An Introduction to Formal 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 moving forward.

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.

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.

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!

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.

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. Examine fascinating cases of each.

Turn from informal fallacies, which are flaws in the premises of an argument, to questions of validity, or the logical integrity of an argument. Here, focus on four fallacies to avoid in inductive reasoning: selective evidence, insufficient sample size, unrepresentative data, and the gambler's fallacy.

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 and fallacies are avoided, the conclusions can be impressively reliable.

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.

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."

Truth-functional logic provides the tools to assess many of the conclusions we make about the world. Previously, 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

See how all that you have learned 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.

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 Godel's incompleteness theorems sounded the death knell for this ambitious project.

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.

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. Finish by considering where you've been and what logic is ultimately about.