Provability. Explicit Proofs. Reflection. · Formulas of LP are built as the usual propositional formulas with an additional formation rule: whenever F is a formula and t is a proof

Provability. Explicit Proofs. Reflection.

Provability. Explicit Proofs. Reflection.

Elena Y. Nogina

The City University of New York

Logic Colloquium 2015. Helsinki.

Elena Y. Nogina Provability. Explicit Proofs. Reflection.

Godel’s Modal Approach to Provability

Godel in 1933 suggested a provability reading of modal logic S4,which is axiomatized over the classical logic by the following list ofpostulates:

2(F → G)→ (2F → 2G) Deductive Closure/Normality2F → 22F Positive Introspection/Transitivity2F → F Reflectionand the Necessitation Rule ` F ⇒ ` 2F .


Godel’s Provability Semantics for Modality

Godel considered the interpretation of 2F as the formal provabilitypredicate

F is provable in Peano Arithmetic PA

and noticed that this semantics is inconsistent with S4.Indeed, 2(2F → F ) can be derived in S4. On the other hand,interpreting 2 as the predicate of formal provability in PA and F asfalsum ⊥, converts this formula into the false statement that theconsistency of PA is provable in PA.


Provability spills over to Non-Standard ”Proofs”

Let Proof (x ,F ) be the Godel proof predicate

x is a proof of F in PA.

Godel numbers’ notations are suppressed when safe.

Let also Provable F be the Godel provability predicate∃xProof (x ,F ) stating that

F is provable in PA.

Peano Arithmetic PA cannot distinguish between standard andnonstandard numbers; given ∃xProof (x ,F ), x may be anonstandard number, hence not a code of any derivation in PA. Itmeans that Provable F → F can fail in a model, and hence be notderivable in PA.


Calculus without Semantics and Provability Semanticswithout Calculus

Indeed, consider a theory T = PA + Provable ⊥. T is consistent,since PA does not prove ¬Provable ⊥. Hence T has a model M inwhich Provable ⊥ holds, but ⊥ does not.So, the formal provability interpretation of S4 did not work; theprovability calculus was left without a semantics and the provabilitysemantics was left without a calculus which opened two problems:

1. Find a precise provability semantics for S4;2. Find a modal logic of formal provability Provable.

Problem 2 was solved in 1976 by Solovay, who established thecompleteness of Godel-Lob logic GL with respect to the formalprovability in arithmetic PA.


Alternative Godel’s Format for Provability

On solving Problem 1.

In 1938, in his Vienna’s lecture Godel outlined a way to provide aprovability semantics for S4; modality there should be readexplicitly as

t is a proof of F in Peano Arithmetic PA.

This Godel’s lecture remained unpublished until 1995. By this time,Problem 1 had found its solution in Artemov’s Logic of Proofs LP.


Godel-Lob Logic GL of Formal Provabilityis given by the following list of postulates:

1. Axioms and rules of classical propositional logic2. 2(F→G)→ (2F→2G) Deductive Closure/Normality3. 2F→22F Verification/Transitivity4. 2(2F→F )→ 2F Lob Axiom

5. Necessitation Rule ` F` 2F

Formal provability interpretation of a modal language is amapping ∗ from the set of modal formulas to the set ofarithmetical sentences such that ∗ agrees with Boolean connectivesand constants and

(2G)∗ = Provable G∗.

Solovay’s Completeness Theorem.GL ` F iff for all provability interpretations ∗, PA ` F ∗.


Artemov’s Logic of Proofs LP

Proof terms in LP are built from proof variables x , y , z , . . . andproof constants a, b, c, . . . by means of two binary operationsapplication ‘·’ and sum ‘+’, and one unary operation proofchecker ‘!.’

Formulas of LP are built as the usual propositional formulas withan additional formation rule: whenever F is a formula and t is aproof term, t:F is a formula which is read as

t is a proof of F .


Axioms and Rules of the Logic of Proofs LP∅

Axioms and rules of classical logics:(F → G) → (t:F → [s ·t]:G) Applicationt:F → !t:(t:F ) Proof Checkers:F → [s +t]:F , t:F → [s +t]:F Sumt:F → F Explicit Reflection

Application. If s is a proof of F → G and t is a proof of F , thens · t is a proof of G .

Proof Checker. Given a proof t, the proof checker returns a proof!t that t is a proof of F .

Sum. ‘s + t’ of proofs s and t is a proof which proves everythingproven by either s or t


All Proof Predicates are assumed Normal.

A proof predicate is a provably decidable formula Proof (x , y) thatenumerates all theorems of PA:

PA ` ϕ iff Proof (n, ϕ) for some n.

A proof predicate Proof (x , y) is normal, if1. for each k the set

T (k) = {l | Proof (k, l)}

is finite and the function from k to T (k) is computable;2. for any k and m there is a natural number n such thatT (k) ∪ T (m) ⊆ T (n).

Prime example: Godel’s proof predicate.


Provability Semantics of the Logic of Proofs

Arithmetical interpretation ∗ is determined by

1. a normal proof predicate Proof (x , y) with natural operations onproofs for ·, +, and !;

2. an interpretation of proof variables and constants by numerals;

3. an interpretation of propositional variables by arithmeticalsentences.

Interpretations respect Boolean connectives and

(p:F )∗ = Proof (p∗,F ∗).


Arithmetical Soundness of LP∅

Postulates of LP∅ are provable in PA under any arithmeticalinterpretation for any normal proof predicate Proof.

Explicit Reflection. If Proof (p,F ) holds, then F is evidently provable inPA, and so is formula Proof (p,F )→ F . If ¬Proof (p,F ) holds, then it isprovable in PA (since ¬Proof (x , y) is decidable) and Proof (p,F )→ F isagain provable.

The validity of Application, Proof Checker, and Sum is assumed forany natural proof system.


Constant Specification CS

is a set of formulas

{c1:A1, c2:A2, c3:A3, . . .}

where each Ai is an axiom and each ci is a proof constant.

LPCS = LP∅ + CS,

LP = LPCS with total CS containing c:A for each c and A.

Validity of an LP-formula F depends on specification of constantsoccurring in F . So, the Soundness Theorem for LPCS is of arelative nature.

LPCS ` F ⇒ for any interpretation ∗ respecting CS, PA ` F ∗


Arithmetical Completeness Theorem for LP

The Logic of Proofs specifies all valid logical principles aboutproofs in its language.

LPCS with a finite Constant Specification CS proves F iff for everyarithmetical interpretation ∗ respecting CS, PA ` F ∗.


Realization of S4

Realization Theorem (Artemov, 1994).

1. The forgetful projection of LP is S4-compliant.

2. For each theorem F of S4 and for each occurrence of 2 in F ,one can recover a witness (proof term) in such a way that theresulting formula is derivable in LP.

This gives a semantics of proofs for S4.


GL and S4/LP complement each other

GL and S4/LP complement each other by addressing differentareas of application.

GL finds applications in traditional proof theory.

LP targets areas of mathematical theories of knowledge andjustification, foundations of verification, typed theories andlambda-calculi, etc.


Joining Languages GL and LP

Certain principles require a mixture of both provability and explicitproofs, e.g., negative introspection. Its purely modal formulation

¬2F → 2¬2F

is not valid as a provability principle. Indeed, let F be ⊥. Then¬2⊥ is Consis PA and the whole formula

Consis PA→ Provable(Consis PA)

is false by Godel’s Second Incompleteness Theorem.


There is No Explicit Negative Introspection either

The principle¬p:S → t:(¬p:S),

where p and t are proof terms and S is a propositional variable, isnot valid. Indeed, fix an interpretation ∗ of p and t and thestandard Godel proof predicate. There are infinitely manyarithmetical instances of S for which the corresponding antecedentholds. Hence t∗ should be a proof of infinitely many theorems,which is impossible.


Mixed Language of Proofs and Provability fits NegativeIntrospection

The principle¬p:F → 2(¬p:F )

is arithmetically provable by Σ-completeness of PA, according towhich for each Σ-formula σ,

PA ` σ → Provable σ.


Godel-Lob-Artemov Logic

We will describe now introduced by the speaker a joint logic offormal provability and explicit proofs GLA (Godel-Lob-Artemovlogic) in the language with provability assertions 2F and proofassertions t:F . We provide GLA with Kripke-style semantics andestablish the arithmetical completeness of this logic.

GLA proved to be useful for applications in formal epistemologywhere it became a template for a family of epistemic logics withjustifications. An elaborate proof theory of GLA was offered byKurokawa (2012).

We will also provide applications of GLA to reflection principles inarithmetic.


Arithmetically Complete Predecessors of GLA

System B (Artemov, 1994) - no operations on proofs.

System LPP (Sidon-Yavorskaya, 1997) - in an extension oflanguages of GL and LP by extra operations on proofs.

Immediate successors of GLA are the Logic GrzA of StrongProvability and Explicit Proofs (Nogina, 2009) and SymmetricLogic of Proofs and Provability (Nogina, 2010).


Godel - Lob - Artemov Logic

Language of GLA.

Following LP’s pattern, proof terms are built from proof variablesx , y , z , . . . and proof constants a, b, c, . . . by means of theoperations application ‘·’ , union ‘+’, and proof checker ‘!’.

Formulas of GLA are defined by the grammar

A = ⊥ | S | A→ A | A ∧ A | A ∨ A | ¬A | 2A | p:A ,

where p and S stand for any proof term and any propositionalvariable correspondingly.


Axioms and Rules of GLA∅

I. Theorems of classical propositional logic

II. Axioms of Provability Logic GLGL1 2(F → G)→ (2F → 2G) Deductive Closure/NormalityGL2 2F → 22F Positive Introspection/TransitivityGL3 2(2F → F )→ 2F Lob Principle

III. Axioms of the Logic of Proofs LPLP1 s:(F → G) → (t:F → [s ·t]:G) ApplicationLP2 t:F → !t:(t:F ) Proof CheckerLP3 s:F → [s +t]:F , t:F → [s +t]:F SumLP4 t:F → F Explicit Reflection


Axioms and Rules of GLA∅ (continued)

IV. Axioms connecting explicit and formal provabilityC1 t:F → 2F Explicit-Implicit ConnectionC2 ¬t:F → 2¬t:F Explicit-Implicit Negative IntrospectionC3 t:2F → F Explicit-Implicit Reflection

V. Rules of inferenceR1 F → G , F ` G Modus PonensR2 ` F ⇒ ` 2F NecessitationR3 ` 2F ⇒ ` F Reflection Rule

Logic GLA∅ is sound w.r.t. the class of normal proof predicates.GLA is obtained from GLA∅ by adding Constant Specifications.


Constant Specification for GLA

Constant Specification CS for GLA is defined similar to the oneof LP.

{c1:A1, c2:A2, c3:A3, . . .}

where each Ai is an axiom of GLA and each ci is a proof constant.

GLACS = GLA∅ + CS,

GLA = GLACS with the total CS.

GLA is closed under substitutions of proof terms for proof variablesand formulas for propositional variables, enjoys the deductiontheorem, and contains both GL and LP.


Internalization Theorem

Theorem 1.If GLA ` F then for some proof term p, GLA ` p:F .Proof. Induction on a derivation of F .Base. F is an axiom. Then use Constant Specification.Induction steps. By internalized rules of GLA.

Internalization of Modus Ponens immediately follows fromApplication LP1.

Internalization of Necessitation rule ` F ⇒ ` 2F .For each F there is t(x) such that GLA ` x:F → t(x):2F

1. x:F → 2F - Explicit-Implicit Connection C1;2. a:(x:F → 2F ) - from 1 by Constant Specification;3. x:F → !x:x:F - Proof Checker LP2;4. !x:x:F → (a·!x):2F - from 2 by Application LP1;5. x:F → (a·!x):2F - from 3, 4 by propositional logic.

Now put t(x) = a·!x .Elena Y. Nogina Provability. Explicit Proofs. Reflection.

Internalization Theorem (continued)

Internalization of Reflection rule ` 2F ⇒ ` FFor each F there is s(x) such that GLA ` x:2F → s(x):F

1. x:2F → F - Explicit-Implicit Reflection C3;2. b:(x:2F → F ) from 1 by Constant Specification;3. x:2F →!x:x:2F - Proof Checker LP2;4. !x:x:2F → (b·!x):F - from 2 by Application LP1;5. x:2F → (b·!x):F - from 3, 4 by propositional logic.

Now put s(x) = b·!x . Note that in 2 we need an internalizedExplicit-Implicit Reflection.


Some Principles of GLA

Positive Introspection. GLA ` t:F → 2t:F1. t:F → !t:t:F - Proof Checker LP2;2. !t:t:F → 2t:F - Explicit-Implicit Connection C1;3. t:F → 2t:F - from 1, 2 by propositional logic.

Stability of Proof Assertions. GLA ` 2 t:F ∨2¬t:F4. ¬t:F → 2¬t:F - Explicit-Implicit Negative Introspection C2;5. 2 t:F ∨2¬t:F - from 3, 4 by propositional logic.


Explicit Version of Lob Principle

In 2(2F → F )→ 2F both modalities of the depth 1 can be readexplicitly as

x:(2F → F )→ l(x):F

for some proof term l(x). Indeed,

1. x:(2F → F )→ t(x):2(2F → F )by Internalized Necessitation Rule;

2. c:(2(2F → F )→ 2F )from Lob Principle GL3 by Constant Specification;

3. t(x):2(2F → F )→ (c · t(x)):2Ffrom 2 by Application LP1;

4. (c · t(x)):2F → s(c · t(x)):Fby Internalized Reflection Rule;

5. x:(2F → F )→ s(c · t(x)):F from 1,3,4.


Lob Principle cannot be Realized in full.Suppose for some proof polynomials u and v ,

GLA ` x:(u:⊥ → ⊥)→ v:⊥,

hence GLA ` x:(u:⊥ → ⊥)→ ⊥ and F = ¬x:(u:⊥ → ⊥) isderivable in GLA. Consider GLA-derivable formula

G = c:(u:⊥ → ⊥).

Let us perform a substitution τ = [c/x ] into both F and G . ThenF becomes ¬c:(τu:⊥ → ⊥) and G yields c:(τu:⊥ → ⊥), which isimpossible.

Realizable Provability Principles. A Franco Montagna’s questionwhich theorems of GL are realizable in GLA, has been answered byEvan Goris.

Theorem. Only those theorems of GL are realizable in GLA whichare from S4.


Provability Semantics for GLA. SoundnessArithmetical interpretation of GLA is the sum of the intendedarithmetical interpretations for GL and LP. In particular,

(2G)∗ = Provable G∗; (p:F )∗ = Proof (p∗,F ∗).

Theorem 2.For any Constant Specification CS and any arithmeticalinterpretation ∗ respecting CS, if GLACS ` F then PA ` F ∗.

Proof.It is immediate that Reflection Rule is valid. If Provable F isderivable in PA, then F is provable.

Validity of C1 and C2 follows from Σ-completeness of PA.

Soundness of Explicit-Implicit Reflection C3 t:2F → F takes placesince it is derivable from other principles of GLA, which is alreadyproved sound.


Derivation of Explicit-Implicit Reflection

Here is a derivation of C3 from other axioms of GLA∅.

Proof.1. ¬2F → ¬t:2F , contrapositive of LP4;2. ¬t:2F → 2(¬t:2F ), axiom C2;3. 2(¬t:2F )→ 2(t:2F → F ) by reasoning in GL;4. ¬2F → 2(t:2F → F ), from 1, 2, and 3;5. 2F → 2(t:2F → F ) by reasoning in GL;6. 2(t:2F → F ), from 4 and 5;7. t:2F → F by R3.

We keep C3 as a basic postulate of GL, since a ConstantSpecification related to C3 is used to guarantee the InternalizationProperty of GLA.


Arithmetical Completeness

Arithmetical completeness of GLA∅ could be established followingprevious arithmetical completeness proofs.

Theorem 3.For any finite Constant Specification CS, if GLACS 6` F , then thereexists a CS-interpretation ∗ such that PA 6` F ∗.

Proof.The claim of the theorem follows from the arithmeticalcompleteness of GLA∅.


Models for GLA

In this section, we build Kripke-style models for GLA.

A frame is a standard GL-frame (W ,≺, root) with the root noderoot, where W is a non-empty set of possible worlds, ≺ is a binarytransitive and conversely well-founded accessibility relation on W(a relation ≺ is conversely well-founded if any increasing chaina1 ≺ a2 ≺ a3 ≺ . . . is finite).

Possible evidence relation (first considered by Mkrtychev andlater by Fitting) is a relation E between proof terms and formulassuch that the following closure conditions are met:

Application. E(s,F → G) and E(t,F ) implies E(s ·t,G).Proof Checker. E(t,F ) implies E(!t, (t:F )).Sum. E(s,F ) or E(t,F ) implies E(s + t,F ).


Models for GLA (continued)

Model is a structure M = (W ,≺, root, E , ); here is arelation between worlds and formulas such that

1. respects Boolean connectives at each world ;2. u 2F iff v F for every v ∈W with u ≺ v ;3. u t:F iff E(t,F ) and v F for every v ∈W .

Following Solovay, we define

H(F ) = {2G → G | 2G is a subformula of F};

for a set of formulas X , H(X ) =⋃H(F ) for F ∈ X .

A model M is called F -sound if root H(F ). For a set offormulas X , M is X-sound if M is F -sound for each F ∈ X .

For a given Constant Specification CS, a model M is a CS-modelif M is CS-sound and CS holds in M.


Models for GLA. Soundness. Completeness.

Theorem 4.Soundness. For any formula F and any Constant Specification CS,if F is derivable in GLACS then F holds in each F-sound CS-model.

Theorem 5.Completeness. For any finite Constant Specification CS if F isnot derivable in GLACS, then there is an F-sound CS-model with afinite frame where F does not hold.Proof goes by a canonical model construction with the use oftechnique developed by Solovay, Artemov, and Fitting. GLA∅exhibits some sort of a finite model property, which also yields thedecidability.

Theorem 6.For any finite Constant Specification CS, the logic GLACS isdecidable.


Implicit-Explicit ReflectionExplicit-Implicit Reflection x:2F → F , as we have already seen, isarithmetically valid, hence could be chosen as an axiom scema.However, the Implicit-Explicit Reflection

IER = 2x:P → P

is not provable.1. Proof via GLA. Use an appropriate Kripke model:W = {1, 2}, 1 ≺ 2, P is false at 1 and 2, E = ∅.

2 ¬P, ¬x:P, 2x:P, ¬(2x:P → P) (i.e., ¬IER)↑1 ¬P, ¬x:P, ¬2x:P, 2x:P → x:P (IER-soundness)

IER is false at node 2 of the model.2. Arithmetical proof. If P = ⊥, then x:P is provably equivalentto ⊥. Therefore, this instance of IER is equivalent to 2⊥ → ⊥,which is the consistency of PA, not provable in PA.


Reflection Principles

Reflection principles were introduced in the 1930s by Rosser andTuring, and later were studied by Feferman, Kreisel & Levi,Schmerl and many others.

Reflection principles for a given theory T are formal schemataexpressing the soundness of T , i.e., schemata formalizing variantsof the statement

if F is provable in T , then F is true.

The technical details may vary significantly.

We study reflection principles of Peano Arithmetic PA based onboth proof and provability predicates.


Implicit Reflection Principle

Let us start with the local reflection principle, i.e., a set of allarithmetical formulas of the type

Provable F → F .

Though all the instances of this reflection are true in the standardmodel of Peano Arithmetic PA, some of them are not provable. Aswe know, when F is falsum ⊥, the local reflection principlebecomes Godel’s consistency formula

¬Provable ⊥.


Explicit Reflection Principle

Another example is given by the explicit reflection principle, the setof formulas of the type

Proof (t,F )→ F

where t is a proof term, and F an arithmetical formula. Here thesituation is quite different: all instances of explicit reflection areprovable.

Indeed, if Proof (t,F ) holds, then F is evidently provable in PA,and so is formula Proof (t,F )→ F . If ¬Proof (t,F ) holds, then itis provable in PA (since ¬Proof (x , y) is decidable) andProof (t,F )→ F is again provable.


Mixed Reflection Principles

We study reflection principles that could be specified usingarbitrary combinations of explicit-implicit provability, for example,

22P → P, u:2P → P, 2u:P → P, etc.

More precisely,


Definition of Reflection Principles

Let P be a propositional letter and each of Q1,Q2, . . . ,Qm iseither 2 or ‘u:’ for some fresh variable u. Formula π

Q1Q2 . . .QmP → P

is called a generator.

An arithmetical instance of a generator is defined by aninterpretation ∗ of P as a sentence of arithmetic, each u as Godelnumbers of proofs in PA and reading 2F as Provable F and t:F asProof (t,F ). The set [π] of all instances of π is a reflectionprinciple corresponding to generator π.

For example, the local reflection principle is generated by 2P → P, andthe explicit reflection principle is generated by u:P → P.

We will refer to reflection principles using their generators.


Comparing Reflection Principles

Definition. Let G and H be generators. We say that

H � G ,

if PA + [G ] proves all formulas from [H]. We say that

H ' G

(is read as “H is equivalent to G”) when both H � G and G � Hhold; H ≺ G stands for H � G and H 6' G .

From what we have already known, it follows that

u:P → P ≺ 2P → P.


Uniqueness of Provability Reflection

It is immediate that all reflection principles without explicit proofs(Qi = 2 for all i) are equivalent to the local reflection principle2P→P.

Theorem 7.For each n ≥ 1,

2nP → P ' 2P → P.


Provability of Leading-Explicit Reflection

Theorem 8.Let u:Q1Q2 . . .QnP → P be a reflection principle generator. Then

u:Q1Q2 . . .QnP → P ' u:P → P.

Theorem 8 follows from the fact that all leading-explicit reflectionprinciples are provable in PA.

General case when explicit provability is present.

Theorem 9.Let k ≥ 0 and 2ku:Q1Q2 . . .QnP → P be a reflection principlegenerator. Then

2ku:Q1Q2 . . .QnP → P ' 2ku:P → P.


Classification of Reflection Principles

Theorems 7 and 9 immediately yield

Theorem 10.Any reflection principle is equivalent to the reflection principlegenerated by either 2P → P or, for some k ≥ 0, to 2ku:P → P.


Hierarchy of Reflection Principles

Theorem 11.For each k ≥ 0, 2ku:P → P ' ¬2k⊥.

From Theorem 11 and a well-known fact that

¬⊥ ≺ ¬2⊥ ≺ ¬22⊥ ≺ . . . ≺ 2P → P,

we immediately get

Theorem 12.Reflection principles form a linear ordering

u:P → P ≺ 2u:P → P ≺ 22u:P → P ≺ . . . ≺ 2P → P.


Uniqueness of Provability Reflection. Proof

Theorem 7 For each n ≥ 1,

2nP → P ' 2P → P.

Proof.In light of the arithmetic soundness of GLA∅,

2P → P � 2nP → P

follows from the fact that GLA∅ ` 2P → 2nP. The converseinequality 2nP → P � 2P → P is implied by the fact that

(2nP → 2n−1P) ∧ . . . ∧ (2P → P)→ (2nP → P)

is derivable in GLA∅.


Provability of Leading-Explicit Reflection. Proof

Theorem 8. Let u:Q1Q2 . . .QnP → P be a reflection principlegenerator. Then

u:Q1Q2 . . .QnP → P ' u:P → P.

Proof. It is an immediate corollary of the arithmetical soundnessof GLA and the following Lemma.


Provability of Leading-Explicit Reflection. Proof (contd)

Lemma 1. For any n ≥ 0, GLA∅ ` u:Q1Q2 . . .QnP → P.

Proof. Induction on n. The base case n = 0 is trivial. For theinduction step consider two cases.

Case 1. Q1 is “v:” for some proof variable v . Then by explicitreflection,

GLA∅ ` u:Q1Q2 . . .QnP → v:Q2 . . .QnP;

by the IH,GLA∅ ` v:Q2 . . .QnP → P;

henceGLA∅ ` u:Q1Q2 . . .QnP → P.


Provability of Leading-Explicit Reflection. Proof (contd)

Case 2. Q1 is 2. Then u:Q1Q2 . . .QnP has a type u:2mF for somem ≥ 1, where F is either P or w:Qn−m−1 . . .QnP. Now we showthat GLA∅ ` u:2mF → F . Indeed,

1. ¬2mF → ¬u:2mF by E-reflection;2. ¬u:2mF → 2(¬u:2mF ) axiom C2;3. 2(¬u:2mF )→ 2(u:2mF → F ) by reasoning in GL;4. ¬2mF → 2(u:2mF → F ) from 1,2, and 3;5. 2(u:2mF → F )→ 2m(u:2mF → F ) from Transitivity;6. ¬2mF → 2m(u:2mF → F ) from 4 and 5;7. 2mF → 2m(u:2mF → F ) by reasoning in GL;8. 2m(u:2mF → F ) from 6 and 7;9. u:2mF → F by Reflection Rule.

If F is P we are done; if F is w:Qn−m−1 . . .QnP, then by the IH,GLA∅ ` F → P which yields the lemma claim as well.


General Case when Explicit Provability is Present. Proof

Theorem 9. Let k ≥ 0 and 2ku:Q1Q2 . . .QnP → P be areflection principle generator. Then

2ku:Q1Q2 . . .QnP → P ' 2ku:P → P.

Proof. Argue in PA. First, we establish “�”, i.e.,

PA′ = PA + [2ku:P → P] ` [2ku:Q1Q2 . . .QnP → P].

Fix an interpretation ∗. By Theorem 8 and Soundness of GLA∅

PA ` u∗:(Q1Q2 . . .QnP)∗ → P∗.

Let s be the corresponding proof in PA. Then,

PA ` s:(u∗:(Q1Q2 . . .QnP)∗ → P∗).


General Case when Explicit Provability is Present.Proof (continued)

By proof checking and internalized Modus Ponens in PA, we canfind an arithmetical proof t such that

PA ` u∗:(Q1Q2 . . .QnP)∗ → t:P∗,

from which we conclude

PA ` 2ku∗:(Q1Q2 . . .QnP)∗ → 2kt:P∗,

PA′ ` 2ku∗:(Q1Q2 . . .QnP)∗ → P∗,

i.e.,PA′ ` (2ku:Q1Q2 . . .QnP → P)∗.

Let us now establish “�”, i.e., that

PA′′ = PA + [2ku:Q1Q2 . . .QnP → P] ` [2ku:P → P].



Lemma 2. For each interpretation ∗ there is an interpretation ]which coincides with ∗ on P such that

PA ` (u:P)∗ → (u:Q1Q2 . . .QnP)].

Proof. By induction on n. The case n = 0 is trivial. Let for someinterpretation [ coinciding with ∗ on P,

PA ` (u:P)∗ → (u:Q2 . . .QnP)[.

By proof-checking, PA ` (u:P)∗ → !u[:u[:(Q2 . . .QnP)[.Case 1. If Q1 is a proof variable v , then define u] as !u[, v ] as u[,set ] to be [ everywhere else, and get the desired

PA ` (u:P)∗ → (u:v:Q2 . . .QnP)].



Case 2. Q1 is 2. By reasoning in PA, find a proof t such that

PA `!u[:u[:(Q2 . . .QnP)[ → t:2(Q2 . . .QnP)[,

therefore, PA ` (u:P)∗ → t:2(Q2 . . .QnP)[. Define u] = t (u isfresh!) and set ] equal [ everywhere else. Then

PA ` (u:P)∗ → (u:Q1Q2 . . .QnP)],

which completes Lemma’s proof. Now by PA-reasoning,

PA ` (2ku:P)∗ → (2ku:Q1Q2 . . .QnP)],

and sincePA′′ ` (2ku:Q1Q2 . . .QnP → P)]

and P] = P∗ we conclude that

PA′′ ` (2ku:P → P)∗.


Classification of Reflection Principles

From Theorems 7 and 9 immediately follows

Theorem 10.Any reflection principle is equivalent to either 2P → P or, forsome k ≥ 0, to 2ku:P → P.


Reflection Principles and Iterated Consistency. Proof

Theorem 11. For each k ≥ 0, 2ku:P → P ' ¬2k⊥.

Proof.Putting P = ⊥ we get 2ku:P → P � ¬2k⊥.

For the converse, argue in GLA∅. Case k = 0 is trivial.

Let k ≥ 1. Assume ¬2k⊥, 2ku:P, and ¬P and look for acontradiction. By Explicit Reflection, from ¬P we derive ¬u:P andby Explicit-Implicit Negative Introspection, 2¬u:P. By Transitivitywe get 2k¬u:P. From this and 2ku:P by the usual modalreasoning we conclude 2k(¬u:P ∧ u:P); hence 2k⊥, acontradiction.


Hierarchy of Reflection Principles

Theorem 11 and a well-known fact that

¬⊥ ≺ ¬2⊥ ≺ ¬22⊥ ≺ . . . ≺ 2P → P,

immediately yield

Theorem 12. Reflection principles form a linear ordering

u:P → P ≺ 2u:P → P ≺ 22u:P → P ≺ . . . ≺ 2P → P.


Thank you!


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Provability. Explicit Proofs. Reflection. · Formulas of LP are built as the usual propositional formulas with an additional formation rule: whenever F is a formula and t is a proof