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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Reviews

An Introduction to Formal Logic is rated 4.6 out of 5 by 40.
Rated 5 out of 5 by from This professor explains very clearly Just in lesson 16. Challenging for me. Love it. Professor's quirky humor and lucid explanations keep my head above water. Thanks Professor Gimbel.
Date published: 2019-07-20
Rated 2 out of 5 by from Beware! This course begins in a user-friendly fashion, but after about Lecture 7, it quickly becomes very challenging. The ordinary person without any background in logic, such as my husband and myself, simply cannot understand it.
Date published: 2019-07-19
Rated 5 out of 5 by from An Introduction to Formal Logic Professor Gimbel is knowledgable and concise in his presentations. Someone without basic knowledge in logic might not comprehend everything in lesson ten concerning truth-functional logic. Much difficult material was covered quickly. The advantage of the course is that one can stop the video and review the problematic parts.
Date published: 2019-07-07
Rated 5 out of 5 by from Having some fun will learning formal logic I like his reference to the wizard of oz and his jokes subtle and otherwise always got me laughing. The few commentaries on formalism, logicism, intuitionism, Hilbert and Godel were especially good, and even a moment of dark comedy related to Godel. Not for anyone who does not enjoy an half hour or so of syntactic detail, so be warned that reference to lines in a natural deductive proof that have scrolled off screen can often be annoying. One thing for sure, I learned a lot about truth functional combinations, fundamental form and then categorical or basic form, and that is certainly saying something, especially coming from a fuzzy prof with an inclination to tell fuzzy jokes while explaining fuzzy logic.
Date published: 2019-05-29
Rated 5 out of 5 by from Excellent Course! This course definitely lives up to its name -- it is truly a great one! Every new idea introduced is clearly illustrated by a thoughtfully chosen example. And the examples are just right -- as simple as they can get, without forfeiting any of the substance of what needs to be explained. (One example of many: DeMorgans identities can be thought of through a little story involving `Bob enjoying his coffee with cream and ...or... sugar by the coffee machine'; think through this example, and you will likely never forget what DeMorgans laws are all about. (And, incidentally, if you ever wondered what the real, real difference between Captain Morgan and DeMorgan is, be sure to watch the lectures on deductive validity.).) Moreover, the exercises are just as carefully chosen as the lecture examples. Be sure to spend time with those - working through them will ensure that you can confidently follow the subsequent lectures. (Finally, I should mention that Dr. Steve clearly has some serious acting talent. Not only is the content of the course right on the spot, but the presentation is also superb. But be glad that the Prof. didn't follow an acting career because that would be a huge loss to philosophy and logic.)
Date published: 2019-05-27
Rated 5 out of 5 by from Complex subject well explained Formal logic is a difficult subject for most of us, including myself with a science degree. Professor Gimbel made this subect overview, both interesting and entertaining. Building from a general description through the stages of logic using clear examples, interspersed with humor, it was of great benefit to me.
Date published: 2019-05-23
Rated 5 out of 5 by from Review #2? Not sure if more than one review is allowed, we shall see, but wish to conclude by saying that it was an absolute privilege and pleasure to observe Professor Gimbell's two dozen lectures on Formal Logic, providing "good rigorous ways of defining what it is that we should believe" and, by extension, "who we are." I was also pleasantly reminded of my college years where it was my experience that classes in the Philosophy Department (e.g. Logic, Linguistics and MetaLogic) were marked by humility and wisdom as opposed to the arrogance within the Computer Science Department, albeit the Department of my Major. Also totally enjoyed the Professor's priceless and ever front-and-center (East Coast?) sense of humor.
Date published: 2019-05-19
Rated 5 out of 5 by from Seemed more than an introduction I purchased this course based on the description. While Professor Gimbel introduced the various methods for proving or disproving arguments, I found the later lessons to be quite complex. Especially when defining parameters as all or some, and the addition of operators. I did enjoy some of his anecdotal arguments whereby he shared some humorous results and warnings. I will say some of what he defined as arguments that could be proved, were not the type I would bother with determining the proof. Unless I really disliked the person presenting the argument, and felt mean spirited. Having the DVD, I will be reviewing the lessons a few more times.
Date published: 2019-05-17
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