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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. - - 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
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Basic Systems - Justification Logic J0
justifications
are represented by proof polynomials (justifications terms) 1 proof variables x, y, z, . . . 2 proof constants a, b, c, . . . 3 binary operation application ”·” 4 binary operation sum (union, choice) ”+” Constants denote atomic justifications which the system no longer analyzes; variables denote unspecified justifications.
J0
is the logic of general (not necessarily factive) justifications for a skeptical agent for whom no formula is provably justified, that is, J0 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
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Basic Systems - Logical Awareness
Logical Awareness
logical axioms are justified ex officio: an agent accepts 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
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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
e1 : A
The way to to postulate that an axiom A is justified for a given agent in Justification Logic (for some evidence
constant e1 with index 1).
e2 : (e1 : A)
The way to to postulate that e1:A is also justified (for the
similar constant e2 with index 2, and so forth).
constant specification
for a given logic L is a set of formulas en:en−1:...:e1:A, n ≥ 1 where A is an axiom of L, and e1, e2, ..., en are similar constants with indices 1, 2, ..., n.
We also assume that constant specification contains all intermediate specifications, that is, whenever en:en−1:...:e1:A then en−1:...:e1:A.
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Basic Systems - Constants Specification
Types of constant specifications (CS):
empty
CS=∅ This corresponds to an absolutely skeptical agent (cf. a
comment after axioms of J0).
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 appropriate
for each axiom A there is a constant e1 such that e1:A is in CS, and if This is necessary for ensuring the Internalization
property.
total (TCS)
for each axiom A and any constants e1, e2, ..., en such that en:en−1:...:e1:A is in CS, and if Naturally, the total
constant specification is axiomatically appropriate.
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Basic Systems - Justification Logic with CS JCS=J0 + CS
Where J0=J0 + R4 R4 is the Axiom Internalization Rule: for each axiom A and any constants e1, e2, ..., en, infer en:en−1:...:e1:A. Unrestricted Logical Awareness for J: J0 is J∅ and J coincides with JTCS. 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
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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:
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Basic Systems - Justification Logic with CS
Deduction Theorem
For each constant specification CS, JCS enjoys the Deduction Theorem, because J0 contains propositional axioms and Modus Ponens as the only rule of inference.
Internalization
For each axiomatically appropriate constant specification CS, JCS enjoys Internalization: If ⊢ F, then ⊢ p : F for some justification term p.
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Basic Systems - Red Barn Example
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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 barn case
The neighborhood has no fake red barns (the fake barns cannot be painted red).
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Basic Systems - R. Nozick’s conditions
Truth-tracking
Robert Nozick’s conditions for knowledge:
1
p is true
2
S believes that p
3
If p weren’t true, S wouldn’t believe that p
4
If p were true, S would believe that p
”Red Barn”
If it happens I see the real (red) barn, then:
1
The statement R: I can see a red barn is knowledge since I wouldn’t believe there was a red barn (via my
red-barn-percepts) if no red barn were there.
2
The statement B: I can see a barn is not knowledge, since I might believe there was a barn (via
blue-barn-percepts) even if no red-barn was there.
3
The statement B can be inferred from R; however R is knowledge and B is not.
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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:
1
B ”I believe that the object in front of me is a barn.”
2
B ∧ R ”I believe that the object in front of me is a red barn.” At the metalevel, we assume that (2) is knowledge, whereas (1) is not knowledge. So we could add factivity: (B ∧ R) → (B ∧ R)
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Basic Systems - Red Barn (Modal Logic of belief)
closure principle
- ne knows everything that one knows to be implied by
what one knows
1
B
2
(B ∧ R)
3
(B ∧ R) → B, logical axiom
4
((B ∧ R) → B), 3, Necessitation (as a logical truth, this
is a case of knowledge too)
5
(B ∧ R) → B, 4, Modal Logic It appears that Closure Principle is violated (2 and 4 is knowledge but 1 is not knowledge).
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Basic Systems - Red Barn (Modal Logic of knowledge)
closure principle
- ne knows everything that one knows to be implied by