PHIL3201: FORMAL AND PHILOSOPHICAL LOGIC

• Fundamentals of Logic (PHIL 2181) or Computational Thinking (COMP1051)

• None.

• None

• To introduce students to philosophically important issues connected to formal and philosophical logic, including a subset of the following: non-classical logics, such as modal and temporal logic; relevance logic; many-valued logics; the developments in early 20th C logic (stemming from Hilbert's programme) leading to Gdel'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 the philosophical issues raised by the formal problems.

• A subset of the following:
• Kripke models for propositional modal and temporal logic.
• Axiomatic proof systems for propositional modal and temporal logic.
• Soundness and completeness results for propositional modal and temporal logic.
• Applications of modal logic to philosophical issues and problems.
• 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.
• Hilbert's problems and the context of Gdel'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.
• Intuitionistic logic and other subclassical logics
• Dynamic logic and other extensions of Modal Logic
• Decidability, translations, and expressive power of logics (first-order, modal, etc.)

Subject-specific Knowledge:

• 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.

Subject-specific Skills:

• 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 Arithmetic.
• Prove meta-level results about non-classical logics.
• Articulate the differences between different foundational approaches to logic and mathematics.
• Explain how modal logic can be applied to philosophical problems and issues
• Present the results of their work to their fellow students.

Key Skills:

• Students will be able to do a selection of the following:
• Present formal logical proofs in a clear, rigorous style.
• Articulate in a clear and concise fashion the historical and philosophical aspects of the material covered.
• Be adequately prepared to go on to do further research in formal logic at the postgraduate level.

• This module will be taught in weekly two-hour seminars, in which core content will be delivered. This content will be supplemented with regular formative and summative assignments, including written reports and short 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 summative exercises.

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