2.1. Reasons

The first concept we need to clarify is the concept of a reason. As emphasized by Armstrong in [Reference Armstrong1], “talk of a man’s reason may be to refer to a (certain sort of) belief-state of his, or it may be to refer to the proposition believed” (p. 79). We handle reasons primarily as state-specific objects, and only derivatively as propositions. For us, a reason is some evidence on the basis of which we come to believe a proposition. A reason therefore does not have the type of a proposition, although it can be associated to a proposition in a systematic manner.Footnote ² For example, my hearing voices outside might be a reason to believe that there are people outside. Semantically, we shall represent reasons by accessibility relations, thereby treating them as akin to belief-states. Given a reason r and a world w, we write $r(w)$ to represent the proposition determined by the reason r in w (that is the set of accessible worlds in virtue of the relation corresponding to r). So, each reason can be associated with a proposition at a world, but reasons per se are not propositions.Footnote ³ Reasons are related to the sort of answers produced by rational agents when they are asked, “Why do you believe this?” or “How do you know that?” If I am asked why I believe that Napoleon I is dead by now, I would answer that it is because I read that he was born in the eighteenth century, and he died in the nineteenth century. If I am asked why I believe that $2+2=4$ , I would respond that my calculations confirm that, or that I accept a certain axiomatic system that I see as consistently deriving that fact. So reasons for us are not bare propositions, like the proposition “Napoleon died in the nineteenth century” or the proposition “ $2+1=3,$ ” or even bare arguments, but we take them to be related to evidential experience of some sort.

2.2. Support

The second concept we deal with is what we call support between a reason and a proposition. Syntactically, we treat the relation of support between a reason and a proposition as primitive: we will write “ $r\,{:}\, \varphi $ ”. Model-theoretically, we represent the support relation between a reason r and a proposition p by the fact that $r(w)$ entails p. This does not mean that we intend the relation of support to model deductive support. Rather, we do intend it to capture a very fundamental form of inductive support. That is, we view each reason as a basic human experience of some kind, creating an inclination to believe a proposition. If I hear voices outside the house, that is a supporting reason for the proposition that there are people outside the house. In principle, that might also be a reason to think there are loudspeakers broadcasting voices outside the house (with no actual person around). In our approach, however, we do not allow for reasons to support contradictory propositions. If r is a supporting reason for $\varphi $ , then we will take r to be a supporting reason for any proposition entailed by $\varphi $ . Consequently, a reason for thinking that there are loudspeakers broadcasting voices outside the house with no one around will have to be different from a reason to think that there are actual people outside the house.

A delicate issue is whether the support relation ought to be treated as subjective (internal to an agent’s beliefs) or as objective (external to the agent’s beliefs). That is, could an agent believe that r supports $\varphi $ without r actually supporting $\varphi $ ? Conversely, could r support $\varphi $ without the agent believing that r supports $\varphi $ ? Regarding the latter, in our system any reason supports every logical truth, but we see it as possible for an agent not to believe that, in particular because an agent may not automatically see the truth of every logical truth or be aware of every reason. Hence, a reason may support a proposition without an agent believing that.

The question whether an agent may believe a reason to support a proposition without that reason supporting the proposition is more delicate. One way in which this could happen is if reasons can be treated as mere propositions. For example, an agent who accepts a fallacious mathematical argument may believe that true premises deductively support a false conclusion. In that case one may say that the agent incorrectly believes the premises to deductively support the conclusion. Again we find it better not to let propositions be reasons per se, but to keep to reasons being evidence of a sort, related to some experience. Take an agent who miscalculates the result of some mathematical equation. He accepts the premise that a bat and a ball cost $1.10, that the bat costs $1 more than the ball, and wrongly computes from that that the ball costs $0.10 (it actually costs $0.05).Footnote ⁴ On an objective reading of reasons, letting $p_1$ and $p_2$ stand for the two premises of the argument, and q for the false conclusion that the ball costs $0.10, we would write: $B((p_{1}\wedge p_{2})\,{:}\, q)$ to express that the agent believes the premises to deductively support the conclusion. But we find more appropriate to write instead that $B(r:(p_{1} \wedge p_{2} \rightarrow q))$ , where r stands for the calculation made by the agent, to represent the fact that the agent sees his calculations to support a certain deductive relation between premises and conclusions. As a result, we treat the relation of deductive support between bare propositions by means of the usual logical resources available in our system, and not in terms of the colon operator.

Because we see reasons as evidence-based propositions, we might wish to endorse the success principle $B(r\,{:}\, \varphi ) \rightarrow r\,{:}\, \varphi $ . That is, the agent believing some kind of experience to support a proposition would suffice to make the experience in question a supporting reason for $\varphi $ . A rational agent may therefore not necessarily be aware of all his or her reasons, but could not be mistaken about reasons that he or she explicitly endorses. One may object that even rational agents may be deluded about their reasons. For example, an agent might be mistaken about her own experiences. She has a memory of an encounter with a famous actor, but that encounter never happened, and the memory is no real experience. When asked to justify why she believes this actor is blond, she gives as a reason that she remembered seeing him to be blond during that encounter. Arguably, the reason does not support the proposition in that case, because the reason is not grounded in any true experience. Such cases invite us to guard against the aforementioned success principle. Because of that, we will propose a weakening of that principle, intended to secure more generality, and to not rule out such cases.

2.3. Adequacy

What about adequacy? Adequacy is by far the most central concept in our approach. There are various ways in which the notion can be thought of. One option is to think of an adequate reason as a reliable reason (see [Reference Goldman and Pappas18]). On that approach, an adequate reason is a reason that produces true beliefs most of the time. Like other critics of reliabilism, we do not view this characterization as strong enough. A stronger characterization might be in terms of Dretske’s notion of conclusive reason. According to Dretske, r is a conclusive reason for p if and only if r would not be the case if p were not the case ([Reference Dretske10]). Another approach is in terms of Chisholm’s or Sosa’s respective conceptions of nondefectiveness [Reference Chisholm8, Reference Sosa26], whereby a justification is nondefective provided it is not the basis of any false proposition. Consistently with those views, our use of the term “adequate” is also related to Spinoza’s understanding in Ethics, where the notions of adequate idea (adaequata idea) and adequate knowledge (adaequata cognitio) are meant to imply truth.Footnote ⁵

We draw inspiration from the latter set of approaches but an important caveat about our approach is that we do not propose to give an explicit definition of adequacy in terms of necessary and sufficient conditions. Instead, we propose to characterize adequacy in terms of two necessary conditions, and in terms of the interaction with the notions of support and belief we have in our ontology. The first central condition we impose on a reason for it to be adequate is very close to Chisholm’s, Sosa’s, or indeed Spinoza’s respective conceptions. We say that a reason is adequate only if the propositions it supports are true (axiom (A) below). But we also see adequacy as putting a requirement on the support relation. In particular, we consider that if an agent wrongly believes a reason r to support a proposition $\varphi $ (that is, believes so although r actually fails to support $\varphi $ ), then r cannot be adequate (axiom (AS) below). So, an adequate reason is such that it secures not only the propositions that it actually supports, but also such that it secures the correctness of the belief in the associated support relation. This second requirement basically corresponds to the weakening of the success condition evoked before, that $B(r\,{:}\, \varphi )$ should always entail $r\,{:}\, \varphi $ . It says that this is so provided r is adequate.

Let us make four important remarks on our treatment of adequacy:

1. 1. Our central requirement on adequacy ensures that if $\varphi $ is not true, and r supports $\varphi $ , then r cannot be adequate. This property of adequacy therefore comes close to Dretske’s, except that we rule out the possibility that a reason might be adequate evidence for one proposition and inadequate for another (we discuss that aspect in §5). That first requirement implies that reasons that do not validly support a given conclusion are inadequate.

2. 2. In our system $B(r\,{:}\, \varphi )$ does not entail that $Br$ , which says that an agent does not automatically deem adequate any reason she has in support of a proposition. For example, in the Müller–Lyer illusion, an agent who trusts her perceptual experience that two lines differ in length has a reason in support of the proposition that the lines are not equal. But she can have a different and overriding reason, her measuring of the lines (reason s), telling her that the lines do not differ in length. We would represent the case as:

   $$ \begin{align*} B(r\,{:}\, p) \wedge B(s\,{:}\, \neg p) \wedge Bs \wedge \neg Br. \end{align*} $$

3. 3. We do not identify the adequacy of a reason with the validity or even the soundness and validity of an argument. Consider the following example of what Sorensen [Reference Sorensen25] calls “paralemmatic reasoning”. Our agent has 10 cents in his pocket (premise 0). He infers from that, and from the information that the bat and ball cost $1.10 (premise 1) and the bat costs $1 more than the ball (premise 2), that he has enough money to buy the ball (conclusion). Let us assume he infers the conclusion really because he infers that the ball costs 10 cents. Let us represent by $p_{0}, p_{1}, p_{2}$ the premises of that argument, by q the correct conclusion that the agent has enough money to buy the ball, and by p the incorrect lemma that the ball costs 10 cents. Let us call r the agent’s calculations. The agent believes that $r\,{:}\, (p_{0} \land p_{1} \land p_{2} \rightarrow q)$ . The argument $p_{0} \land p_{1} \land p_{2} \rightarrow q$ has sound premises, and the conclusion validly follows. In effect, however, what r supports in the agent’s mind is a more specific argument, namely the argument that $p_{1} \land p_{2} \rightarrow p$ , combined with the argument that $p \wedge p_{0}\rightarrow q$ . Because of our adequacy constraint, r cannot be an adequate reason here, because r supports an incorrect argument. Sorensen in [Reference Sorensen25] writes that “paralemmas are precise counterexamples to bare evidentialism—the view that we know a proposition simply by virtue of having adequate evidence for it.” By adequate evidence we take Sorensen to refer to some intrinsic property of the evidence, holding irrespective of the agent’s beliefs. This is not our characterization of adequacy, precisely because for us a reason can support a logically perfect argument without being adequate. We are not bare evidentialists as a consequence.

4. 4. Finally, we do not view the two conditions we impose on adequacy, namely (A) and (AS), as sufficient conditions for a reason being adequate, but only as necessary conditions. For us, and as our semantics will show, adequacy is fundamentally a property of a reason and a world. Or put otherwise, the adequacy of a reason represents the property of a world being a good case for the reason in question (see [Reference Williamson33]). The same reason, with the same intrinsic properties, could turn out to be adequate in a good case and inadequate in a different case that is bad. Being a good case is not a notion we think we can explicitly define and exhaustively characterize. Intuitively, being a good case (relative to a reason) means being a case at which the reason is used in a way that is not lucky, that is safe, or that has the right fit, whatever those notions might mean. We consider our two axioms on adequacy to capture part of that intuition, but some of the cases we discuss (particularly Goldman cases, see below) are arguably cases in which both of our conditions are fulfilled, but which we do not want to characterize as adequate in the externalist sense we think is relevant.

Our axiomatic treatment of adequacy therefore implies that the adequacy of a reason does not lead to any false conclusion, and it also implies that the agent is not deluded about the experience reported in the reason. But to rehearse our main point, those are only two necessary conditions on adequacy. We view adequacy as an even stronger property of a reason and a world, namely as the property for a world to secure that the reason is not merely internally but also externally warranted.

3.1. Reason-based belief

Fix nonempty sets P of propositional letters and R of reason symbols (also called “reasons”). F is the set of formulas $\varphi $ defined by the following grammar:

$$ \begin{align*} \varphi ::=& \; p \mid \lnot\varphi \mid (\varphi\lor\varphi) \mid (r\,{:}\,\varphi) \mid r \mid B\varphi \\ & \; { p\in P, r\in R} \end{align*} $$

Other Boolean connectives are defined as abbreviations.

• • $r\,{:}\,\varphi $ says, “r supports $\varphi $ ”.

• • r says, “r is an adequate reason.”

  For the sake of clarity, we might have introduced a unary operator A in the language, and written $Ar$ to express “r is an adequate reason.” We choose to write r instead of $Ar$ to save symbols. Our logic will guarantee that every proposition supported by an adequate reason is true (i.e., we will have $r\,{:}\,\varphi \to (r\to \varphi )$ for all formulas $\varphi $ ). Note that, as we define it, adequacy is not relativized to a specific proposition. This makes our treatment of adequacy different from Dretske’s treatment of conclusiveness. For Dretske, a reason is not conclusive per se but conclusive for a proposition [Reference Dretske10]. In our setting, when a reason is adequate, it makes every proposition it supports true. Hence no reason can be such that it is adequate relative to one proposition it supports, and inadequate relative to another. This feature of our account may eventually prove too strong (we return to this issue in the last section of the paper), but our proposal is to think of adequacy as a fundamental property of truth-conduciveness.

• • $B\varphi $ says, “the agent believes $\varphi $ .”

  Table 1. The theory ${\mathsf {RBB}}$

  

  The formula $Br$ is therefore read, “the agent believes r is an adequate reason.” Sometimes it will be convenient to say that “the agent accepts reason r” to mean that $Br$ holds. So believing reason r is adequate and accepting reason r will be considered to mean the same thing.

To reduce the number of parentheses while ensuring unique readability of formulas, we adopt the convention that the colon operator binds more strongly than any Boolean connective. For example,

$$ \begin{align*} r\,{:}\,\varphi\to\psi &\quad \text{denotes}& (r\,{:}\,\varphi)\to\psi \quad \bigl[\text{and not } r\,{:}\,(\varphi\to\psi) \bigr]. \end{align*} $$

The theory ${\mathsf {RBB}}$ of reason-based belief appears in Table 1. We write $\vdash \varphi $ to mean that $\varphi $ is derivable in ${\mathsf {RBB}}$ . The negation is written $\nvdash \varphi $ .

Regarding the axioms and rules of ${\mathsf {RBB}}$ from Table 1, classical logic (CL) and modus ponens (MP) say that ${\mathsf {RBB}}$ is an extension of classical propositional logic. Kripke’s axiom (RK) says that reasons are closed under material implication, and reason necessitation (RN) says that reasons support all derivable formulas. Adequacy (A) says that if r supports $\varphi $ and r is an adequate reason, then $\varphi $ is true. Belief version of RK (BRK) says that if the agent believes r supports a conditional, and believes r supports the antecedent, then the agent believes r supports the consequent. Belief version of adequacy (BA) says that if the agent believes r supports $\varphi $ and the agent believes that r is an adequate reason, then the agent believes $\varphi $ . Adequate support (AS) says that if the agent believes r to support a proposition, then r does not support the proposition unless r is adequate. Well-known axiom from modal logic (D) says that the agent’s beliefs are consistent: she cannot have contradictory beliefs (i.e., believe both $\varphi $ and $\lnot \varphi $ for some $\varphi $ ). Finally, well-known rule from minimal modal logic (E) says that the agent’s beliefs do not distinguish between provably equivalent formulas.

As for mnemonics, (CL) is “classical logic,” (MP) is “modus ponens,” (RK) is Kripke’s axiom (K) of modal logic (used here for reasons), (RN) is “reason necessitation,” (A) is “adequacy,” (BRK) is “the belief version of RK,” (BA) is “the belief version of adequacy,” (AS) is “adequate support,” (D) is a well-known axiom from modal logic [Reference Chellas7], and (E) is a well-known rule from minimal modal logic [Reference Chellas7].

3.2. Semantics for ${\mathsf {RBB}}$

We now present a semantics for our system. The semantics is based on two main components: a neighborhood semantics for belief, intended to make belief as weak as possible, and a Kripke semantics for reasons, this time intended to capture the closure properties on reasons.Footnote ⁶ We justify both desiderata in the next section.

We construct the models of ${\mathsf {RBB}}$ in a couple of stages. We first begin with premodels, which are structures $M=(W\!,[\cdot ],N,V)$ having:

• • a nonempty set W of “possible worlds,”

• • a function $[\cdot ]:R\to \wp (W\times W)$ mapping each reason $r\in R$ to a binary relation $[r]\subseteq W\times W$ on the set of possible worlds,

• • a “neighborhood function” $N:W\to \wp (\wp (W))$ mapping each world w to a collection $N(w)$ of sets of worlds (“propositions”) that the agent believes at w,

• • a propositional valuation $V:W\to \wp (P)$ mapping each world w to the set $V(w)$ of propositional letters that are true at w.

To indicate that the components W, $[\cdot ]$ , N, and V come from premodel M, we may  write $W^M$ , $[\cdot ]^M$ , $N^M$ , and $V^M$ (respectively). Letting $w\in W$ be a world, $r\in R$ be a reason, and $X\subseteq W$ be a set of reasons (i.e., a “proposition”), we introduce the following abbreviations:

• • $r(w) := \{v\in W\mid (w,v)\in [r]\}$ is the set of all worlds that are r-accessible from w (i.e., accessible by the relation $[r]$ ).

• • $r^\circ := \{ v\in W \mid (v,v)\in [r]\}$ is the set of all worlds that are r-accessible from themselves. These are the worlds at which the reason r is said to be reflexive. As we will see shortly, a reason that is reflexive at a world will be adequate at that world. So reflexivity and adequacy are equivalent notions, and hence we may conflate the two, which ought not cause confusion.

To indicate the sets $r(w)$ and $r^\circ $ arise from worlds in premodel M, we may write $r^M(w)$ and $r^{M\circ }$ (respectively). Though not mentioned above, we do require that all pre-models satisfy the following property:

• (pr) For each $x\in P\cap R$ , we have $x\in V(w)$ if and only if $w\in x(w)$ .

This says that for propositional letters that are also reasons, the truth assignment given to x by the valuation V agrees with the reflexivity of x. This ensures that there is no ambiguity in the assignment of truth to propositional letters that are also reasons.Footnote ⁷

A pointed premodel is a pair $(M,w)$ consisting of a premodel M and a world w in M. We write $M,w\models \varphi $ to say that $\varphi $ is true at the pointed premodel $(M,w)$ , and we write $M,w\not \models \varphi $ for the negation. We define the satisfaction relation $\models $ and the set

$$ \begin{align*} [\![{\chi}]\!]_M:=\{v\in W\mid M,v\models\chi\} \end{align*} $$

of worlds in the premodel M at which the formula $\chi $ is satisfied as follows.

• • $M,w\models p$ means that $p\in V(w)$ .

• • $M,w\models \lnot \varphi $ means that $M,w\not \models \varphi $ .

• • $M,w\models \varphi \lor \psi $ means that $M,w\models \varphi $ or $M,w\models \psi $ .

• • $M,w\models r\,{:}\,\varphi $ means that $r(w)\subseteq [\![{\varphi }]\!]_M$ .

• • $M,w\models r$ means that $w\in r(w)$ .

• • $M,w\models B\varphi $ means that $[\![{\varphi }]\!]_M\in N(w)$ .

We note that $\models $ is well-defined: for each $x\in P\cap R$ , we have $x\in V(w)$ if and only $w\in x(w)$ by (pr), and therefore $M,w\models x$ is well-defined.

We say that a premodel $M=(W,[\cdot ],N,v)$ is a model if and only if M satisfies each of the following additional properties:

1. (brk) If $[\![{r\,{:}\,(\varphi \to \psi )}]\!]_M\in N(w)$ and $[\![{r\,{:}\,\varphi }]\!]_M\in N(w)$ , then $[\![{r\,{:}\,\psi }]\!]_M\in N(w)$ .

   This says that if the agent believes r supports the implication $\varphi \to \psi $ and its antecedent $\varphi $ , then she believes r supports the consequent $\psi $ as well.

2. (ba) If $[\![{r\,{:}\,\varphi }]\!]_M\in N(w)$ and $r^\circ \in N(w)$ , then $[\![{\varphi }]\!]_M\in N(w)$ .

   This says that if the agent believes r supports $\varphi $ and she believes r is reflexive, then she also believes $\varphi $ .

3. (as) If $[\![{r\,{:}\,\varphi }]\!]_M\in N(w)$ and $w\in r(w)$ , then $r(w)\subseteq [\![{\varphi }]\!]_M$ .

   This says that if the agent believes r supports $\varphi $ and r is reflexive, then r does in fact support $\varphi $ .

4. (d) $X\in N(w)$ implies $W-X\notin N(w)$ .

   This says that if the agent believes X at world w, then she does not believe the complement $W-X$ at world w.

(brk), (ba), (as), and (d) are the model-theoretic analogs of the axioms (BRK), (BA), (AS), and (D), respectively. A pointed model is a pair $(M,w)$ consisting of a model M and a world w in M.

Given a premodel M, to say that $\varphi $ is valid in M, written $M\models \varphi $ , means that $[\![{\varphi }]\!]_M=W$ . To say that $\varphi $ is valid, written $\models \varphi $ , means that $M\models \varphi $ for every model M. That is, validity is taken over the class of models (and not the larger class of pre-models). It is shown in Theorem A.1 that ${\mathsf {RBB}}$ is sound and complete for this semantics: we have $\vdash \varphi $ if and only if $\models \varphi $ . Unless explicitly noted otherwise, our focus in what follows will generally be on the concept of model (and not the concept of premodel).

The following terminology will be useful for what follows: given a reason r and a pointed model $(M,w)$ representing the key features of a particular situation of reason-based belief, to say

• • “r is adequate at $(M,w)$ ” means $M,w\models r$ ;

• • “r is veridical for $\varphi $ at $(M,w)$ ” means $M,w\models r\,{:}\,\varphi \to \varphi $ ; and

• • “r is veridical at $(M,w)$ ” means $M,w\models r\,{:}\,\varphi \to \varphi $ for each formula $\varphi $ .

In using the above terminology, we may omit mention of $(M,w)$ if it should be clear from context which pointed model is meant. It follows by the semantics that every adequate reason is veridical;Footnote ⁸ however, a reason may be veridical for a proposition without being thereby adequate. Note that being veridical is only a necessary condition for a reason to be adequate. This means we do not take veridicality to provide an explicit definition of adequacy, but only to put a constraint on what it is for a reason to be adequate.

3.3. Weak belief but strong reasons

${\mathsf {RBB}}$ posits a weak notion of belief. In particular, as reflected in the semantics, beliefs are not necessarily closed under material implication. That is, it is consistent to have

$$ \begin{align*} B(\varphi\to\psi)\land B\varphi \land \lnot B\psi , \end{align*} $$

which says that the agent believes an implication and the antecedent of the implication but not the consequent.

Reasons, on the other hand, are strong. First, as just seen, they encompass all derivable statements by (RN). Second, they are closed under implication (and hence under (MP)) by (RK). Third, if adequate, then (A) says that they are veridical: everything they support is true. Reasons therefore support many assertions (infinitely many, in fact, because each reason supports each of the infinitely many theorems of ${\mathsf {RBB}}$ by (RN)). This puts more requirements on reasons (i.e., they must do more things), which makes them stronger.

Reasons are governed by what is essentially the normal multi-modal logic $\mathsf {KT}$ (with one modal operator “ $r\,{:}\,$ ” for each reason r), except that the $\mathsf {T}$ axiom $r\,{:}\,\varphi \to \varphi $ (sometimes called “veridicality”) is not guaranteed to hold unless, according to (A), we make the additional assumption that r is adequate. This way of having a multi-modal logic like $\mathsf {KT}$ but with the “modal operator” r itself a formula (whose truth implies veridicality) is, to our knowledge, new.Footnote ⁹

We have chosen a theory of weak belief but strong reasons in order to keep things as simple as possible but still address some of the major trends in the epistemological study of knowledge as JTB. In all of the examples from epistemology we consider in this paper, we need some way to track an agent’s logical inferences and some way to link these inferences to what the agent believes. Our theory ${\mathsf {RBB}}$ is a rather minimalistic way of doing just this: reasons are used to handle the relevant inferencing, the agent can “accept” a reason (or not) by believing it to be adequate (or not), the agent comes to believe things supported by reasons she accepts, and the agent’s beliefs are always consistent. This way we can encode inferencing using a reason, indicate whether the agent accepts this inferencing, and thereby infer whether she believes some statement based on a reason. We also allow the possibility that she believes something without a reason, by which we mean that the set

$$ \begin{align*} \{B\varphi\}\cup\{B(r\,{:}\,\varphi)\to\lnot Br\mid r\in R\} \end{align*} $$

is consistent with our theory. This assumption is metaphysically disputable, however, and we may provide for every belief to be accompanied by a reason if we wish.

3.4. Consistency of reasons

One interesting theorem of ${\mathsf {RBB}}$ is the principle

(RC) $$ \begin{align} \vdash(Br\land Bs)\to( B(r\,{:}\,\varphi)\to\lnot B(s\,{:}\,\lnot\varphi)) \end{align} $$

of reason consistency. This principle says that if an agent believes reasons r and s are adequate, then she cannot believe that r and s support contradictory assertions. Intuitively, the derivability of (RC) follows because (BA) requires that an agent who accepts a reason and believes that reason to support a formula must also believe that formula, and (D) requires that an agent not have contradictory beliefs. Notice that if we take $r=s$ in (RC), then we obtain a statement provably equivalent to

(IC) $$ \begin{align} \vdash Br\to(B(r\,{:}\,\varphi)\to\lnot B(r\,{:}\,\lnot\varphi)), \end{align} $$

which says that a reason believed to be adequate is believed to be internally consistent.

If the agent does not believe r is an adequate reason, then she can believe that r is internally inconsistent. Put another way, the formula

$$ \begin{align*} \lnot Br \land B(r\,{:}\,\varphi) \land B(r\,{:}\,\lnot\varphi) \end{align*} $$

is consistent with our theory. Similarly, if the agent believes r is an adequate reason but does not believe s is an adequate reason, then our theory does not rule out the possibility that she believes r and s are inconsistent. That is,

$$ \begin{align*} Br\land \lnot Bs \land B(r\,{:}\,\varphi)\land B(s\,{:}\,\lnot\varphi) \end{align*} $$

is also consistent with our theory.

Finally, since the theory ${\mathsf {RBB}}$ is consistent (and hence does not prove both $\varphi $ and $\lnot \varphi $ for some formula $\varphi $ ),Footnote ¹⁰ any adequate reason is internally consistent. That is,

(AIC) $$ \begin{align} \vdash r\,{:}\,\varphi\to(r\to\lnot (r\,{:}\,\lnot\varphi)) , \end{align} $$

which says that a reason r that supports $\varphi $ and is adequate cannot also support $\lnot \varphi $ . It follows from (AIC) that any internally inconsistent reason is not adequate.

3.5. Logical closure and combination of reasons

If $\psi $ is a logical consequence of $\varphi $ , meaning $\vdash \varphi \to \psi $ , then our theory says that r is a reason to believe the consequent $\psi $ whenever r supports the antecedent $\varphi $ . That is,

(RCLC) $$ \begin{align} \vdash \varphi\to\psi \quad\text{implies}\quad \vdash r\,{:}\,\varphi\to r\,{:}\,\psi . \end{align} $$

The proof: from $\varphi \to \psi $ , we obtain $r\,{:}\,(\varphi \to \psi )$ by (RN). This is the antecedent of an instance of (RK), so the consequent $r\,{:}\,\varphi \to r\,{:}\,\psi $ of this instance is derivable by an application of (MP). This completes the proof. In examining this proof, we see that (RCLC) is a consequence of the stronger logical principle encompassed by our axiom

(RK) $$ \begin{align} r\,{:}\,(\varphi\to\psi)\to(r\,{:}\,\varphi\to r\,{:}\,\psi) , \end{align} $$

which says that reasons are closed under material implication.

The principle (RCLC) says that reasons are closed under logical consequence. It is unclear whether this is a desirable principle. For example, it may make more sense to say that if $\psi $ is a logical consequence of $\varphi $ and r supports $\varphi $ , then it is not r itself that supports the consequence $\psi $ . Instead, what is wanted is some more complicated reason $r'$ that not only references r but also provides some reason s as to why $\psi $ obtains from $\varphi $ . That is, we might seek a principle like this:

(App) $$ \begin{align} s\,{:}\,(\varphi\to\psi)\to(r\,{:}\,\varphi\to(s\cdot r)\,{:}\,\psi) . \end{align} $$

This is the principle of Application from justification logic (JL) [Reference Artemov, S., Fitting and Zalta3]. It says: if s supports the implication $\varphi \to \psi $ and r supports the antecedent $\varphi $ , then a new object $s\cdot r$ obtained by combining s and r supports the consequent $\psi $ . In essence, the more complex reason $s\cdot r$ keeps track of everything we would need to check to see that $\psi $ indeed obtains: the initial reason r for the antecedent $\varphi $ and a reason s for the implication $\varphi \to \psi $ . Further, the syntactic structure of $s\cdot r$ , with s to the left and r to the right, tells us what kind of a reason we have: based on the form of (App), it is suggested that s is some implication, r is the antecedent of that implication, and $s\cdot r$ is a reason for the consequent. In essence, we are describing specific witnesses for an instance of the rule (MP) of modus ponens:

$$ \begin{align*} \frac{\varphi\to\psi\quad\varphi}{\psi}\;\text{(MP)} \quad\text{is encoded by}\quad \frac{s\quad r}{s\cdot r} . \end{align*} $$

(App) is a more nuanced version of (RK): if we have $r\,{:}\,(\varphi \to \psi )$ and $r\,{:}\,\varphi $ , then we do not obtain $r\,{:}\,\psi $ straightaway using (App). Instead, we must construct the reason $r\cdot r$ in support of $\psi $ . The single instance of “ $\,\cdot \,$ ” in the syntactic structure of the latter reason reflects our use of one derivational step (i.e., one instance of (MP)) to obtain $\psi $ .

To do away with (RCLC), we must do away with (RK) and modify (RN). In particular, let $R_0$ be a nonempty set of “basic” reasons, define R to be the smallest extension of $R_0$ satisfying the property that $s,t\in R$ implies $s\cdot t\in R$ , replace scheme (RK) by scheme (App), and restrict (RN) by requiring that $r\in R_0$ . (It is assumed that all other schemes and rules can use reasons coming from the full set R.) Call the resulting theory ${\mathsf {RBB}}{+}$ (App). In ${\mathsf {RBB}}{+}$ (App), it is consistent to have

(1) $$ \begin{align} r\,{:}\,(\varphi\to\psi)\land r\,{:}\,\varphi\land\lnot (r\,{:}\,\psi) , \end{align} $$

which says that r is not closed under implication. As a result, it can be shown that (RCLC) no longer obtains. But note that (RK) and (RCLC) do not fail in ${\mathsf {RBB}}{+}$ (App) because logical or materially implied consequences of assertions are no longer “accessible” by some reason. Indeed, in the situation (1) under the theory ${\mathsf {RBB}}{+}$ (App), the logical consequence $\psi $ of $\varphi $ is still “accessible” by the reason $r\cdot r$ . However, this reason $r\cdot r$ is more “complex” than the original reason r (in terms of the number of instances of the (MP)-encoding Application operator “ $\cdot $ ” that appear inside it). In general, distant consequences that would require many repetitions of (App) are still accessible; it is just that the reasons that access these consequences may be very “complex.”

Justification logic (JL) [Reference Artemov, S., Fitting and Zalta3] is the study of logics of reason-based belief (with reasons thought of as “justifications”). Defining ${\mathsf {JL}}_0$ to be the fragment of ${\mathsf {RBB}}{+}$ (App) obtained by omitting all belief formulas and belief axioms from the theory, JLs may be thought of as extensions of ${\mathsf {JL}}_0$ .Footnote ¹¹ Many JLs permit other kinds of combinations of reasons than what we saw with (App). Our logic ${\mathsf {RBB}}$ is closely related to the JL tradition, though we conspicuously omit (App), retain (RK) (and thereby endorse (RCLC)), leave (RN) without the additional restriction, and maintain a set R of primitive reasons that cannot be combined to form more complex reasons. In so doing, we lose the ability to have the syntactic structure of reasons reflect the structure of derivations in the theory, and thereby forgo a more nuanced tracking of the interaction between logical consequence and the complexity of reasons. We accept these consequences in the interest of developing a system that is simple and yet still of use to the formal epistemologist. Nevertheless, we recognize that a reader may be interested to see a thorough study of more sophisticated extensions of our theory that allow for the combination of reasons along the lines of (App) (and perhaps for other features as well). We advise such a reader to consult the JL literature directly [Reference Artemov, S., Fitting and Zalta3].

Since our theory does not allow the agent to combine reasons in the sense of (App) and beliefs are not closed under implication, it is consistent for us to have a situation wherein the agent has the requisite information to draw a belief but simply does not draw it. For example, assuming $s\neq r$ , it is consistent to have

(2) $$ \begin{align} Bs\land Br \land B(s\,{:}\,(\varphi\to\psi))\land B(r\,{:}\,\varphi) \land \lnot B\psi , \end{align} $$

which says that the agent believes s is adequate and believes s supports an implication (and hence believes the implication by (BA)), believes r is adequate and believes r supports the antecedent of the implication (and hence believes the antecedent by (BA)), and yet the agent does not believe the consequent (even though the believes the implication and its antecedent). The problem is that her beliefs are not closed under (MP). This is so despite the fact that, by (RCLC) and (RB), beliefs coming from the same reason are closed under (MP):

$$ \begin{align*} \vdash (Bt\land B(t\,{:}\,(\varphi\to\psi))\land B(t\,{:}\,\varphi))\to B\psi . \end{align*} $$

So long as the implication and its antecedent come from separate beliefs as in (2), the agent need not believe the consequent.

The consistency of (2) is a consequence of our design: we use reason operators to encode inferencing, and we use belief operators to encode the particular inferencing and the individual assertions that the agent accepts (in terms of her affirmed beliefs). As such, the situation (2) is one in which the agent has not yet performed sufficient inferencing to accept the conclusion $\psi $ , even though she is very close (after all, she has done enough to accept the implication $\varphi \to \psi $ and its antecedent $\varphi $ ). In essence, this lack of closure allows us to place one kind of constraint on the agent’s inferencing powers. If desired, one can place even more severe constraints as in [Reference Baltag, Renne and Smets5]; however, this seems to require more syntax and additional axioms. One can also go the other way and lift these constraints entirely; in §3.6 we will suggest one natural way to do this in our setting. But for now we retain what we hope is a “happy medium” in the form of our theory ${\mathsf {RBB}}$ .

3.6. Implication-closed and purely reason-based beliefs

We saw in (2) that reason-based beliefs in ${\mathsf {RBB}}$ need not be closed under implication if the source reasons are different. If we would like to ensure that reason-based beliefs are always closed under implication, even if the beliefs come from separate reasons, then a simple solution is to introduce a “master reason” $\sigma $ that encodes the combined information of all reasons the agent accepts. This requires us to expand our reason set R to include a new symbol $\sigma $ not already present and then add the following additional schemes to ${\mathsf {RBB}}$ :

$$ \begin{align*} & {\mathrm{(MA)}} \quad \sigma\to(Br\to r) \\ & {\mathrm{(MB)}} \quad B\sigma \\ & {\mathrm{(MR)}} \quad B(r\,{:}\,\varphi)\to(Br\to B(\sigma\,{:}\,\varphi)) \end{align*} $$

(MA) says that every accepted reason is adequate if the master reason is adequate, (MB) says that the agent always accepts the master reason, and (MR) says that anything the agent believes is supported by an accepted reason the agent will also believe to be supported by the master reason. Adding these principles makes it so that $\sigma $ is the sum of the agent’s evidence. Calling ${\mathsf {RBB}}_\sigma $ the theory obtained from ${\mathsf {RBB}}$ by expanding the set R of reasons to include a new symbol $\sigma $ and adding (MA), (MB), and (MR), it follows that

(RCL2) $$ \begin{align} {\mathsf{RBB}}_\sigma \vdash (Bs\land Br\land B(s\,{:}\,(\varphi\to\psi)) \land B(r\,{:}\,\varphi)) \to B\psi . \end{align} $$

Indeed, if the agent accepts s, believes s supports an implication, accepts r, and believes r supports the antecedent of the implication, then it follows by (MR) that the agent believes $\sigma $ supports the implication and its antecedent. Applying (BRK), it follows that the agent believes $\sigma $ supports the consequent. But the agent also accepts $\sigma $ by (MB), so it follows by an application of (BA) that the agent believes the consequent. It is in this sense that the agent “combines” the information conveyed by reasons the agent accepts into the master reason $\sigma $ .

While we have shown that the ${\mathsf {RBB}}_\sigma $ -agent may combine the information from two reasons to derive her beliefs, there is no need to restrict the number of reasons to two. Indeed, according to (MB), the agent implicitly combines into $\sigma $ the information from every reason she believes to be adequate, no matter how many of these there may be.

In ${\mathsf {RBB}}_\sigma $ , the master reason $\sigma $ serves as a witness for an existential quantifier over the believed support of accepted reasons.Footnote ¹² In particular, (MR) tells us that if there exists an accepted reason r that the agent believes supports $\varphi $ , then the agent believes that $\sigma $ supports $\varphi $ . Hence by (MB), if there exists such an r, then the agent believes $\sigma $ supports $\varphi $ and is itself accepted. If we were to add quantifiers to the language (something we do later in §3.7), we could express this as:

$$ \begin{align*} (\exists r)(Br\land B(r\,{:}\,\varphi))\to(B\sigma\land B(\sigma\,{:}\,\varphi)) . \end{align*} $$

That is, if there exists an accepted reason that the agent believes supports $\varphi $ , then $\sigma $ is a witness to the existential quantifier.

Both ${\mathsf {RBB}}_\sigma $ and the basic theory ${\mathsf {RBB}}$ allow for the possibility that the agent believes a formula $\varphi $ without any supporting reasons (i.e., she does not believe to be adequate any reason that she believes supports $\varphi $ ). This is the same as saying that the set

$$ \begin{align*} \{B\varphi\} \cup \{ B(r\,{:}\,\varphi)\to\lnot Br \mid r\in R\} \end{align*} $$

is consistent with both ${\mathsf {RBB}}_\sigma $ and ${\mathsf {RBB}}$ . If this situation is undesirable, then a simple remedy is to extend the theory ${\mathsf {RBB}}_\sigma $ by adding a principle that says all beliefs are believed to be supported by the master reason:

$$ \begin{align*} {\mathrm{(MT)}} \quad B\varphi\to B(\sigma\,{:}\,\varphi). \end{align*} $$

With (MT) in place, the agent believes only those things she believes are supported by $\sigma $ and hence, by (MR), she believes only those things she believes are supported by some reason. In short, every belief is “reason-based.” Defining ${\mathsf {RBB}}_\sigma ^+$ to be the theory obtained from ${\mathsf {RBB}}_\sigma $ by adding (MT), it follows by (MT), (RB), and (MB) that

$$ \begin{align*} {\mathsf{RBB}}_\sigma^+\vdash B\varphi \leftrightarrow B(\sigma\,{:}\,\varphi) , \end{align*} $$

which says that the agent believes something just in case it is supported by the master reason. But then belief can be conflated with the master reason. As a result, we have by (RCLC) that the beliefs of the ${\mathsf {RBB}}_\sigma ^+$ -agent are always closed under (MP). Belief in ${\mathsf {RBB}}_\sigma ^+$ is therefore governed by the normal modal logic $\mathsf {KD}$ .

Semantics for ${\mathsf {RBB}}_\sigma $ and for ${\mathsf {RBB}}_\sigma ^+$ may be found in §A.1. It is shown in Theorem A.2 that each of ${\mathsf {RBB}}_\sigma $ and ${\mathsf {RBB}}_\sigma ^+$ is sound and complete for its semantics.

3.7. Quantification over reasons

We have observed that the theory ${\mathsf {RBB}}$ does not require that every belief arise from a reason: it is consistent with ${\mathsf {RBB}}$ for the agent to believe $\varphi $ and yet have no accepted reason r she believes supports $\varphi $ . If we were to introduce quantifiers over reasons into our language, then we could express this situation by saying that the following formula is consistent:

$$ \begin{align*} B\varphi \land (\forall r)(B(r\,{:}\,\varphi)\to\lnot Br) . \end{align*} $$

Another example: we might like to say that r is the unique accepted reason the agent believes supports $\varphi $ :

$$ \begin{align*} B(r\,{:}\,\varphi) \land Br \land (\forall s)( (s\neq r \land B(s\,{:}\,\varphi)) \to \lnot Bs) . \end{align*} $$

To allow such expressions as formulas, we extend our set of formulas F to the larger set $F^\forall $ consisting of all formulas $\varphi $ that may be formed by the following grammar:

$$ \begin{align*} \varphi &::= p \mid \lnot\varphi \mid (\varphi\lor\varphi) \mid (r\,{:}\,\varphi) \mid r \mid B\varphi \mid r=r \mid (\forall r)\varphi \\ & \qquad\!\! {p\in P, r\in R} \end{align*} $$

We adopt usual Boolean connective abbreviations along with the following:

$$ \begin{align*} r\neq s &:= \lnot(r=s) \\ (\exists r)\varphi &:= \lnot(\forall r)\lnot\varphi \\ (\forall r\neq s)\varphi &:= (\forall r)(r\neq s\to\varphi) \\ (\exists r\neq s)\varphi &:= (\exists r)(r\neq s\land\varphi) \end{align*} $$

Note that in this language, an element $r\in R$ can act both as a reason (as in the formula $r\,{:}\, p$ ) and as a quantifier variable (as in the formula $(\forall r)(r\,{:}\, p)$ ). Therefore, reasons may appear either bound or free in formulas, with the notion of bound and free defined in the usual way. For reasons s and r and a formula $\varphi $ , we say that s is free for r in $\varphi $ to mean that r has no free occurrence in $\varphi $ within the scope of a quantifier $(\forall s)$ . Put another way, if s is free for r in $\varphi $ , then in the formula

$$ \begin{align*} \varphi[s/r] \end{align*} $$

obtained by substituting all free occurrences of r in $\varphi $ by s, no newly replaced occurrence is bound. Examples: s is free for r in $(\forall t)(t\neq r)$ but not in $(\forall s)(s\neq r)$ .

The theory ${\mathsf {QRBB}}$ of quantified reason-based belief is defined by adding to the axiomatization of ${\mathsf {RBB}}$ the axioms and rules in Table 2. (UD), (UI), and (Gen) are standard principles of first-order quantification. (EP) and (EN) say that two reasons are considered to be the same if and only if they are syntactically identical.

Table 2. The theory ${\mathsf {QRBB}}$ consists of ${\mathsf {RBB}}$ augmented with the above axioms and rule

It is shown in Corollary A.11 that for each $\varphi \in F$ not containing quantifiers, we  have ${\mathsf {QRBB}}\vdash \varphi $ if and only if ${\mathsf {RBB}}\vdash \varphi $ . It is therefore unproblematic for us to simply write $\vdash \varphi $ to say that $\varphi $ is provable.

We can extend ${\mathsf {QRBB}}$ to the theory ${\mathsf {QRBB}}_\sigma $ obtained by extending R to include a new master reason $\sigma $ and adding the schemes (MA), (MB), and (MR) for $\sigma $ . We can further extend ${\mathsf {QRBB}}_\sigma $ to the theory ${\mathsf {QRBB}}_\sigma ^+$ obtained by adding the additional scheme (MT) to guarantee all beliefs are reason-based.

Semantics for ${\mathsf {QRBB}}$ , for ${\mathsf {QRBB}}_\sigma $ , and for ${\mathsf {QRBB}}_\sigma ^+$ may be found in §A.2. It is shown in Theorems A.5 and A.12 that each of ${\mathsf {QRBB}}$ , ${\mathsf {QRBB}}_\sigma $ , and ${\mathsf {QRBB}}_\sigma ^+$ is sound and complete for its semantics. However, for the completeness results, there is one caveat: our proofs require that the set R of reasons be at least countably infinite.

4.1. Two notions of justification

In our theory, there are (at least) two natural ways to define JTB:

• • ${\mathsf {JTB}}^e_r(\varphi ):=B(r\,{:}\,\varphi )\land Br\land r$ , and

• • ${\mathsf {JTB}}^i_r(\varphi ):=B(r\,{:}\,\varphi )\land Br\land \varphi $ .

Both imply that the agent has a true belief that $\varphi $ . However, ${\mathsf {JTB}}^e_r(\varphi )$ suggests that the agent has a true belief justified by an adequate reason, whereas ${\mathsf {JTB}}^i_r(\varphi )$ suggests that the agent only has a true belief justified by a prima facie reason (that may not be adequate). ${\mathsf {JTB}}^e_r$ is thus externalist, while ${\mathsf {JTB}}^i_r$ is internalist.

Gettier’s achievement was to deny that ${\mathsf {JTB}}^i_r(\varphi )$ is the same as knowledge of $\varphi $ . Thus, if we assume that

(G2) $$ \begin{align} B(r\,{:}\, p) \land Br \land B(r\,{:}\, (p \rightarrow p \vee q)) \land (\lnot p\land q) , \end{align} $$

then we have Gettier’s second case. This is the case where the agent named Smith has a reason to believe p (“Jones owns a Ford”) and “realizes” that $p\lor q$ (“Jones owns a Ford or Brown is in Barcelona”) follows from p on that basis; however, unknown to Smith, p is false and q is true, hence $p \vee q$ is premised on a false lemma. Let us call r the reason Smith has to believe p. We can safely assume that r does indeed support p in that case $(r\,{:}\, p$ ) (i.e., the past evidence adduced by the agent does correspond to a real experience of his) and we leave this premise implicit in (G2) above, since delusion is not the problem in this case. Since p is false, and $r\,{:}\, p$ by assumption, r cannot be adequate, by (A). However, since r supports p, we have by (RCLC) that r supports $p\lor q$ . Smith is said by Gettier to realize that p entails $p\vee q$ on the basis of his reason, and by (BRK) it follows that $B(r\,{:}\,(p\vee q))$ . Moreover, Smith has no reason supporting q that she believes is adequate (indeed, in the scenario, “Brown is in Barcelona” is consciously picked at random by Smith). So we are in a situation where Smith has an internally justified true belief that p $({\mathsf {JTB}}^i_r p$ ), and also an internally justified true belief that $p \vee q$ ( ${\mathsf {JTB}}^i_r(p\lor q)$ ), but he fails to have an externally justified true belief of either proposition $(\neg {\mathsf {JTB}}^e_r p$ , and $\neg {\mathsf {JTB}}^e_r(p\lor q))$ .

In contrast, if we assume that ${\mathsf {JTB}}^e_r(p)$ , which is

(G2^′) $$\begin{align} B(r\,{:}\, p)\land Br \land r , \end{align} $$

this time r is adequate. Since r supports p, it also supports $p\lor q$ by (RCLC). By (A) both p and $p\lor q$ are true. Since the agent believes r is an adequate reason supporting p (and therefore also supporting $p\lor q$ ), she believes both p and $p\lor q$ , and in this case her belief is based on an adequate (and hence veridical) reason.

In general, it is easy to see that ${\mathsf {JTB}}^e_r(\varphi )$ satisfies:

• • $\vdash {\mathsf {JTB}}^e_r(\varphi \to \psi )\to ({\mathsf {JTB}}^e_r(\varphi ) \to {\mathsf {JTB}}^e_r(\psi ))$ ,

  which says that external JTB based on a reason r is closed under implication;

• • $\vdash {\mathsf {JTB}}^e_r(\varphi )\to \varphi $ ,

  which says that external JTB is veridical; and

• • $\vdash {\mathsf {JTB}}^e_r(\varphi )\to (r\,{:}\,\psi \to \psi )$ ,

  which says that if an agent has an external JTB based on reason r, then r cannot support any false assertions (so-called “false lemmas”).

To compare with internal JTB, one can show that ${\mathsf {JTB}}^i_r(\varphi )$ satisfies:

• • $\vdash {\mathsf {JTB}}^i_r(\varphi \to \psi )\to ({\mathsf {JTB}}^i_r(\varphi ) \to {\mathsf {JTB}}^i_r(\psi ))$ ,

  which says that internal JTB based on a reason r is closed under implication;

• • $\vdash {\mathsf {JTB}}^i_r(\varphi )\to \varphi $ , which says that internal JTB is veridical; and

• • $\nvdash {\mathsf {JTB}}^i_r(\varphi )\to (r\,{:}\,\psi \to \psi )$ ,

  which says that if an agent has an internal JTB based on reason r, then r might support false assertions (so-called “false lemmas”).

The differences between ${\mathsf {JTB}}^e_r$ and ${\mathsf {JTB}}^i_r$ are in the last property. So we see that the main difference between external and internal JTB is in the adequacy of the reason on which the JTB is based.

Using our quantified language, we adopt the following abbreviations:

$$ \begin{align*} {\mathsf{JTB}}^e(\varphi):=(\exists r){\mathsf{JTB}}^e_r(\varphi) \quad\text{and}\quad {\mathsf{JTB}}^i(\varphi):=(\exists r){\mathsf{JTB}}^i_r(\varphi) . \end{align*} $$

${\mathsf {JTB}}^e(\varphi )$ says that the agent has an external JTB for $\varphi $ (based on some reason), and ${\mathsf {JTB}}^i(\varphi )$ says the same but for internal JTB.

4.2. Is knowledge ${\mathsf {JTB}}^e$ ?

${\mathsf {JTB}}^i$ falls prey to Gettier’s examples because the supporting reason need not be veridical (i.e., it admits “false lemmas”). ${\mathsf {JTB}}^e$ , however, requires an adequate supporting reason, and hence this reason is necessarily veridical (i.e., it admits “no false lemmas”). This suggests we examine the equation

(KJTBe) $$ \begin{align} K\varphi := {\mathsf{JTB}}^e(\varphi) , \end{align} $$

which defines knowledge as external JTB. What should we think of this equation?

Consider the “fake barn county” situation [Reference Goldman17]: the agent is in a county that has numerous fake barns that look exactly like real barns. Not knowing she is in this county, she sees what she thinks is a barn and concludes that it is indeed a barn. It turns out she is correct because, by chance, she happens to be looking at the only real barn in the entire county. Obviously, she has an internal JTB that she sees a barn, though most philosophers argue that she does not know she sees a barn.Footnote ¹³ But does she have an external JTB in this case? One reason to answer affirmatively: the agent’s reason is veridical, unlike in Gettier’s original examples.

However, veridicality does not imply adequacy. Take r and p so that

$$ \begin{align*} r & \quad \text{is read,} & & \text{``the agent sees what looks to her like a barn,'' and} \\ p & \quad \text{is read,} & & \text{``what the agent sees is a barn.''} \end{align*} $$

Our agent is in the situation:

(Barn) $$ \begin{align} B(r\,{:}\, p)\land Br \land p . \end{align} $$

That is, the agent believes that her seeing what looks like a barn supports the assertion that what she sees is a barn, she believes r is adequate to guarantee the truth of what it supports, and the agent does actually see a barn. But is r in fact adequate? If we say it is, then we run into the following problem: had the agent picked a different barn-looking structure that turned out to be a fake, we would have

(Barn^′) $$\begin{align} B(r\,{:}\, p)\land Br\land\lnot p , \end{align} $$

from which it would follow by (AS) and the assumed adequacy of r that p holds (i.e., we have $(r\,{:}\, p)\land p$ ), but this contradicts the assumed adequacy of r (since in fact $\neg p$ ). This suggests to us that r is not necessarily adequate; that is, each of (Barn) $\land\; r$ and (Barn) $\land\; \lnot r$ is consistent with our intuitions about the “fake barn county” example. Conclusion: the agent need not have external JTB in this case.

We take it that the “fake barn county” example seeks to challenge the agent’s acumen in determining when it is safe to reason according to the principle

(WSWG) $$ \begin{align} \text{(what I see looks like an} X) \to \text{(what I see is an} X) , \end{align} $$

which has the colloquial reading “what I see is what I get.”Footnote ¹⁴ Since adequacy implies veridicality, one could use our notion of adequacy to indicate that the agent’s use of (WSWG) is licensed. In particular, if we assume that (Barn) $\land\;r$ , then we might construe this as a case in which the agent is in “normal barn county” (where there are no fake barns) and so her use of (WSWG) is licensed: r is an externally valid reason for the agent to infer that she sees a barn, so the agent knows that she sees a barn. In contrast, if we assume that (Barn) $\land \lnot r$ , then we might construe this as a case in which the agent is back in “fake barn county” and not licensed to draw the conclusion: r is not an externally valid reason for her to infer that she sees a barn, so she does not know that she sees a barn.

So let us assume that our definition (KJTBe) of knowledge as external JTB is correct. Is it then possible to define knowledge (i.e., external JTB) in terms of internal JTB plus some other condition? Indeed it is:

$$\begin{align*}\vdash {\mathsf{JTB}}^e(\varphi) \leftrightarrow (\exists r)(r \land {\mathsf{JTB}}^i_r(\varphi)). \end{align*}$$

In words: to have external justification it is necessary and sufficient to have an adequate justification that serves as the basis for an internal JTB. Zagzebski’s criticism of a JTB-based analysis of knowledge [Reference Zagzebski34] might apply here: we either must sever the link between truth and justification (thereby going so far as to concede that there is knowledge in Gettier cases) or else assert that “there is no degree of independence at all between truth and justification” (in order to avoid Gettier problems). Zagzebski’s position is that neither horn of her dilemma is satisfactory, and so the proper way to avoid the dilemma is to reject the possibility of analyzing knowledge in terms of JTB plus  some extra component (i.e., reject the “knowledge is $\text {JTB}+x$ ” approach all together); see also [Reference Williamson33]. We accept that our approach is close to endorsing the second horn of Zagzebski’s dilemma. However, by distinguishing adequacy from veridicality, we can still maintain a notion of independence between truth and justification. In particular, pace Zagzebski, our semantic analysis distinguishes between “adequate belief” (i.e., ${\mathsf {JTB}}^e$ ) and “lucky true belief” (i.e., ${\mathsf {JTB}}^i$ ).

5.1. Knowledge from mixed reasons?

Let us start with the worry that our account might be too weak. The ${\mathsf {JTB}}^e$ analysis of knowledge raises the issue of the force of the quantifier on the right side of the equivalence. To appreciate the problem, it is worth reminding ourselves of one of the first responses to Gettier’s examples: the so-called “no false lemmas” (hereafter “NFL”) requirement (see [Reference Clark9, Reference Harman19, Reference Sosa26]). The NFL requirement is meant to rule out situations like Gettier’s second case, wherein the agent starts from a mistaken belief that p to obtain a correct belief that $p\lor q$ . Thinking of the reasoning sequence of beliefs $\langle {p,p\lor q}\rangle $ as a “proof,” the initial “lemma” (i.e., assumption) p is false, but then a perfectly legitimate inference step to a logical consequence $p\lor q$ ends up on a formula that just so happens to be true.

In our framework, the obvious counterpart to the NFL requirement is the “no inadequate lemmas” (henceforth “NIL”) requirement:

$$ \begin{align*} {\mathsf{NIL}}(\varphi) := (\forall s)({\mathsf{JTB}}^i_s(\varphi)\to s) . \end{align*} $$

This says that every reason that supports an internal JTB of $\varphi $ is adequate. Since adequate reasons support only true formulas (by axiom scheme (A)), the NIL requirement guarantees that no false “lemma” (i.e., formula) intrudes on a reason justifying a potential internal JTB of $\varphi $ . This gives rise to the following notion of JTB with no inadequate lemmas:

$$ \begin{align*} {\mathsf{JTB}}{+}{\mathsf{NIL}}(\varphi) := {\mathsf{JTB}}^i(\varphi)\land{\mathsf{NIL}}(\varphi) . \end{align*} $$

This notion of JTB is logically stronger than external JTB: ${\mathsf {JTB}}^e(\varphi )$ only requires that there be one adequate reason supporting an internal JTB of $\varphi $ , whereas ${\mathsf {JTB}}{+}{\mathsf {NIL}}$ requires adequacy of every reason supporting an internal JTB of $\varphi $ . Thus

$$ \begin{align*} \vdash{\mathsf{JTB}}{+}{\mathsf{NIL}}(\varphi)\to{\mathsf{JTB}}^e(\varphi) \end{align*} $$

but not the other way around.

These considerations raise a potential worry for the ${\mathsf {JTB}}^e$ analysis of knowledge: what happens when the agent rests her beliefs in a proposition (such as $p\lor q$ ) on multiple sources? For example,Footnote ¹⁶ suppose our agent, who has excellent eyesight, sees someone in the distance but cannot quite make out who it is. Nevertheless, based on what she can see (represented by reason s), she correctly believes that the person in the distance is either Tweedle Dee or Tweedle Dum (represented respectively by $p\lor q$ ). Further, she has another reason r to believe that the person is Tweedle Dee (i.e., p). For example, a friend might have told her that Tweedle Dum is on vacation in some faraway country. Now, unknown to our agent, the person in the distance is actually Tweedle Dum. Put formally:

(TDTD) $$ \begin{align} B(s\,{:}\,(p\lor q))\land \lnot &B(s\,{:}\, p) \land \lnot B(s\,{:}\, q) \land B(r\,{:}\, p) \;\land \nonumber\\ &\quad Bs \land Br \land s \land \lnot r\land (\lnot p\land q) . \end{align} $$

That is, the agent believes s supports the disjunction (that the person is Dee or Dum) but no disjunct, she believes r supports the claim it is Dee, she believes s and r to be adequate, s is adequate (by hypothesis, because the agent’s eyesight is excellent, and it could not possibly be someone other than Dee or Dum), r is inadequate (since the friend’s information is not reliable), and the person is actually not Dee but Dum. Now suppose we add to (TDTD) the assumption

(NoR) $$ \begin{align} (\forall t)((t\neq s \land t\neq r)\to \lnot Bt) \end{align} $$

that the agent believes no other reasons to be adequate. It can be shown that

$$ \begin{align*} \begin{array}{ll} \vdash[(\mathrm{TDTD})\land(\mathrm{NoR})]]\to{\mathsf{JTB}}^e(p\lor q) & \text{but } \\ \nvdash[(\mathrm{TDTD})\land(\mathrm{NoR})]\to{\mathsf{JTB}}{+}{\mathsf{NIL}}(p\lor q) . \end{array} \end{align*} $$

In words: the agent has external JTB that $p\lor q$ (because the adequate reason s supports the disjunction); however, she does not have JTB with NIL of $p\lor q$ (because the inadequate reason r supports the disjunction). But is it a mistake to equate knowledge with ${\mathsf {JTB}}^e$ instead of with ${\mathsf {JTB}}{+}{\mathsf {NIL}}$ ?

One reaction is to deny that there is knowledge when the universal condition is not satisfied. For an example supporting this reaction, suppose the agent proves that a certain Mersenne number $m=2^n-1$ is prime. Later, she bolsters her belief in the primality of m by coming to believe (incorrectly) that all Mersenne numbers are prime (i.e., all numbers of the form $2^k-1$ are prime, which is false). Can the agent still be said to know that m is prime? On one account, it seems not. Such situations of mixed reasons, where an agent has both adequate and inadequate reasons supporting the same proposition, arguably occur often in everyday life.

We are inclined to the opposite view: in a situation of mixed reasons, the agent can still have knowledge. Returning to the primality example, if the agent learns that not all Mersenne numbers are prime, then she will still believe that m is prime on the basis of her adequate “backup” reason (that she proved m is prime). So she could still be said to know that m is prime.Footnote ¹⁷ The Dee/Dum case is arguably similar: if the agent were to learn that r is unreliable, then she would still have an external JTB of the disjunction based on the “backup” reason s.

Perhaps the most difficult challenge to the claim that (KJTBe) is correct even in the case of mixed reasons comes when the quantity of inadequate reasons vastly exceeds the quantity of adequate reasons. For example, suppose our agent has an adequate reason s (based on an assertion in some recent official document) that one of the 20 members of the faculty of department D is a logician; further, suppose she has inadequate reasons $r_1,\dots ,r_{19}$ (based on a mistaken understanding of which specialties imply competence in logic) that the first 19 names listed on the department D faculty roster are logicians. We might be hard-pressed to say that our agent knows that department D has a logician on staff.

Perhaps this suggests that the agent in a case of mixed reasons can only be said to know the proposition if she also knows that her reasons are adequate. We resist this move, basically because we accept that an agent may have an adequate reason without necessarily knowing that that reason is adequate (more on this below in the conclusion). Therefore, if we assume for the sake of argument that our agent values all reasons equally, then a tiny island of adequacy within an ocean of inadequacy is sufficient for the defender of mixed-reason knowledge. This grants that the agent’s reasons are in some sense confused or that an agent who has only adequate reasons (and hence satisfies ${\mathsf {JTB}}{+}{\mathsf {NIL}}$ ) seems to “know better” than the agent with mixed reasons. But if one agent “knows better,” it does not follow that the other does not know at all.

In our view, an account of knowledge that would not allow for mixed reasons would in fact be too demanding. We consider it a virtue of our account that it allows for mixed reasons, precisely because we think it gives us a more realistic picture of the process of acquiring and managing reasons. We say that an agent knows a proposition if he believes that proposition based on at least one adequate reason. But knowing is more than passively believing propositions on the basis of reasons. It obviously also involves comparing and relating reasons. Consider an agent who believes the true proposition p on the basis of r and $r'$ , but who eventually realizes that $r'$ supports a false proposition. The agent ought to revise her belief in $r'$ , and also to reconsider her reasons for p. Hence, while our account of knowledge commits us to what Dutant calls method infallibilism, it does make room for errors and revisions in how reasons are acquired. It contains, in that sense, a measure of fallibilism.

5.2. Knowledge from inadequate reasons?

Although our account of knowledge allows for mixed reasons, a converse objection is that it may turn out to be too strong relative to ordinary knowledge ascriptions. The problem in this case is even more radical, and concerns whether one can have knowledge on the basis of a single, inadequate reason.

Here is an example:Footnote ¹⁸ based on his seeing Jones driving a Ford in the past (let us call that reason r), Smith comes to wrongly believe that Jones owns a Ford. Let us modify the scenario and suppose that Jones does in fact own a car (say, a Mazda). Based on his seeing Jones drive a Ford in the past, Smith also comes to believe that Jones owns a car. Could it not happen, intuitively, that although Smith fails to have knowledge on the basis of r that Jones owns a Ford (p), he nevertheless has knowledge on the basis of r that Smith owns a car (q)? In our system, this is not possible, for by (A), if r is adequate for q, then r must be adequate for any other proposition that r supports, hence for p as well. Dretske’s treatment of conclusive reasons would be able to address this problem: a reason r can be conclusive for q without being conclusive for p, even when p entails q. Our approach does not have this flexibility.Footnote ¹⁹

The question, more generally, is whether the same reason can adequately justify one to believe q without adequately justifying one to believe a stronger proposition p. Similar cases have been discussed by Warfield [Reference Warfield32], Fitelson [Reference Fitelson14], and Sorensen [Reference Sorensen25]. Warfield argues that I may know that I am not late for the meeting if I believe that it is currently 2:58pm, when in fact it is 2:56pm, assuming the meeting is at 7pm. On our account, my reason to believe that it is currently less than 7pm is inadequate, simply because it also supports the false proposition that it is 2:58pm. This is a case in which I have ${\mathsf {JTB}}^i$ that it is less than 7pm, without having ${\mathsf {JTB}}^e$ that it is less than 7pm. For anyone whose intuition is that I do hold knowledge that it is is less than 7pm on the basis of my observing “2:58pm” on the watch, our equation between knowledge and ${\mathsf {JTB}}^e$ is too strong in this case.

One option in the face of such examples is to bite the bullet and to resist the intuition that I know I am not late for the meeting, or that I know that Jones owns a car. But we think this is not the right response. My evidence “2:58pm” is obviously wrong regarding the actual time, but still close enough to the actual time to be relevantly used. The case would be different, it seems to us, if the agent’s watch indicated 6pm when it is 2:56pm, or even 9am. For the latter cases, our intuition is that I merely have a luckily true belief. More generally, we think the problem concerns how much approximation is tolerated in forming beliefs based on one’s evidence. If, when I see “2.58pm” (r) on my watch, I form the belief “it is around 2:58pm” (p), and from that proposition I infer “it is less than 7pm” (q), then my reason r now is veridical for both p and q. A way out, therefore, might be to relativize the adequacy of a reason to the selection of an appropriate domain of propositions supported by that reason.

This nevertheless puts pressure on us to clarify the relation of support between a reason and a proposition. In our statement of the axiom (A), we include no restriction on the support relation. We think it is better to be normative, and not to include any such restriction in the definition of knowledge in terms of ${\mathsf {JTB}}^e$ . On the other hand, we are ready to accept that in actual ascriptions of knowledge, the definition of adequate evidence is relativized to various domains of propositions. Consider the Warfield example again: this is not a perfect case of knowledge. This still counts as evidence that comes close to adequate, though not perfectly adequate. But it is adequate given the relevant domain of propositions. Our account, therefore, does not rule out the familiar mechanisms of contextualization at play in ordinary knowledge ascriptions, despite being fundamentally more normative.