Logical reasoning : a first course

LogicalReasoning_1
	1 • What is \'logic\'?
		1.1 - Aristotle
		1.2 - Formal Logic
		1.3 - Exercises
	2 • Abstract Propositions
		2.1 - Propositions
		2.2 - Abstract propositions and connectives
		2.3 - Recursive definition of propositions
		2.4 - The structure of abstract propositions
		2.5 - Dropping parentheses
		2.6 - Exercises
	3 • Truth Tables
		3.1 - The conjunction
		3.2 - The disjunction
		3.3 - The negation
		3.4 - The implication
		3.5 - The bi-implication
		3.6 - Other notations
		3.7 - Exercises
	4 • The Boolean behaviour of propositions
		4.1 - Truth functions
		4.2 - Classes of equivalent propositions
		4.3 - Equivalency of propositions
		4.4 - Tautologies and Contradictions
		4.5 - Exercises
	5 • Standard Equivalence
		5.1 - Commutativity, Associativity
		5.2 - Intermezzo: ⇒ and ⇔ as meta-symbols
		5.3 - Idempotence, Double negation
		5.4 - Rules with True and False
		5.5 - Distributivity, De Morgan
		5.6 - Rules with ⇒
		5.7 - Rules with ⇔
		5.8 - Exercises
	6 • Working with equivalent propositions
		6.1 - Basic properties of Equivalence
		6.2 - Substitution, Leibniz
		6.3 - Calculations with Equivalence
		6.4 - Equivalence in Mathematics
		6.5 - Exercises
	7 • Strengthening and weakening propositions
		7.1 - Stronger and Weaker
		7.2 - Standard weakenings
		7.3 - Basic properties of \'stronger\'
		7.4 - Calculations with weakening
		7.5 - Exercises
	8 • Predicates and Quantifiers
		8.1 - Sorts of variables
		8.2 - Predicates
		8.3 - Quantifiers
		8.4 - Quantifying many-place predicates
		8.5 - The structure of quantified formulas
		8.6 - Exercises
	9 • Standard equivalence with Quantifiers
		9.1 - Equivalence of predicates
		9.2 - The renaming of bound variables
		9.3 - Domain splitting
		9.4 - One or zero element domains
		9.5 - Domain weakening
		9.6 - De Morgan for ∀ and ∃
		9.7 - Substitution and Leibniz for quantifications
		9.8 - Other equivalences with ∀ and ∃
		9.9 - Tautologies and Contradictions with quantifiers
		9.10 - Exercises
	10 • Other binders of Variables
		10.1 - Predicates vs. Abstract function values
		10.2 - The set binder
		10.3 - The sum binder
		10.4 - The symbol #
		10.5 - Scopes of binders
		10.6 - Exercises
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	11 • Reasoning
		11.1 - The strength and weakness of calculations
		11.2 - \'Calculating\' against \'Reasoning\'
		11.3 - An example from mathematics
		11.4 - Inference
		11.5 - Hypotheses
		11.6 - The use of hypotheses
		11.7 - Exercises
	12 • Reasoning with ∧ and ⇒
		12.1 - Flags
		12.2 - Introduction and Elimination Rules
		12.3 - The construction of an abstract reasoning
		12.4 - The setting up of a reasoning
		12.5 - Exercises
	13 • The structure of the context
		13.1 - Validity
		13.2 - Nested contexts
		13.3 - Other notations
		13.4 - Exercises
	14 • Reasoning with other connectives
		14.1 - Reasoning with ¬
		14.2 - An example with ⇒ and ¬
		14.3 - Reasoning with False
		14.4 - Reasoning with ¬¬
		14.5 - Reasoning by contradiction
		14.6 - Reasoning with ∨
		14.7 - A more difficult example
		14.8 - Case distinction
		14.9 - Reasoning with ⇔
		14.10 - Exercises
	15 • Reasoning with quantifiers
		15.1 - Reasoning with ∀
		15.2 - An example with ∀-intro and ∀-elim
		15.3 - Reasoning with ∃
		15.4 - Alternatives for ∃
		15.5 - An example concerning the ∃-rules
		15.6 - Exercises
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	16 • Sets
		16.1 - Set construction
		16.2 - Universal set and subset
		16.3 - Equality of Sets
		16.4 - Intersection and Union
		16.5 - Complement
		16.6 - Difference
		16.7 - The empty set
		16.8 - Powerset
		16.9 - Cartesian Product
		16.10 - Exercises
	17 • Relations
		17.1 - Relations between sets
		17.2 - Relations on a set
		17.3 - Sepcial relations on a set
		17.4 - Equivalence Relations
		17.5 - Equivalence classes
		17.6 - Composing relations
		17.7 - Equality of Relations
		17.8 - Exercises
	18 • Mappings
		18.1 - Mappings from one set to another
		18.2 - The characteristics of a mapping
		18.3 - Image and source
		18.4 - Special mappings
		18.5 - The inverse function
		18.6 - Composite mappings
		18.7 - Equality of mappings
		18.8 - Exercises
	19 • Numbers and Structures
		19.1 - Sorts of numbers
		19.2 - The structure of the natural numbers
		19.3 - Inductive proofs
		19.4 - Inductive definition of sets of numbers
		19.5 - Strong induction
		19.6 - Inductive definition of sets of formulas
		19.7 - Structural induction
		19.8 - Cardinality
		19.9 - Denumerability
		19.10 - Uncountability
		19.11 - Exercises
	20 • Ordered Sets
		20.1 - Quasi-ordering
		20.2 - Orderings
		20.3 - Linear orderings
		20.4 - Lexicographic orderings
		20.5 - Hasse diagrams
		20.6 - Extreme elements
		20.7 - Upper and lower bounds
		20.8 - Well-ordering and well-foundedness
		20.9 - Exercises