAT;
• a reasonable price (a third the cost of some competitors); and
• exercises that correspond to the free LogiCola instructional program.
This Third Edition:
• improves explanations, especially on areas that students find difficult;
• has a fuller explanation of traditional Copi proofs and of truth trees; and
• updates the companion LogiCola software, which now is touch friendly (for use on Windows tablets and touch monitors), installs more easily on Windows and Macintosh, and adds exercises on Copi proofs and on truth trees. You can still install LogiCola for free (from http://www.harryhiker.com/lc or http://www.routledge.com/cw/gensler).
Harry J. Gensler, S.J., is Professor of Philosophy at Loyola University Chicago. His fourteen earlier books include Gödel’s Theorem Simplified (1984), Formal Ethics (1996), Catholic Philosophy Anthology (2005), Historical Dictionary of Logic (2006), Historical Dictionary of Ethics (2008), Ethics: A Contemporary Introduction (1998 & 2011), Ethics and the Golden Rule (2013), and EthicsandReligion(2016).
00ii
“Equal parts eloquent and instructive, Gensler has once again provided an invaluable resource for those looking to master the fundamental principles of logic. The Third Edition improves upon an already exceptional text by infusing the introduction of new concepts with enhanced clarity, rendering even the most challenging material a joy to teach. The updated LogicCola program is sure to become an indispensable component of my own introductory course.”
ChristopherHaley , Waynesburg University, USA
“This Third Edition improves on a book that was already superb. I have used Gensler’s book to teach introductory courses in logic to undergraduate philosophers and linguists, and the response from the students has always been positive. They appreciate its clear explanation and the wealth of examples and practice opportunities it provides. In particular, the translation exercises help to refine logico-semantic intuitions. The supporting LogiCola software, which is feely downloadable, is a great support tool.”
MarkJary , University of Roehampton, UK
“The Third Edition is an improved version of an already excellent introduction to logic. Gensler’s Reductio proof procedure enables a seamless transition from elementary propositional logic to quantification theory and more advanced modal logics. Many of the exercises involve formulations of philosophical problems. The explanations of advanced topics have been greatly improved. The upgraded LogiCola program now supports alternative proof procedures. This book is a winner!”
MichaelBradie , Bowling Green State University, USA
Introduction to Logic Third Edition
HarryJ.Gensler
Third edition published 2017 by Routledge
711 Third Avenue, New York, NY 10017
and in the UK by Routledge
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The right of Harry J. Gensler to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.
Trademarknotice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.
First edition published by Routledge 2002
Second edition published by Routledge 2010
Library of Congress Cataloging in Publication Data
Gensler, Harry J., 1945 –
Introduction to logic / Harry J. Gensler. – 3rd ed. p. cm.
Includes index.
1. Logic. I. Title. BC71.G37 2017 160–dc22 2009039539
ISBN: 9781138910584 (hbk)
ISBN: 9781138910591 (pbk)
ISBN: 9781315693361 (ebk)
Typeset in Aldus LT Roman by the author.
Visit the companion website: http://www.harryhiker.com/lc or http://www.routledge.com/cw/gensler
Note
to E-book Users
Contents
Preface
1 Introduction
1.1 Logic
1.2 Valid arguments
1.3 Sound arguments
1.4 The plan of this book
2 Syllogistic Logic
2.1 Easier translations
2.2 The star test
2.3 English arguments
2.4 Harder translations
2.5 Deriving conclusions
2.6 Venn diagrams
2.7 Idiomatic arguments
2.8 The Aristotelian view
3 Meaning and Definitions
3.1 Uses of language
3.2 Lexical definitions
3.3 Stipulative definitions
3.4 Explaining meaning
3.5 Making distinctions
3.6 Analytic and synthetic
3.7 Aprioriand aposteriori
4 Fallacies and Argumentation
4.1 Good arguments
4.2 Informal fallacies
4.3 Inconsistency
4.4 Constructing arguments
4.5 Analyzing arguments
5 Inductive Reasoning
5.1 The statistical syllogism
5.2 Probability calculations
5.3 Philosophical questions
5.4 Reasoning from a sample
5.5 Analogical reasoning
5.6 Analogy and other minds
5.7 Mill’s methods
5.8 Scientific laws
5.9 Best-explanation reasoning
5.10 Problems with induction
6 Basic Propositional Logic
6.1 Easier translations
6.2 Basic truth tables
6.3 Truth evaluations
6.4 Unknown evaluations
6.5 Complex truth tables
6.6 The truth-table test
6.7 The truth-assignment test
6.8 Harder translations
6.9 Idiomatic arguments
6.10 S-rules
6.11 I-rules
6.12 Mixing S- and I-rules
6.13 Extended inferences
6.14 Logic and computers
7 Propositional Proofs
7.1 Easier proofs
7.2 Easier refutations
7.3 Harder proofs
7.4 Harder refutations
7.5 Copi proofs
7.6 Truth trees
8 Basic Quantificational Logic
8.1 Easier translations
8.2 Easier proofs
8.3 Easier refutations
8.4 Harder translations
8.5 Harder proofs
8.6 Copi proofs
9 Relations and Identity
9.1 Identity translations
9.2 Identity proofs
9.3 Easier relations
9.4 Harder relations
9.5 Relational proofs
9.6 Definite descriptions
9.7 Copi proofs
10 Basic Modal Logic
10.1 Translations
10.2 Proofs
10.3 Refutations
11 Further Modal Systems
11.1 Galactic travel
11.2 Quantified translations
11.3 Quantified proofs
11.4 A sophisticated system
12 Deontic and Imperative Logic
12.1 Imperative translations
12.2 Imperative proofs
12.3 Deontic translations
12.4 Deontic proofs
13 Belief Logic
13.1 Belief translations
13.2 Belief proofs
13.3 Believing and willing
13.4 Willing proofs
13.5 Rationality translations
13.6 Rationality proofs
13.7 A sophisticated system
14 A Formalized Ethical Theory
14.1 Practical reason
14.2 Consistency
14.3 The golden rule
14.4 Starting the GR proof
14.5 GR logical machinery
14.6 The symbolic GR proof
15 Metalogic
15.1 Metalogical questions
15.2 Symbols
15.3 Soundness
15.4 Completeness
15.5 An axiomatic system
15.6 Gödel’s theorem
16 History of Logic
16.1 Ancient logic
16.2 Medieval logic
16.3 Enlightenment logic
16.4 Frege and Russell
16.5 After Principia
17 Deviant Logics
17.1 Many-valued logic
17.2 Paraconsistent logic
17.3 Intuitionist logic
17.4 Relevance logic
18 Philosophy of Logic
18.1 Abstract entities
18.2 Metaphysical structures
18.3 The basis for logical laws
18.4 Truth and paradoxes
18.5 Logic’s scope
For Further Reading
Answers to Selected Problems
Chapter 2 answers
Chapter 3 answers
Chapter 4 answers
Chapter 5 answers
Chapter 6 answers
Chapter 7 answers
Chapter 8 answers
Chapter 9 answers
Chapter 10 answers
Chapter 11 answers
Chapter 12 answers
Chapter 13 answers
Chapter 14 answers
Index
Preface
This very comprehensive IntroductiontoLogiccovers:
• syllogisms;
• informal aspects of reasoning (like meaning and fallacies);
• inductive reasoning;
• propositional and quantificational logic;
• modal, deontic, and belief logic;
• the formalization of an ethical view about the golden rule; and
• metalogic, history of logic, deviant logic, and philosophy of logic.
Different parts can be used in a range of logic courses, from basic introductions to graduate courses. The teachers manual and the end of Chapter 1 both talk about which chapters fit which type of course.
Earlier Routledge editions appeared in 2002 and 2011. Features included (a) clear, concise writing; (b) engaging arguments from philosophy and everyday life; (c) simpler ways to test arguments, including an innovative proof method and the syllogism star-test; (d) the widest range of materials of any logic text; (e) high suitability for self-study and preparation for tests like the LSAT; (f) a reasonable price (a third that of some competitors); and (g) the free companion LogiCola instructional program (which randomly generates problems, gives feedback on answers, provides help and explanations, and records progress). I’m happy with how earlier editions were received, often with lavish praise.
I improved this third edition in many ways. I went through the book, making explanations clearer and more concise. I especially worked on areas that students find difficult, such as (to give a few examples) why “all A is B” and “some A is not B” are contradictories (§2.4), deriving syllogistic conclusions (§2.5), the transition from inference rules to formal proofs (§§6.10–13 & 7.1), how to evaluate formulas in quantificational logic (§§8.3 & 8.5), how to translate “exactly one” and “exactly two” in identity logic (§9.1), multiplequantifier translations and endless-loop refutations in relational logic (§§9.4–9.5), when to drop a necessary formula into the actual world in modal logic (§10.2), and how inference rules work in belief logic (§13.2). I expanded sections on traditional Copi proofs (§§7.5, 8.6, and 9.7, urged on by reviewers) and truth trees (§7.6, urged on by my friend Séamus Murphy), for teachers who might also want to teach these methods or have students learn them on their own for additional credit (as I do). “For Further Reading” now
mentions further sections of the book that an advanced student might want to pursue while doing specific chapters; for example, the Basic Propositional Logic chapter goes well with sections on metalogic, deviant logic, and 000x philosophy of logic. I didn’t substantially change exercise sections. Despite additions, the book is now six pages shorter.
The book now has a very nice Kindle e-book version, with real page numbers, based on a second version of the manuscript that I made with simplified formatting. And yes, you can add your own highlighting and notes.
I improved the companion LogiCola software, which runs on Windows, Macintosh, and Linux. Cloud Sync allows syncing scores between various computers. Proofs have a Training Wheels option; this gives hints about what to derive (it might bold lines 4 and 7 and ask “4 is an IF-THEN; do you have the first part true or the second part false?”) – hints disappear as your score builds up. Touch features let LogiCola be done using only touch, only mouse and keyboard, or any combination of these; touch works nicely on Windows tablets or touch-screen monitors. Quantificational translations have a Hints option; this gives Loglish hints about how to translate English sentences (for “All Italians are lovers” it might say “For all x, if x is Italian then x is a lover”) –hints disappear as your score builds up. There are exercises for Copi proofs and truth trees; to process scores from these, your LogiSkor program needs a version date of at least January 2016. And the Macintosh setup is easier. LogiCola (with a score-processing program, teachers manual, class slides, flash cards, and sample quizzes) can be downloaded for free from any of these Web addresses:
All supplementary materials are conveniently accessible from LogiCola’s HELP menu; so I suggest that you just install LogiCola (teachers should check the option to install the score processor too).
I wish to thank all who have somehow contributed to this third edition. I thank Andy Beck at Routledge and his staff and reviewers, who made good suggestions. I thank my logic students, especially those whose puzzled looks pushed me to make things clearer. And I thank the many teachers, students, and self-learners who e-mailed me, often saying things like “I love the book and software, but there’s one thing I have trouble with ….” If this third edition is a genuine improvement, then there are many people to thank besides me. Long live logic!
Harry J. Gensler Philosophy Department
Loyola University Chicago, IL 60660 USA
http://www.harryhiker.com
1 Introduction
1.1 Logic
Logic 1 is theanalysisandappraisalofarguments . Here we’ll examine reasoning on philosophical areas (like God, free will, and morality) and on other areas (like backpacking, water pollution, and football). Logic is a useful tool to clarify and evaluate reasoning, whether on deeper questions or on everyday topics.
Why study logic? First, logic builds our minds. Logic develops analytical skills essential in law, politics, journalism, education, medicine, business, science, math, computer science, and most other areas. The exercises in this book are designed to help us think more clearly (so people can better understand what we’re saying) and logically (so we can better support our conclusions).
Second, logic deepens our understanding of philosophy – which can be defined as reasoning about the ultimate questions of life . Philosophers ask questions like “Why accept or reject free will?” or “Can one prove or disprove God’s existence?” or “How can one justify a moral belief?” Logic gives tools to deal with such questions. If you’ve studied philosophy, you’ll likely recognize some of the philosophical reasoning in this book. If you haven’t studied philosophy, you’ll find this book a good introduction to the subject. In either case, you’ll get better at recognizing, understanding, and appraising philosophical reasoning.
Finally, logic can be fun. Logic will challenge your thinking in new ways and will likely fascinate you. Most people find logic enjoyable.
1.2 Valid arguments
I begin my basic logic course with a multiple-choice test. The test has ten problems; each gives information and asks what conclusion necessarily follows. The problems are fairly easy, but most students get about half wrong. 2 0002
Here’s a problem that almost everyone gets right:
1 Key terms (like “logic”) are introduced in bold. Learn each key term and its definition.
2 Http://www.harryhiker.com/logic.htm has my pretest in an interactive format. I suggest that you try it. I developed this test to help a psychologist friend test the idea that males are more logical than females; both groups, of course, did equally well on the problems.
If you overslept, you’ll be late. You aren’t late.
(a) You did oversleep.
Therefore
(b) You didn’t oversleep. ⇐ correct
(c) You’re late.
(d) None of these follows.
With this next one, many wrongly pick answer “(b)”:
If you overslept, you’ll be late. You didn’t oversleep.
(a) You’re late.
(b) You aren’t late.
(c) You did oversleep.
Therefore
(d) None of these follows. ⇐ correct
Here “You aren’t late” doesn’t necessary follow, since you might be late for another reason; maybe your car didn’t start. 1 The pretest shows that untrained logical intuitions are often unreliable. But logical intuitions can be developed; yours will likely improve as you work through this book. You’ll also learn techniques for testing arguments.
In logic, an argument is a set of statements consisting of premises (supporting evidence) and a conclusion (based on this evidence). Arguments put reasoning into words. Here’s an example (“∴” is for “therefore”):
Validargument
If you overslept, you’ll be late. You aren’t late.
∴ You didn’t oversleep.
An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. “Valid” doesn’t say that the premises are true, but only that the conclusion followsfromthem: if the premises were all true, then the conclusion would have to be true. Here we implicitly assume that there’s no shift in the meaning or reference of the terms; hence we must use “overslept,” “late,” and “you” the same way throughout the argument. 2 Our argument is valid because of its logical form: how it arranges logical
1 These two arguments were taken from Matthew Lipman’s fifth-grade logic textbook: Harry Stottlemeier’sDiscovery(Caldwell, NJ: Universal Diversified Services, 1974).
2 It’s convenient to allow arguments with zero premises; such arguments (like “∴ x = x”) are valid if and only if the conclusion is a necessary truth (couldn’t have been false).
notions like “if-then” and content like “You overslept.” We can display the form using words or symbols for logical notions and letters for content phrases:
If you overslept, you’ll be late. You aren’t late.
∴ You didn’t oversleep.
If A then B Valid
Not-B
∴ Not-A
Our argument is valid because its form is correct. Replacing “A” and “B” with other content yields another valid argument of the same form: 0003
If you’re in France, you’re in Europe. You aren’t in Europe.
∴ You aren’t in France.
If A then B Valid
Not-B
∴ Not-A
Logic studies forms of reasoning. The content can deal with anything –backpacking, math, cooking, physics, ethics, or whatever. When you learn logic, you’re learning tools of reasoning that can be applied to any subject. Consider our invalid example:
If you overslept, you’ll be late. You didn’t oversleep.
∴ You aren’t late.
If A then B Invalid
Not-A
∴ Not-B
Here the second premise denies the first part of the if-then; this makes it invalid. Intuitively, you might be late for some other reason – just as, in this similar argument, you might be in Europe because you’re in Italy:
If you’re in France, you’re in Europe. You aren’t in France.
∴ You aren’t in Europe.
If A then B Invalid Not-A
∴ Not-B
1.3 Sound arguments
Logicians distinguish validarguments from soundarguments:
An argument is valid if it would be contradictory to have the premises all true and conclusion false.
An argument is sound if it’s valid and every premise is true.
Calling an argument “valid” says nothing about whether its premises are true. But calling it “sound” says that it’s valid (the conclusion follows from the premises) andhas all premises true. Here’s a soundargument:
Validandtruepremises
If you’re reading this, you aren’t illiterate. You’re reading this.
∴ You aren’t illiterate.
When we try to prove a conclusion, we try to give a sound argument: valid and true premises. With these two things, we have a sound argument and our conclusion has to be true.
An argument could be unsound in either of two ways: (1) it might have a false premise or (2) its conclusion might not follow from the premises: 0004
Firstpremisefalse
All logicians are millionaires. Gensler is a logician.
∴ Gensler is a millionaire.
Conclusiondoesn’tfollow
All millionaires eat well. Gensler eats well.
∴ Gensler is a millionaire.
When we criticize an opponent’s argument, we try to show that it’s unsound .
We try to show that one of the premises is false or that the conclusion doesn’t follow. If the argument has a false premise or is invalid, then our opponent hasn’t proved the conclusion. But the conclusion still might be true – and our opponent might later discover a better argument for it. To show a view to be false, we must do more than just refute an argument for it; we must give an argument that shows the view to be false.
Besides asking whether premises are true, we can ask how certain they are, to ourselves or to others. We’d like our premises to be certain and obvious to everyone. We usually have to settle for less; our premises are often educated guesses or personal convictions. Our arguments are only as strong as their premises. This suggests a third strategy for criticizing an argument; we could try to show that one or more of the premises are very uncertain.
Here’s another example of an argument. In fall 2008, before Barack Obama was elected US president, he was ahead in the polls. But some thought he’d be defeated by the “Bradley effect,” whereby many whites say they’ll vote for a black candidate but in fact don’t. Barack’s wife Michelle, in an interview with Larry King, argued that there wouldn’t be a Bradley effect:
Barack Obama is the Democratic nominee.
If there’s going to be a Bradley effect, then Barack wouldn’t be the nominee [because the effect would have shown up in the primaries].
∴ There isn’t going to be a Bradley effect.
Once she gives this argument, we can’t just say “Well, my opinion is that there will be a Bradley effect.” Instead, we have to respond to her reasoning. It’s clearly valid – the conclusion follows from the premises. Are the premises true? The first premise was undeniable. To dispute the second premise, we’d have to argue that the Bradley effect would appear in the final election but not in the primaries. So this argument changes the discussion. (By the way, there was no Bradley effect when Obama was elected president a month later.)
Logic, while not itself resolving substantive issues, gives us intellectual tools to reason better about such issues. It can help us to be more aware of reasoning, to express reasoning clearly, to determine whether a conclusion follows from the premises, and to focus on key premises to defend or criticize.
Logicians call statements true or false (not valid or invalid). And they call arguments valid or invalid (not true or false). While this is conventional usage, it pains a logician’s ears to hear “invalid statement” or “false argument.”0005
Our arguments so far have been deductive. With inductive arguments, the conclusion is only claimed to follow with probability (not with necessity):
Deductivelyvalid
All who live in France live in Europe.
Pierre lives in France.
∴ Pierre lives in Europe.
Inductivelystrong
Most who live in France speak French.
Pierre lives in France.
This is all we know about the matter.
∴ Pierre speaks French (probably).
The first argument has a tight connection between premises and conclusion; it would be impossible for the premises to all be true but the conclusion false. The second has a looser premise–conclusion connection. Relative to the premises, the conclusion is only a good guess; it’s likely true but could be false (perhaps Pierre is the son of the Polish ambassador and speaks no French)
1.4 The plan of this book
This book starts simply and doesn’t presume any previous study of logic. Its four parts cover a range of topics, from basic to rather advanced:
• Chapters 2 to 5 cover syllogistic logic (an ancient branch of logic that focuses on “all,” “no,” and “ some”), meaning and definitions, informal fallacies, and inductive reasoning.
• Chapters 6 to 9 cover classical symbolic logic, including propositional logic (about “if-then,” “and,” “or,” and “not”) and quantificational logic (which adds “all,” “no,” and “ some”). Each chapter here builds on previous ones.
• Chapters 10 to 14 cover advanced symbolic systems of philosophical interest: modal logic (about “necessary” and “possible”), deontic logic (about “ought” and “permissible”), belief logic (about consistent believing and willing), and a formalized ethical theory (featuring the golden rule). Each chapter here presumes the previous symbolic ones (except that Chapter 10 depends only on 6 and 7, and Chapter 11 isn’t required for 12 to 14).
• Chapters 15 to 18 cover metalogic (analyzing logical systems), history of logic, deviant logics, and philosophy of logic (further philosophical issues). These all assume Chapter 6.
Chapters 2–8 and 10 are for basic logic courses, while other chapters are more advanced. Since this book is so comprehensive, it has much more material than can be covered in one semester.
Logic requires careful reading, and sometimes rereading. Since most ideas build on previous ideas, you need to keep up with readings and problems. The companion LogiCola software (see Preface) is a great help.
2 Syllogistic Logic
Aristotle, the first logician (§16.1), invented syllogistic logic, which features arguments using “all,” “no,” and “some.” This logic, which we’ll take in a nontraditional way, provides a fine preliminary to modern logic (Chapters 6–14).
2.1 Easier translations
We’ll now create a “syllogistic language,” with rules for constructing arguments and testing validity. Here’s how an English argument goes into our language:
All logicians are charming. Gensler is a logician.
∴ Gensler is charming.
all L is C g is L
∴ g is C
Our language uses capital letters for general categories (like “logician”) and small letters for specific individuals (like “Gensler”). It uses five words: “all,” “no,” “some,” “is,” and “not.” Its grammatical sentences are called wffs, or well-formed formulas. Wffs are sequences having any of these eight forms, where other capital letters and other small letters may be used instead: 1
all A is B
no A is B
some A is B
some A is not B
x is A
x is not A
x is y
x is not y
You must use one of these exact forms (but perhaps using other capitals for “A” and “B,” and other small letters for “x” and “y”). Here are examples of wffs (correct formulas) and non-wffs (misformed formulas):
1 Pronounce “wff” as “woof” (as in “wood”). We’ll take upper and lower case forms (like A and a) to be different letters, and letters with primes (like A´ and A´´) to be additional letters.
Wffs: “all L is C,” “no R is S,” “some C is D,” “g is C”
Non-wffs: “only L is C,” “all R is not S,” “some c is d,” “G is C” 0007
Our wff rule has implications about whether to use small or capital letters:
Wffs beginning with a word(not a letter) use two capital letters:
Correct: “some C is D”
Incorrect: “some c is d”
Wffs beginning with a letter(not a word) begin with a small letter:
Correct: “g is C”
Incorrect: “G is C”
A wff beginning with a small letter could use a capital-or-small second letter (as in “a is B” or “a is b”). Which to use depends on the second term’s meaning:
Use capital letters for general terms, which describeor put in a category:
B = a cute baby
C = charming
F = drives a Ford
Use capitals for “a so and so,” adjectives, and verbs.
Use small letters for singular terms, which pick out a specificperson or thing:
b = the world’s cutest baby
t = this child
d = David
Use small letters for “the so and so,” “this so and so,” and proper names.
Will Gensler is a cute baby = w is B
Will Gensler is the world’s cutest baby = w is b
An argument’s validity can depend on whether upper or lower case is used. Be consistent when you translate English terms into logic; use the same letter for the same idea and different letters for different ideas. It matters little which letters you use; “a cute baby” could be “B” or “C” or any other
capital. I suggest that you use letters that remind you of the English terms. Syllogistic wffs all use “is.” English sentences with a different verb should be rephrased to make “is” the main verb, and then translated. So “All dogs bark” is “all D is B” (“All dogs is [are] barkers”); and “Al drove the car” is “a is D” (“Al is apersonwhodrovethecar”).
2.1a Exercise: LogiCola A (EM & ET) 1
Translate these English sentences into wffs.
John left the room.
j is L
1. This is a sentence.
2. This isn’t the first sentence.
3. No logical positivist believes in God.
4. The book on your desk is green. 0008
5. All dogs hate cats.
6. Kant is the greatest philosopher.
7. Ralph was born in Detroit.
8. Detroit is the birthplace of Ralph.
9. Alaska is a state.
10. Alaska is the biggest state.
11. Carol is my only sister.
12. Carol lives in Big Pine Key.
13. The idea of goodness is itself good.
14. All Michigan players are intelligent.
15. Michigan’s team is awesome.
16. Donna is Ralph’s wife.
1 Exercise sections have a boxed sample problem that’s worked out. They also refer to LogiCola computer exercises (see Preface), which give a fun and effective way to master the material. Problems 1, 3, 5, 10, 15, and so on are worked out in the answer section at the back of the book
2.2 The star test
Syllogisms, roughly, are arguments using syllogistic wffs. Here’s an English argument and its translation into a syllogism (the Cuyahoga is a Cleveland river that used to be so polluted that it caught on fire):
No pure water is burnable.
Some Cuyahoga River water is burnable.
∴ Some Cuyahoga River water isn’t pure water.
no P is B
some C is B
∴ some C is not P
More precisely, syllogisms are vertical sequences of one or more wffs in which each letter occurs twice and the letters “form a chain” (each wff has at least one letter in common with the wff just below it, if there is one, and the first wff has at least one letter in common with the last wff):
(If you imagine the two instances of each letter being joined, it’s like a chain.)
no P is B
some C is B
∴ some C is not P
The last wff is the conclusion; other wffs are premises . Here are three more syllogisms:
a is C
b is not C
∴ a is not b
some G is F
∴ some F is G
∴ all A is A
The last example is a premise-less syllogism; it’s valid if and only if it’s impossible for the conclusion to be false.
Before doing the star test, we need to learn the technical term “distribut-
ed”: 1
An instance of a letter is distributed in a wff if it occurs just after “all” or anywhere after “ no ” or “not.”
0009 The distributed letters below are underlined and bolded:
all A is B no A is B some A is B some A is not B
By our definition:
x is A
x is not A
x is y
x is not y
• The first letter after “all” is distributed, but not the second.
• Both letters after “no” are distributed.
• Any letter after “not” is distributed.
Once you know which letters are distributed, you’re ready to learn the star test for validity. The star test is a gimmick, but a quick and effective one; for now, it’s best just to learn the test and not worry about why it works.
The star test for syllogisms goes as follows:
Star premise letters that are distributed and conclusion letters that aren’t distributed. Then the syllogism is valid if and only if every capital letter is starred exactlyonce and there is exactlyone star on the right-hand side.
As you learn the star test, use three steps: (1) underline distributed letters, (2) star, and (3) count the stars. Here are two examples:
(1) Underline distributed letters (here only the first “A” is distributed): all A is B
some C is A
∴ some C is B
(2) Star premise letters that are underlined and conclusion letters that aren’t
1 §16.2 mentions the meaning of “distributed” in medieval logic. Here I suggest that you take a distributedtermto be one that occurs just after “all” or anywhere after “no” or “not.”
underlined:
all A* is B Valid some C is A
∴ some C* is B*
(3) Count the stars. Here every capital letter is starred exactly once and there is exactly one star on the right-hand side. So the first argument is VALID.
(1) For our next argument, again underline distributed letters (here all the letters are distributed – since all occur after “no”):
no A is B no C is A
∴ no C is B
(2) Star premise letters that are underlined and conclusion letters that aren’t underlined:
no A* is B* Invalid no C* is A*
∴ no C is B
(3) Count the stars. Here capital “A” is starred twice and there are two stars on the right-hand side. So the second argument is INVALID.
A valid syllogism must satisfy two conditions: (a) each capital letter is starred in one and only one of its instances (small letters can be starred any number of times); and (b) one and only one right-hand letter (letter after “is” or “is not”) 0010 is starred. Here’s an example using only small letters:
(1) Underline distributed letters (here just ones after “not” are distributed):
a is not b
∴ b is not a
(2) Star premise letters that are underlined and conclusion letters that aren’t underlined:
a is not b* Valid
∴ b* is not a
(3) Count the stars. Since there are no capitals, that part is automatically satisfied; small letters can be starred any number of times. There’s exactly one right-hand star. So the argument is VALID.
Here’s an example without premises:
(1) Underline distributed letters:
∴ all A is A
(2) Star conclusion letters that aren’t underlined:
∴ all A is A* Valid
(3) Count the stars. Each capital is starred exactly once and there’s exactly one right-hand star. So the argument is VALID.
When you master this, you can skip the underlining and just star premise letters that are distributed and conclusion letters that aren’t. After practice, the star test takes about five seconds to do. 1
Logic takes “some” to mean “ one or more” – and so takes this to be valid: 2
Gensler is a logician. Gensler is mean.
∴ Some logicians are mean.
g is L Valid g is M
∴ some L* is M*
Similarly, logic takes this next argument to be invalid:
Some logicians are mean.
∴ Some logicians are not mean.
some L is M Invalid
∴ some L* is not M
If one or more logicians are mean, it needn’t be that one or more aren’t mean; maybe alllogicians are mean.
2.2a Exercise – No LogiCola exercise
Which of these are syllogisms?
1 The star test is my invention. For why it works, see http://www.harryhiker.com/star.htm or my “A simplified decision procedure for categorical syllogisms,” NotreDameJournalofFormal Logic14 (1973): pp. 457–66.
2 In English, “some” can also mean “two or more,” “several,” “one or more but not all,” “two or more but not all,” or “several but not all.” Only the one-or-more sense makes our argument valid.
