Logical reasoning : a first course

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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