- The 'Equivalence of Things' (齐物论) chapter of the Zhuangzi (庄子) (c. 3rd c. B.C.E.)

Once upon a time I, Zhuang Zhou (庄周), dreamt I was a butterfly.
Flapping my wings in true butterfly fashion, I was happy as could be, and I knew nothing of any person named Zhuang Zhou.
But suddenly I awakened, astonished to be Zhuang Zhou.
I still don't know whether as Zhuang Zhou I was dreaming I was a butterfly or whether as a butterfly I was dreaming I was Zhuang Zhou.
There ought to be a difference between Zhuang Zhou and a butterfly, but this is called the transformation of things.
- David K. Jordan translation (adapted)

## Justification Logic

### Epistêmê is the Greek word for 'knowledge'

1. Justification logic allows the belief and knowledge modalities in epistemic logic to be unfolded into justification terms
2. Epistemic logic allows us to capture knowledge as true belief
3. Justification logic provides us with the missing component of knowledge as justified true belief from the JTB analysis of knowledge
4. ∴ Justification logic allows us to bridge the gap between epistemic logic (absence of the notion of justification) and mainstream epistemology (presence of the notion of justification)

1. In justification logic (Artemov, 1995, 2001, Mkrtychev, 1997, Kuznets, 2000):

2. # t:F

3. means the following:
4. 't is a justification of F';
5. 'F is justified by reason t'; or
6. (more strictly) 't is accepted by the agent as a justification of F'

# (C1) t:F → □F

1. Let □ denote the knowledge operator in epistemic logic
2. ∴ □F will denote that F is known

3. We will be able to convert the axioms of epistemic logic in the following fashion:
Axioms of epistemic logic:
Formal notation Translation
(T) □F → F Truth or veridicality
(K) □(F → G) → (□F → □G) Closure under implication
(RN) ⊦F → ⊦□F Rule of necessitation
(4) □F → □ □F Positive introspection

### ⋮

We now have an additional set of axioms from justification logic:
Axioms of justification logic:
Formal notation Translation
(LP1) s:(F → G) → (t:F → (s·t):G) A justification of the implication relation (F → G), applied to any justification of the antecedent F, returns a justification of the consequent G (application)
(LP2) t:F → !t:(t:F) For any evidence t of F, the result of applying a checker (viz. !t) to t provides a justification of t:F (inspection)
LP2 captures the verifiability property of evidence
(LP3) s:F → (s + t):F, t:F → (s + t):F A justification for F remains a justification after adding any additional evidence (union)
LP3 reflects the monotonicity principle with respect to evidence
(LP4) t:F → F If F is justified by t, then F (reflexivity)
(C1) t:F → □F C1 is the principle that connects implicit and explicit knowledge (justification-knowledge connection)

### ⋮

NOTE:
1. '!' denotes an evidence checker (unary)
2. '·' denotes application (binary)
3. '+' denotes union (binary)

1. The justification component of the JTB analysis of knowledge now has a formal representation
2. Justification logic introduces the notion of justification into formal epistemology
3. Epistemic logic, when combined with justification logic, can formally represent the following assertions:
1. □F: F is known
2. t:F : t is a justification for F
3. NOTE: By Axiom T of epistemic logic, F will be true