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