Logic Ladder · High school · Grade 11–12

Symbolic Logic

Logic in symbols: translation, truth tables, natural deduction with conditional and indirect proof, then predicate logic with quantifiers, relations and identity, ending with a first look at soundness and completeness.

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55 topics in 9 units. Builds on Traditional Logic. Not sure this is the right course? Find your rung.

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Unit 1 Symbols and Translation

7 topics

Statements as letters joined by five connectives (¬ ∧ ∨ → ↔): reading formulas, finding the main connective, and translating English to symbols and back.

  1. Simple and Compound Statements
  2. The Five Connectives
  3. The Main Connective and Scope
  4. Translating Not, And and Or
  5. Translating Conditionals
  6. Biconditionals and Longer Sentences
  7. From Symbols Back to English

Unit 2 Truth Tables

11 topics

Truth values, full truth tables, tautologies and equivalences, and deciding validity by a full table or a short one.

  1. Truth Values of the Connectives
  2. Computing a Truth Value
  3. Building a Truth Table
  4. Tautologies, Contradictions and Contingent Statements
  5. Logical Equivalence
  6. De Morgan's Laws
  7. Converse, Inverse and Contrapositive
  8. Consistent, Inconsistent and Contradictory
  9. Validity by Truth Table
  10. The Short Truth Table Method
  11. Valid and Invalid Argument Forms

Unit 3 Natural Deduction: Rules of Inference

9 topics

Formal proofs line by line with the eight rules of inference: MP, MT, HS, DS, CD, Simp, Conj and Add.

  1. What a Formal Proof Is
  2. Modus Ponens and Modus Tollens (MP, MT)
  3. Hypothetical and Disjunctive Syllogism (HS, DS)
  4. Simplification, Conjunction and Addition (Simp, Conj, Add)
  5. Constructive Dilemma (CD)
  6. Matching a Rule to Whole Lines
  7. Justifying Proof Lines
  8. Building Short Proofs
  9. Proof Strategy: Working Backward

Unit 4 Rules of Replacement

6 topics

Equivalences that can rewrite any part of a line: DN, DM, Com, Assoc, Dist, Trans, Impl, Equiv, Exp and Taut.

  1. Rules of Replacement
  2. De Morgan and Double Negation (DM, DN)
  3. Commutation, Association and Distribution (Com, Assoc, Dist)
  4. Transposition and Material Implication (Trans, Impl)
  5. Equivalence, Exportation and Tautology (Equiv, Exp, Taut)
  6. Proofs with All Eighteen Rules

Unit 5 Conditional and Indirect Proof

3 topics

Proofs with assumptions: conditional proof (CP), indirect proof (IP), and nested subproofs, which can prove theorems from no premises at all.

  1. Conditional Proof (CP)
  2. Indirect Proof (IP)
  3. Nested Subproofs

Unit 6 Predicate Logic: Translation and Truth

7 topics

Inside the statement: predicates, names and quantifiers (∀ ∃), translating categorical and harder sentences, quantifier negation, and truth in a small world.

  1. Predicates and Names
  2. The Quantifiers ∀ and ∃
  3. Translating A, E, I and O
  4. Translating Harder Quantified Sentences
  5. Quantifier Negation
  6. Truth in a Small World
  7. Counterexample Worlds

Unit 7 Predicate Logic: Proofs

5 topics

The four quantifier rules (UI, EI, UG, EG) and their restrictions, and proofs that use them with CP, IP and QN.

  1. Universal Instantiation and Existential Generalization (UI, EG)
  2. Existential Instantiation (EI)
  3. Universal Generalization (UG)
  4. Proofs with Quantifiers
  5. Conditional and Indirect Proof with Quantifiers

Unit 8 Relations and Identity

6 topics

Predicates of two or more places, the order of quantifiers, properties of relations, and identity for "only", "at least", "at most" and "exactly".

  1. Relational Predicates
  2. Quantifier Order
  3. Properties of Relations
  4. Identity
  5. At Least, At Most and Exactly
  6. Proofs with Identity

Unit 9 Soundness and Completeness

1 topics

Stepping back: the difference between provable and valid, and why a good proof system is both sound and complete.

  1. Soundness and Completeness
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