Justification Logics Pushing at the Edges Melvin Fitting City University of New York Doxastic Agency and Epistemic Logic Bochum — December, 2017 1
Background 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. 2
The subject started with one 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. 3
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. 4
It is based on an abstract notion of proof . ⊥ has no proof. A proof of X ∧ Y consists of a proof of X and a proof of Y . A proof of X ∨ Y consists of a proof of X or a proof of Y . A proof of X ⊃ Y consists of an algorithm converting any proof of X into a proof of Y . 5
But, what is a proof? Can this be given an arithmetic interpretation? In 1933 Gödel made a first step. One can can characterize intuitionistic “truth” using classical validity plus informal provability. 6
G¨ odel proposed that informal provability should meet the following conditions (writing ⇤ X for X is “provable”). classical tautologies ⇤ ( A � B ) � ( ⇤ A � ⇤ B ) ⇤ A � A ⇤ A � ⇤⇤ A ` A and ` A � B implies ` B ` A implies ` ⇤ A This is the well-known modal logic S4. 7
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. 8
Gödel himself noted that S4 does not embed into arithmetic. At least not by using his provability predicate ( ∃ y )( y is the G¨ odel number of a proof of x ) to interpret ⇤ . 9
In 1938 G¨ odel 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. 10
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. 11
The really new things are proof terms (now usually called justification terms) 12
Variables, v 1 , v 2 , . . . are proof terms. Constant symbols, c 1 , c 2 , . . . 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 . 13
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, or t is a justification for X . 14
The informal ideas: t · u justifies X whenever u justifies some formula Y , and t justifies Y ⊃ X . ( modus ponens ) t + u justifies X whenever t justifies X , (weakening) or u justifies X . If t justifies X , ! t justifies that fact. (Justification checker) 15
Constants justify formulas that we do not further analyze; that is, axioms. Variables stand for arbitrary justifications. 16
LP Axiom Schemes 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 ) 17
Constants Each axiom X has a constant, say c , assigned to it so that c : X is another axiom. How this is done is called a constant specification . It is a parameter of the axiomatization. We don’t need more details for now. 18
LP Rules Modus Ponens ` X, ` X � Y = ) ` Y 19
What follows is an abbreviated example of a proof in LP . 20
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 ) 21
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 ) 22
Internalization We have that if X is an axiom, a : X for some constant a . in fact, if X is a theorem, t : X is provable for some justification term t . The structure of t internalizes the proof of X . 23
The Basic Picture Int , → S4 , → LP , → Arith Gödel’s Arithmetic translation Embedding Realization Theorem Theorem (Artemov) (Artemov) Intuitionistic logic has an arithmetic interpretation. 24
The Basic Picture Int , → S4 , → LP , → Arith Realization Theorem (Artemov) Central for this talk. 25
What is Realization? For any LP formula X let X � be the result of replacing every justification term with ⇤ . This is the forgetful functor. 26
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 ) . 27
The rules preserve this property. That’s all there is to this. Importantly, there is a converse too. 28
If X is a theorem of S4, there is a theorem Y of LP so that Y � = X . We say Y realizes X . Then, the image of the set of LP theorems is exactly the set of S4 theorems. We say S4 and LP correspond . 29
Better yet, if Y realizes X , Y can have distinct justification variables where X has negative ⇤ . Positive ⇤ occurrences become terms computed from these variables. There is a kind of hidden input/output structure to S4 theorems. 30
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. 31
The Simplest Justification Logic 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. 32
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. 33
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. 34
Thus There Are Many Justification Logics 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. 35
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. 36
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. 37
Realization Proofs There are two families of realization proofs. 38
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. 39
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 of modal logics. 40
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 . 41
In addition to putting conditions on the accessibility relation, one can put conditions on the evidence function, and on the relations between the evidence function and the accessibility relation. 42
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. 43
Some Peculiar Justification Examples 44
Some Interesting Justification Examples 45
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. 46
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