Propositional and Predicate Logic

Winter semester 2024/25

Previous years: ZS 2023/24 EN, ZS 2022/23 EN, ZS 2021/22 EN (by Jakub Bulín), ZS 2018/19 EN (by Martin Pilát), ZS 2015/16 EN, ZS 2014/15 EN.
Lecture in Czech: ZS 2024/24 CZ (Jakub Bulín).

Schedule: Thursday 10:40-12:10, room S3.
Lecturer: Petr Gregor (gregor(at)ktiml.mff.cuni.cz)
Information on the lecture: 2/2 Ex/C NAIL062 in SIS

Plan of the lecture
The plan is only preliminary. I will try to synchronize with the parallel lecture in Czech.

Introduction

03.10. Lecture 1: A brief history, paradoxes, the language of mathematics, relation of syntax and semantics. Basic syntax and semantics of propositional logic. Appendix.

Propositional Logic

10.10. Lecture 2: Semantics in a theory, universality of logical connectives, normal forms. Properties and extensions of theories. Algebra of propositions.
17.10. Lecture 3: Satisfiability problem, SAT-solvers. 2-SAT, Horn-SAT, DPLL. Tableau method: introduction.
24.10. Lecture 4: Tableau method: proofs, systematic tableau. Soundness and completeness. Compactness.
31.10. Lecture 5: Resolution method. Linear resolution, resolution in Prolog.

Predicate Logic

07.11. Lecture 6: Hilbert's calculus in propositional logic. Predicate logic: basic syntax, instances and variants, semantics.
14.11. Lecture 7: Semantics, models, theories. Substructures, expansions, reducts. Theorem on constants.
22.11. Lecture 8: Extensions by definitions. Definability. Tableau method in predicate logic.
28.11. Lecture 9: Systematic tableau, equality. Soundness. Canonical model, completeness.
05.12. Lecture 10: Löwenheim-Skolem theorem, compactness. Prenex form, Skolemisation, Herbrand's theorem. Resolution in predicate logic.
12.12. Lecture 11: Resolution, unification, soundness and completeness. LI-resolution.

Model theory, decidability, incompleteness.

19.12. Lecture 12: Hilbert's calculus in predicate logic. Elementary equivalence, completeness. Isomorphisms, categoricity. Axiomatizability.
09.01. Lecture 13: Decidable theories, recursive axiomatizations. Undecidability of predicate logic. Fixed-point theorem, undefinability of truth. Incompleteness theorems, corollaries.

Recommended literature

J. Bulín, Lecture Notes on Propositional and Predicate Logic, 2024.
M. Pilát, Lecture Notes on Propositional and Predicate Logic, 2020.
A. Nerode, R. A. Shore, Logic for Applications, Springer, 2nd edition, 1997.
P. Pudlák, Logical Foundations of Mathematics and Computational Complexity - A Gentle Introduction, Springer, 2013.
J. R. Shoenfield, Mathematical Logic, A. K. Peters, 2001.
W. Hodges, Shorter Model Theory, Cambridge University Press, 1997.
W. Rautenberg, A concise introduction to mathematical logic, Springer, 2009.

Tutorials
The lecture is accompanied by tutorials on Thursday 12:20-13:50, room S11 (taught by me), or on Monday 10:40-12:10, room S11 (taught by Martin Pilát).

To obtain a credit from tutorials you need at least 140 points out of max 250 points. There will be two tests during the semester (each for 45 minutes). The first one (in the mid of semester) will cover propositional logic, the other (at the end of semester) will cover predicate logic. For each test you can obtain max 100 points, and in each test you must obtain at least 40 points. Apart from that you can obtain max 40 points for SAT solver project and 2x5 points for two homeworks. At the end of semester it will be possible to write a remedial test for each test (during the first week of the exam period). If you did not score at least 40 points, you will need to take these remedial test. There will be no other chances for remedy.

03.10. Tutorial 1: Statements expressed in formulas of various orders.
10.10. Tutorial 2: Statements into formulas (cont.), semantics in a theory, universality, DNF and CNF.
17.10. Tutorial 3: Semantics in a theory. Analysis of theories over finite languages.
24.10. Tutorial 4: Tableau method in propositional logic.
31.10. Tutorial 5: Resolution method in propositional logic.
07.11. Tutorial 6: Examples for the midterm test.
14.11. Tutorial 7: Midterm test. Basic syntax and semantics of predicate logic.
21.11. Tutorial 8: Semantics, substructures, theories, extensions.
28.11. Tutorial 9: Extensions of theories.
05.12. Tutorial 10: Definability. Tableau method in predicate logic.
12.12. Tutorial 11: Tableau method (cont.), prenex form, skolemization, resolution.
20.12. Tutorial 12: Resolution (cont.). Examples of test problems.

Exam
The exam is oral with written preparation. Requirements for the exam correspond to the syllabus of the course in the extent that has been covered in the lecture. A prerequisite for the exam is the credit from the tutorials. Exam dates will be available in SIS. Details about the exam (questions, format, grading) are specified here.

Consultation hours
Thursday from 14:30 or by an (email) appointment.