An Introduction to Formal Logic

Course No. 4215
Professor Steven Gimbel, Ph.D.
Gettysburg College
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Course No. 4215
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What Will You Learn?

  • Become familiar with common logical fallacies such as circular reasoning, slippery slope, and causal oversimplification.
  • Examine what makes deductive arguments valid.
  • Use truth tables to test the validity of famous forms of argument called modus ponens and affirming the consequent.
  • Follow the work of Hilbert, Cantor, Frege, Russell, and Godel to prove that the logical consistency of mathematics can be reduced to basic arithmetic.

Course Overview

Flawed, misleading, and false arguments are everywhere. From advertisers trying to separate you from your money, to politicians trying to sway your vote, to friends who want you to agree with them, your belief structure is constantly under attack.

Logic is intellectual self-defense against such assaults on reason and also a method of quality control for checking the validity of your own views. But beyond these very practical benefits, informal logic—the kind we apply in daily life—is the gateway to an elegant and fascinating branch of philosophy known as formal logic, which is philosophy’s equivalent to calculus. Formal logic is a breathtakingly versatile tool. Much like a Swiss army knife for the incisive mind, it is a powerful mode of inquiry that can lead to surprising and worldview-shifting conclusions.

Award-winning Professor of Philosophy Steven Gimbel of Gettysburg College guides you with wit and charm through the full scope of this immensely rewarding subject in An Introduction to Formal Logic, 24 engaging half-hour lectures that teach you logic from the ground up—from the fallacies of everyday thinking to cutting edge ideas on the frontiers of the discipline. Professor Gimbel’s research explores the nature of scientific reasoning and the ways in which science and culture interact, which positions him perfectly to make advanced abstract concepts clear and concrete.

Packed with real-world examples and thought-provoking exercises, this course is suitable for everyone from beginners to veteran logicians. Plentiful on-screen graphics, together with abundant explanations of symbols and proofs, make the concepts crystal clear.

For the Logician in All of Us

You will find that the same rational skills that help you spot the weaknesses in a sales pitch or your child’s excuse for skipping homework will also put you on the road to some of the most profound discoveries of our times, such as Kurt Gödel’s incompleteness theorems, which shook the foundations of philosophy and mathematics in the 20th century and can only be compared to revolutions in thought such as quantum mechanics. But Gödel didn’t need a lab to make his discovery—only logic.

A course with a surprising breadth and depth of applications, An Introduction to Formal Logic will appeal to:

  • critical thinkers who aspire to make better decisions, whether as doctors, lawyers, investors, managers, or others faced with the task of weighing conflicting options
  • lovers of intellectual history, who wish to trace one of the most influential and underappreciated currents of thought from antiquity to the present day
  • students of philosophy, for whom logic is the gold standard for evaluating philosophical arguments and a required course for mastery of the discipline
  • students of mathematics, who want to understand the foundations of their field and glimpse the machinery that drives every mathematical equation ever written
  • anyone curious about how computers work, for programs know nothing about words, sentences, or even numbers—they only comprehend logic
  • those fascinated with language, the brain, and other topics in cognitive science, since logic models grammar, meaning, and thought better than any other tool

Logic Is Your Ally

Professor Gimbel begins by noting that humans are wired to accept false beliefs. For example, we have a strong compulsion to change our view to match the opinion of a group, particularly if we are the lone holdout—even if we feel certain that we are right. From these and other cases of cognitive bias where our instincts work against sound reasoning, you begin to see how logic is a marvelous corrective that protects us from ourselves. With this intriguing start, An Introduction to Formal Logic unfolds as follows:

  • Logical concepts: You are introduced to deductive and inductive arguments and the criteria used to assess them—validity and well-groundedness. Then you learn that arguments have two parts: conclusions (that which is being argued for) and premises (the support given for the conclusion).
  • Informal logic: Often called critical thinking, this type of logical analysis looks at features other than the form of an argument—hence “informal.” Here, you focus on establishing the truth of the premises, as well as spotting standard rhetorical tricks and logical fallacies.
  • Inductive reasoning: Next you learn to assess the validity of an argument using induction, which examines different cases and then forms a general conclusion. Inductive arguments are typical of science, taking what we already know and giving us logical permission to believe something new.
  • Formal symbolic deductive logic: Known as “formal” logic because it focuses on the form of arguments, this family of techniques uses symbolic language to assess the validity of a wide range of deductive arguments, which infer particulars from general laws or principles.
  • Modal logic: After an intensive exploration of formal logic, you venture into modal logic, learning to handle sentences that deal with possibility and necessity—called modalities. Modal logic has been very influential in the philosophy of ethics.
  • Current advances: You close the course by looking at recent developments, such as three-valued logical systems and fuzzy logic, which extend our ability to reason by denying what seems to be the basis of all logic—that sentences must be either true or false.

Learn the Language of Logic

For many people, one of the most daunting aspects of formal logic is its use of symbols. You may have seen logical arguments expressed with these arrows, v’s, backwards E’s, upside down A’s, and other inscrutable signs, which can seem as bewildering as higher math or an ancient language. But An Introduction to Formal Logic shows that the symbols convey simple ideas compactly and become second nature with use. In case after case, Professor Gimbel explains how to analyze an ambiguous sentence in English into its component propositions, expressed in symbols. This makes what is being asserted transparently clear.

Consider these two sentences: (1) “A dog is a man’s best friend.” (2) “A dog is in the front yard.” Initially, they look very similar. Both say “A dog is x” and seem to differ only in the property ascribed to the dog. However, the noun phrase “a dog” means two completely different things in these two cases. In the first, it means dogs in general. In the second, it denotes a specific dog. These contrasting ideas are symbolized like so:

1. "x(Dx→Bx)

2. $x(Dx&Fx)

You will discover that many consequential arguments in daily life hinge on a similar ambiguity, which dissolves away when translated into the clear language of logic.

Professor Gimbel notes that logical thinking is like riding a bicycle; it takes skill and practice, and once you learn you can really go places! Logic is the key to philosophy, mathematics, and science. Without it, there would be no electronic computers or data processing. In social science, it identifies patterns of behavior and uncovers societal blind spots—assumptions we all make that are completely false. Logic can help you win an argument, run a meeting, draft a contract, raise a child, be a juror, or buy a shirt and keep from losing it at a casino. Logic says that you should take this course.

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24 lectures
 |  Average 31 minutes each
  • 1
    Why Study Logic?
    Influential philosophers throughout history have argued that humans are purely rational beings. But cognitive studies show we are wired to accept false beliefs. Review some of our built-in biases, and discover that logic is the perfect corrective. Then survey what you will learn in the course. x
  • 2
    Introduction to Logical Concepts
    Practice finding the logical arguments hidden in statements by looking for indicator words that either appear explicitly or are implied-such as therefore" and "because." Then see how to identify the structure of an argument, focusing on whether it is deductive or inductive." x
  • 3
    Informal Logic and Fallacies
    Explore four common logical fallacies. Circular reasoning uses a conclusion as a premise. Begging the question invokes the connotative power of language as a substitute for evidence. Equivocation changes the meaning of terms in the middle of an argument. And distinction without a difference attempts to contrast two positions that are identical. x
  • 4
    Fallacies of Faulty Authority
    Deepen your understanding of the fallacies of informal logic by examining five additional reasoning errors: appeal to authority, appeal to common opinion, appeal to tradition, fallacy of novelty, and arguing by analogy. Then test yourself with a series of examples, and try to name that fallacy! x
  • 5
    Fallacies of Cause and Effect
    Consider five fallacies that often arise when trying to reason your way from cause to effect. Begin with the post hoc fallacy, which asserts cause and effect based on nothing more than time order. Continue with neglect of a common cause, causal oversimplification, confusion between necessary and sufficient conditions, and the slippery slope fallacy. x
  • 6
    Fallacies of Irrelevance
    Learn how to keep a discussion focused by recognizing common diversionary fallacies. Ad hominem attacks try to undermine the arguer instead of the argument. Straw man tactics substitute a weaker argument for a stronger one. And red herrings introduce an irrelevant subject. As in other lectures, examine fascinating cases of each. x
  • 7
    Inductive Reasoning
    Turn from informal fallacies, which are flaws in the premises of an argument, to questions of validity, or the logical integrity of an argument. In this lecture, focus on four fallacies to avoid in inductive reasoning: selective evidence, insufficient sample size, unrepresentative data, and the gambler's fallacy. x
  • 8
    Induction in Polls and Science
    Probe two activities that could not exist without induction: polling and scientific reasoning. Neither provides absolute proof in its field of analysis, but if faults such as those in Lecture 7 are avoided, the conclusions can be impressively reliable. x
  • 9
    Introduction to Formal Logic
    Having looked at validity in inductive arguments, now examine what makes deductive arguments valid. Learn that it all started with Aristotle, who devised rigorous methods for determining with absolute certainty whether a conclusion must be true given the truth of its premises. x
  • 10
    Truth-Functional Logic
    Take a step beyond Aristotle to evaluate sentences whose truth cannot be proved by his system. Learn about truth-functional logic, pioneered in the late 19th and early 20th centuries by the German philosopher Gottlob Frege. This approach addresses the behavior of truth-functional connectives, such as not," "and," "or," and "if" —and that is the basis of computer logic, the way computers "think."" x
  • 11
    Truth Tables
    Truth-functional logic provides the tools to assess many of the conclusions we make about the world. In the previous lecture, you were introduced to truth tables, which map out the implications of an argument's premises. Deepen your proficiency with this technique, which has almost magical versatility. x
  • 12
    Truth Tables and Validity
    Using truth tables, test the validity of famous forms of argument called modus ponens and its fallacious twin, affirming the consequent. Then untangle the logic of increasingly more complex arguments, always remembering that the point of logic is to discover what it is rational to believe. x
  • 13
    Natural Deduction
    Truth tables are not consistently user-friendly, and some arguments defy their analytical power. Learn about another technique, natural deduction proofs, which mirrors the way we think. Treat this style of proof like a game-with a playing board, a defined goal, rules, and strategies for successful play. x
  • 14
    Logical Proofs with Equivalences
    Enlarge your ability to prove arguments with natural deduction by studying nine equivalences-sentences that are truth-functionally the same. For example, double negation asserts that a sentence and its double negation are equivalent. It is not the case that I didn't call my mother," means that I did call my mother." x
  • 15
    Conditional and Indirect Proofs
    Complete the system of natural deduction by adding a new category of justification-a justified assumption. Then see how this concept is used in conditional and indirect proofs. With these additions, you are now fully equipped to evaluate the validity of arguments from everyday life. x
  • 16
    First-Order Predicate Logic
    So far, you have learned two approaches to logic: Aristotle's categorical method and truth-functional logic. Now add a third, hybrid approach, first-order predicate logic, which allows you to get inside sentences to map the logical structure within them. x
  • 17
    Validity in First-Order Predicate Logic
    For all of their power, truth tables won't work to demonstrate validity in first-order predicate arguments. For that, you need natural deduction proofs-plus four additional rules of inference and one new equivalence. Review these procedures and then try several examples. x
  • 18
    Demonstrating Invalidity
    Study two techniques for demonstrating that an argument in first-order predicate logic is invalid. The method of counter-example involves scrupulous attention to the full meaning of the words in a sentence, which is an unusual requirement, given the symbolic nature of logic. The method of expansion has no such requirement. x
  • 19
    Relational Logic
    Hone your skill with first-order predicate logic by expanding into relations. An example: If I am taller than my son and my son is taller than my wife, then I am taller than my wife." This relation is obvious, but the techniques you learn allow you to prove subtler cases." x
  • 20
    Introducing Logical Identity
    Still missing from our logical toolkit is the ability to validate identity. Known as equivalence relations, these proofs have three important criteria: equivalence is reflexive, symmetric, and transitive. Test the techniques by validating the identity of an unknown party in an office romance. x
  • 21
    Logic and Mathematics
    See how all that you have learned in the course relates to mathematics-and vice versa. Trace the origin of deductive logic to the ancient geometrician Euclid. Then consider the development of non-Euclidean geometries in the 19th century and the puzzle this posed for mathematicians. x
  • 22
    Proof and Paradox
    Delve deeper into the effort to prove that the logical consistency of mathematics can be reduced to basic arithmetic. Follow the work of David Hilbert, Georg Cantor, Gottlob Frege, Bertrand Russell, and others. Learn how Kurt Gödel’s incompleteness theorems sounded the death knell for this ambitious project. x
  • 23
    Modal Logic
    Add two new operators to your first-order predicate vocabulary: a symbol for possibility and another for necessity. These allow you to deal with modal concepts, which are contingent or necessary truths. See how philosophers have used modal logic to investigate ethical obligations. x
  • 24
    Three-Valued and Fuzzy Logic
    See what happens if we deny the central claim of classical logic, that a proposition is either true or false. This step leads to new and useful types of reasoning called multi-valued logic and fuzzy logic. Wind up the course by considering where you've been and what logic is ultimately about. x

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Your professor

Steven Gimbel

About Your Professor

Steven Gimbel, Ph.D.
Gettysburg College
Professor Steven Gimbel holds the Edwin T. Johnson and Cynthia Shearer Johnson Distinguished Teaching Chair in the Humanities at Gettysburg College in Pennsylvania, where he also serves as Chair of the Philosophy Department. He received his bachelor's degree in Physics and Philosophy from the University of Maryland, Baltimore County, and his doctoral degree in Philosophy from the Johns Hopkins University, where he wrote his...
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An Introduction to Formal Logic is rated 4.5 out of 5 by 31.
Rated 5 out of 5 by from What a Great Teacher! This is the second Great Course I've purchased taught by Dr. Steven Gimbel (see also "Take My Course, Please, The Philosophy of Humor") and he continues to be my favorite teacher. Dr. Gimbel presents subject matter in a style that enables all viewers to understand the most abstract of academic principles. He's a brilliant academician who is also a spectacular performer!
Date published: 2019-02-11
Rated 5 out of 5 by from Wish I had a few Dr. Gimbel's when I was a student Dr. Gimbel is an excellent presenter. He keeps the lecture interesting and does an great job presenting the material. I found it easy to follow and most enjoyable. My one recommendation would be to suggest a workbook that would allow the student to work through an assortment of statements. There weren't enough in the course book.
Date published: 2018-12-11
Rated 4 out of 5 by from Thick, but delectable Prof. Guelzo is animated, and very long-winded. At first I found it too hard to keep up, and so I started over, re-hearing the beginning of the first "class" numerous times until I started to feel I had actually heard and understood what he was saying. Once I got to that point, then I found that not only is he setting forth a fundamental definition of history and using the history of history to illustrate the various stages of its development, but he is also inviting us to swoon at the delectable details of this magnificent thing, this human thing, history. And he is profoundly prosaic, and at times I have to stop and do something else. But I love this class, and I have only studied the first class. I give it only four stars because, for me, without digging in deeply, I was not enjoying the surface of his presentation. It was too challenging to follow, at first. Here's a tip: Prof. Guelzo is actually a lot of fun, and is quite funny. But you have to listen closely to catch it...
Date published: 2018-12-05
Rated 5 out of 5 by from Dr. Gimbel is Awesome I found that Dr. Gimbel's combination of subject knowledge and humour made for a very enlightening course which held my attention at all times. I will definitely watch it many times.
Date published: 2018-07-27
Rated 5 out of 5 by from Great examples! Gimbel's presentation is lucid and keeps you interested! There are certainly places where it gets a little dense, but I'm not sure how you'd avoid that in a course this short. Possibly there should be more lessons in the sections toward the end -- that is, they should be covered more slowly.
Date published: 2018-06-04
Rated 5 out of 5 by from This course is superb! With his flawless presentation Professor Gimbel takes you through – step by step – the basics , first of informal logic and then the intricacies of formal logic. This course provides a very accessible introduction for the first time student of logic and is also a great review for the more experienced. I especially enjoyed his two later lectures on logic and mathematics. This course is excellent! Professor Gimbel’s other philosophy course called Redefining Reality is equally as well done.
Date published: 2018-02-19
Rated 1 out of 5 by from Stop the Madness! STOP THE MADNESS! This is really a bad course, I had to stop before the 11th minute mark! Let me start at the very beginning. "IN the beginning God created heaven, and earth." (Genesis 1:1) God created all things. Things didn't evolve themselves from nothing by their own effort. The Bible REJECTS evolution. "You shall be as Gods, knowing (gnostically deciding) good and evil." (Genesis 3:5) "And to Adam he said: Because thou hast hearkened to the voice of thy wife, and hast eaten of the tree, whereof I commanded thee that thou shouldst not eat, cursed is the earth in thy work; with labor and toil shalt thou eat thereof all the days of thy life." (Genesis 3:17) "LET every soul be subject to higher powers. For there is no power but from God: and those that are ordained of God." (Romans 13:1) In other words, the individuals, states and governments legislated laws are subjected to Christ. The allegation that "from the people" or the "consent of the governed" is a claim of authority, is a refusal of being subject to Christ. Psychology doesn't mean the study of human behavior and psychology isn't a science, it is more like a religion with different sects. Psychology: Greek - Psyche and logos Psyche means soul. Logos generally means, any organized body of knowledge contains logos in its name. Hegelian dialectics: Thesis meets antithesis to produce a new thesis (synthesis) which, in turn, will meet another antithesis, and so on. Bertrand Russell's first cause argument fails! God is independent of causes and is independent of the LIMITATION which causes impose. Therefore, God the first cause is FREE from limitation; in other words, God is infinite. There can be only ONE NECESSARY BEING, because a necessary being is infinite. Therefore, the necessary first cause MUST be ONE and INFINITE. (RT. REV. MSGR. PAUL J. GLENN, Ph.D., S.T.D.) LOGIC: is the science and art of correct reasoning. THE TWO MAIN BRANCHES OF LOGIC: FORMAL (minor Logic) and MATERIAL (major logic). In formal logic, the purpose is not to discover truth, but to lead us from one truth to another. THREE KINDS of LOGICAL PROCESSES (Formal Logic): MENTAL ACT: 1) Simple Apprehension VERBAL EXPRESSION: 1) Term MENTAL ACT: 2) Judgment VERBAL EXPRESSION: 2) Proposition MENTAL ACT: 3) Deductive Inference VERBAL EXPRESSION: 3) Syllogism ARGUMENT: 1) Premise 2) Premise 3) Conclusion What is man? A substance that is material, living, sentient and rational. What is the extension of man? All the men who have ever lived, who are now living and who will live in the future. What is an animal? A substance that is material, living and sentient. What is the extension of animal? All the animals (including men, lions, dogs, fish, insects, etc.) that have ever lived, are now living, and that ever will live. It is called the Porphrian Tree because it was invented by the 3rd-century logician Porphyry.
Date published: 2018-02-13
Rated 5 out of 5 by from Informative and convincing presentation I am just a few lessons into the course but so far am very pleased with it.
Date published: 2017-12-17
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