Part 1: Sets, Relations, Functions
Sets, intension vs. extension, definition by enumeration, abstraction,
recursive definitions, union, intersection, complement, difference,
powerset, ordered tuples, relations, common properties of relations,
functions, common properties of functions, generalized quantifiers
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Part 2: Propositional Logic
Syntax and model-theoretic semantics of propositional logic,
truth-tables, semantic tableaux (tree calculus), common notions
(validity, satisfiability, tautology, contradiction, provability, ...),
deductively-valid argument schemes, fallacies, doing proofs using
truth-tables and tableaux, translating from natural language to
propositional logic, informal understanding of relevant metatheorems
(soundness, completeness, decidability, compactness)
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Part 3: First-Order Predicate Logic
Syntax and model-theoretic semantics of first-order predicate logic
with identity, semantic tableaux for first-order predicate logic with
identity, common notions (validity, satisfiability, tautology,
contradiction, provability, ...), deductively-valid argument schemes,
fallacies, translating from natural language to first-order logic, iota
operator and corresponding quantifier, reified situations and time
intervals, informal understanding of limitations of first-order logic,
informal understanding of relevant metatheorems (soundness, completeness, semi-decidability, compactness)
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Part 4: λ-Calculus, Higher-Order Logic and Applicative Categorial Grammar
Conversion rules of λ-calculus, simple type theory, syntax and
model-theoretic semantics higher-order logic (generalized Henkin
models), applicative fragment of categorial grammar, application to
natural language semantics: generalized quantifiers/quantifying
determiners, modification, analysis of small fragments of English or
Portuguese, type-shifting, informal understanding of relevant
metatheorems (soundness, completeness vs. incompleteness, lack of
compactness, Church-Rosser)
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Lecture Notes