Logic: lecture notes

Basic notions
	What is logic?
	Arguments and their formulation
	Complex reasoning
	Truth and falsity
	Ambiguity, vagueness, context sensitivity
Validity
	Some set-theoretical notions
	True premises
	Validity as necessary truth preservation
	Other logical properties and relations
	Some important facts about validity
	Validity is not everything
Formality
	Validity and formal validity
	Validity and formal invalidity
	Formal language
	Formal system
	Object language and metalanguage
	Some further set-theoretical notions
The symbols of propositional logic
	Sentence letters
	Sentential connectives
	Brackets
	Expressive completeness
	Truth-functionality and substitutivity
	Formalization in a propositional language
The language L
	Formation rules
	Syntactic trees
	Some basic syntactic notions
	Interpretation
	Truth tables
Logical consequence in L
	Definition of logical consequence
	Some important facts about logical consequence
	Logical consequence as a test for validity
	Other logical properties and relations
	Effective computability
The system Sn
	Derivation
	Rules for negation
	Rules for the conditional
	Rules for conjunction
	Rules for disjunction
Derivability in Sn
	Definition of derivability
	Some important facts about derivability
	Some tips
	Derived rules
	Other natural deduction systems
The system Sa
	Axioms and inference rule
	Deduction theorem
	Some syntactic properties of Sa
	Deductive equivalence between S-p and Sa
	Systems and theories
Consistency, soundness, completeness
	Consistency of Sa
	Correspondence between syntax and semantics
	Soundness of Sa
	Completeness of Sa
	Extension to S-n
Quantification
	Quantified sentences
	A brief historical survey
	Existential import
	Multiple generality
	Definite descriptions
The symbols of predicate logic
	non-logical expressions
	Logical constants and auxiliary symbols
	Other symbols
	Numerical expressions
	Multiple generality and scope ambiguity
	Existence
The language Lp
	Syntax
	Basic semantic notions
	Satisfaction
	Truth
	Logical consequence
	Undecidability
The system Sp
	Axioms and inference rule
	Derivability in Sp
	Validity and derivability
	Deduction theorem and other syntactic results
	Alphabetic variants
Consistency, soundness, completeness
	Consistency of Sp
	Equivalent formulations of soundness and completeness
	Soundness of Sp
	Completeness of Sp
	Final remarks
Undecidability and related results
	Undecidability of Sp
	Gödel numbering
	Effective enumerability of the theorems of Sp
	Effective enumerability of the valid formulas of Lp
	A further corollary
	Recursive axiomatization and recursive decidability
First-order logic
	First-order language
	First-order logic with identity
	First-order theory
	The language of basic arithmetic
	Peano Arithmetic
Theories and Models
	Cardinality
	Compactness theorem
	Löwenheim-Skolem theorem
	Isomorphism
	Categoricity
Gödel\'s incompleteness theorems
	Overview
	The arithmetization of syntax
	The Gödel sentence
	First incompleteness theorem: semantic version
	First incompleteness theorem: syntactic version
	Second incompleteness theorem
Rudiments of modal logic
	Modal operators
	A modal propositional language
	Systems of modal propositional logic
	A modal predicate language
	Systems of modal predicate logic
	Soundness and completeness