An Introduction to Formal Logic

Influential philosophers argue that humans are purely rational beings; cognitive studies show people are wired to accept false beliefs; reviewing some built-in biases; learning that logic is the perfect corrective; what one will learn moving forward.

Practicing finding the logical arguments hidden in statements by looking for indicator words that either appear explicitly or are implied; how to identify the structure of an argument; focusing on whether it is deductive or inductive.

Exploring four common logical fallacies; circular reasoning uses a conclusion as a premise; begging the question invokes the connotative power of language as a substitute; equivocation changes the meaning of terms; distinction without a difference.

Deepening the understanding of the fallacies of informal logic; examining five additional reasoning errors that include appeal to authority, appeal to common opinion and appeal to tradition; testing oneself with a series of examples.

Considering five fallacies that often arise when one tries to reason their way from cause to effect; the post hoc fallacy; continuing with other fallacies such as neglect of a common cause and causal oversimplification.

Recognizing common diversionary fallacies in a discussion; ad hominem attacks try to undermine the arguer; straw man tactics substitute a weaker argument for a stronger one; red herrings introduce an irrelevant subject; examining cases of each.

Turning from informal fallacies to questions of validity; focusing on four fallacies to avoid in inductive reasoning; these include selective evidence, insufficient sample size, unrepresentative data and the gambler's fallacy.

Probing two activities that cannot exist without induction, which are polling and scientific reasoning; neither provides absolute proof in its field of analysis; if faults and fallacies are avoided, the conclusions can be reliable.

Examining what makes deductive arguments valid; learning that it all starts with Aristotle, who devises rigorous methods for determining with absolute certainty whether a conclusion must be true given the truth of its premises.

Evaluating sentences whose truth cannot be proved by Aristotle's system; learning about truth-functional logic; it was pioneered in the 19th and 20th centuries by Gottlob Frege; this approach addresses the behavior of truth-functional connectives.

Truth-functional logic provides the tools to assess many of the conclusions humans make about the world; truth tables map out the implications of an argument's premises; this technique has a lot of versatility.

Using truth tables to test the validity of famous forms of argument called modus ponens and its fallacious twin, affirming the consequent; untangling the logic of more complex arguments; the point of logic is to discover what is rational.

Truth tables are not consistently user-friendly; some arguments defy their analytical power; natural deduction proofs are another technique; treating this proof as a game; strategies for a successful play.

Proving arguments with natural deduction by studying nine equivalences; these are sentences that are truth-functionally the same; double negation asserts that a sentence and its double negation are equivalent; examples.

Completing the system of natural deduction; adding a new category of justification; justified assumption; how this concept is used in conditional and indirect proofs; evaluating the validity of arguments from everyday life.

Learning about the two approaches to logic; Aristotle's categorical method and truth-functional logic; the hybrid approach; first-order predicate logic; getting inside sentences to map the logical structure within them.

Truth tables cannot demonstrate validity in first-order predicate arguments; natural deduction proofs; four additional rules of inference and one new equivalence; reviewing these procedures and examples.

Studying 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; the method of expansion.

First-order predicate logic; expanding into relations; considering an example where the relation is obvious; proving subtler cases.

The ability to validate identity; these proofs are known as equivalence relations; they have three important criteria; equivalence is reflexive, symmetric and transitive; testing the techniques by validating the identity of an unknown party.

How logic relates to mathematics and vice versa; tracing the origin of deductive logic to the ancient geometrician Euclid; the development of non-Euclidean geometries in the 19th century; the puzzle this posed for mathematicians.

The logical consistency of mathematics can be reduced to basic arithmetic; the work of David Hilbert, Georg Cantor, Gottlob Frege, Bertrand Russell and others; how Kurt Godel's incompleteness theorems ended a project.

Two new operators; first-order predicate vocabulary; a symbol for possibility and for necessity; dealing with modal concepts; they are contingent or necessary truths; how philosophers have used modal logic to investigate ethical obligations.

Denying the central claim of classical logic which states that a proposition is either true or false; new and useful types of reasoning called multivalued logic and fuzzy logic; what logic is ultimately about.