The logic of formulas Andre Kornell UC Davis BLAST August 10, - - PowerPoint PPT Presentation

The logic of Σ formulas

Andre Kornell

UC Davis

BLAST August 10, 2018

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 1 / 22

the Vienna Circle

The meaning of a proposition is the method of its verification.

• Moritz Schlick, in Meaning and Verification

Say: a proposition is verifiable iff it admits a method of verification. The negation of a verifiable proposition need not be verifiable.

examples (ignoring experimental error)

1 “a3 + b3 = c3 is solvable” vs “a3 + b3 = c3 is not solvable” 2 “ZFC is inconsistent” vs “ZFC is consistent” 3 “α = 1/137” vs “α = 1/137”

Verifiable propositions are closed under conjunction and disjunction. Verifiable propositions are not closed under negation.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 2 / 22

propositional positivistic logic

connectives

∧ ∨ ⊤ ⊥

definition

A positivistic propositional theory T consists of implications α = ⇒ β, with α and β positivistic. A valuation m models T just in case m | = α implies m | = β, for all α = ⇒ β in T.

completeness theorem

Let T be a positivistic propositional theory, and let φ = ⇒ ψ be an implication between positivistic formulas. Then, T proves φ ⇒ ψ if and

• nly if every valuation that models T also models φ ⇒ ψ.

Proof systems: Hilbert style, sequent calculus, etc.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 3 / 22

meaningful deduction

For φ and ψ positivistic formulas in verifiable propositional constants: φ and ψ are always both verifiable φ ⇒ ψ is generally not verifiable A deduction establishes nothing unless it consists of meaningful formulas. To a finitist: “meaningful” means “verifiable by finite computation” To a realist: “meaningful” means “verifiable by transfinite computation” (Tarski’s definition of truth.) An axiom α = ⇒ β of should be interpreted as a rule of inference.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 4 / 22

deep inference

Example. propositional constants P, Q1, Q2, R. theory T = {P ⇒ Q1 ∨ Q2, Q1 ⇒ R, Q2 ⇒ ⊥}. A deduction from P To R: P Q1 ∨ Q2 R ∨ Q2 R ∨ ⊥ R ∨ R R We define the deductive system RK(T). For each implication α ⇒ β that is a logical axiom or an axiom of T, we have the rule: Φ(α) Φ(β)

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 5 / 22

logical axioms of propositional positivistic logic

⊥ ⇒ ψ φ ⇒ ⊤ φ ∧ ψ ⇒ φ φ ∧ ψ ⇒ ψ φ ⇒ φ ∧ φ ψ ∨ ψ ⇒ ψ φ ⇒ φ ∨ ψ ψ ⇒ φ ∨ ψ (φ ∨ ψ) ∧ χ ⇒ (φ ∧ χ) ∨ (ψ ∧ χ)

Theorem (K)

For positivistic propositional theories T, and implications φ ⇒ ψ, TFAE:

1 every valuation modeling T models φ ⇒ ψ 2 there is a classical derivation of T ⊢ φ ⇒ ψ 3 there is an intuisionistic derivation of T ⊢ φ ⇒ ψ 4 there is a deduction of φ ⇒ ψ in RK(T) 5 every bounded distributive lattice modeling T models φ ≤ ψ Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 6 / 22

coherent logic

∧ ∨ ⊤ ⊥ ∃ A coherent theory T is consist of implications between coherent formulas. φ(t, w) = ⇒ ∃v : φ(v, w) ∃v : ψ(w) = ⇒ ψ(w) φ(w) ∧ ∃v : ψ(v, w) = ⇒ ∃v : φ(w) ∧ ψ(v, w) We define the deductive system RK∃(T). For each implication α ⇒ β that is a logical axiom or an axiom of T, we have the rule: Φ(α(t1, . . . , tn)) Φ(β(t1, . . . , tn))

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 7 / 22

completeness for coherent logic

Theorem (K)

For coherent theories T, and implications φ ⇒ ψ, TFAE:

1 every model of T models φ ⇒ ψ 2 there is a classical derivation of T ⊢ φ ⇒ ψ 3 there is an intuisionistic derivation of T ⊢ φ ⇒ ψ 4 there is a deduction of φ ⇒ ψ in RK∃(T)

The free variables in a deduction can be treated as constant symbols. Compare to the induction rule of primitive recursive arithmetic: φ(0) φ(v, w) ⇒ φ(S(v), w) φ(v′, w)

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 8 / 22

completed, surveyable, definite

Universal quantification is suspect whenever we have an infinite universe. “potential infinity” vs. “completed infinity” Even if the natural numbers form a completed totality, the totality of all sets need not be. (Russell’s paradox.) In the computational framework: A class is “surveyable” just in case there is a (transfinite) process that surveys, i. e., sorts through the totality. (Weaver) Similarly, a class is “definite” just in case quantification takes bivalent propositions to bivalent proposition. (Feferman) completed ≈ surveyable ≈ definite ≈ realist universal quantification

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 9 / 22

universal quantification

∧ ∨ ⊤ ⊥ ∃ ∀ ψ(w) = ⇒ ∀v : ψ(w) ∀v : φ(v, w) = ⇒ φ(t, w) We define the system RI∃,∀ analogously to RK∃.

Theorem (K)

Let T be a theory in coherent logic with universal quantification. For each implication φ ⇒ ψ, TFAE:

1 There is an intuitionistic derivation of T ⊢ φ ⇒ ψ 2 there is a deduction of φ ⇒ ψ in RI∃,∀(T)

We do not have a completeness result for RI∃,∀!

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 10 / 22

surveyability axiom

We express realist universal quantification: ∀v : φ(w) ∨ ψ(v, w) = ⇒ φ(w) ∨ ∀v : ψ(v, w) If we can survey the universe, then we can either verify φ(w) or verify ψ(v, w) for each value of v. We define the deductive system RK∃,∀(T) to be the system RI∃,∀(T) together with the above schema of logical axioms.

Theorem (K)

Let T be a theory in coherent logic with universal quantification. For each implication φ ⇒ ψ, TFAE:

1 every model of T models φ ⇒ ψ 2 there is a classical derivation of T ⊢ φ ⇒ ψ 3 there is a deduction of φ ⇒ ψ in RK∃,∀(T) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 11 / 22

negation

In applications, our primitive predicates are usually decidable. ⊤ ⇒ P(w) ∨ ˜ P(w) P(w) ∧ ˜ P(w) ⇒ ⊥ In this case, RK∃,∀(T) is essentially a classical system. classical logic ⇐ ⇒ decidable primitive predicates ∧ surveyable universe In applications, the universe is usually not surveyable/definite/completed, but the universe is rather the union of surveyable subclasses. (sets) ∀v : ∀v ∈ t :

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 12 / 22

positivistic logic

definition

The class of Σ formulas is the closure of the atomic formulas for the forms φ ∧ ψ φ ∨ ψ ∃v : φ ∀v ∈ t : ψ The system RKΣ(T) has the axiom schemes as RK∃(T) together with the following logical axioms: ⊤ ⇒ x ∈ y ∨ x ∈ y x ∈ y ∧ y ∈ x ⇒ ⊥ ψ(w) = ⇒ ∀v ∈ t : ψ(w) ∀v ∈ t : φ(v, w) = ⇒ φ(s, w) ∀v ∈ t : φ(w) ∨ ψ(v, w) = ⇒ φ(w) ∨ ∀v ∈ t : ψ(v, w) ∀v ∈ t : v ∈ t ∨ φ(v, w) = ⇒ ∀v ∈ t : φ(v, w)

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 13 / 22

completeness

Theorem (K)

Let T be a set of implications between Σ formulas, and let φ and ψ be Σ

• formulas. TFAE:

1 every model of T is a model of φ ⇒ ψ 2 there is a classical deduction of T ⊢ φ ⇒ ψ. 3 there is a deduction of φ ⇒ ψ in RKΣ(T).

The equivalence (2) ⇔ (3) is a theorem of IΣ1.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 14 / 22

equality

In applications, equality is usually available.

1 ⊤ =

⇒ x = x

2 x = y =

⇒ y = x

3 x = y ∧ y = z =

⇒ x = z

4 φ(x, w) ∧ x = y =

⇒ φ(y, w)

5 ⊤ =

⇒ x = y ∨ x = y

6 x = y y = x =

⇒ ⊥

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 15 / 22

set theory T1

1 (∀z ∈ x : z ∈ y) ∧ (∀z ∈ y : z ∈ x) =

⇒ x = y

2 z ∈ {x, y} ⇐

⇒ z = x ∨ z = y

3 z ∈ x ⇐

⇒ ∃y ∈ x : z ∈ x

4 y ∈ ℘(x) ⇐

⇒ ∀z ∈ y : z ∈ x

5 ⊤ ⇒ ∃Y ∈ ℘(X): ∀x ∈ X : x ∈ Y ↔ φ 6 ∃z ∈ x : ⊤ =

⇒ ∃z ∈ x : ∀y ∈ x : z ∈ y

7 ∀x ∈ X : ∃y : φ =

⇒ ∃Y : ∀x ∈ X : ∃y ∈ Y : φ

8 ⊤ =

⇒ ∃x : Ord(x) ∧ x ≈ z

9 ⊤ =

⇒ ∃V : Mod(V ) ∧ z ∈ V

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 16 / 22

subuniverses

Mod(V ) abbreviates: (∀x ∈ V : ∀y ∈ x : y ∈ V ) ∧ (∀x ∈ V : ∀y ∈ V : {x, y} ∈ V ) ∧ (∀x ∈ V :

• x ∈ V ) ∧ (∀x ∈ V : ℘(x) ∈ V )

∧ (∀x ∈ V : ∀y ∈ ℘(V ): x ≈ y → y ∈ V )

corollary*

The deductive system RKΣ(T1) deduces Mod(V ) ⇒ φV for every axiom φ of ZFC.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 17 / 22

the truth predicate

Well-known Σ truth predicate for Σ formulas: T(φ ∧ ψ) ⇐ ⇒ ∃w : Wit(w, φ)

1 T(φ ∧ ψ) ⇐

⇒ T(φ) ∧ T(ψ)

2 T(φ ∨ ψ) ⇐

⇒ T(φ) ∨ T(ψ)

3 T(∃v : ψ(v)) ⇐

⇒ ∃a: T(ψ(a))

4 T(∀v ∈ b: φ(v)) ⇐

⇒ ∀a ∈ b: T(φ(a))

5 T(P(t1, . . . , tn))

⇐ ⇒ ∃a1 : · · · ∃an : P(a1, . . . , an)∧T(a1 = t1)∧. . .∧T(an = tn)

6 T(a = F(t1, . . . , tn))

⇐ ⇒ ∃b1 : · · · ∃bn : a = F(b1, . . . , bn) ∧ T(b1 = t1) ∧ . . . ∧ T(bn = tn)

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 18 / 22

metamathematics is mathematics

Define the theory T2 to be the theory T1 together with the axiom T(φ) ∧ DT1(φ ⇒ ψ) = ⇒ T(ψ)

proposition (K)

The theory T2 is finitely axiomatizable, and RKΣ(T2) deduces T(φ) ∧ DT2(φ ⇒ ψ) = ⇒ T(ψ). Thus, T2 describes a self-contained incompletable universe of pure sets, that includes nested transitive models of ZFC. The object language and the metalanguage coincide.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 19 / 22

primitive recursive set theory

The primitive recursive set functions (Jensen and Karp) are obtained using the following recursion scheme: F(v, w) = H

• {F(r, w) | r ∈ v}, v, w
• Rathjen defined a theory PRS of primitive recursive set functions,

analogous to Skolem’s theory PRA of primitive recursive functions. Both PRS and PRA can be formulated as positivistic theories. Define a positivistic theory PRS+ by adding to PRS the Σ-collection schema, and the well-ordering principle. PRS+ is essentially a weak fragment of KP + WOP. It is strong enough to formalize elementary constructions.

Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 20 / 22

the system LK∞ω

Introduction Rules for and (∃φ ∈ K) Γ, φ ⊢ ∆ Γ, K ⊢ ∆ (∀φ ∈ K) Γ ⊢ ∆, φ Γ ⊢ ∆, K (∀φ ∈ K) Γ, φ ⊢ ∆ Γ, K ⊢ ∆ (∃φ ∈ K) Γ ⊢ ∆, φ Γ ⊢ ∆, K Cut Rule Γ ⊢ ∆, φ Γ′, φ ⊢ ∆′ Γ, Γ′ ⊢ ∆, ∆′

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model completeness principle

Let T be a set of LK∞ω formulas. Conclude that T is inconsistent for LK∞ω or that T has a model.

set completeness principle

Let A be a transitive set. Let T be a set of LK∞ω formulas in the vocabulary {=, ∈, S} and parameters from A. Assume that T includes basic axioms describing the structure (A, =, ∈). Conclude that T is inconsistent for LK∞ω or that there exists a set B ⊆ A such that (A, =, ∈, B) is a model of T.

theorem (K)

Work in PRS+ together with cut-elimination* for LK∞ω. Each may be proved using the other as an axiom:

1 the model completeness principle 2 the set completeness principle together with the axiom of infinity Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 22 / 22