Logic

Logic studies which arguments are good in virtue of their form alone. The beginner tier builds the vocabulary of argument, validity, and truth-functional connectives from nothing, using truth tables as the only formal machinery. The intermediate tier is the standard first course in symbolic logic: the formal language, natural deduction, tableaux, Hilbert systems, predicate logic with identity, the set theory metatheory needs, and modal logic through the normal systems. The advanced tier is metatheory and philosophy of logic proper, from the Henkin construction and the incompleteness theorems through provability logic, model theory, reverse mathematics, and the substructural and non-classical families.

Beginner

No prior exposure assumed. Every term is defined where it first appears.

What an Argument Is
Premises, conclusions, and the difference between arguing and asserting
6 min
Validity and Soundness
Why a valid argument can have a false conclusion, and what soundness adds
7 min
Needs firstWhat an Argument Is
Deductive and Inductive Support
Two ways premises can back a conclusion, and why only one of them is the subject of this ladder
8 min
Needs firstWhat an Argument Is Validity and Soundness
Statements and Connectives
Negation, conjunction, disjunction, the conditional, and the biconditional
9 min
Needs firstWhat an Argument Is
Truth Tables
Building the table for any formula, and reading validity off it
10 min
Needs firstValidity and Soundness Statements and Connectives
Common Formal Fallacies
Affirming the consequent, denying the antecedent, and why they fail
7 min
Needs firstStatements and Connectives Truth Tables
Basic Inference Rules
Modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism
9 min
Needs firstStatements and Connectives Truth Tables
Translating English into Symbols
Where ordinary English and the truth-functional connectives come apart
11 min
Needs firstStatements and Connectives Truth Tables

Intermediate

Undergraduate level. Formal machinery, standard results, and the objections that stuck.

Propositional Logic and Well-Formed Formulas
The formal grammar of the language, stated recursively, and why recursion is the right tool
12 min
Needs firstTranslating English into Symbols
Semantic and Syntactic Consequence
Two different relations that beginners run together, and the notation that keeps them apart
12 min
Needs firstPropositional Logic and Well-Formed Formulas
Truth-Functional Completeness
Showing that a small set of connectives can express every truth function
13 min
Needs firstSemantic and Syntactic Consequence
The Sixteen Binary Connectives
Enumerating every binary truth function, and showing that the Sheffer stroke and the Peirce arrow are each complete alone
15 min
Needs firstTruth-Functional Completeness
Natural Deduction
Introduction and elimination rules, conditional proof, and reductio
18 min
Needs firstThe Sixteen Binary Connectives
Semantic Tableaux
A decision procedure that searches for a countermodel and reports failure as proof
14 min
Needs firstNatural Deduction
Axiomatic Systems and the Deduction Theorem
Hilbert-style proof, why it is painful to use, and the theorem that makes it usable
18 min
Needs firstSemantic Tableaux
Predicate Logic
Quantifiers, variable binding, scope, and multiply quantified sentences
20 min
Needs firstAxiomatic Systems and the Deduction Theorem
Identity and Definite Descriptions
Adding the identity predicate, numerical claims, and Russell's analysis of 'the'
15 min
Needs firstPredicate Logic
Basic Set Theory for Logicians
Membership, functions, relations, and cardinality, at the depth metatheory needs
16 min
Needs firstIdentity and Definite Descriptions
Probability as an Extension of Logic
The Kolmogorov axioms read as a generalisation of the truth table, with Cox's derivation of them from constraints on plausible reasoning
18 min
Needs firstBasic Set Theory for Logicians
Soundness and Completeness, Informally
What the two metatheorems claim, before anyone tries to prove them
12 min
Needs firstProbability as an Extension of Logic
Modal Logic: Possible Worlds Semantics
The box and diamond, frames, models, and the accessibility relation
22 min
Needs firstSoundness and Completeness, Informally
The Normal Systems K, D, T, B, S4, S5
Each system's characteristic axiom and the frame condition it corresponds to
24 min
Needs firstModal Logic: Possible Worlds Semantics

Advanced

Graduate level. Metatheory, technical variants, and disputes that are still open.

Soundness and Completeness Proofs
Full proofs for propositional logic and the Henkin construction for first-order logic
30 min
Compactness and Löwenheim-Skolem
Two corollaries of completeness, and the limits they place on first-order description
22 min
Needs firstSoundness and Completeness Proofs
Computability, Decidability, and Church's Theorem
Recursive functions, the halting problem, and the proof that first-order validity has no decision procedure
26 min
Needs firstCompactness and Löwenheim-Skolem
Sequent Calculus and Cut Elimination
Gentzen's LK and LJ, the structural rules, and the Hauptsatz that makes proof search possible
28 min
Needs firstComputability, Decidability, and Church's Theorem
Gödel's Incompleteness Theorems
Arithmetization, the fixed-point lemma, and what the theorems do and do not show
32 min
Needs firstSequent Calculus and Cut Elimination
Undefinability and the Theories of Truth
Tarski's theorem, the hierarchy it forces, and Kripke's fixed-point alternative
28 min
Needs firstGödel's Incompleteness Theorems
Provability Logic and Löb's Theorem
Reading the box as "is provable in PA", and the modal system that turns out to be exactly right
30 min
Needs firstUndefinability and the Theories of Truth
Nonstandard Models of Arithmetic
Structures that satisfy every truth of arithmetic and contain numbers larger than every number
26 min
Needs firstProvability Logic and Löb's Theorem
Model Theory: Types, Quantifier Elimination, Categoricity
How the models of a theory can be counted, classified, and in rare cases pinned down
30 min
Needs firstNonstandard Models of Arithmetic
Second-Order and Higher-Order Logic
Quantifying over properties, the categoricity it buys, and the metatheory it destroys
24 min
Needs firstModel Theory: Types, Quantifier Elimination, Categoricity
Reverse Mathematics
Finding the weakest axioms that prove a theorem, by proving the axioms from the theorem
28 min
Needs firstSecond-Order and Higher-Order Logic
Correspondence Theory
Proving that a modal formula is valid on exactly the frames with a given property
20 min
Needs firstReverse Mathematics
Quantified Modal Logic and the Barcan Formulas
What happens when quantifiers meet the box: constant versus variable domains, and the necessity of identity
26 min
Needs firstCorrespondence Theory
Non-Normal Modal Logics
The Lewis systems below K, neighbourhood semantics, and impossible worlds
24 min
Needs firstQuantified Modal Logic and the Barcan Formulas
Intuitionistic Logic and Kripke Semantics
Dropping excluded middle, the BHK reading of the connectives, and monotone Kripke models
24 min
Needs firstNon-Normal Modal Logics
The Curry-Howard Correspondence
Proofs as programs: the isomorphism between natural deduction and the typed lambda calculus, and what normalization turns out to be
28 min
Needs firstIntuitionistic Logic and Kripke Semantics
Substructural Logics
Dropping weakening, contraction, or exchange, and what linear logic and the Lambek calculus buy by doing it
26 min
Needs firstThe Curry-Howard Correspondence
Relevance Logic and Paraconsistency
Rejecting explosion, the variable sharing constraint, and dialetheism
22 min
Needs firstSubstructural Logics
Many-Valued Logics
Łukasiewicz three-valued logic, the Kleene schemes, and the logic of paradox
22 min
Needs firstRelevance Logic and Paraconsistency
Vagueness and the Sorites
Supervaluationism, degree theories, epistemicism, and the problem of higher-order vagueness
26 min
Needs firstMany-Valued Logics
Conditionals and the Triviality Results
Indicative against subjunctive, the Stalnaker-Lewis semantics, Adams' thesis, and why probability cannot be a conditional's meaning
28 min
Needs firstVagueness and the Sorites
Probability Logic and Uncertain Premises
Validity fails to preserve certainty you never had, so the real question is how much uncertainty a valid inference can accumulate
26 min
Needs firstConditionals and the Triviality Results
Inductive Logic and the Carnap Programme
The attempt to make confirmation a matter of logical form alone, the parameter it needed, and the result that universal laws end up confirmed to degree zero
28 min
Needs firstProbability Logic and Uncertain Premises
Free Logic
Quantifying without existence assumptions, and the positive, negative, and neutral variants
18 min
Needs firstInductive Logic and the Carnap Programme
Temporal, Deontic, and Epistemic Logic
Three applied modal families, their characteristic axioms, and their characteristic paradoxes
26 min
Needs firstFree Logic
What Makes a Constant Logical?
Tarski's analysis of consequence, the permutation invariance criterion, and the case for logical pluralism
26 min
Needs firstTemporal, Deontic, and Epistemic Logic