Epistemic Tradition

The properties of knowledge and belief have been a subject for formal logic at least since von Wright and Hintikka (Hintikka, 1962; von Wright, 1951). Knowledge and belief are both treated as modalities in a way that is now very familiar—Epistemic Logic.

The modal approach to the logic of knowledge is, in a sense, built around the universal quantifier: X is known in a situation if X is true in all situations indistinguishable from that one. Justifications, on the other hand, bring an ex­istential quantifier into the picture: X is known in a situation if there exists a justification for X in that situation. This universal/existential dichotomy is a familiar one to logicians—in formal logics there exists a proof for a formula X if and only if X is true in all models for the logic. One thinks of models as inherently nonconstructive, and proofs as constructive things. One will not go far wrong in thinking of justifications in general as much like mathematical proofs. Indeed, the first justification logic was explicitly designed to capture mathematical proofs in arithmetic, something that will be discussed later.

In justification logic, in addition to the category of formulas, there is a sec­ond category of justifications.

t is a justification for X.

One operation, common to all justification logics, is application, written like multiplication. The idea is, if s is a justification for A → B and t is a justification for A, thenis a justification for B.^[2] That is, the validity of the

following is generally assumed

This is the explicit version of the usual distributivity of knowledge operators, and modal operators generally, across implication

How adequately does the traditional modal form (1.2) embody epistemic closure? We argue that it does so poorly! In the classical logic context, (1.2) only claims that it is impossible to have both K(A → B) and KA true, but KB false. However, because (1.2), unlike (1.1), does not specify dependencies between K(A → B), KA, and KB, the purely modal formulation leaves room for a counterexample.

The distinction between (1.1) and (1.2) can be exploited in a discussion of the paradigmatic Red Barn Example of Goldman and Kripke; here is a simpli­fied version of the story taken from Dretske (2005).

Suppose I am driving through a neighborhood in which, unbeknownst to me, papier- maclie 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.

In the first formalization of the Red Barn Example, logical derivation will be performed in a basic modal logic in which ? is interpreted as the “belief” modality. Then some of the occurrences of ? will be externally interpreted as a knowledge modality K according to the problem’s description. Let B be the sentence “the object in front of me is a barn,” and let R be the sentence “the object in front of me is red.”

(1) ?B, “I believe that the object in front of me is a barn.” At the metalevel, by the problem description, this is not knowledge, and we cannot claim KB.

(2) ?(B Λ R), “I believe that the object in front of me is a red barn.” At the metalevel, this is actually knowledge, e.g., K(B Λ R) holds.

(3) ?(B ΛR → B), a knowledge assertion of a logical axiom. This is obviously knowledge, i.e., K(B Λ R → B).

Within this formalization, it appears that epistemic closure in its modal form (1.2) is violated: K(B ΛR), and K(B ΛR → B) hold, whereas, by (1), we cannot claim KB. The modal language here does not seem to help resolving this issue.

Next consider the Red Barn Example in justification logic where t:F is inter­preted as “I believe F for reason t” Let u be a specific individual justification for belief that B, and v for belief that B Λ R. In addition, let a be a justification for the logical truth B Λ R → B. Then the list of assumptions is

(i) u:B, “u is a reason to believe that the object in front of me is a barn”;

(ii) v:(B Λ R), “v is a reason to believe that the object in front of me is a red barn”;

(iii) a:(B Λ R → B).

On the metalevel, the problem description states that (ii) and (iii) are cases of knowledge, and not merely belief, whereas (i) is belief, which is not knowl­edge.

(iv) a:(B Λ R → B) → (v:(B Λ R) → [a·v]:B), by principle (1.1);

(v) v:(B Λ R) → [a·v]:B, from 3 and 4, by propositional logic;

(vi) [a · v]: B, from 2 and 5, by propositional logic.

Notice that conclusion (vi) is [a ■ v]:B, and not u:B; epistemic closure holds. By reasoning in justification logic it was concluded that [a·ν]:Β is a case of knowledge, i.e., “I know B for reason a ■ v.” The fact that u:B is not a case of knowledge does not spoil the closure principle because the latter claims knowl­edge specifically for [a ■ v]:B. Hence after observing a red facade, I indeed know B, but this knowledge has nothing to do with (i), which remains a case of belief rather than of knowledge. The justification logic formalization represents the situation fairly.

Tracking justifications represents the structure of the Red Barn Example in a way that is not captured by traditional epistemic modal tools. The justification logic formalization models what seems to be happening in such a case; closure of knowledge under logical entailment is maintained even though “barn” is not perceptually known.

One could devise a formalization of the Red Barn Example in a bimodal lan­guage with distinct modalities forknowledge and belief. However, it seems that such a resolution must involve reproducing justification tracking arguments in a way that obscures, rather than reveals, the truth. Such a bimodal formal­ization would distinguish u:B from [a ■ v]:B not because they have different reasons (which reflects the true epistemic structure of the problem), but rather because the former is labeled “belief” and the latter “knowledge.” But what if one needs to keep track of a larger number of different unrelated reasons? By introducing a multiplicity of distinct modalities and then imposing var­ious assumptions governing the interrelationships between these modalities, one would essentially end up with a reformulation of the language of justi­fication logic itself (with distinct terms replaced by distinct modalities). This suggests that there may not be a satisfactory “halfway point” between a modal language and the language of justification logic, at least inasmuch as one tries to capture the essential structure of examples involving the deductive nature of knowledge.

1.2