Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Semantics and Proof Theory of the Epsilon Calculus Richard Zach University of Calgary, Canada richardzach.org ICLA 2017 January 6, 2017 Richard Zach Epsilon Calculus ICLA 2017 1 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Outline Introduction 1 The Epsilon Calculus 2 Subclassical Logics (joint work with M. Baaz) 3 Proof Theory for Epsilon Calculus 4 Conclusion 5 Richard Zach Epsilon Calculus ICLA 2017 2 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion What is the Epsilon Calculus? Formalization of logic without quantifiers but with the ε -operator. If A(x) is a formula, then ε x A(x) is an ε -term. Intuitively, ε x A(x) is an indefinite description: ε x A(x) is some x for which A(x) is true. ε can replace ∃ : ∃ x A(x) ⇔ A(ε x A(x)) Axioms of ε -calculus: ◮ Propositional tautologies ◮ (Equality schemata) ◮ A(t) → A(ε x A(x)) Predicate logic can be embedded in ε -calculus. Richard Zach Epsilon Calculus ICLA 2017 3 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Why Should You Care? Epsilon calculus is of significant historical interest. ◮ Origins of proof theory ◮ Hilbert’s Program Alternative basis for fruitful proof-theoretic research. ◮ Epsilon Theorems and Herbrand’s Theorem: proof theory without sequents ◮ Epsilon Substitution Method: yields functionals, e.g., ⊢ ∀ x ∃ y A(x, y) ⇝ ∀ n : ⊢ A(n, f (n)) Interesting Logical Formalism ◮ Trade logical structure for term structure. ◮ Suitable for proof formalization. Other Applications: ◮ Applications in linguistics (choice functions, anaphora). ◮ Connections to Fine’s “arbitrary object” theory. ◮ Propositions-as-types for dynamic linking. Richard Zach Epsilon Calculus ICLA 2017 4 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Epsilon Substitution and Epsilon Theorems Two approaches to consistency proofs in the ε -calculus: Epsilon Substitution: For every epsilon term ε x A(x) , find a 1 numerical substitution; i.e., interpret ε s as particular numbers. ◮ Specific to arithmetical theories. ◮ Developed by Ackermann (1924, 1940), von Neumann (1927) Epsilon Theorems: Eliminate epsilon terms “directly” from a 2 proof using proof transformations. ◮ Can be applied to any quantifier-free theory. ◮ Difficult to extend to arithmetic (induction). ◮ Epsilon theorems have other applications as well (e.g., Herbrand’s theorem) Richard Zach Epsilon Calculus ICLA 2017 5 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion The Epsilon Calculus: Syntax ∧ , ∨ , → , … If A(x) is a formula then ∀ x A(x) and ∃ x A(x) are formulas. If A(x) is a formula, then ε x A(x) is a term. An ε -term p ≡ ε x A(x ; y 1 , . . . , y n ) is the ε -type of an ε -term e if ◮ the y i are all immediate subterms, ◮ every y i has exactly one occurrence, and ◮ e ≡ ε x A(x ; t 1 , . . . , t n ) . Every ε -term a substitution instance of an ε -matrix. Richard Zach Epsilon Calculus ICLA 2017 6 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Extensional Semantics Interpretation: M = � D, Φ , s � ◮ D ≠ ∅ is the domain ◮ M : interpretation of function and predicate symbols ◮ s : Var → D : variable assignment ◮ Φ an extensional choice function Extensional choice function: Φ (S) ∈ S if S ≠ ∅ Valuation of ε -terms ε x A(x) val M, Φ ,s (ε x A(x)) = Φ ( ˆ x[A(x)] M, Φ ,s ) where ˆ x[A(x)] M, Φ ,s = { d ∈ D : M, Φ , s[d/x] ⊨ A(x) } . Richard Zach Epsilon Calculus ICLA 2017 7 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Intensional Semantics Interpretation: M = � D, Φ , s � ◮ D ≠ ∅ is the domain ◮ M : interpretation of function and predicate symbols ◮ s : Var → D : variable assignment ◮ Ψ an intensional choice function Intensional choice function: Ψ (S, p, d 1 , . . . , d n ) ∈ S if S ≠ ∅ Valuation of ε -terms ε x A(x) = p(t 1 , . . . , t n ) with type p = ε x A ′ (x ; y 1 , . . . , y n ) : ε x A(x) M = Φ ( ˆ x[A(x)] M, Ψ ,s , p, t M 1 , . . . , t M n ) Richard Zach Epsilon Calculus ICLA 2017 8 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Axiomatisation of the Epsilon Calculus EC (axioms of the elementary calculus): all propositional tautologies EC ε (the pure epsilon calculus): add to EC all substitution instances of A(t) → A(ε x A(x)) . (1) An axiom of the form (1) is called a critical formula. PC (the predicate calculus), PC ε (extended predicate calculus): EC and EC ε , respectively, together with all instances of A(t) → ∃ x A(x) and ∀ x A(x) → A(t) in the respective language. Richard Zach Epsilon Calculus ICLA 2017 9 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Completeness Elementary calculus/extended predicate calculus complete for intensional semantics EC ε with identity axioms plus ε -identity schema t = u → ε x A(x ; s 1 . . . t . . . s n ) = ε x A(x ; s 1 . . . u . . . s n ) complete for intensional semantics including = EC ε with identity, ε -identity, and ε -extensionality schema ∀ x(A(x) ↔ B(x)) → ε x A(x) = ε x B(x) complete for extensional semantics. Richard Zach Epsilon Calculus ICLA 2017 10 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Embedding PC in EC ε Map ε of expressions in L( PC ε ) to expressions in L( EC ε ) as follows: x ε = x P(t 1 , . . . , t n ) ε = P(t ε 1 , . . . , t ε n ) ( ¬ A) ε = ¬ A ε (A ∨ B) ε = A ε ∨ B ε (A ∧ B) ε = A ε ∧ B ε (A → B) ε = A ε → B ε (ε x A(x)) ε = ε x A(x) ε ( ∃ x A(x)) ε = A ε (ε x A(x) ε ) ( ∀ x A(x)) ε = A ε (ε x ¬ A(x) ε ) Richard Zach Epsilon Calculus ICLA 2017 11 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion The Embedding Lemma A ε is of the form: [A(t) → ∃ x A(x)] ε ≡ A ε (t ε ) → A ε (ε x A(x) ε ) , which is a critical formula. A ε is of the form: [ ∀ x A(x) → A(t)] ε ≡ A ε (ε x ¬ A(x)) → A ε (t ε ) This is the contrapositive of, and hence provable from, the critical formula ¬ A ε (t ε ) → ¬ A ε (ε x ¬ A(x)) Richard Zach Epsilon Calculus ICLA 2017 12 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion The First Epsilon Theorem First Epsilon Theorem If A is a formula without bound variables (no quantifiers, no epsilons) and PC ε ⊢ A then EC ⊢ A . Extended First Epsilon Theorem If ∃ x 1 . . . ∃ x n A(x 1 , . . . , x n ) is a purely existential formula containing only the bound variables x 1 , …, x n , and PC ε ⊢ ∃ x 1 . . . ∃ x n A(x 1 , . . . , x n ), then there are terms t ij such that � EC ⊢ A(t i 1 , . . . , t in ). i Richard Zach Epsilon Calculus ICLA 2017 13 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Herbrand Theorem Herbrand Theorem for ∃ 1 If ∃ x 1 . . . ∃ x n A(x 1 , . . . , x n ) is a purely existential formula PC ⊢ ∃ x 1 . . . ∃ x n A(x 1 , . . . , x n ), then there are terms t ij such that � EC ⊢ A(t i 1 , . . . , t in ). i From the last formula, the original formula can be proved in PC. Can be extended to prenex formulas (by “Herbrandization”) Can be extended to all formulas, since PC proves every formula equivalent to prenex form. Richard Zach Epsilon Calculus ICLA 2017 14 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Extended First Epsilon Theorem Extended First Epsilon Theorem Suppose E(e 1 , . . . , e m ) is a quantifier-free formula containing only the ε -terms e 1 , …, e m , and EC ε ⊢ π E(e 1 , . . . , e m ) , then there are ε -free terms t i j such that n � E(t i 1 , . . . , t i m ) EC ⊢ i = 1 where n ≤ 2 2 ... 23 · cc (π) � stack of 3 · cc (π) 2’s. Richard Zach Epsilon Calculus ICLA 2017 15 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Superintuitionistic Logics In classical logic, ∃ and ∀ are interdefinable Not true in subclassical logics such as intuitionistic logic Epsilon operator seems intuitively related to choice, so intuitionistically suspect So: what happens when ε added to a superintuitionistic logic? Richard Zach Epsilon Calculus ICLA 2017 16 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Interdefinability of ∀ and ∃ In classical logic: ¬∃ x ¬ A(x) ↔ ∀ x A(x) ¬¬ A(ε x ¬ A(x)) ↔ A(ε x ¬ A(x)) → fails in intuitionistic logic Richard Zach Epsilon Calculus ICLA 2017 17 / 39
Introduction Classical Logic Subclassical Logics Proof Theory Conclusion Solution: ε and τ Introduce dual operator τ : τ x A(x) Critical formulas now: A(t) → A(ε x A(x)) and A(τ x A(x)) → A(t) ετ -translation just like ε -translation, except for: ∃ x A(x) ⇔ A(ε x A(x)) ∀ x A(x) ⇔ A(τ x A(x)) Richard Zach Epsilon Calculus ICLA 2017 18 / 39
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