Justification Logic Who? Natalia Kotsani - based on the work of S. - PowerPoint PPT Presentation

Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of Justification, 2008) When? 28 November 2012

Table of contents 3.Basic Systems Justification Logic J 0 Logical Awareness Constants Specification

Basic Systems - Justification Logic J 0 justifications are represented by proof polynomials (justifications terms) 1 proof variables x, y, z, . . . 2 proof constants a, b, c, . . . binary operation application ” · ” 3 4 binary operation sum (union, choice) ”+” Constants denote atomic justifications which the system no longer analyzes; variables denote unspecified justifications. J 0 is the logic of general (not necessarily factive) justifications for a skeptical agent for whom no formula is provably justified, that is, J 0 does not derive t : F for any t and F A1. Classical propositional axioms and rule Modus Ponens A2. s : ( F → G ) → ( t : F → ( s · t ) : G ) (Application Axiom) A3. F → ( s + t ) : F , s : F → ( t + s ) : F (Monotonicity Axiom) The agent can make relative justification conclusions: if x:A, y:B, . . ., z:C hold, then t:F

Basic Systems - Logical Awareness logical axioms are justified ex officio: an agent accepts Logical Awareness logical axioms (including the ones concerning justifications) as justified Logical Awareness is too restrictive Justification Logic offers a flexible mechanism of Constant Specifications to represent all shades of logical awareness

Basic Systems - Constants Specification distinction between an assumption and a justified assumption constants are used to denote justifications of assumptions in situations when we don’t analyze these justifications any further e 1 : A The way to to postulate that an axiom A is justified for a given agent in Justification Logic (for some evidence constant e 1 with index 1). e 2 : ( e 1 : A ) The way to to postulate that e 1 :A is also justified (for the similar constant e 2 with index 2, and so forth). constant for a given logic L is a set of formulas e n :e n − 1 :...:e 1 :A, specification n ≥ 1 where A is an axiom of L , and e 1 , e 2 , ..., e n are similar constants with indices 1, 2, ..., n. We also assume that constant specification contains all intermediate specifications, that is, whenever e n :e n − 1 :...:e 1 :A then e n − 1 :...:e 1 :A .

Basic Systems - Constants Specification Types of constant specifications (CS): empty CS= ∅ This corresponds to an absolutely skeptical agent (cf. a comment after axioms of J 0 ). finite CS is a finite set of formulas. Any specific derivation in Justification Logic concerns only finite sets of constants and constant specifications (representative case). axiomatically for each axiom A there is a constant e 1 such that e 1 :A is appropriate in CS, and if This is necessary for ensuring the Internalization property. total (TCS) for each axiom A and any constants e 1 , e 2 , ..., e n such that e n :e n − 1 :...:e 1 :A is in CS, and if Naturally, the total constant specification is axiomatically appropriate.

Basic Systems - Justification Logic with CS J CS = J 0 + CS Where J 0 = J 0 + R 4 R4 is the Axiom Internalization Rule: for each axiom A and any constants e 1 , e 2 , ..., e n , infer e n :e n − 1 :...:e 1 :A. Unrestricted Logical Awareness for J : J 0 is J ∅ and J coincides with J TCS . any specific derivation in J may be regarded as a derivation in JCS for a corresponding finite constant specification CS , hence finite CS’s constitute an important representative class of constant specifications. Logical Awareness expressed by axiomatically appropriate CS is an explicit incarnation of the Necessitation Rule in modal epistemic logic: ⊢ F ⇒⊢ K F

Basic Systems - Justification Logic with CS Example 3.1 This example shows how to build a justification of a conjunction from justifications of the conjuncts. In the traditional modal language, this principle is formalized as:

Basic Systems - Justification Logic with CS Deduction For each constant specification CS, J CS enjoys the Theorem Deduction Theorem, because J 0 contains propositional axioms and Modus Ponens as the only rule of inference. Internalization For each axiomatically appropriate constant specification CS, J CS enjoys Internalization: If ⊢ F, then ⊢ p : F for some justification term p.

Basic Systems - Red Barn Example

Basic Systems - Red Barn Example Suppose I am driving through a neighborhood in which, unbeknownst to me, papier-mache barns are scattered, and I see that the object in front of me is a barn. Because I have barn-before-me percepts, I believe that the object in front of me is a barn. Our intuitions suggest that I fail to know barn. But now suppose that the neighborhood has no fake red barns, and I also notice that the object in front of me is red, so I know a red barn is there. This juxtaposition, being a red barn, which I know, entails there being a barn, which I do not, ”is an embarrassment”. Kripkesque The neighborhood has no fake red barns (the fake barns barn case cannot be painted red).

Basic Systems - R. Nozick’s conditions Truth-tracking Robert Nozick’s conditions for knowledge: p is true 1 S believes that p 2 If p weren’t true, S wouldn’t believe that p 3 If p were true, S would believe that p 4 ”Red Barn” If it happens I see the real (red) barn, then: The statement R: I can see a red barn is knowledge 1 since I wouldn’t believe there was a red barn (via my red-barn-percepts) if no red barn were there. The statement B: I can see a barn is not knowledge, 2 since I might believe there was a barn (via blue-barn-percepts) even if no red-barn was there. The statement B can be inferred from R; however R is 3 knowledge and B is not.

Basic Systems - Red Barn in Modal Logic the logical derivation will be made in epistemic modal logic with my belief modality � . We then interpret some of the occurrences of � as knowledge according to the problems description. The formulation claims logical dependencies between the statements R and B. The natural formalization of these assumptions in the epistemic modal logic of belief is: � B ”I believe that the object in front of me is a barn.” 1 � B ∧ R ”I believe that the object in front of me is a red 2 barn.” At the metalevel, we assume that (2) is knowledge, whereas (1) is not knowledge. So we could add factivity: � ( B ∧ R ) → ( B ∧ R )

Basic Systems - Red Barn (Modal Logic of belief) closure one knows everything that one knows to be implied by principle what one knows � B 1 � ( B ∧ R ) 2 ( B ∧ R ) → B , logical axiom 3 � (( B ∧ R ) → B ), 3, Necessitation (as a logical truth, this 4 is a case of knowledge too) � ( B ∧ R ) → � B , 4, Modal Logic 5 It appears that Closure Principle is violated (2 and 4 is knowledge but 1 is not knowledge).

Basic Systems - Red Barn (Modal Logic of knowledge) closure one knows everything that one knows to be implied by principle what one knows � B 1 � ( B ∧ R ) 2 ( B ∧ R ) → B , logical axiom 3 � (( B ∧ R ) → B ), 3, Necessitation (as a logical truth, this 4 is a case of knowledge too) � ( B ∧ R ) → � B , 4, Modal Logic 5 It appears that Closure Principle is violated (2 and 4 is knowledge but 1 is not knowledge).