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 “ a 3 + b 3 = c 3 is solvable” vs “ a 3 + b 3 = c 3 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 only 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 , Q 1 , Q 2 , R . theory T = { P ⇒ Q 1 ∨ Q 2 , Q 1 ⇒ R , Q 2 ⇒ ⊥} . A deduction from P To R : P Q 1 ∨ Q 2 R ∨ Q 2 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 ( t 1 , . . . , t n )) Φ( β ( t 1 , . . . , t n )) 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 RK Σ ( 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 RK Σ ( T ) . The equivalence (2) ⇔ (3) is a theorem of IΣ 1 . 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 T 1 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 RK Σ ( T 1 ) deduces Mod ( V ) ⇒ φ V for every axiom φ of ZFC. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 17 / 22