| Area of Mathematics: | Analysis (EA) | ||
| Semester: | 5ο | ||
| Course ID: | 52103 | ||
| Course Type: | Elective | ||
| Teaching hours per week: | Theory: 4 | Practice: 0 | Laboratory: 0 |
| ECTS : | 5 | ||
| Eclass: | |||
| Instructors |
Description
- Classical Propositional Logic
– CPL syntax, propositional variables, propositions and sub-propositions, syntactic tree diagrams and unique readability.
– Truth and evaluation tables, Boolean algebras and algebraic interpretation, general interpretation. Tautologies and contradictions. Logical consistency and logical equivalence. Classical logical equivalences (law of double negation, De Morgan laws, definitions of logical connections by others). Excluded third party principle. Truth functions and logical links. Adequate sets of logical links. Normal forms (CNF, DNF, NNF). Logical validity and satisfiability of propositions.
– Evidence systems for CPL: Hilbert system, Gentzen system and cut rule. Evidence in the Hilbert and Gentzen systems. The production theorem for the Hilbert system. Elimination of the cut rule in the Gentzen system and Decidability. Propositional Theorem Provers. Decidability of the SAT problem for the CPL. The Tableaux method.
– Compactness theorem for CPL
– Algebraization of CPL – Lindenbaum-Tarski algebra. Filters and ideals, first filters and maximum filters. Stone’s Representation for Boole algebra. Coherence and Completeness Theorems.
– Brief reference to non-classical propositional systems and their applications. - Primary Logic (Predicate Logic, PL)
Quantities, atomic variables, predicates and functional symbols. Familiarity with the syntax of the primary Predicate Logic (PL) and translations from natural language. Well-formed formulas. Quantifiers Standpoint and Variable Binding. Bound and free variable display. Open formulas and propositions. PL with an equality Predicate.
Axioms (Hilbert system) and rules (Gentzen system) for primary logic. Skolem normal form and Skolem theorem.
– The production theorem for primary logic.
– Primary structures (models) and interpretations. Logic validity and satisfiability. Coherence and completeness theorems of PL. - Elements of Model Theory & Metatheorems for PL
– Consequences of coherence-completeness of PL: The compactness theorem of primary logic. The Lowenheim-Skolem theorem.
– Primary structures (models) and primary theories – Examples.
– Peano arithmetic and unintended models
– Undecidable theories – Examples.
Bibliography
- Enderton , A Mathematical introduction to Logic , Amazon .
- Mendelson E., Introduction to Mathematical Logic, Chapman & Hall, 6th edition, 2015.
