An accessible introduction to the study of logic (parts 1 & 2), as well as an in-depth treatment of the discipline (parts 3 & 4), built on a robust Christian worldview. Includes helpful charts, diagrams, and review questions.
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About the Author
Vern S. Poythress (PhD, Harvard University; ThD, University of Stellenbosch) is professor of New Testament interpretation at Westminster Theological Seminary in Philadelphia, Pennsylvania, where he has taught for nearly four decades. In addition to earning six academic degrees, he is the author of numerous books and articles on biblical interpretation, language, and science.
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Logic in Tension
In the original Star Trek TV series, the characters Spock and Leonard McCoy are opposites. Spock is logical; McCoy is passionate. Spock is cold; McCoy is hot. The contrast raises lots of questions. How does logic fit in with our humanity? Is logic opposite to emotion? What should we be like as human beings — logical or emotional or both?
Logic and Humanity
The Star Trek series gained popularity not only because it had entertaining plots but also because it laid out in narrative form some of the big questions about man and his relation to the cosmos. Who are we? What is the meaning of life? What is the cosmic purpose of humanity? Why do logic and emotion struggle within us?
Viewers' reactions to Spock reveal different attitudes toward logic. To some people, Spock's logic is an ideal. To others, he may be either admirable or pitiable, but he lacks something. The creators of the show make their own comment by revealing that, while McCoy is human, Spock is the offspring from a Vulcan father and a human mother. He is only half human. A deeper look at Spock reveals further complexity: though Spock endeavors to follow logic, he sometimes struggles with inner emotions because of his human side. Does this fictional portrayal hint that logic is not enough?
What about us? How do we relate to logic? Does it appeal to us? Or do we feel that by itself it is too "cold"?
Some people are more logical, some more emotional. Some people think that we have problems because we are not logical enough. Others think that we are much too logical. In their view, devotion to logic creates difficulties, and we ought to move beyond logic to something else — to nature or mysticism or art. Science, in the minds of some, is driven by logic and by a tightly defined, cold rationality. Human beings in their full personality are driven by warmth: they have desires and emotions and imagination, which are aptly expressed in the arts, in leisure, in entertainment, and in the humanities. Science, according to this view, is at odds with the humanities and with what is most precious to us.
So what is logic? Is it important? How do we understand its relation to emotion, intuition, and other aspects of human life? How do we use it? Does it have limits?
I believe that common conceptions about logic do not provide healthy answers to these questions. We need a new approach to the subject — we need a distinctively Christian approach.
Is there such a thing as a Christian view of logic? We would not be surprised to find a distinctively Christian approach to theology or ethics, because the Bible has much to say about God and ethics. But could there be a distinctively Christian approach to logic? Many people would say no. They would say that logic is what it is, irrespective of religious belief. I think that the reality is more complicated. There is a Christian view of logic. But it will take some time to see why.1 Readers may, if they wish, treat this book as a general introduction to logic. Our discussion does not assume any previous acquaintance with the subject. We try to make the ideas accessible by including simple explanations with each new concept. But the discussion also has pertinence for experts, because we do not take a conventional approach. We develop a distinctively Christian approach. Human thinking about logic needs redeeming. As a result, it will take us some time to come to the point of discussing details that typically become the focus of logic textbooks.
For Further Reflection
1. What makes the difference between Spock and McCoy so fascinating?
2. What different reactions are there to Spock as a character, and what do they say about people's views about logic?
3. When people think about an ideal for humanity, what role do they assign to logic?
4. How might human beings deal with the apparent tension between logic and emotion? What implications are there for the nature of our humanity?
5. Why might some people think that a distinctively Christian approach to logic makes no sense?CHAPTER 2
Why Study Logic?
Why should we bother to study logic? Spock exemplifies one part of its importance. On the one hand, Spock's rational analysis gives the Star Trek crew valuable advice. On the other hand, we struggle with an apparent conflict between logic and emotion, or even between logic and humanness. We need a remedy.
We can find other reasons for studying logic. Some people find logic intrinsically interesting. For them, it is fun. Others study it for practical purposes. They hope that studying logic can help them sharpen their ability to reason carefully. Practice in logic can help us detect logical errors in reasoning, which have been called logical fallacies.
The Influence of Logic
Logic is important for another, historical reason. Logic has had a profound influence on the whole history of Western thought. In the Western world, the formal study of logic began largely with the Greek philosopher Aristotle — though Aristotle built to some extent on his philosophical predecessors, Socrates and Plato.1 Plato and Aristotle hoped to find deep truths about the nature of the world by careful reasoning. Aristotle's study of logic tried to codify the most basic forms of reasoning. This codification could then serve as a solid foundation for philosophical investigations trying to answer the big questions about the nature of reality and the meaning of life.
Western philosophy ever since Aristotle's time has followed in the steps of Plato and Aristotle. Philosophers have reasoned. They have used logic. Up until the nineteenth century, with few exceptions, they built on the foundation of Aristotle's logic. The nineteenth and twentieth centuries have seen further, more technical developments in logic, which have gone well beyond what Aristotle achieved. But for the most part these developments have enhanced rather than overthrown the classical logic developed by Aristotle. (For more detailed discussion of logic and philosophy, see part IV.F, especially appendix F2. For some alternatives to classical logic, see chapters 63 and 64.)
Logic and philosophy have had a broad influence on intellectual culture in the West. Philosophy has directly influenced intellectual life, because it has seemed to many people to offer the most profound and far-reaching kind of knowledge. Science has taken a leading role in more recent times, but for centuries reasoning in intellectual centers was influenced and guided by ideas from philosophy.
In addition, logic has had indirect influence. People engage in reasoning in every area of serious study, not just in philosophy. In almost every sphere, universities today rely on reasoning — in natural sciences, medicine, historical studies, law, economics, political science, language study, literary analysis, mathematics. Academic work aspires to conduct its reasoning rigorously. And logic is a model for rigor. Reasoning in universities today still has underneath it the foundation for logic that Aristotle laid.
Though Aristotle's logic functions as a foundation for Western thought, we should not exaggerate its role. In both the past and the present, much influential reasoning takes the form of informal reasoning and does not explicitly invoke Aristotelian logic or any kind of formally organized logic. Appropriately, logicians themselves distinguish between the formal logic that Aristotle developed and the informal logic involved in more ordinary instances of reasoning. Yet rigorous formal logic offers an ideal that can still influence what people expect and how people evaluate informal reasoning. Logic has an influence far wider than its core.
Logic has also influenced perceptions about the contrast between rationality on the one hand and emotion, desire, and imagination on the other. The historical movement called the Enlightenment championed reason. But soon people became restless. They sensed that reason was not enough. Reason gave us only half of humanity — or less. The Enlightenment stimulated a reaction, the Romantic movement, which depreciated reason and championed the imaginative, the spontaneous, the natural, and the pre-rational aspects of humanity. Like the opposition between sciences and humanities, the opposition between the Enlightenment and the Romantic movement expresses the contrast between logic and emotion, or between Spock and McCoy. Thus, the contrast between Spock and McCoy has analogues that play out in culture and history.
At the foundation of this cultural opposition lies logic. It feeds into the Enlightenment's conception of reason, and it shapes the Romantic opposition to the Enlightenment as well, because the opposition defines itself in reaction to reason.
This foundation for Western thought in logic needs to be redone. And that means that the whole of Western thought has to be redone. It is a most serious issue.
What do we mean by logic? One textbook on logic defines it as "the analysis and appraisal of arguments." When we hear the word argument, we may picture a situation where two people are having a dispute with each other — perhaps a bitter, heated dispute. They are fighting verbally, each person vigorously defending his own view. But the word argument can be used not only to describe quarrels but to describe any reasoning in support of a conclusion.
Arguments of this kind may crop up in friendly settings. An advertisement for a car may present arguments to persuade you to buy one. The advertisement tells you that its car gives you good gas mileage. It is durable. It has special computerized features to play your favorite songs. It has a luxurious interior. It looks cool. And so on. These are informal arguments in favor of buying the car.
We meet arguments not only when someone else is trying to lay out the desirable features of a product, but when we are quietly trying to decide something for ourselves. For example, Irene may be "arguing with herself" about which college to attend. College A is closer to home. College B has lower tuition. College A is reputed to have a better program in economics. College B has a beautiful rural campus. College A is right in the middle of exciting city life. College B has a larger student body. Irene formulates arguments in her own mind in favor of each of the options. Arguments are useful not only for small purchases, but also for major decisions like choosing a college or deciding what kind of job to pursue.
We also meet arguments in academic settings. A university class may lay out reasoning to reach conclusions in chemistry or in the history of World War I. When a class considers disputed ideas, the class members may study arguments both for and against the ideas. Underneath the particular arguments lies a foundation in logic, which analyzes general principles of argument.
Arguments can help to lead us to a wise conclusion. But they can also lead us astray. For example, a student says, "Either you get an A in the course or you show that you are an idiot." But might there be a third alternative? The presentation of two extreme alternatives as if they were the only alternatives is called the fallacy of bifurcation. There are other forms of fallacy as well. A fallacy is a kind of argument that may sound plausible but that uses tricks rather than solid reasoning. Logic includes the study of various kinds of fallacies. People hope that by studying fallacies they may more easily detect them in the future.
Arguments in the Bible
Arguments occur in the Bible. We should not be surprised, because the Bible describes human life in all its ups and downs. For example, a major argument takes place in 2 Samuel 17:1–14. Absalom, the son of David, has just mounted a rebellion against the kingship of his father David. He has forced David out of Jerusalem, the capital city. But as long as David is alive, Absalom's own position in power remains in jeopardy. Absalom asks for advice from Ahithophel, who has a reputation for giving shrewd counsel (2 Sam. 16:23). Absalom also consults Hushai, who gives opposite advice. Ahithophel says Absalom should attack David right away with a small force of select troops (17:1). Hushai advises Absalom to wait in order to assemble a large army. Both Ahithophel and Hushai give supporting reasons in favor of their stratagems.
Absalom and his supporters think that Hushai's advice is better. Hushai's arguments are convincing; but they lead to disaster. Absalom is killed in the battle that eventually takes place (2 Sam. 18:15). Clearly an argument can be a major turning point in a person's life, and even in the life of a whole kingdom — in this case, the kingdom of Israel.
The arguments from Ahithophel and Hushai are even more striking because the reader of 2 Samuel receives some information that Absalom and Ahithophel did not know. Hushai is pretending to serve Absalom, but secretly he is loyal to David. In fact, David has earlier told Hushai to go to Absalom and to try to interfere by dissuading Absalom from following Ahithophel's advice (2 Sam. 15:34). Hushai appears to Absalom to give his advice sincerely, and the arguments that he offers are plausible and attractive. But the reader can infer that Hushai does not believe in these arguments himself. He is acting out a role. Hushai's arguments therefore have two layers: what he intends Absalom to understand and what he himself understands and intends. In fact, the arguments have a third layer, because God the Lord is active behind the scenes: "For the LORD had ordained to defeat the good counsel of Ahithophel, so that the LORD might bring harm upon Absalom" (17:14).
Arguments can be used to deceive and manipulate. But they can also become part of wise counsel. At one point David has decided to order his men to attack Nabal and kill him. Abigail, Nabal's wife, comes out and dissuades him with her arguments (1 Sam. 25:23–31). David is persuaded, and blesses Abigail for having kept him back from sin (v. 33). The story has a further happy ending because after Nabal dies — by God's act rather than David's — David and Abigail marry (v. 42). Abigail's arguments have steered David toward righteous action and away from sin.
We meet still further arguments within the Bible, including arguments that address all-important religious decisions. The serpent in Genesis 3 gives arguments to try to induce Adam and Eve to sin. Elijah in 1 Kings 18 gives arguments (and a demonstration) to try to turn the people of Israel away from worshiping Baal and toward worshiping the Lord, the true God of Israel. Since Elijah presents himself as a prophet of God, his arguments claim to be not merely human but also divine. Elijah claims that God is presenting the arguments to Israel through him.
The New Testament indicates that God continues to speak, and it includes arguments to call people to come to Christ for salvation. The apostle Peter presents arguments in his sermon in Acts 2:14–36. Since Peter is an apostle, commissioned by Christ, these arguments also present themselves as divine arguments. The apostle Paul presents arguments in his sermons here and there in Acts. Acts 13:16–41; 14:15–17; and 17:22–31 give examples. In addition, some of the summaries of Paul's preaching mention argument and reasoning:
And Paul went in, as was his custom, and on three Sabbath days he reasoned with them from the Scriptures, explaining and proving that it was necessary for the Christ to suffer and to rise from the dead, and saying, "This Jesus, whom I proclaim to you, is the Christ." (Acts 17:3)
So he [Paul] reasoned in the synagogue with the Jews and the devout persons, and in the marketplace every day with those who happened to be there. (Acts 17:17)
And he [Paul] reasoned in the synagogue every Sabbath, and tried to persuade Jews and Greeks. (Acts 18:4)
And he [Paul] entered the synagogue and for three months spoke boldly, reasoning and persuading them about the kingdom of God. (Acts 19:8)
We also hear of arguments within the church when controversies arose:
And after Paul and Barnabas had no small dissension and debate with them, Paul and Barnabas and some of the others were appointed to go up to Jerusalem to the apostles and the elders about this question. (Acts 15:2)
The apostles and the elders were gathered together to consider this matter. And after there had been much debate, Peter stood up ... (Acts 15:6–7)
In 1 Corinthians 15 Paul presents an extended argument to try to correct wavering in the Corinthian church over the question of whether there will be a future resurrection of the body.
The Bible contains many other types of communication in addition to arguments. It has songs, historical reports, prophecies, and so on. But we can use the idea of argument and persuasion as a perspective on everything the Bible does. In a looser sense, we can say that the whole of the Bible functions as an argument to induce us to change ourselves, our beliefs, and our behavior.(Continues…)
Excerpted from "Logic"
Copyright © 2013 Vern Sheridan Poythress.
Excerpted by permission of Good News Publishers.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Tables and Illustrations 13
Symbols Used in Parts II-IV 17
Part I Elementary Logic
Part I.A Introducing Logic and Argument
1 Logic in Tension 24
2 Why Study Logic? 27
3 What Do We Trust? 34
4 Formal Logic 45
5 Inductive Logic 51
6 The Importance of Formal Logic 55
Part I.B God in Logic
7 Logic Revealing God 62
8 Logic as Personal 68
9 Logic within Language 73
10 Suppressing the Truth 80
11 Logic and the Trinity 86
12 The Absoluteness of God 93
13 Logic and Necessity 101
14 Transcendence and Immanence 108
15 Reflections on the Mediation of Human Knowledge of Logic 117
16 Fallacies and God 123
Part I.C The Problem of Classification
17 Analogy 134
18 Unity and Diversity 143
19 Stability of Meaning 152
20 Form and Meaning 158
21 Context for Meaning 164
22 Persons and Logic 168
23 Logic and Religious Antithesis 173
24 Theistic Proofs 176
25 Rethinking Western Thought 187
Part I.D Aristotelian Syllogisms
26 Theistic Foundations for a Syllogism 192
27 Venn Diagrams 196
28 Syllogisms of the First Figure 208
29 Checking Validity by Venn Diagrams 221
Part II Aspects of Propositional Logic
Part II.A Truth in Logic
30 Truth in Logic: Truth Functions 230
31 Divine Origin of Logical Functions 240
32 Complex Expressions 247
Part II.B Perspectives on Truth in Logic
33 Venn Diagrams for Truth Functions 256
34 Other Representations of Logical Truth and Falsehood 261
35 Boolean Algebra 270
36 Truth-functional Equivalence 281
37 Harmony in Truth 287
38 Perspectives on Truth Functions 295
Part II.C Propositional Logic
39 Introducing Propositional Logic 302
40 Axioms of Propositional Logic 309
41 Alternative Axioms 318
42 Dispensing with Axioms 324
43 Perspectives on Propositional Logic 336
44 Soundness and Completeness of Propositional Logic 346
45 Imitations of Transcendence 354
Part III Enriching Logic
Part III.A Predicate Logic
46 Introducing Predicate Logic 364
47 Theistic Foundations for Predicates 368
Part III.B Quantification
48 Quantification 376
49 The Theistic Foundation for Quantification 381
50 Axioms and Deductions for Quantification 389
51 Soundness of Quantification 400
Part III.C Including Equality and Functions
52 Equality 404
53 Functions 408
Part III.D Introducing Formal Systems
54 Troubles in Mathematics 418
55 Axiomatizing Mathematics 423
56 Studying Proofs 429
57 Theistic Foundations for Proof Theory 436
58 A Computational Perspective 446
59 Theistic Foundations of Computation 453
60 Models 457
61 Theistic Foundations for Models 463
Part III.E Special Logics and More Enriched Logics
62 Higher-order Quantification 468
63 Multivalued Logic 474
64 Intuitionistic Logic 487
65 Modal Logic 494
66 Theistic Foundations for Modal Logic 499
67 Models for Modal Logic 507
68 Conclusion 516
Part IV Supplements
Part IV.A Supplements to Elementary Logic
A1 Antinomies with Sets: The Set of All Sets and Russell's Paradox 522
A2 Deriving Syllogisms of the First Figure 528
A3 Syllogisms of the Second Figure 533
A4 Syllogisms of the Third and Fourth Figures 537
Part IV.B Supplementary Proofs for Propositional Logic
B1 Some Proofs for Boolean Algebra 550
B2 Deriving Whitehead and Russell's Axioms 556
B3 Practice in Proofs 560
B4 The Rule of Replacement 572
B5 Reasoning toward the Completeness of Propositional Logic 575
Part IV.C Proofs for Quantification
C1 Deductions of Rules for Quantification 584
C2 Natural Deduction of Syllogisms 586
Part IV.D Proofs for Formal Systems
D1 Introducing Gödel's First Incompleteness Theorem 592
D2 Simple Proofs within a Formal System 599
D3 Deriving Natural Deduction and the Associative Axiom 605
D4 Helping Lemmas 613
Part IV.E Other Proofs
E1 The Halting Problem for Computer Programs 616
E2 Diagonalization 620
Part IV.F Philosophy and Logic
F1 Kantian Subjectivism 632
F2 The Role of Logic in Philosophy 645
F3 A View of Modern Logic 657
F4 Modal Ontological Argument 664
F5 Reforming Ontology and Logic 669
General Index 717
Scripture Index 729
What People are Saying About This
“Most books on logic content themselves with trying to fix our problems in logic. But in this book, Vern Poythress takes a ‘God-centered’ approach to the subject, which addresses a more foundational problem we haveour problem about logic. What is it? Why is it authoritative? If logic is part of the creation, could God have made a world that had round squares in it? And if logic is untouchable and mysteriously somehow ‘over’ God, then doesn’t that make logic God? In this book, Poythress does a masterful job of showing how our ability to think rationally is grounded in the very nature of God Himself. I heartily recommend this book to every student of the subject.”
Douglas Wilson, Senior Fellow of Theology, New St. Andrews College; Pastor, Christ Church, Moscow, Idaho