Justification Logics Pushing at the Edges Melvin Fitting City - - PowerPoint PPT Presentation

Justification Logics Pushing at the Edges

Melvin Fitting City University of New York Doxastic Agency and Epistemic Logic Bochum — December, 2017

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Background

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Justification Logics are like modal logics, except they involve explicit knowledge (proofs, reasons, justifications) within the language itself. They can reason about the reasons for things.

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The subject started with

• ne well-motivated example.

The range of justification logics has been expanding ever since. Today I want to give you some idea of this range. Show you something of what is out there. But it might be best to begin at the beginning.

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Intuitionistic logic was intended to be constructive. And it is, in a precise sense. The well-known Brouwer, Heyting, Kolmogorov (BHK) semantics has a constructive flavor.

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It is based on an abstract notion of proof. A proof of X ∧ Y consists of a proof of X and a proof of Y . A proof of X ⊃ Y consists of an algorithm converting any proof of X into a proof of Y . ⊥ has no proof. A proof of X ∨ Y consists of a proof of X or a proof of Y .

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In 1933 Gödel made a first step. But, what is a proof? Can this be given an arithmetic interpretation? One can can characterize intuitionistic “truth” using classical validity plus informal provability.

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G¨

• del proposed that

informal provability should meet the following conditions (writing ⇤X for X is “provable”). This is the well-known modal logic S4. classical tautologies ⇤(A B) (⇤A ⇤B) ⇤A A ⇤A ⇤⇤A ` A and ` A B implies ` B ` A implies ` ⇤A

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Translate intuitionistic formulas by putting ⇤ before every subformula. For example, (A ∧ B) ⊃ A becomes ⇤((⇤(⇤A ∧ ⇤B) ⊃ ⇤A) Then, X is an intuitionistic theorem if and only if the translate of X is a theorem of S4.

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Gödel himself noted that S4 does not embed into arithmetic. At least not by using his provability predicate (∃y)(y is the G¨

• del number of a proof of x)

to interpret ⇤.

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In 1938 G¨

• del had another proposal,

interpret ⇤ as explicit provability. (This moves the existential quantifier to the metalevel.) This was not published during Gödel’s lifetime, but was independently rediscovered by Sergei Artemov in the 1990’s.

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Artemov introduced a logic he called LP, for logic of proofs. It is a kind of explicit modal logic. Here it is, axiomatically, beginning with the language.

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The really new things are proof terms (now usually called justification terms)

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Variables, v1, v2, . . . are proof terms. Constant symbols, c1, c2, . . . are proof terms. If t and u are proof terms, so are t + u and t · u. If t is a proof term, so is !t.

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Formulas are built up from propositional letters, P, Q, . . . , and ⊥. Using ⊃ and maybe other connectives. And, if t is a proof term, and X is a formula, t:X is a formula. Think of t:X as asserting: X is so, with t as a proof,

• r t is a justification for X.

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The informal ideas: t · u justifies X whenever u justifies some formula Y , and t justifies Y ⊃ X. t + u justifies X whenever t justifies X,

• r u justifies X.

If t justifies X, !t justifies that fact.

(modus ponens) (weakening) (Justification checker)

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Constants justify formulas that we do not further analyze; that is, axioms. Variables stand for arbitrary justifications.

LP Axiom Schemes

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Classical Tautologies Application t:(X ⊃ Y ) ⊃ (s:X ⊃ [t · s]:Y ) Weakening s:X ⊃ [s + t]:X t:X ⊃ [s + t]:X Factivity t:X ⊃ X Justification Checker t:X ⊃!t:(t:X)

Constants

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How this is done is called a constant specification. It is a parameter of the axiomatization.

Each axiom X has a constant, say c, assigned to it so that c:X is another axiom.

We don’t need more details for now.

LP Rules

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Modus Ponens ` X, ` X Y = ) ` Y

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What follows is an abbreviated example

• f a proof in LP

.

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• 1. x:P ⊃ (x:P ∨ y:Q)
• 2. a:(x:P ⊃ (x:P ∨ y:Q)) using axiom nec.
• 3. a:(x:P ⊃ (x:P ∨ y:Q)) ⊃ (!x:x:P ⊃ [a·!x]:(x:P ∨ y:Q))
• 4. !x:x:P ⊃ [a·!x]:(x:P ∨ y:Q)
• 5. x:P ⊃!x:x:P
• 6. x:P ⊃ [a·!x]:(x:P ∨ y:Q)
• 7. y:Q ⊃ [b·!y]:(x:P ∨ y:Q) similarly
• 8. x:P ⊃ [a·!x + b·!y]:(x:P ∨ y:Q) weakening
• 9. y:Q ⊃ [a·!x + b·!y]:(x:P ∨ y:Q) similarly
• 10. (x:P ∨ y:Q) ⊃ [a·!x + b·!y]:(x:P ∨ y:Q)

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So we have (x:P ∨ y:Q) ⊃ [a·!x + b·!y]:(x:P ∨ y:Q) where a justifies the tautology x:P ⊃ (x:P ∨ y:Q) and b justifies the tautology y:Q ⊃ (x:P ∨ y:Q)

Internalization

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We have that if X is an axiom, a:X for some constant a. The structure of t internalizes the proof of X.

in fact, if X is a theorem, t:X is provable for some justification term t.

The Basic Picture

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Int , → S4 , → LP , → Arith Gödel’s translation Arithmetic Embedding Theorem (Artemov) Realization Theorem (Artemov) Intuitionistic logic has an arithmetic interpretation.

The Basic Picture

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Int , → S4 , → LP , → Arith Realization Theorem (Artemov) Central for this talk.

What is Realization?

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For any LP formula X let X be the result of replacing every justification term with ⇤. This is the forgetful functor.

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If X is an LP theorem, X is an S4 theorem. True for axioms. For example, s:(X ⊃ Y ) ⊃ (t:X ⊃ [s · t]:Y ) becomes ⇤(X ⊃ Y ) ⊃ (⇤X ⊃ ⇤Y ).

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The rules preserve this property. That’s all there is to this. Importantly, there is a converse too.

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If X is a theorem of S4, there is a theorem Y of LP so that Y = X. Then, the image of the set of LP theorems is exactly the set of S4 theorems. We say S4 and LP correspond. We say Y realizes X.

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Positive ⇤ occurrences become terms computed from these variables. There is a kind of hidden input/output structure to S4 theorems.

Better yet, if Y realizes X, Y can have distinct justification variables where X has negative ⇤.

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This is called a normal realization. For example, the S4 theorem (⇤P ∨ ⇤Q) ⊃ ⇤(⇤P ∨ ⇤Q) has the normal realization (x:P ∨ y:Q) ⊃ [a·!x + b·!y]:(x:P ∨ y:Q). Realizations are not unique.

The Simplest Justification Logic

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Reduce LP to a minimum.

Keep + and ·, get rid of !. Keep Application: t:(X ⊃ Y ) ⊃ (s:X ⊃ [t · s]:Y ) and Weakening: (s:X ∨ t:X) ⊃ [s + t]:X Get rid of Factivity and Justification Checker.

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This gives the smallest justification logic J. It corresponds to modal K. One can build bigger justification logics on J. Add other function symbols, and axioms, to J.

Examples soon.

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We will still have internalization. Loosely, this is because the only rule is modus ponens, and the Application axiom gives us what we need for it. Adding other rules is hard. We won’t consider it today.

Thus There Are Many Justification Logics

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Modal logic KL and justification logic JL correspond if the set of theorems of KL is exactly the image of the set of theorems of JL under the forgetful functor.

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In fact, infinitely many normal modal logics have justification counterparts. I don’t actually know of an example of one that does not. But that may just be a limitation on my part.

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Showing the forgetful functor maps theorems of a justification logic JL to a modal logic KL is generally easy. The hard part is the other direction.

Realization.

Each theorem of KL should have an ‘analysis’ in JL.

Realization Proofs

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There are two families of realization proofs.

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Some realization proofs are algorithmic. They need, as input, not a modal validity, but a modal cut free proof. It seems that cut free can be in sequent calculi, tableaus, nested sequents, prefixed tableaus, hypersequents.

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But, most modal logics don’t have cut free proof systems. The best known ones do, but there are lots of others. In 2003 I introduced a non-constructive way of proving realization for S4 and LP . It turns out to apply to a very wide range

• f modal logics.

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It is semantic based. Possible world models for justification logics are introduced. They are Kripke style, with one more piece of machinery: an Evidence Function.

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In addition to putting conditions

• n the accessibility relation,
• ne can put conditions on the evidence function,

and on the relations between the evidence function and the accessibility relation.

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I’m going to skip the details here. The important fact is that there is a canonical model construction for justification logics. It covers most known ones.

Some Peculiar Justification Examples

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Some Interesting Justification Examples

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Some Special Justification Examples

The subject of justification logic began with intuitionistic logic. The Gödel embedding maps it into S4. We can think of S4 as a logic of implicit proofs. S4 maps into LP . We can think of LP as a logic of explicit proof representations.

What About Superintuitionistic Logics?

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The top superintuitionistic logic is classical. The Gödel translation embeds it into S5. A justification counterpart to S5 is well-known. Let’s call it JT45.

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Build on LP. Add a unary function symbol ?, and an axiom schema ¬t:X ⊃?t:¬t:X

(Note: this is not the only way of coming up with an S5 counterpart.) This is JT45.

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This was initially proved semantically. (The non-constructive proof introduced important new ideas.) S5 realizes into JT45. It has also been proved constructively, in several ways.

And Intermediate Logics?

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At the moment, I only know how to handle two intermediate logics, but this is already illustrative.

Weak Excluded Middle

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Add to intuitionistic logic the scheme ¬X ∨ ¬¬X

Also known as KC or Jankov’s logic. The smallest modal companion for this is S4.2

S4 plus ⌃⇤X ⊃ ⇤⌃X

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Semantically, modal models are S4 models that are convergent.

A justification counterpart called JT4.2 is LP plus the following axiom scheme: f(t, u):¬t:X ∨ g(t, u):¬u:¬X

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Very Informal Idea

Using factivity, (t:X ∧ u:¬X) ⊃ ⊥ So ¬t:X ∨ ¬u:¬X is provable. In any context we have one of ¬t:X or ¬u:¬X. f(t, u):¬t:X ∨ g(t, u):¬u:¬X says we have a reason for the one that holds

An Example

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[⌃⇤A ∧ ⌃⇤B] ⊃ ⌃⇤(A ∧ B) is a theorem of S4.2

negative occurrences must become variables

We rewrite it as [¬⇤¬⇤A ∧ ¬⇤¬⇤B] ⊃ ¬⇤¬⇤(A ∧ B)

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A realization of it, provable in JT4.2 is

{¬[j4 · j3·!v5 · g(!v3, j2 · v5·!v9)]:¬v9:A ∧ ¬[j5 · f(!v3, j2 · v5·!v9)]:¬v3:B} ⊃ ¬v5:¬[j1 · v9 · v3]:(A ∧ B)

j1, j2, j3, j4, j5 are given using Internalization. and j2 internalizes a proof of ¬[j1 · v1 · v3]:(A ∧ B) ⊃ (v1:A ⊃ ¬v3:B) For instance j1 internalizes a proof of A ⊃ (B ⊃ (A ∧ B))

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No constructive proof of realization is known for S4.2. (However, there is a hypersequent calculus for S4.2 due to Hidenori Kurosawa, and it looks like it will work here. Details need checking.)

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Non-constructively we have the following very general result.

A Geach logic is one axiomatized over K using schemas of the form ⌃k⇤lX ⊃ ⇤m⌃nX

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All Geach logics have justification counterparts, connected via a Realization theorem. In particular, infinitely many modal logics have justification counterparts. In particular, S4.2.

Gödel-Dummet Logic

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Add to intuitionistic logic the scheme (X ⊃ Y ) ∨ (Y ⊃ X) The smallest modal companion for this is S4.3 S4 plus ⇤(⇤X ⊃ Y ) ∨ ⇤(⇤Y ⊃ X)

Also known as LC.

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Semantically, modal models are S4 models that are linear.

A justification counterpart called J4.3 is LP plus the following axiom scheme: f(t, u):(t:X ⊃ Y ) ∨ g(t, u):(u:Y ⊃ X)

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Again, no constructive proof of realization is known. And S4.3 is not a Geach logic. But we do have an even more general semantic result that applies. (Or perhaps one is on the horizon.)

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Theorem: if KL is a canonical modal logic and JL is a candidate for a justification counterpart and the canonical justification model for JL is built on a KL frame, then KL realizes into JL. This is how the result about Geach logics was proved, but it is broader than that.

Justifications That Could be Wrong

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Drop factivity from S4, ⇤X ⊃ X. Replace it with the weaker ⇤⇤ ⊃ ⇤X. If there is a reason for a reason, we have a reason. Call this KX4.

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We can think of KX4 as a logic

• f implicit justifications that

might be wrong, since factivity is missing. Interestingly, the Gödel embedding also maps positive intuitionistic logic into KX4.

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This does not extend to negation.

For example, (A ⊃ ⊥) ⊃ ¬A maps to ⇤[⇤(⇤A ⊃ ⇤⊥) ⊃ ⇤¬⇤A] which isn’t always a validity. Consider the special case ⇤[⇤(⇤⊥ ⊃ ⇤⊥) ⊃ ⇤¬⇤⊥].

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Semantics for KX4 has reflexive and dense accessibility. Here is a counter-model (details left to you). Strong negation can be introduced. But I won’t go into that now.

Possible worlds: (0, 1] Accessibility: < ⊥ true nowhere ⇤[⇤(⇤⊥ ⊃ ⇤⊥) ⊃ ⇤¬⇤⊥] is false at 0.

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Call this JX4.

From LP drop factivity, t:X ⊃ X. Replace it with t:u:X ⊃ [t c u]:X where c is a new binary operation symbol.

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KX4 is a Geach logic, so Realization follows from the general theorem earlier. KX4 and JX4 are counterparts. There is a Realization result connecting them.

A Few Different Kinds

• f Examples

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I recently found a very simple justification counterpart for Gödel-Löb logic. It isn’t the first version of something like this. Daniyar Shamkanov has a different such logic. I don’t know the relationships between them yet.

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First, recall the forgetful functor, recursively replace t:X with ⇤X. This maps justification Z to modal Z.

And also rather unexpectedly, here realization has a constructive proof, but I don’t know how to prove it semantically.

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We build on justification logic J.

We add the axiom scheme t:(u:X1 ⊃ X2) ⊃ g(t):X3 where X1, X2, and X3 are justification formulas such that X

1 = X 2 = X 3

Note that X1, X2, X3 don’t have to be the same!

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A Realization Example

The GL theorem ⇤(⇤(X ∨ Y ) ⊃ (X ∧ Z)) ⊃ ⇤X is realized by v1:([v2 + g(a)]:(X ∨ Y ) ⊃ (X ∧ Z)) ⊃ g(b):X Where v4:X `JGL a:(v5:(X _ Y ) (X _ Y )) and v1:[g(a):(X _ Y ) (X ^ Z)] `JGL b:(v4:X X)

(both from internalization)

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Second comment: I don’t have a justification semantics for this logic. Canonical machinery won’t work, but I don't know what will. Realization is proved constructively, using a tableau system for GL.

Going Further

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Similar ideas work for Grzegorczyk logic. Add to LP

s:(t:(X1 ⊃ u:X2) ⊃ X3) ⊃ [g(s)]:X4, where X

1 = X 2 = X 3 = X 4

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Again, realization is constructive, but a semantics is missing. And for both GL and Grz, what benefit does a justification counterpart bring?

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That’s where things are at the moment. The justification phenomenon is certainly very broad. How broad?

Thank You

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