2.1 Standard algorithmic models

The standard and most sophisticated existing algorithmic model is due to Halpern et al. [Reference Halpern, Moses and Vardi17]. In the single-agent and static setting (i.e., where we don’t consider the evolution of belief and knowledge over time), Halpern and Pucella [Reference Halpern and Pucella19] work with standard Kripke structures of the form $\langle \mathscr {W}, \mathscr {W}', \pi \rangle $ , where $\mathscr {W}$ is a set of possible worlds, $\mathscr {W}'$ is the set of possible worlds that are accessible to the agent, and $\pi $ is an interpretation function that associates each possible world $w \in \mathscr {W}$ with a truth assignment $\pi (w)$ to the primitive propositions of the language.Footnote ¹³ An algorithmic knowledge structure is defined as a tuple $\mathscr {M} = \langle \mathscr {W}, \mathscr {W}', \pi , \mathscr {A} \rangle $ where $\langle \mathscr {W}, \mathscr {W}', \pi \rangle $ is a Kripke structure and $\mathscr {A}$ is a knowledge algorithm that takes as input a formula and returns either “Yes,” “No,” or “?” (knowledge algorithms are thus assumed to terminate). “The agent knows $\phi $ ,” symbolized standardly as “ $K\phi $ ,” then gets the following satisfaction (or truth) conditions:

$$\begin{align*}\langle \mathscr{M}, w \rangle \vDash K\phi \ \ \ \ \Leftrightarrow \ \ \ \ \mathscr{A}(\phi)=` `\textrm{Yes."}\end{align*}$$

To capture the evolution of knowledge over time, Halpern et al. [Reference Halpern, Moses and Vardi17] give a run-based semantics on which agents can use different algorithms in different states. The local state of an agent is modeled as consisting of some local data and a local algorithm. The agent then knows $\phi $ at a state if her local algorithm outputs “Yes” when given both $\phi $ and the local data as inputs.Footnote ¹⁴

On this semantics, whether one knows $\phi $ has nothing to do with the truth-value of $\phi $ in any accessible world, and there are no constraints on the algorithm $\mathscr {A}$ . Halpern and Pucella [Reference Halpern and Pucella19, p. 223] propose the following constraint: a knowledge algorithm $\mathscr {A}$ is sound for $\mathscr {M}$ if and only if for all $\phi $ , $\mathscr {A}(\phi )=` `\textrm {Yes"}$ implies $\langle \mathscr {M}, w \rangle \vDash \phi $ for all $w \in \mathscr {W}'$ and $\mathscr {A}(\phi )=` `\textrm {No"}$ implies $\langle \mathscr {M}, w \rangle \vDash \neg \phi $ for some $w \in \mathscr {W}'$ .Footnote ¹⁵ That is, an algorithm is sound just in case an output of “Yes” implies that $\phi $ true in all epistemically accessible worlds (and thus “known” in the sense of the standard possible-worlds model), while an output of “No” implies that $\phi $ is false in some epistemically accessible world (and thus “unknown” in the standard model’s sense). This is the sense in which knowledge algorithms “compute knowledge” on Halpern et al.’s [Reference Halpern, Moses and Vardi17] construal: they compute whether the input formula is true in all accessible worlds. According to Halpern et al. [Reference Halpern, Moses and Vardi17, p. 259], what is defined without the soundness constraint is a notion of belief, i.e., using “ $B\phi $ ” to formalize “the agent believes $\phi $ ”:

$$\begin{align*}\langle \mathscr{M}, w \rangle \vDash B\phi \ \ \ \ \Leftrightarrow \ \ \ \ \mathscr{A}(\phi)=` `\textrm{Yes."}\end{align*}$$

Knowledge is thus defined as above but with the additional constraint that $\mathscr {M}$ ’s algorithm is sound.

The standard algorithmic model avoids the problem of logical omniscience: since there are no constraints on the knowledge algorithms, one’s knowledge algorithm can output “Yes” given $\phi $ but either “No” or “?” given $\psi $ even if $\phi $ entails $\psi $ or is logically equivalent to it. But given that one’s knowledge algorithms can be extremely weak, we don’t yet have a model that captures agents who are logically (or conceptually) competent. For instance, an agent can have a sound knowledge algorithm that never outputs “Yes.” As it stands, the standard algorithmic model thus isn’t an adequate middle-ground model.

There are other disadvantages of the standard algorithmic model given our purposes. The first is that the model is overly linguistic. Belief and knowledge are assumed to manifest always and only in linguistic action: the agent is given a sentence and in return provides a verbal response. But this is too narrow, as people can have knowledge that they aren’t able to verbally articulate. Stalnaker provides many such examples: the “shrewd but inarticulate” chess player who can access information for choosing a move but not for answering questions [Reference Stalnaker48, p. 439], or the experienced outfielder who knows exactly when and where the ball will come down for the purpose of catching the ball but not for the purpose of answering the question “Exactly when and where is the ball going to come down?” [Reference Stalnaker50, p. 263]. More generally, on the dispositional understanding, belief and knowledge are supposed to help explain people’s overall behavior—whether linguistic or otherwise.

A related disadvantage of the algorithmic model is that the objects of belief and knowledge are taken to be sentences, because the algorithms operate on sentences. Parikh [Reference Parikh35, pp. 472–474], whose algorithmic notion of knowledge is even called “linguistic knowledge” [Reference Parikh, Ras and Zemankova34, p. 4], explains that this choice is made so that belief and knowledge on the resulting account aren’t closed under necessary equivalence (if $\phi $ and $\psi $ are possible-worlds propositions and necessarily equivalent, then $\phi =\psi $ , and thus the algorithm’s output is the same for $\phi $ and $\psi $ .) As we will see in Section 2.2, one can maintain that the objects of belief and knowledge are propositions while avoiding closure under necessary equivalence by moving to an impossible-worlds framework.

Finally, the most important gap in the standard algorithmic model given our purposes is that it doesn’t address limitations of computational resources. On this model, an agent knows $\phi $ if her knowledge algorithm will eventually output “Yes” given $\phi $ , but that could take an unlimited amount of computational resources. For our purposes, we need to model bounds on the resources that the algorithms can use: In particular, if it would take an agent too long to compute $\phi $ , then she is unable to act on the information that $\phi $ and thus neither knows nor believes $\phi $ . For instance, an outfielder who would need to sit down for 3 hours to calculate the trajectory of the ball clearly neither knows nor believes that the ball will come down at the relevant location and time before they perform the calculations.

Halpern et al. [Reference Halpern, Moses and Vardi17, pp. 260f.] briefly discuss this problem and mention that one could put constraints on local algorithms so that they have to complete their run within a given unit of time.Footnote ¹⁶ Duc [Reference Duc15, pp. 39–51] develops an algorithmic system for knowledge that involves this idea. He introduces the formula “ $K^n_i\phi $ ” to stand for “if asked about $\phi $ , i is able to derive reliably the answer ‘yes’ within n units of time,” and the formula “ $K^\exists _i\phi $ ” to stand for “agent i can infer $\phi $ reliably in finite time” [Reference Duc15, p. 41]. The latter captures the spirit of Halpern et al.’s notion of knowledge: $K^\exists _i\phi $ holds just in case the agent has an algorithm that computes that $\phi $ is true within a finite but unbounded amount of time. In these definitions, the qualification “reliably” is supposed to imply both that the agent’s computation is correct (i.e., if either $K^n_i\phi $ or $K^\exists _i\phi $ hold, then $\phi $ is true) and that the agent doesn’t choose a procedure that correctly computes $\phi $ by chance, but is able to “select deterministically a suitable procedure for the input” [Reference Duc15, p. 41]. The idea is that the agent has a general procedure or algorithm that, given $\phi $ , correctly computes that $\phi $ is true within n units of time, and this general algorithm might involve steps for choosing the right sub-algorithms to run on $\phi $ .

Duc [Reference Duc15] thus defines infinitely many “time-stamped” notions of knowledge, one for each time unit n. This is slightly counterintuitive, given that we presumably only have one non-time-stamped notion of knowledge. As I will explain in Section 2.2, however, it is highly plausible to define knowledge in terms of such time-stamped notions by using a threshold.

Duc [Reference Duc15, pp. 45–48] goes on to provide a derivation system for his language of algorithmic knowledge. Some of his assumptions are too strong for our purposes. He assumes that both $K^n_i\phi $ and $K^\exists _i\phi $ imply that the agent is able to prove $\phi $ . On this view, agents employ a decidable axiom system extending propositional logic, and thus “all proofs can be generated algorithmically” by a general-purpose theorem prover [Reference Duc15, p. 44]. Duc only considers agents who have such a general-purpose theorem prover. After analysing a query and trying out special algorithms, these agents revert to using the general-purpose theorem prover. Thus, if $\phi $ is a theorem of the agent’s axiom system, the agent will eventually find its proof. In other words, the following rule of inference is valid in Duc’s system:

(NEC ^A) $$\begin{align} K^\exists_i\phi \text{ may be inferred from } \phi. \end{align}$$

Similarly, Duc assumes that if formulae $\phi _1, \dots , \phi _n$ can all be derived in the agent’s system and $\phi _1 \wedge \cdots \wedge \phi _n \rightarrow \psi $ is a theorem, then the agent will also eventually output a proof of $\psi $ if queried about it [Reference Duc15, p. 44]. A special case of this principle is that agents can use modus ponens in their reasoning, which yields the following as an axiom in Duc’s system:

(K ^A) $$\begin{align} K^\exists_i\phi \wedge K^\exists_i(\phi \rightarrow \psi) \rightarrow K^\exists_i\psi. \end{align}$$

As I will argue in Section 2.2, principles such as ( NEC ^A ) and ( K ^A ) can form plausible constraints on logically or conceptually competent agents. But we should avoid the extremely strong assumption that knowing $\phi $ always requires being able to find a proof of $\phi $ in some derivation system. After all, an agent might know even mathematical or logical truths by retrieving them from a memory base that was reliably stored (for instance, via expert testimony), without having any ability to produce proofs. Generally speaking, it is overly restrictive to think of all belief and knowledge that $\phi $ in terms of features of a proof of $\phi $ in some derivation system. (I come back to this point in Section 3.)

2.2 An algorithmic impossible-worlds model

My proposal is to build an algorithmic model on the basis of an impossible-worlds model. An important advantage of impossible-worlds models is that they preserve some core aspects of the standard possible-worlds model, including the idea that one’s belief and information states are sets of worlds, that belief and knowledge are truth in all doxastically or epistemically accessible worlds, and that acquiring beliefs and learning are ruling out of doxastic or epistemic possibilities. Moreover, impossible worlds allow for more fine-grained constructions of content: sentences that are true in all the same possible worlds correspond to the same possible-worlds proposition, but most often (if not always) differ in truth-values at impossible worlds and thus correspond to different propositions construed as sets of possible and impossible worlds.Footnote ¹⁷ As such, the impossible-worlds framework will enable us to avoid three disadvantages of the standard algorithmic framework discussed above, viz., that it gives up the standard worlds-based framework for modeling belief and knowledge, that belief and knowledge are assumed to manifest always and only in linguistic action, and that the objects of belief and knowledge are sentences and not propositions. Impossible-worlds frameworks also have some disadvantages, for instance, because the account of content that they yield is extremely fine-grained.Footnote ¹⁸ But they are widely and increasingly used for solving problems of hyperintensionality in philosophy and, as we will see in Section 3, they are used in the most important competitor approaches to developing a middle-ground model.Footnote ¹⁹

Let us turn to some of the details of impossible-worlds models. Let $\mathscr {L}$ be the language of our model, defined as follows:

where “ $B\phi $ ” formalizes “the agent believes $\phi $ ” and “ $K\phi $ ” formalizes “the agent knows $\phi $ .” An impossible-worlds model is a tuple $\mathscr {M} = \langle \mathscr {W}, \mathscr {P}, d, e, v \rangle $ , where $\mathscr {W}$ is a non-empty set of worlds; $\mathscr {P} \subseteq \mathscr {W}$ is a non-empty set of possible worlds (thus

is the set of impossible worlds); $d : \mathscr {W} \to 2^{\mathscr {W}}$ is a doxastic accessibility function that assigns each world $w \in \mathscr {W}$ to the set of worlds doxastically accessible from $w$ ; $e : \mathscr {W} \to 2^{\mathscr {W}}$ is an epistemic accessibility function that assigns each world $w \in \mathscr {W}$ to the set of worlds epistemically accessible from $w$ ; and $v$ is a valuation function that maps each atomic sentence $p \in At$ and world $w \in \mathscr {P}$ to either 0 or 1, and maps each sentence $\phi \in \mathscr {L}$ and world $w \in \mathscr {I}$ to either 0 or 1.Footnote ²⁰ Since knowledge entails belief, it is standard to assume that doxastically accessible worlds are also epistemically accessible, i.e., that $d(w) \subseteq e(w)$ for all $w \in \mathscr {W}$ . I also assume throughout that worlds are centered, i.e., they are worlds with a marked individual at a time. As Lewis [Reference Lewis30, pp. 27–30] explains, this should be assumed in all doxastic and epistemic models because agents can obviously have different beliefs and knowledge at different times.Footnote ²¹ Finally, I add to the satisfaction relation (written “ $\vDash $ ” as usual) the dissatisfaction (or making false) relation, written “

.” If $w \in \mathscr {P}$ , $\vDash $ and

are defined recursively:

If $w \in \mathscr {I}$ , then $\vDash $ and

are defined as follows:

The idea here is that a sentence $\phi $ is dissatisfied (or false) at a world if and only if its negation $\neg \phi $ is satisfied (or true) at that world, for both possible and impossible worlds. Given a possible world $w \in \mathscr {P}$ , only one of $\phi $ and $\neg \phi $ is satisfied at $w$ and the other is dissatisfied at $w$ . But this needn’t be the case at impossible worlds: given an impossible world $w \in \mathscr {I}$ , both $\phi $ and $\neg \phi $ could be satisfied at $w$ (if $v(\phi , w) = 1$ and $v(\neg \phi , w) = 1$ ), and neither $\phi $ nor $\neg \phi $ could be satisfied (if $v(\phi , w) = 0$ and $v(\neg \phi , w) = 0$ ). Impossible worlds can thus be both inconsistent and incomplete entities. I further adopt the maximally permissive construal of impossible worlds on which for any incomplete and/or inconsistent set of sentences $\Gamma \subseteq \mathscr {L}$ , there is an impossible world $w \in \mathscr {I}$ such that for all $\phi \in \mathscr {L}$ , $v(\phi , w) = 1$ if and only if $\phi \in \Gamma $ .Footnote ²² There are thus no constraints on what sentences impossible worlds can satisfy, and there are at least as many impossible worlds as there are sets of sentences of our base language $\mathscr {L}$ .Footnote ²³

As I explained in Section 1, permissive impossible-worlds models don’t face the problem of logical omniscience. For instance, assume that $\langle \mathscr {M}, w \rangle \vDash B\phi $ , and that $\phi $ logically entails $\psi $ , i.e., that $\psi $ is true in all the possible worlds in which $\phi $ is true. For all that we have said about the doxastic accessibility function d, it might be that $d(w)$ includes a world $w' \in \mathscr {I}$ such that $\langle \mathscr {M}, w' \rangle \nvDash \psi $ , and thus that $\langle \mathscr {M}, w \rangle \nvDash B\psi $ . On the other hand, the model as it stands allows any combination of beliefs and knowledge, and thus doesn’t capture logical or conceptual competence. It also doesn’t capture the idea that one already believes (knows) what one can easily (and correctly) compute.

Here is the main idea of my proposed supplementation of the impossible-worlds model. I propose to add the following constraints on the accessibility functions: a world $w'$ is accessible from $w$ if and only if $w'$ respects what is “easily computable” from $w$ , i.e., $w'$ satisfies all the sentences that the agent can easily compute in $w$ to be true, and doesn’t satisfy any sentence that the agent can easily compute in $w$ not to be true (which might not be equivalent to computing that the sentence is false). At a first pass, an agent “computes” $\phi $ to be true if she would answer affirmatively when asked whether $\phi $ is true, given that she desires to give the correct answer. The answer could be incorrect in the doxastic case, but not in the epistemic case. In either case, an agent can compute $\phi $ to be true without having a proof (or what they take to be a proof) of $\phi $ . As we will see, we can generalize this understanding of “compute $\phi $ to be true” to cover non-linguistic actions as well.

The qualifier “easily” is supposed to bound the algorithmic notion of belief and knowledge as discussed in Section 2.1. In general, an agent who believes (knows) $\phi $ is disposed to act (capable of acting) upon $\phi $ —including to answer questions about $\phi $ —using only a small amount of resources. For instance, we wouldn’t say that Ola believes that $38,629$ is prime if she would have to think for 3 weeks before answering “Yes” when asked about it. But Ola knows that $38,624$ is composite even though she has never considered the question, because it only takes her a very small amount of computational resources to answer “Yes” to the question “Is $38,624$ composite?” For simplicity, here I will only consider the resource of time. I will thus model “easily” as “in $\leq \epsilon $ units of time,” where $\epsilon $ is a small “threshold” natural number (assuming also that units of time can be counted in natural numbers). It is clear from the examples above that a couple of seconds count as “in $\leq \epsilon $ units of time,” while 3 weeks or 3 hours don’t. But there are plausibly indeterminate cases in between. As Stalnaker notes in a similar context, this fits with the plausible view that attributions of belief and knowledge are context-sensitive:

I will henceforth assume that the threshold $\epsilon $ is a small enough unit of time in the range of a few seconds, but follow Stalnaker in allowing the value of $\epsilon $ to be sensitive to the context of attribution of belief or knowledge.

On this algorithmic impossible-worlds model, it will turn out that an agent believes (knows) $\phi $ if and only if she has an algorithm that would (correctly) output an affirmative answer if asked “Is $\phi $ true?” in less than or equal to $\epsilon $ units of time. Our worlds-based algorithmic model will thus turn out to be a combination of the non-worlds-based algorithmic models of Halpern et al. [Reference Halpern, Moses and Vardi17] and Duc [Reference Duc15], but where belief and knowledge are bounded (unlike Halpern et al.’s notions of belief and knowledge and Duc’s notion of knowledge $K^\exists _i$ ) and not themselves time-stamped notions (unlike Duc’s notions of knowledge $K^n_i$ ). The model will obviously satisfy the third desideratum on middle-ground models outlined in Section 1: an agent already believes (knows) what is easily (and correctly) computationally accessible to her. But further constraints will be needed to satisfy the other two desiderata, viz., to capture logically competent agents, and to capture conceptually competent agents.

Here, then, is the proposal in more detail. Let $\mathscr {L}$ , as defined above, be the language of our model, and let an algorithmic impossible-worlds model be a tuple $\mathscr {M} = \langle \mathscr {W}, \mathscr {P}, d, e, v, \mathscr {A}, A \rangle $ , where $\mathscr {M} = \langle \mathscr {W}, \mathscr {P}, d, e, v \rangle $ is an impossible-worlds model; $\mathscr {A}$ is a set of algorithms that take as input a sentence $\phi \in \mathscr {L}$ and output either “Yes,” “No,” or “?”;Footnote ²⁴ and A is a local algorithm function that assigns each $w \in \mathscr {W}$ to a local algorithm in $\mathscr {A}$ , which I denote “ $A_w$ .” I abbreviate “ $A_w$ outputs ‘Yes’ given $\phi $ in less than or equal to n units of time” as “ $A^{\leq n}_w(\phi ) = `\textrm {Yes'}$ ,” and “ $A_w$ outputs ‘Yes’ given $\phi $ in some finite unit of time” as “ $A_w(\phi ) = `\textrm {Yes'}$ ” (and do the same for the other outputs). Thus $A^{\leq n}_w(\phi ) = ` `\textrm {Yes"}$ implies $A_w(\phi ) = ` `\textrm {Yes"}$ and $A_w(\phi ) = ` `\textrm {Yes"}$ implies that there is an $n \in \mathbb {N}$ such that $A^{\leq n}_w(\phi ) = ` `\textrm {Yes"}$ (and the same holds for the other outputs).

Let me first explain what I take the local algorithms to capture. As I mentioned above, on a first-pass understanding, $A_w$ captures the agent’s dispositions in state $w$ to answer questions about the truth-values of certain sentences. Thus $A_w(\phi ) = ` `\textrm {Yes"}$ captures that in state $w$ , the agent is such that if she were asked whether $\phi $ is true and wanted to give the correct answer, she would engage in a certain chain of reasoning and answer affirmatively within some finite unit of time. “No” corresponds to a negative answer, and “?” to “I don’t know” or “I give up.”Footnote ²⁵ I assume that agents can have different local algorithms in different states, and that a local algorithm in a state can run different sub-algorithms given different inputs.Footnote ²⁶ For instance, it might be that 10 years ago, Ola had no local algorithm that would check for primality if asked “Is n prime?” for any n, and that in her current state, Ola has a local algorithm that would run a trial division if asked “Is $55,579$ prime?” but directly output “No” if asked whether a number ending in 4 is prime. As in [Reference Duc15], I thus assume that a local algorithm captures all the steps for choosing different sub-algorithms for different inputs.

On the first-pass understanding, local algorithms capture linguistic dispositions—just as in the standard algorithmic models from Section 2.1. This understanding is simpler to work with, which is why I will adopt it in discussing the model. Importantly, however, this understanding can be generalized now that we have adopted an impossible-worlds framework. For instance, following Stalnaker’s [Reference Stalnaker47] definition of belief, we can take $A_w(\phi ) = ` `\textrm {Yes"}$ to capture that in state $w$ , the agent is such that, all else being equal, she would output behavior that would tend to satisfy her desires in worlds in which , together with her other beliefs, are true, within some finite unit of time (where “ ” stands for the proposition expressed by $\phi $ ). “No” could then correspond to behavior that would tend to satisfy her desires in worlds in which is not true, and “?” to behavior that would tend to satisfy her desires irrespective of whether is true. On this generalized understanding, outputting the correct answer to the question of whether $\phi $ is true is only one example of behavior that would tend to satisfy the agent’s desires in worlds in which , together with her other beliefs, are true; thus, $A_w(\phi ) = ` `\textrm {Yes"}$ on the generalized understanding doesn’t require that the agent in state $w$ would answer affirmatively if asked about $\phi $ (she can misspeak, for instance) or that she has any linguistic dispositions at all. On a possible-worlds framework, the first-pass understanding of the local algorithms can’t be generalized in this way, since on a possible-worlds construal of propositions, Stalnaker’s definitions of belief and knowledge imply that they are closed under entailment. In the case of single-premise entailment, this is because for possible-worlds propositions A and B, if A entails B, then $A \cap B = A$ , and belief distributes over intersections on Stalnaker’s definition, i.e., if S believes $A \cap B$ , then she believes A and she believes B. (This is because if S is disposed to satisfy desires in worlds in which $A \cap B$ , together with her other beliefs, is true, she is also disposed to satisfy desires in worlds in which A, together with her other beliefs—which include $A \cap B$ —is true.)Footnote ²⁷

On the algorithmic impossible-worlds model as I have defined it, the objects of knowledge, the objects of belief, and the inputs of local algorithms are all sentences and not propositions—just as in the standard algorithmic models from Section 2.1. But because we have adopted an impossible-worlds framework, we can now also easily modify this definition and give an alternative interpretation of the formalism: we can instead let the inputs of the algorithms in $\mathscr {A}$ be propositions, , and interpret “ $K\phi $ ” as the formalization of the proposition that the agent knows . The reason this is possible is that unlike in the construction of propositions as sets of possible worlds, for any two sentences $\phi \neq \psi \in \mathscr {L}$ , the respective propositions as sets of worlds and aren’t identical. (This follows from the maximally permissive construal of impossible worlds on which there is some $w' \in \mathscr {I}$ such that for all $\alpha \in \mathscr {L}$ , $v(\alpha , w') = 1$ if and only if $\alpha \in \{\phi \}$ , and this .) I will use the original definitions here for ease of notation, but officially adopt the modified definition and interpretation.

I retain the standard definitions of satisfaction and dissatisfaction given above. The key remaining task is to define the accessibility functions d and e. We first need to contrast the epistemic case with the doxastic case. Recall that on the first-pass understanding, the general idea is that one believes $\phi $ just in case one would affirmatively answer the question of whether $\phi $ is true. Since it is standardly assumed that knowledge entails belief and is factive, a natural addition for the case of knowledge is to require that the answer would be correct. Accordingly, given an algorithmic impossible-worlds model, $\mathscr {M}$ , let an algorithm $X \in \mathscr {A}$ be veridical about $\phi $ at $\langle \mathscr {M}, w \rangle $ if and only if, if $X(\phi )=` `\textrm {Yes,"}$ then $\langle \mathscr {M}, w \rangle \vDash \phi $ , and if $X(\phi )=` `\textrm {No,"}$ then $\langle \mathscr {M}, w \rangle \nvDash \phi $ . That is, an algorithm is veridical about $\phi $ at $w$ just in case if it answers “Yes” to whether $\phi $ is true, then $\phi $ is true at $w$ , and if it answers “No” to whether $\phi $ is true, then $\phi $ is not true at $w$ (we relativize veridicality to a world to leave it open that different worlds can have the same local algorithm).Footnote ²⁸ In the following, I will adopt veridicality as a minimal condition on knowledge. Plausibly, knowledge requires more than true belief. Other potential conditions could be integrated into the algorithmic framework as well. Take, for instance, the condition that knowledge is safe true belief, i.e., belief that is true in all nearby possible worlds.Footnote ²⁹ Our model could incorporate this idea by requiring that a local algorithm isn’t just veridical about $\phi $ locally but in all the nearby worlds in which that algorithm is local.Footnote ³⁰ Or take the sensitivity condition on knowledge, i.e., that the agent wouldn’t believe $\phi $ if it were false.Footnote ³¹ Our model could incorporate this idea by requiring that a local algorithm doesn’t output “Yes” in the nearest world(s) in which it is local and $\phi $ is false. In similar ways, we could require the algorithms to take information as input, or to not rely on false lemmas, and so on.Footnote ³² The algorithmic approach thus yields novel and potentially fruitful ways of formally representing conditions on knowledge, by making these conditions on agents’ algorithms.Footnote ³³

We can now define the accessibility functions d and e. Following the main idea explained above, for any $w \in \mathscr {W}$ :

As before, $\epsilon $ is our threshold unit of time, which we assume to be fixed by the context we are modeling.

On our definition, the worlds accessible from $w$ satisfy all the sentences that the agent can easily (and correctly) compute in $w$ to be true, and don’t satisfy any of the sentences that the agent can easily (and correctly) compute not to be true (there are no constraints on sentences for which the agent outputs “?.”)Footnote ³⁴ As desired, it follows from these definitions that for any $w \in \mathscr {P}$ , (i) and (ii) are equivalent, and (iii) and (iv) are equivalent:

1. (i) $\langle \mathscr {M}, w \rangle \vDash B\phi $ .

2. (ii) $A^{\leq \epsilon }_w(\phi )=` `\textrm {Yes."}$

3. (iii) $\langle \mathscr {M}, w \rangle \vDash K\phi $ .

4. (iv) $A^{\leq \epsilon }_w(\phi )=` `\textrm {Yes"}$ and $A_w$ is veridical about $\phi $ at $\langle \mathscr {M}, w \rangle $ .

To see that these equivalences hold, consider, first, the equivalence between (i) and (ii). Let $w \in \mathscr {P}$ , and assume (i). Let $Y_w = \{ \alpha \in \mathscr {L} \mid A^{\leq \epsilon }_w(\alpha )=` `\textrm {Yes"}\}$ . By the maximally permissive construction of impossible worlds, there is some $y \in \mathscr {I}$ such that for all $\alpha \in \mathscr {L}$ , $v(\alpha , y) = 1$ if and only if $\alpha \in Y_w$ . Thus, for all $\alpha \in \mathscr {L}$ , if $A^{\leq \epsilon }_w(\alpha )=` `\textrm {Yes"}$ then $\langle \mathscr {M}, y \rangle \vDash \alpha $ , and if $A^{\leq \epsilon }_w(\alpha )=` `\textrm {No"}$ then $\langle \mathscr {M}, y \rangle \nvDash \alpha $ . Thus, by the definition of d, $y \in d(w)$ . Thus, by (i), $\langle \mathscr {M}, y \rangle \vDash \phi $ . By the choice of y, $\phi \in Y_w$ , and thus (ii) follows. For the converse, assume (ii). Let $w' \in \mathscr {W}$ be such that $w' \in d(w)$ . Thus, by the definition of d, for all $\gamma \in \mathscr {L}$ , if $A^{\leq \epsilon }_w(\gamma )=` `\textrm {Yes"}$ then $\langle \mathscr {M}, w' \rangle \vDash \gamma $ . Thus, by (ii), $\langle \mathscr {M}, w' \rangle \vDash \phi $ . Thus, (i) follows. A parallel argument establishes the equivalence of (iii) and (iv). It is easy to see that our definitions of d and e entail the factivity of knowledge (since e is reflexive, i.e., for all $w \in \mathscr {W}, w \in e(w)$ ) and that knowledge entails belief (since $d(w) \subseteq e(w)$ for all $w \in \mathscr {W}$ ).Footnote ³⁵

Let me now turn to the three desiderata on middle-ground models from Section 1 and explain how this algorithmic impossible-worlds models can be supplemented to satisfy the first and second desiderata, and how it already satisfies the third desideratum. Consider the first desideratum, viz., that our model should capture agents who are logically competent. At an intuitive level, there are different ways to understand what it is for an agent to be logically competent. For one, and as Duc [Reference Duc, Pinto-Ferreira and Mamede13, p. 6; Reference Duc14, p. 637] mentions, one might want a logically competent agent to “know at least a (sufficiently) large class of logical truths.” For instance, it might be that any logically competent agent will know all propositions of the form $\phi \rightarrow \phi $ (although perhaps only as long as $\phi $ is simple enough). Similarly, and as captured by Duc’s ( NEC ^A ), one might want a logically competent agent to be capable of eventually giving the right answer to the question of whether some sentence is a tautology in some basic logical formal system, if they have enough time to think about it. Our algorithmic framework can capture this first intuitive sense of logical competence by putting both time-sensitive and non-time-sensitive constraints on logically competent agents’ local algorithms. Let F be some formal system such that all the tautologies of F are true in all possible worlds $w \in \mathscr {P}$ (for instance, F might be propositional logic), and let the basic tautologies of F be some set of tautologies we deem to be required for logical competence (what counts as “basic” in this sense might be context-sensitive). We can then require that for any world $w \in \mathscr {P}$ and sentence $\phi \in \mathscr {L}$ :

1. (a) If $\phi $ is a tautology of F, then $A_w(\phi ) = ` `\textrm {Yes"}$ and $A_w$ is veridical about $\phi $ at $\langle \mathscr {M}, w \rangle $ .

2. (b) If $\phi $ is a basic tautology of F, then $A^{\leq \epsilon }_w(\phi ) = ` `\textrm {Yes"}$ and $A_w$ is veridical about $\phi $ at $\langle \mathscr {M}, w \rangle $ .

Given the equivalence of (iii) and (iv), (b) entails that logically competent agents know all the basic tautologies.

As we saw in Section 2.1, we can also follow Duc [Reference Duc15] and introduce new formulas of the form “ $K^{\exists }\phi $ ” in our object language $\mathscr {L}$ to formalize propositions such as “the agent would correctly answer ‘Yes’ when asked whether $\phi $ is true in finite time” or “the agent is in a position to know $\phi $ ” (assuming that after having correctly computed that $\phi $ is true, an agent knows $\phi $ ). By defining the relevant kinds of accessibility relations, we can then just as above get the equivalence between (v) and (vi) for all $w \in \mathscr {P}$ :

1. (v) $\langle \mathscr {M}, w \rangle \vDash K^\exists \phi $ .

2. (vi) $A_w(\phi )=` `\textrm {Yes"}$ and $A_w$ is veridical about $\phi $ at $\langle \mathscr {M}, w \rangle $ .

We can thus in the object language capture that for any tautology, $\phi $ , the logically competent agent is “in a position to know $\phi $ ” in that she would give the right answer to whether $\phi $ is true within a finite unit of time. Given our purposes here, however, putting constraints directly on the local algorithms suffices to constrain our model to capture logically competent agents.

On a second intuitive understanding, logical competence is a conditional achievement. This is captured by the statements that, for instance, a logically competent agent “can draw sufficiently many conclusions from their knowledge” [Reference Duc, Pinto-Ferreira and Mamede13, p. 6], “does not miss out on any trivial logical consequences of what she believes” [Reference Bjerring and Skipper8, pp. 502f.], or that “rational agents seemingly know the trivial consequences of what they know” [Reference Jago24, p. 1152]. On one way of understanding them, these statements suggest a closure principle of the form: if the agent believes (knows) certain propositions $\Phi $ , then she also believes (knows) the “trivial” consequences of $\Phi $ . However, as Bjerring and Skipper [Reference Bjerring and Skipper8, pp. 506–508] point out, a “collapse argument” shows that such closure principles entail or “collapse into” Full Logical Omniscience: Assume that $\phi $ is a trivial consequence of $\Phi $ if $\phi $ is derivable from $\Phi $ in one application of a standard rule of inference. Since any logical consequence of $\Phi $ is derivable from $\Phi $ via a chain of trivial consequences, if one fails to know some logical consequence of what one knows, then one must also fail to know some trivial consequence of what one knows.Footnote ³⁶ The challenge for capturing the second intuitive sense of logical competence, then, is to formulate some conditional constraint(s) that doesn’t (don’t) face a collapse argument; let me call this “the collapse challenge.” In Section 3, I will outline how Bjerring and Skipper [Reference Bjerring and Skipper8] propose to meet the collapse challenge. In our algorithmic impossible-worlds approach, we can meet the collapse challenge by adopting (some of) the following conditional constraints on logically competent agents’ local algorithms. For simplicity, I only formulate these constraints for the primitive logical expressions of our base language $\mathscr {L}$ , viz. “ $\wedge $ ” and “ $\neg $ ,” but similar constraints can be formulated for other logical constants. I also omit the veridicality condition, but similar constraints that include it can be formulated.Footnote ³⁷

Consider, first, the following non-time-sensitive constraints. For any $w \in \mathscr {P}$ and $\phi , \psi \in \mathscr {L}$ :

1. (c) If $A_w(\phi ) = ` `\textrm {Yes"}$ and $A_w(\psi ) = ` `\textrm {Yes,"}$ then $A_w((\phi \wedge \psi )) = ` `\textrm {Yes."}$

2. (d) If $A_w((\phi \wedge \psi )) = ` `\textrm {Yes,"}$ then $A_w(\phi ) = ` `\textrm {Yes"}$ and $A_w(\psi ) = ` `\textrm {Yes."}$

3. (e) $A_w(\neg \phi ) = ` `\textrm {Yes"}$ if and only if $A_w(\phi ) = ` `\textrm {No."}$

4. (f) $A_w(\neg \phi ) = ` `\textrm {No"}$ if and only if $A_w(\phi ) = ` `\textrm {Yes."}$

In outline, constraints (c)–(f) capture the intuitive idea that a logically competent agent should respect the introduction and elimination rules for the logical connectives, even though it might take her a long time to do so. For instance, constraint (c) states that if a logically competent agent would eventually assent to $\phi $ and to $\psi $ , then she would also eventually assent to their conjunction. Similarly, constraint (d) states that if a logically competent agent would eventually assent to a conjunction, then she would also eventually assent to its conjuncts. Constraints (e) and (f) capture that a logically competent agent computes something to be not true just in case she computes it to be false (i.e., computes its negation to be true), and vice versa. Constraints (c)–(f), together with analogous constraints we could formulate for other logical connectives and with veridicality, thus entail that if a logically competent agent believes (knows) certain propositions $\Phi $ , and $\phi $ can be derived from $\Phi $ in one application of a standard rule of inference, then the agent would also eventually, though perhaps not immediately, (correctly) compute that $\phi $ is true. Constraints (c)–(f) thus closely capture the second intuitive understanding of logical competence.

A useful consequence of (e) and (f) is that for any $w \in \mathscr {P}$ , if $\langle \mathscr {M}, w \rangle \vDash B\phi $ , then $\langle \mathscr {M}, w \rangle \vDash \neg B\neg \phi $ and if $\langle \mathscr {M}, w \rangle \vDash B\neg \phi $ , then $\langle \mathscr {M}, w \rangle \vDash \neg B\phi $ . That is, if an agent believes $\phi $ , then she doesn’t believe $\neg \phi $ , and if she believes $\neg \phi $ , then she doesn’t believe $\phi $ . To see this, consider the former conditional (the second is equivalent to the first given the classical truth-conditions for “ $\neg $ ”). Let $w \in \mathscr {P}$ , and assume that $\langle \mathscr {M}, w \rangle \vDash B\phi $ . By the equivalence of (i) and (ii), it follows that $A^{\leq \epsilon }_w(\phi ) = ` `\textrm {Yes."}$ It thus follows that $A_w(\phi ) = ` `\textrm {Yes."}$ Thus, by (f), $A_w(\neg \phi ) = ` `\textrm {No."}$ It thus follows that $A^{\leq \epsilon }_w(\neg \phi ) \neq ` `\textrm {Yes."}$ By the equivalence of (i) and (ii), it follows that $\langle \mathscr {M}, w \rangle \nvDash B\neg \phi $ , and thus that $\langle \mathscr {M}, w \rangle \vDash \neg B\neg \phi $ . Similarly, (c)–(f) together entail that if an agent believes $(\phi \wedge \psi )$ , then she doesn’t believe $\neg \phi $ or $\neg \psi $ , and if an agent believes $\phi $ and $\psi $ , she doesn’t believe $\neg (\phi \wedge \psi )$ . The following three closure principles thereby follow from (c)–(f): for any $\alpha , \beta , \gamma \in \mathscr {L}$ and $w \in \mathscr {P}$ , if $\langle \mathscr {M}, w \rangle \vDash B\alpha $ and $\langle \mathscr {M}, w \rangle \vDash B\neg \alpha $ , then $\langle \mathscr {M}, w \rangle \vDash B\beta $ ; if $\langle \mathscr {M}, w \rangle \vDash B(\alpha \wedge \beta )$ and $\langle \mathscr {M}, w \rangle \vDash B\neg \alpha $ , then $\langle \mathscr {M}, w \rangle \vDash B\gamma $ ; and if $\langle \mathscr {M}, w \rangle \vDash B\alpha $ , $\langle \mathscr {M}, w \rangle \vDash B\beta $ , and $\langle \mathscr {M}, w \rangle \vDash B\neg (\alpha \wedge \beta )$ , then $\langle \mathscr {M}, w \rangle \vDash B\gamma $ . As desired, these closure principles are too weak to entail Full Logical Omniscience. Constraints (c)–(f) thus don’t face a collapse argument, and we have thus met the collapse challenge for capturing the second intuitive sense of logical competence.

We can also consider the result of adding time-sensitivity to (c) and (d) as possible candidate conditional constraints on logical competence:

1. (g) If $A^{\leq i}_w(\phi ) = ` `\textrm {Yes"}$ and $A^{\leq j}_w(\psi ) = ` `\textrm {Yes,"}$ then $A^{\leq i + j + k}_w((\phi \wedge \psi )) = ` `\textrm {Yes"}$ for some small $k \in \mathbb {N}$ , for any $i, j \in \mathbb{N}$ .

2. (h) If $A^{\leq i}_w((\phi \wedge \psi )) = ` `\textrm {Yes,"}$ then $A^{\leq j}_w(\phi ) = ` `\textrm {Yes"}$ and $A^{\leq k}_w(\psi ) = ` `\textrm {Yes"}$ for some $j, k \in \mathbb {N}$ such that $j, k \leq i$ for any $i \in \mathbb{N}$ .

Constraint (g) puts a limit on how long it should take a logically competent agent to say “Yes” to a conjunction if she believes each of its conjuncts. This constraint leaves open that logically competent agents can believe some conjuncts without believing their conjunction. I think this is a plausible conclusion: if the computational resources it takes for an agent to compute each of the conjuncts are close to the threshold for what counts as believing something, then it might be that the agent would just take too long to compute that a conjunction is true if asked about it, in which case we wouldn’t intuitively say that the agent believes the conjunction before she performed the computation. In the case of knowledge, accepting pragmatic encroachment can further strengthen this intuition: imagine that the stakes are such that an agent doesn’t count as knowing $\phi $ unless she can make a split-second decision about whether $\phi $ (say, she needs to answer a timed quiz). The agent can then know $\phi $ and know $\psi $ , but, assuming that it would take her longer than the allowed time to answer whether $(\phi \wedge \psi )$ , fail to know $(\phi \wedge \psi )$ .Footnote ³⁸

Constraint (h), in turn, states that a logically competent agent’s belief algorithm outputs “Yes” to conjuncts in less or equal time than it outputs “Yes” to their conjunction. Typicality effects such as those involved in the conjunction fallacy might provide counterexamples to (h): an agent might be quicker to judge that a conjunction such as “Linda is a bank teller and is active in the feminist movement” is true than to judge that its conjunct “Linda is active in the feminist movement” is true, because the conjunction is more “typical.”Footnote ³⁹ One might perhaps maintain that logically competent agents would judge the conjunct and the conjunction to be true at least equally fast. In any case, given the equivalence of (i) and (ii), (h) entails a fourth closure principle, viz., distribution over conjunction (i.e., if a logically competent agent believes a conjunction, then she also believes its conjuncts). Although the case for distribution over conjunction isn’t decisive either,Footnote ⁴⁰ it is widely assumed and it doesn’t by itself imply anything like Full Logical Omniscience. Constraint (h) could thus also be an additional constraint on logically competent agents, depending on how strong one wants such constraints to be.

Next, consider the second desideratum on middle-ground models, viz., that our model should capture agents who are conceptually competent with the logical connectives. In my view, the constraints we have laid out in (c)–(h) can serve just as well as constraints on conceptual competence with conjunction and negation. Let us take conjunction as our main example. On traditional inferentialist metasemantic theories, possessing the concept of conjunction requires inferring according to the rules of conjunction-introduction and conjunction-elimination, or, at least, inferring according to these rules when “given a chance,” for instance, “when someone or something brings the conclusion to your direct attention, perhaps by querying you on the matter” [Reference Warren55, p. 46].Footnote ⁴¹ Our constraints (c)–(h) capture precisely such a view: they entail that an agent who believes a conjunction would assent to its conjuncts if queried about them, and that an agent who believes the conjuncts of a conjunction would eventually (though perhaps not immediately) assent to the conjunction if queried about it. Different versions of inferentialism could impose slightly different interpretations on what exactly it is to have a belief algorithm that outputs “Yes” given a proposition, but the general inferentialist picture is nicely captured by constraints along the lines of (c)–(h).

Finally, consider the third desideratum on middle-ground models, viz., that our model should entail that whatever an agent can computationally easily access is already part of what she believes or knows. Our algorithmic impossible-worlds model satisfies this desideratum because it entails the equivalences between (i) and (ii) and between (iii) and (iv): on our model, an agent believes (knows) $\phi $ just in case she can easily (and correctly) compute $\phi $ to be true. In light of our discussion of the collapse challenge above, it is worth noting that this third desideratum doesn’t require a closure principle of the form: agents believe (know) whatever is easily computationally accessible from what they believe (know) (as opposed to whatever is easily computationally accessible “simpliciter”). “Easily accessing $\psi $ from $\Phi $ ” means that we ignore any potential computational costs of accessing $\Phi $ itself: we only consider the computational costs of accessing $\psi $ once one has already accessed $\Phi $ . But on the dispositional understanding of belief and knowledge, believing and knowing propositions $\Phi $ are compatible with the existence of computational costs of accessing $\Phi $ : as we saw in Section 1, one can believe (know) $\phi $ even if it takes a short computation to access and thus be able to act on $\phi $ . This idea is reflected in our algorithmic impossible-worlds model. But then some proposition, $\psi $ , can be easily accessible from what one knows, $\Phi $ , while not being easily accessible simpliciter: the computational costs can add up, or the agent might not even consider $\Phi $ in situations where she needs to act on $\psi $ (e.g., if she is asked about whether $\psi $ ). So, one won’t necessarily believe (know) every proposition that is easily accessible from what one believes (knows): the dispositional understanding of belief and knowledge that motivate our third desideratum is incompatible with closure principles of the type that face collapse arguments.