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Foundations of Mathematical Logic
A rigorous introduction to propositional and predicate logic, the methods of mathematical proof, and the foundations on which all later mathematics rests.
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Learning Objectives
On completing this course, the student will be able to:
- Distinguish a proposition from a non-proposition and determine its truth value.
- State and apply the principles of bivalence, non-contradiction, and tertium non datur (the excluded middle).
- Use predicates and the universal (∀) and existential (∃) quantifiers to express mathematical claims with precision.
- Negate quantified statements correctly, including statements with nested quantifiers.
- Define the five logical connectives (¬, ∧, ∨, →, ↔) and articulate their truth conditions.
- Distinguish between necessary and sufficient conditions in a conditional statement.
- Construct truth tables for arbitrary propositional formulas and identify tautologies, contradictions, and contingent formulas.
- Establish logical equivalences using truth tables and the standard equivalence laws (De Morgan, distributivity, contrapositive, etc.).
- Translate between the language of predicates and the language of sets, and exploit the duality between predicate operations and set operations.
- Construct and critique proofs by contradiction (reductio ad absurdum).
- Construct proofs by mathematical induction, formulating the base case and the inductive hypothesis correctly.
- Compute the cardinality of finite sets formed by union, intersection, and complement, using the inclusion–exclusion principle.
- Apply the sum rule and the product rule to count alternatives and structured arrangements.
About the course
This course develops the language of mathematical reasoning from first principles. Beginning with the notion of a proposition and its truth value, it builds the syntax and semantics of propositional logic, extends the formalism to predicates and quantifiers, and culminates in the principal methods of proof: reductio ad absurdum and mathematical induction. The deep correspondence between predicate operations and set operations is made explicit, and the resulting formalism is applied to the cardinality of finite sets and to elementary counting. The course assumes no prior exposure to formal logic; it presupposes only attentive reading and a willingness to argue by precise rules. By its end the student should be able to read, write, and verify mathematical statements with the discipline expected at university level.
