PH21015 Logic II: Metatheory
Introductory description
This module will develop the metatheory of propositional and firstorder logic. Our primary goal will be to show that a proof system similar to that presented in Logic I is sound (i.e. proves only logically true sentences) and complete (proves all logically true sentences). In order to better understand how we prove things about (as opposed to within) a proof system, we will first study the syntax, semantics, and proof theory of propositional logic. We will then consider Tarski's definitions of satisfaction and truth in a model and proceed to develop the Henkin completeness proof for firstorder logic. Other topics covered along the way will include countable versus uncountable sets, the compactness theorem, and the expressive limitations of firstorder logic. PH210 is recommended as a prerequisite for PH340 (Logic III: Incompleteness and Undecidability), PH341 (Modal Logic), and PH345 (Computability Theory).
Module aims
To expose students to the basic metalogical notions of soundness and completeness. A natural deduction system for propositional and first order logic will be introduced and proven to be sound and complete. Along the way, basic mathematical tools needed for proving these results will be developed. These include elementary set theory and inductive proofs and definitions.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Week 1: Introduction, set theory, inductive definitions
 Week 2: Syntax and semantics of propositional logic
 Week 3: Natural deduction
 Week 4: Soundness and completeness of propositional logic
 Week 5: Syntax and semantics of firstorder logic
 Week 7: Semantic notions, theories and models
 Week 8: Natural deduction and soundness for firstorder logic
 Week 9: Completeness for firstorder logic
 Week 10: The compactness theorem and applications
Learning outcomes
By the end of the module, students should be able to:
 demonstrate knowledge of the Soundness and Completeness Theorems for propositional and firstorder logic and related technical results and definitions (subject knowledge and understanding)
 understand the significance these concepts and results have for logic and mathematics (cognitive skills)
 use and define concepts with precision, both within formal and discursive contexts (key skills)
 write precise mathematical proofs (subject specific skills)
Indicative reading list
Our primary text will be a version of the Open Logic text customised for PH210.
The same material is also covered in a number of other sources including:
Logic and Structure, 5th edition by Dirk van Dalen, Springer Verlag, 2008.
Much of the material we will be covering is also presented at a more elementary level in chapters 15–19 of the textbook for PH136 (Logic I):
Language, Proof, and Logic by Jon Barwise and John Etchemendy, CSLI Publications, 2002.
Students desiring additional background on problem solving techniques are also encouraged to
obtain:
How to Prove It by Daniel J. Velleman, Cambridge University Press, 2006.
View reading list on Talis Aspire
Subject specific skills
Demonstrate knowledge of the Soundness and Completeness Theorems for propositional and firstorder logic and related technical results and definitions
Understand the significance these concepts and results have for logic and mathematics
Use and define concepts with precision, within both formal and discursive contexts
Transferable skills
Understand definitions and proof techniques from metalogic applicable in other domains—e.g. such as mathematical and structural induction, basic set theoretic operations and constructions
The ability to comprehend and construct precise mathematical proofs
Study time
Type  Required 

Lectures  9 sessions of 3 hours (18%) 
Seminars  9 sessions of 1 hour (6%) 
Private study  114 hours (76%) 
Total  150 hours 
Private study description
No private study requirements defined for this module.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group B7
Weighting  Study time  

Inperson Examination  100%  
2 hour exam

Feedback on assessment
Discussion and feedback on exercises during seminar.
Courses
This module is Core for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy

UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 2 of GV18 Mathematics and Philosophy with Intercalated Year
 Year 2 of GV18 Mathematics and Philosophy with Intercalated Year
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Optional for:

UECALM1D Undergraduate Economics, Politics and International Studies
 Year 2 of LM1D Economics, Politics and International Studies
 Year 2 of LM1D Economics, Politics and International Studies

UHIAV1V8 Undergraduate History and Philosophy (with Year Abroad and a term in Venice)
 Year 3 of V1V8 History and Philosophy (with Year Abroad and a term in Venice)
 Year 4 of V1V8 History and Philosophy (with Year Abroad and a term in Venice)
 Year 3 of UHIAV1V7 Undergraduate History and Philosophy (with a term in Venice)

UPHAV700 Undergraduate Philosophy
 Year 2 of V700 Philosophy
 Year 2 of V700 Philosophy
 Year 3 of V700 Philosophy
 Year 3 of V700 Philosophy
 Year 4 of UPHAV701 Undergraduate Philosophy (wiith Intercalated year)
 Year 4 of UPHAV702 Undergraduate Philosophy (with Work Placement)

UPHAVQ72 Undergraduate Philosophy and Literature
 Year 2 of VQ72 Philosophy and Literature
 Year 3 of VQ72 Philosophy and Literature

UPHAV7ML Undergraduate Philosophy, Politics and Economics
 Year 2 of V7ML Philosophy, Politics and Economics (Tripartite)
 Year 2 of V7ML Philosophy, Politics and Economics (Tripartite)
 Year 2 of V7ML Philosophy, Politics and Economics (Tripartite)

UPHAV7MM Undergraduate Philosophy, Politics and Economics (with Intercalated year)
 Year 4 of V7MQ Philosophy, Politics and Economics (Bipartite) with Intercalated Year
 Year 4 of V7MH Philosophy, Politics and Economics  Economics/Philosophy Bipartite (Economics Major) (with Intercalated year)
 Year 4 of V7MF Philosophy, Politics and Economics  Economics/Politics Bipartite (Economics Major) (with Intercalated year)
 Year 4 of V7MI Philosophy, Politics and Economics  Philosophy/Economics Bipartite (Philosophy Major) (with Intercalated year)
 Year 4 of V7MJ Philosophy, Politics and Economics  Philosophy/Politics Bipartite (with Intercalated year)
 Year 4 of V7MG Philosophy, Politics and Economics  Politics/Economics Bipartite (Politics Major) (with Intercalated year)
This module is Option list A for:

UPHAVL78 BA in Philosophy with Psychology
 Year 2 of VL78 Philosophy with Psychology
 Year 3 of VL78 Philosophy with Psychology
 Year 4 of UPHAVL79 BA in Philosophy with Psychology (with Intercalated year)
This module is Option list B for:
 Year 2 of UHIAV1V5 Undergraduate History and Philosophy

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 2 of G105 Mathematics (MMath) with Intercalated Year
 Year 4 of G105 Mathematics (MMath) with Intercalated Year

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G100 Mathematics
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 3 of G100 Mathematics
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 2 of G106 Mathematics (MMath) with Study in Europe
 Year 3 of G106 Mathematics (MMath) with Study in Europe
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

UMAAG101 Undergraduate Mathematics with Intercalated Year
 Year 2 of G101 Mathematics with Intercalated Year
 Year 4 of G101 Mathematics with Intercalated Year

UPHAVQ72 Undergraduate Philosophy and Literature
 Year 2 of VQ72 Philosophy and Literature
 Year 3 of VQ72 Philosophy and Literature
 Year 4 of UPHAVQ73 Undergraduate Philosophy and Literature with Intercalated Year
This module is Option list C for:
 Year 3 of UHIAV1V5 Undergraduate History and Philosophy
 Year 4 of UHIAV1V6 Undergraduate History and Philosophy (with Year Abroad)
This module is Option list D for:

UHIAV1V5 Undergraduate History and Philosophy
 Year 2 of V1V5 History and Philosophy
 Year 3 of V1V5 History and Philosophy
 Year 4 of UHIAV1V6 Undergraduate History and Philosophy (with Year Abroad)
 Year 2 of UHIAV1V7 Undergraduate History and Philosophy (with a term in Venice)
This module is Option list E for:

UPHAV7MW Undergraduate Politics, Philosophy and Law
 Year 2 of V7MW Politics, Philosophy and Law
 Year 2 of V7MW Politics, Philosophy and Law