Lecture: Elementary Metamathematics

Objective

To reason about natural deduction itself (metamathematics). In particular, we show that every sequent (argument) that can be proven is true in the truth table sense (soundness) and every sequent that is true can be proven (completeness).

The proof uses several other metamathematical theorems. Namely that any boolean expression can be converted to an equivalent one in conjunctive normal form and that derived rules may be used in proofs.

The proofs given are informal since the formal proofs require induction which will be covered later in the term.

Background Material

Propositional Calculus
Natural Deduction
Equivalence of Boolean expressions
Disjunctive normal form
Derived rules
Law of the excluded middle

Reading

Chapter 6 of CMU OLI text.

Topics

Derived Rules
Validity of sequents (arguments) as a tautology
Normal Forms
- Definition of disjunctive normal form (DNF) and its construction from a truth table.
- Negative normal form
- Algorithm for converting a boolean expression to an equivalent one in negative normal form (NNF)
- Algorithm for converting a boolean expression in NNF to an equivalent one in DNF
- Conjunctive normal form (CNF)
Soundness
Completeness

Lecture Notes

meta.pptx or meta.pdf

Programs

None.

Resources

Assignments

Do the practice problems in Chapter 6 of the OLI text.
Convert several formulas to DNF using the algorithm in the lecture.
Derive an analogous algorithm for converting to CNF.
Following the example in the lecture, use a truth table to prove the validity of several other rules from natural deduction.
Trace through an example of the proof of completeness.

Created: Apr. 15, 2016 by jjohnson@cs.drexel.edu