Module PHIL3201: FORMAL AND PHILOSOPHICAL LOGIC

Department: Philosophy

PHIL3201: FORMAL AND PHILOSOPHICAL LOGIC

Prerequisites

• Introduction to Logic (PHIL1031), Mathematics for Computer Science (COMP1021) and Theory of Computation (COMP2181)

Corequisites

• None

Excluded Combination of Modules

• None

Aims

• To introduce students to philosophical important issues connected to formal logic, including a subset of the following: non-classical logics, such as modal (including temporal logic) logic and relevance logic; the developments in early 20th-century logic (stemming from Hilbert’s programme) leading to Gödel’s proofs of the completeness of first-order logic and the incompleteness of Peano Arithmetic; axiomatizations of set theory, including the independence of the Axiom of Choice and the Continuum Hypothesis; different approaches to philosophy of mathematics and the foundations of mathematics.
• To provide them with the technical means necessary to prove these results for themselves, and the philosophical skills to engage with current philosphical issues raised by the formal problems.
• Aimed at: Students in Philosophy, Mathematics, and Computer Scienc

Content

• Kripke models for propositional modal logic.
• Axiomatic proof systems for propositional modal and temporal logic
• Theoretical and philosophical issues related to quantified modal logic
• Motivations for other non-classical systems
• Proof systems for first-order logic
• Model theory for first-order logic
• A brief history of Hilbert’s problems and the context of Gödel’s theorems
• Completeness Theorems for first-order logic
• Peano Arithmetic and proof by mathematical induction
• Incompleteness Theorems for Peano Arithmetic
• Axiomatizations of set theory
• The independence of the Axiom of Choice and the Continuum Hypothesis
• Platonist, Intuitionist, Formalist, and Structuralists Philosophies of Mathematics

Learning Outcomes

• At the end of the module students should have a grasp of the philosophical significance of various developments in logic and mathematics, such as completeness and incompleteness phenomena; the historical context in which these issues first arose, and the relevant proof and model theory for proving the necessary technical results.

• By the end of the module students should be able to do a selection of the following:
• Prove completeness and canonicity of specific propositional modal logics
• Prove correspondence results between properties of models and specific modal axioms.
• Prove theorems of first-order logic using mathematical induction
• Prove the completeness theorem for first-order logic
• Explain the incompleteness theorem for Peano Arithmatic
• Prove meta-level results about non-classical logics
• Understand independence results in set theory and logic
• Articulate the differences between different foundational approaches to logic and mathematics.

• Students will be:
• Able to write formal logic proofs in a clear, rigorious style.
• Able to articulate in a clear and concise fashion the historical and philosophical aspects of the material covered.
• Adequately prepared to go on to do further research in formal logic at master's level.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• This module will be taught in weekly two-hour seminars, with regular homework assignments and seminar presentations allowing the students to practice the technical skills they are being taught. Teaching and learning methods will support students in achieving the Subject-Specific Skills above. The Subject-Specific Skills will be formally assessed by the end of year exam.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Regular homework assignments

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University