Introduction to Proof Theory Lecture 1 - Propositional logic, semantics and proofs Anupam Das & Thomas Powell University of Copenhagen & Technische Universität Darmstadt European Summer School on Logic, Language, and Information Sofia University 13 August 2018 These slides are available at http://www.anupamdas.com/esslli18. 1 / 51 Outline 1 Introduction, motivation and this course 2 Language of propositional logic 3 Truth table semantics 4 Some puzzles 5 Deductive reasoning 6 A Hilbert-Frege style deductive system 7 The deduction theorem 8 Questions and exercises 9 References 2 / 51 Formally, a proof system defines what a proof may be. This allows not only the study of proofs themselves, but also of what it means to be provable. But why do we study proofs? • Foundational results, which tell us something fundamental about reasoning itself. These lead to... • Applications, which use insights and techniques from proof theory to accomplish something concrete in another discipline altogether. This lecture course: A comprehensive introduction to the field, focussing on classical results but hinting at powerful applications. Wherefore proof theory? Proof theory is the study of mathematical proofs as formal objects. 3 / 51 But why do we study proofs? • Foundational results, which tell us something fundamental about reasoning itself. These lead to... • Applications, which use insights and techniques from proof theory to accomplish something concrete in another discipline altogether. This lecture course: A comprehensive introduction to the field, focussing on classical results but hinting at powerful applications. Wherefore proof theory? Proof theory is the study of mathematical proofs as formal objects. Formally, a proof system defines what a proof may be. This allows not only the study of proofs themselves, but also of what it means to be provable. 3 / 51 • Foundational results, which tell us something fundamental about reasoning itself. These lead to... • Applications, which use insights and techniques from proof theory to accomplish something concrete in another discipline altogether. This lecture course: A comprehensive introduction to the field, focussing on classical results but hinting at powerful applications. Wherefore proof theory? Proof theory is the study of mathematical proofs as formal objects. Formally, a proof system defines what a proof may be. This allows not only the study of proofs themselves, but also of what it means to be provable. But why do we study proofs? 3 / 51 This lecture course: A comprehensive introduction to the field, focussing on classical results but hinting at powerful applications. Wherefore proof theory? Proof theory is the study of mathematical proofs as formal objects. Formally, a proof system defines what a proof may be. This allows not only the study of proofs themselves, but also of what it means to be provable. But why do we study proofs? • Foundational results, which tell us something fundamental about reasoning itself. These lead to... • Applications, which use insights and techniques from proof theory to accomplish something concrete in another discipline altogether. 3 / 51 Wherefore proof theory? Proof theory is the study of mathematical proofs as formal objects. Formally, a proof system defines what a proof may be. This allows not only the study of proofs themselves, but also of what it means to be provable. But why do we study proofs? • Foundational results, which tell us something fundamental about reasoning itself. These lead to... • Applications, which use insights and techniques from proof theory to accomplish something concrete in another discipline altogether. This lecture course: A comprehensive introduction to the field, focussing on classical results but hinting at powerful applications. 3 / 51 Foundational results from the early days Gödel’s incompleteness theorem (1931), informally We cannot prove every true sentence in a given proof system. This result fundamentally reoriented the direction of logic research in the 20th century. Gentzen’s Hauptsatz (1934), informally Every purely logical theorem can be proved analytically, i.e. without making ‘guesses’. The translation of proofs to analytic form, cut-elimination, has become one of the most powerful tools in all of logic. Since then we have come very far... 4 / 51 Applications of proof theory today It is impossible to list the many roles that proof theory plays in the areas relevant to the ESSLLI community, but we list some of the highlights: Computer science • Proof normalisation as the execution of functional programs. • Proof search as logic programming. Linguistics • Language grammars and substructural logics. • Proof-theoretic semantics. Mathematics • Relative strength of mathematical theories. • Extracting bounds and computational content from proofs. Philosophy • Theories of meaning in logic. • Formal accounts of modalities from epistemology. 5 / 51 Who are we? Anupam • Marie Skłodowska-Curie research fellow at the University of Copenhagen. • Computational proof theorist with background in mathematics and philosophy. • Specialises in structural proof theory and interactions with computational complexity. Thomas • Postdoctoral researcher at the Technical University of Darmstadt. • Computational proof theorist with background in mathematics and computer science. • Specialises in proof interpretations, the extraction of computational content from proofs, and higher-type computability theory. 6 / 51 Overview of the course We assume you have seen some logic, but there are no formal prerequisites. • Lecture 1: Propositional logic, semantics and proofs. Syntax of propositional logic, truth table semantics, Hilbert-Frege style proof systems, the deduction theorem. (Anupam and Tom) • Lecture 2: Soundness and completeness. Structural induction, soundness, Lindenbaum’s construction, completeness, the compactness theorem. (Anupam) • Lecture 3: First-order logic and formal theories. Syntax of predicate logic, structures, and Hilbert-Frege style proof systems. First order theories and their models. Peano arithmetic. (Tom) • Lecture 4: Cut-elimination and its consequences. The problem of proof search, the sequent calculus, the cut-elimination theorem, Herbrand’s theorem, Craig interpolation. (Anupam) • Lecture 5: Nonclassical logics and some perspectives. Constructivism and intuitionistic logic, linear logic, modal logic, Lambek calculus, current trends. (Anupam and Tom) 7 / 51 References for this course Proof theory is an incredibly big field (both deep and broad). The following are popular broad-spectrum references: 1 [Buss, 1998] Handbook of Proof Theory (a mathematical approach) 2 [Troelstra and Schwichtenberg, 1996] Basic Proof Theory. (a computational approach) 3 [Schoenfeld, 1967] Mathematical Logic. (a broader overview of logic) 4 Stanford Encyclopedia of Philosophy. (an excellent high-level general reference) http://plato.stanford.edu/ NB: While 1, 2 and 3 are a little dated in terms of topics, the material still remains standard initiation to proof theory. 8 / 51 Sprechen Sie deutsch? We also highly recommend taking a look at original texts: • [Frege, 1879] Begrifsschrit. (the first serious attempt to formalise mathematics in logic, unfortunately erroneously) • [Whitehead and Russell, 1927] Principia Mathematica. (the second serious attempt, notably error-free) • [Gödel, 1931] Über Formal Unentscheidbare Sätze der Principia Mathematica Und Verwandter Systeme I. (the incompleteness theorems) • [Szabo, 1972] The collected papers of Gerhard Gentzen. English translations. (the foundation of structural proof theory) (For those of you who are interested, we will also provide pointers to accessible research articles showcasing what is going on in proof theory today.) 9 / 51 Outline 1 Introduction, motivation and this course 2 Language of propositional logic 3 Truth table semantics 4 Some puzzles 5 Deductive reasoning 6 A Hilbert-Frege style deductive system 7 The deduction theorem 8 Questions and exercises 9 References 10 / 51 Definition (Formulae) The set Form of formulae, written A; B; C etc., are generated as follows: • Any propositional variable p is a formula. • ? is a formula. (falsum) • > is a formula. (veritum) • For a formula A, the expression :A is a formula. (negation) • For formulae A; B, the expression (A _ B) is a formula. (disjunction) • For formulae A; B, the expression (A ^ B) is a formula. (conjunction) • For formulae A; B, the expression (A ! B) is a formula. (implication) Examples of formulae: :(p _ >) ; (p !:q) ; :? Examples of non-formulae: (_p ! ( ; p _ q ! p0 ^ q0 ; ?p> Propositional logic syntax Let us fix a countable set Prop of propositional variables, written p; q; r etc. 11 / 51 Examples of formulae: :(p _ >) ; (p !:q) ; :? Examples of non-formulae: (_p ! ( ; p _ q ! p0 ^ q0 ; ?p> Propositional logic syntax Let us fix a countable set Prop of propositional variables, written p; q; r etc. Definition (Formulae) The set Form of formulae, written A; B; C etc., are generated as follows: • Any propositional variable p is a formula. • ? is a formula. (falsum) • > is a formula. (veritum) • For a formula A, the expression :A is a formula. (negation) • For formulae A; B, the expression (A _ B) is a formula. (disjunction) • For formulae A; B, the expression (A ^ B) is a formula. (conjunction) • For formulae A; B, the expression (A ! B) is a formula. (implication) 11 / 51 Propositional logic syntax Let us fix a countable set Prop of propositional variables, written p; q; r etc. Definition (Formulae) The set Form of formulae,