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Epistemic modals and mathematics

Craige Roberts and Stewart Shapiro November 19, 2015

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The cases

There is, of course, a large literature in philosophy and linguistics on epistemic modals. It is common, at least in philosophy, to introduce the topic with examples from mathematics, in order to distinguish the targeted epistemic modals from other kinds of modals. John MacFarlane [2011, 144], for example, writes: It is hard to say exactly what makes a word modal, or what makes a use of a modal epistemic, . . . but some examples should get the idea across. If I say “Goldbach’s conjecture might be true, and it might be false”, I am not endorsing the Cartesian view that God could have made the truths of arithmetic come

• ut differently. I make the claim not because I believe in the

metaphysical contingency of mathematics, but because I know that Goldbach’s conjecture has not yet been proved or refuted.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The cases

In short, this modal is not metaphysical, saying what the world might be

• like. Similarly, the modal here is not deontic, saying that the conjecture is

somehow morally or pragmatically permissible. The statement has something to do with what is known, what is believed, what there is evidence for, or the like.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The current state

Despite the controversies, there is broad agreement on some aspects of the semantics of epistemic modals. But those agreed aspects are all incorrect. Virtually all articulated accounts give the wrong truth conditions for these particular epistemic modals. Consider an utterance in the form “It might be that P” or “It must be that Q”, both understood epistemically. The embedded proposition, P or Q is called the prejacent of the

• utterance. In McFarlane’s first example, the prejacent is a statement of

the truth of the famous Goldbach conjecture, that every even number greater than two is the sum of two primes; in his second example, the prejacent is that the Goldbach conjecture is false.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The current state

On virtually all accounts, the context of utterance will determine an agent,

• r a group of agents, and a body of information related in a relevant way

to the agents. The first utterance, “It might be that P” is true just in case P is compatible with, or somehow not ruled out, by the indicated body of information The second utterance, “it must be that Q” is true if Q is entailed by the body of information, or, equivalently, if ¬Q is ruled out by the information.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The controversies

As noted the controversies over epistemic modals are many. First, who are the agents? Is the agent always the speaker only? Or the speaker plus those addressed? Or the speaker plus those who happen to be listening (including eavesdroppers)? Or humanity at large, perhaps at a given time,

• r at all times?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The controversies

Second, how are the agents determined? Is it from the context of utterance (as with typical indexicals)? Or is it from a separate context of assessment? A matter of pragmatics, or semantics?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The controversies

Third, what goes into the body of information? Is it what the agent or agents know? Or is it what they know that is relevant to the topic at hand? Or is it what they strongly believe? Or what they are supposing in that context, the common ground? Or is there some other contextually supplied body of evidence that is relevant?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The controversies

Fourth, what sort of semantics is appropriate for epistemic modals? Are we in for a variety of contextualism, where the content of the utterance varies from context to context? If so, of what kind? Are epistemic modals like indexicals? Or is there some sort of elided constituent in the utterance? Or a variable in logical form? Or is relativism a better option?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

We will not engage any of those issues here (or not much). Our concern is with what is meant by the indicated body of information being “compatible with” the prejacent (P or Q) or by the information “ruling

• ut”, or “not ruling out”, or “guaranteeing” the prejacent.

Typically, the relevant relation is said to be logical consistency, or

• entailment. No other relations to serve this purpose have been articulated.

Consistency and entailment, however, give the wrong truth-conditions for the mathematical examples–no matter how the other matters are resolved.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

Let us modify McFarlane’s example a little. Go back to 1990 (or 1900, or just about any date between 1640 and 1990), and suppose that a mathematician says, then, that Fermat’s Last Theorem might be true and it might be false. The intuitive verdict, apparently in line with McFarlane’s suggestion, is that, in that context (or as assessed from that context), the mathematician’s utterance is true. No proof, of course, and no refutation was known then–by anyone.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

However, the standard Dedekind-Peano axioms for the natural numbers were known then, at least tacitly, by the relevant agent or agents–or so we stipulate. One might go so far as to argue that the Dedekind-Peano axioms are somehow implicit in the very notion of being a natural number. They are perhaps believed, and maybe known–tacitly–by anyone who understands the content of Fermat’s last theorem, or any other sufficiently rich statement about the natural numbers.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

And the negation of Fermat’s Last Theorem is decidedly not consistent with the Dedekind-Peano axioms (or other things in the epistemic basis). That is what Wiles established (possibly, sort of). So, according to the standard truth-conditions for epistemic modals, it was not true then that Fermat’s Last Theorem might be false. It was not true at any time when the relevant epistemic basis–the Dedekind-Peano axioms–were known, for the simple reason that (thanks to Wiles) the negation of Fermat’s Last Theorem is not consistent with those basic principles–all of which were known at the time.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

aside

Aside: It is highly likely that Fermat’s Last Theorem has an elementary proof (i.e., from the Dedekind-Peano axioms). See, for example, McLarty [2010]. If it does not, then just expand the epistemic base–what is presumed known by the agent(s) in question–to include the basics of elliptical function theory (the background of Wiles’s proof). Of course, if we have to do that, it is not correct to date our scenario as early as the seventeenth century. The year 1990 will still do. Notice, incidentally, that our above statement that it is likely that Fermat’s Last Theorem has an elementary proof is itself an epistemic

• modal. One of our (ultimate) goals here is to give truth conditions for it.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

It seems that an utterance in the form, “it might be that P” is false if the negation of the prejacent P can be readily determined by the relevant agents, or at least if it can be easily deduced from what they know. To paraphrase, using our example: “there might be a counterexample to the Goldbach conjecture” is false if the Goldbach conjecture can be readily deduced from what is known (however it is known).

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

Not quite. Consider, for example, a much discussed scenario in Hacking [1967, 148]: Imagine a salvage crew searching for a ship that sank a long time

• ago. The mate of the salvage ship works from an old log, makes

some mistakes in his calculations, and concludes that the wreck may be in a certain bay. It is possible, he says, that the hulk is in these waters. No one knows anything to the contrary. But in fact, as it turns out later, it simply was not possible for the vessel to be in that bay; more careful examination of the log shows that the boat must have gone down at least thirty miles further south. The mate said something false when he said, “It is possible that we shall find the treasure here,” but the falsehood did not arise from what anyone actually knew at the time.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

We would get the same verdict if we modified the scenario so that the mate did not actually make any mistakes, but simply ignored bits of the log–bits that were readily available to him. It must be admitted, however, that matters are a bit delicate here. MacFarlane [2011, 153] modifies the scenario a little. Suppose the mate has not yet considered the log. He then says: It’s possible that we shall find the treasure here, and it is possible that we shall find it farther south. Let’s examine the log before we dive: maybe we can eliminate one of these locations. The epistemic modal seems correct here, even if the log is readily available, and even if it contains information that rules out one of the

• possibilities. The mate had not gotten around to considering it yet.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

Our suggestion, roughly and provisionally, is that the context determines an evidential base (see von Fintel and Gillies [2010] and Roberts [forthcoming]). A statement in the form “it might be that P” is false if P can be easily inferred from the evidential base. In the original Hacking scenario, the evidential base includes the log, since the mate had examined it before coming to his verdict concerning the possible location of the ship (and he made a mistake in calculation from the information). In MacFarlane’s modification, the evidential base does not include the log, as the mate had not looked it over yet (or not with sufficient care). With the mathematical examples, the evidential base includes the basic principles known at the time, including the Dedekind-Peano postulates.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

The evidential base can be given explicitly, in which case it need not be determined from the context. And there can be more then one such in the same context. Consider, for example, the following (due to von Fintel and Gillies): Given the results of the DNA tests, John might be the thief. But if we take the eyewitness seriously, John can’t have been the thief. In this scenario, there are two separate evidential bases, one including the DNA data and the other including an eyewitness testimony.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

We suggest that, according to the semantics for epistemic modals, speakers are responsible for what they can readily come to know, from the evidential base. Or at least for what they can easily deduce from the evidential base. Suppose that someone says that 23 might be the largest prime number. Intuitively, the utterance–the epistemic modal–is false, even if the speaker never bothered to determine whether 29 is prime (or if she did try and made a mistake along the way). It is an easy calculation, and an

• bvious case to try.

We do not let agents off the hook for being lazy (or stupid).

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

In contrast, it seems that speakers are not responsible for missing long and complex chains of reasoning, say deductions that go on for hundreds of pages and require a mathematical genius to discover. The verdict, in 1990 (or 1900 or 1640) that Fermat’s Last Theorem might be false seems correct–at least when assessed from the relevant context. And it seems correct now to say that the Goldbach conjecture might be true and it might be false.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

We might add that speakers are not (always) responsible for long and tedious deductions or calculations from the evidential base, even if they are basic and the obvious thing to try. The mate’s original utterance, that the ship might be in a certain place, would be true if the calculation that shows the ship to be far away took, say, 5,000 steps on a calculator. Similarly, if the Goldbach conjecture is false, then it has a counterexample, a very very large even number that is not the sum of two primes. This fact (if it is a fact) can be discovered by a routine calculation, albeit one involving large numbers. The conjecture has been verified for all numbers smaller than 1017, i.e., 100,000,000,000,000,000. In any possible counterexample, there are a lot of steps in the calculation, but each one is very elementary, and routine to carry out.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

So we seem to be at least in the general neighborhood of matters related to P = NP. Here is another, closely related example. It is based on a scene in “Mr. Monk and the marathon man”. A young and, let us say, inexperienced detective is examining the body of a woman who fell from a balcony 21 stories up. He says, “it might be murder and it might be suicide”. Monk comes on the scene and says, “No, it cannot be suicide. Look at her toenails. She was painting them and did not finish that one. So she was interrupted.”

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

The problem

Our intuitive verdict is that the epistemic modal uttered by the inexperienced detective, “it might be murder and it might be suicide” is true (at the time). He cannot be expected to notice the relevance of the unpainted (or partially painted) toenail. But once those facts are made salient, and shown to be relevant, suicide is ruled out (or let us assume). In the expanded epistemic base, it is false that it might be suicide. This suggests that the crucial notion of what is “available” evidence is

• subtle. The unpainted toenail was, in some sense, available to the

inexperienced detective–it was in his visual field. But, at any time, there are a lot of things in someone’s visual field. No one is expected to be consciously aware of all of them at all times. Again, loosely related to the P = NP matter.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Idealizations

The notions of consistency, logical consequence, and deducibility developed in formal logic are highly idealized. We assume that agents doing deductions do not make mistakes and, more important here, that they have unlimited (but still finite) time, materials, and attention. We do not count an argument as invalid just because no one could deduce its conclusion from its premises using all the available material in the universe, or before the sun goes cold. Such idealizations are common in mathematics, and always have been. Euclid’s first postulates reads “to draw a straight line from any point to any point” and the third is “to describe a circle with any center and distance”. There are no bounds on how close the two points have to be to each other, or how small or large the radius must be.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Idealizations

These idealizations, however, are decidedly not appropriate when it comes to the semantics of epistemic modals, even for the natural languages spoken by professional mathematicians (given that they utter epistemic modals from time to time). We need a more “realistic” model of what is and is not ruled out by an evidential base. What matters is not what follows from the evidential base (in the sense of logic), but what a given agent or agents can be expected to determine, given the evidential base.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

There is, of course, a large body of work on the semantics of modals, of all

• kinds. Many of the more successful accounts invoke possible worlds.

Typically, a sentence in the form “P is possible”, is taken to be true, at a given world, just in case there is a relevant, accessible possible world in which P is true. The kind of modality invoked, plus various contextual factors, determine which possible worlds are relevant, and what the “accessibility relation” on possible worlds is.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

The program is highly successful, accounting for a wealth of linguistic

• data. One would be loathe to give up the whole thing.

In the case of deontic modals, the accessibility relation gives those worlds in which things are as they should be. In the case of metaphysical modality, the worlds are those that are metaphysically possible. In the case of epistemic modals, the relevant and accessible worlds are those that are consistent with what the agents know (or believe). In other words, a world is epistemically possible if it is a way things could be, given the evidential basis of the modal. So a sentence in the form “it might be that P” is true if there is such a world in which P is true.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Nevertheless, it seems that possible worlds are not the right tool, by themselves, for epistemic modals, for at least two related reasons. First, however they are construed, possible worlds are usually taken to be closed under logical consequence. So every logical truth is true in every world. More important, every logical consequence of the evidential base is true in every epistemic alternative of the information in the base. We thus broach the old problem of logical omniscience, for epistemic logic and the philosophical analysis of belief and knowledge: possible worlds are not sufficiently fine-grained for present purposes–thus the well-known problems of hyper-intensionality.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

The second problem with the possible worlds framework follows from what was said just above: What matters for epistemic modals is not (only) what is true in all epistemic alternatives, but what the relevant agents can readily come to know on the basis of the evidential base. And that depends on what the agent has access to.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Agents may grasp propositions, on some sense of “proposition”, but they do not grasp sets of possible worlds, at least not directly. And when knowledge is extended, by deduction, observation, abduction, etc., agents are not directly manipulating sets of possible worlds. So what are they (we) doing when we extend knowledge? This takes us into deep waters, concerning how the mind represents the world–both the way the world is and the ways it might be, the ways we’d like it to be, etc. And what can the mind do with these representations, when reasoning?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

MacFarlane [2011, 145], at least, is aware of the issue: “Joe might be running” expresses a truth just in case what the [agent] knows does not rule out that Joe is running . . . For present purposes, we can leave the notion of “ruling out” schematic: we need not decide, for instance, whether knowledge that p rules out everything logically inconsistent with p.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Our goal is to make some progress on what MacFarlane “leaves schematic”. We are after a notion of proposition, and notion of “ruling out”, according to which, in most situations, a basic knowledge of arithmetic does rule out 23 being the largest prime number, but it does not rule out that Fermat’s last theorem is false.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

It is common, at least informally, to allude to “impossible worlds” for various purposes. It is argued that impossible worlds are needed, anyway, to understand

• belief. Clearly, one’s beliefs may not be consistent, in which case, no

possible world is consistent with them. There simply are no possible worlds that are doxastic alternatives (dialetheism aside) to inconsistent beliefs, no matter how subtle the inconsistency may be.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

We suppose that, depending on how the framework is set up (and the devil lies in these details), impossible worlds may resolve issues of hyperintensionality. Sure every logical and mathematical truth holds in all possible worlds, but we have impossible worlds where, say, the Dedekind-Peano postulates are true but Fermat’s Last Theorem is false.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

However, the move to impossible worlds only postpones the problem, at best. First, impossible worlds seem to be just as abstract, idealized, and inaccessible as possible worlds. How do agents grasp and manipulate them, as they extend their beliefs?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

And it is not clear that the move even solves our problem, at least not in general. Presumably even impossible worlds are closed under some logic, presumably a paraconsistent one. So start with the items in the relevant evidential base, and consider anything that follows from them via a long and complicated chain of deductive reasoning in the favored weak, paraconsistent logic. An agent might think that the negation of that consequence is possible, and such a verdict seems correct (or as correct as our standard cases involving Fermat’s Last Theorem and Goldbach’s conjecture).

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

One option, we suppose, is to insist there is no non-trivial consequence relation on impossible words. There is a world in which seven is the largest prime (and in which the Dedekind-Peano axioms hold). But then we fail to accommodate the intuition that agents are responsible for readily available consequences of what it is in the evidential base. On this view, it seems, a sentence in the form “it might be that P” is true just in case the agent does not explicitly know P, or perhaps just in case P is not a member of the evidential base.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

Consider a 1990 utterance, “there might be a counterexample to Fermat’s last theorem”. We take this epistemic modal to be true. The friend of impossible worlds will say this is because there is a world–an impossible world–in which the Dedekind-Peano axioms (and the basic principles of elliptical function theory) are true and in which there is a counterexample to Fermat’s last theorem. But if there was such a world then, there is one

• now. So why is it not true now that there might be a counterexample to

Fermat’s last theorem.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

The friend of impossible worlds might reply that this impossible world was “accessible” then–it was an epistemic alternative–but it is no longer accessible (thanks to Wiles). This just pushes the problem over to the issue of accessibility. Just what are the epistemic alternatives to various possible and impossible worlds? What, among the impossible worlds, gets “ruled out” after successful chains of deductive (or inductive) reasoning are executed. This is essentially the same problem we started with.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

Impossible worlds

Jens Christian Bjerring [2013] shows that if we assume that each world (possible or otherwise) gives a truth value–truth or falsehood–to every sentence in the language, then any inconsistent world will contain an

• bvious inconsistency.

It seems best to look elsewhere for a solution.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

situations

Perhaps the problems is the “completeness” of worlds, whether they be possible or otherwise. Each sentence is true or false in each world. Situations have been used to get around this feature, for various purposes. A situation is a part of a possible world, containing some objects and some properties and relations on the objects of the situation.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Standard frameworks – possible worlds and the like

situations

Situations do not seem to help here. The key mathematical prejacents–the Goldbach conjecture and Fermat’s last theorem–both have a universal quantifier ranging over the natural numbers. So any situation that resolves them must contain all of the natural numbers. Presumably, any such situation makes the Dedekind-Peano postulates true, and thus resolves both propositions in only one way: Fermat’s last theorem is true and the Goldbach conjecture has the same truth value it has in reality. The next move might be to impossible situations. But what are those? Parts of impossible worlds? We thus seem to inherit the problems of both situations and impossible worlds.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

Robert Stalnaker [1991], [1999] is one of the best discussions of the problem of logical omniscience. He does not address epistemic modals, even indirectly, but virtually everything he says bears on our issues. The key task is not (or not only) to figure out what knowledge (or belief) is, in general, but to determine what it is we do with our knowledge (and beliefs). To be useful for action (including the action of proving theorems), a piece of knowledge must be available. That is the key notion. The distinction between available and unavailable knowledge is different from the distinction between overt and tacit knowledge.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

We are after a rather special, and somewhat technical, notion of “available” knowledge or information. Presumably, the proof of Fermat’s last theorem was not “available” in 1990, even though Wiles found it a few years later. What of the information, in the Monk example, that the victim was painting her toenails, and the relevance of that to the matter at issue? Was that “available” to the inexperienced detective?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

sidetrack on epistemology

There is a group of accounts that hold that the verb “know” is context-sensitive. According to these contextualists, whether an agent knows a given proposition depends on what standards are in play. The view is, in part, a response to skeptical arguments. In normal contexts, I know that I am not a brain in a vat, and that there is no Cartesian Demon working to deceive me. In skeptical contexts, where the standards are higher, I do not know these things. Some hold that the contextualism is more general, and independently

• motivated. The view remains controversial.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

sidetrack on epistemology

Jason Stanley’s Knowledge and practical interests [2005] rejects contextualism, but argues that whether an agent knows a given proposition can depend on not only her epistemic state toward the proposition–how good her evidence is–but also on what is at stake. His view is called “subject-sensitive invariantism”. To modify one of Stanley’s examples, suppose that two people, Jack and Hannah, have essentially the same evidence that a given bank will be open the next day, a Saturday. They remember that it was open the Saturday before, and did not hear anything concerning any change in the bank’s

• hours. For Jack, the stakes are low, since he does not have any pressing

business that depends on the bank being open the next day. The intuitive verdict is that he knows that the bank will be open then.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

sidetrack on epistemology

For Hannah, however, the stakes are high. She has a check to deposit and a bill that is due. And her partner reminds her that banks do sometimes change their hours. According to Stanley, Hannah does not know that the bank is open the next day. To be sure, this account of knowledge remains controversial.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

sidetrack on epistemology

We can recast the examples used to support contextualism and Stanley’s subject-sensitive invariantism as epistemic modals. And there, it seems, the intuitive verdicts seem at a bit clearer. For example, in a conversation over skepticism, one can say, felicitously, that he might be a brain in a vat, or that there might be a Cartesian Demon at work, or that the universe might have been created one hour ago, or . . . . It would be infelicitous to say this in a different context, say where one is trying to figure out the best way to Boston or trying to prove the Goldbach conjecture. It might even be correct to say, then, that one cannot be a brain in a vat–although one may be pressed to show why one would bother saying this.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

sidetrack on epistemology

The Stanley examples can also be cast in terms of epistemic modals. For Hannah, where the stakes are high, it seems correct to say that the bank might be closed the next day. While Jack, who has less at stake, can say, “don’t worry; the bank will be open” or even “the bank must be open tomorrow–it was so last Saturday”. The moral, so far: Knowledge, in general, may or may not be context sensitive, varying with standards; and knowledge, in general, may or may not depend on what is at stake. But available knowledge–the knowledge relevant for evaluating epistemic modals–does seem to vary, both with prevailing standards and according to what is at stake.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

Stalnaker [1991, 254] writes: I know whether P if I have the capacity to make my actions depend on whether P. But I may have this capacity for some actions, and not for others . . . the accessibility of knowledge and belief can be understood only relative to the actions they are being used to guide. The problem of logical omniscience, I am suggesting, is the problem of accessibility ([1991, 254]). we need to understand knowledge and belief as capacities and dispositions–states that involve the capacity to access information, and not just its storage–in order to distinguish what we actually know and believe . . . from what we know and believe only implicitly. ([1991, 353])

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

The following representational-intentional systems are central features of the bridge connecting the-way-things-are to action: (i) desire (motivation to aim for, the-way-the-agent-wants-things-to-be) (ii) knowledge (a capacity for action, an ability or power)

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

Here is a preliminary stab at characterizing capacity (ii): (iia) knowledge of P is the ability to access the information that P in case one’s actions depend on whether P, and to do so in such a way that one can use it to (determine how to) act effectively in those cases. (iib) knowledge of P is the ability to access the information that P when whether-P is relevant to one’s choice of action, where a choice of action is the answer to a specific instantiation of the Question-Under-Discussion ‘what should I do?’ In this case, we have a strategy of inquiry where whether-P is a relevant sub-question of ‘what should I do?’

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

As usual, relevance may be partly a function of the implicit background assumptions that constrain and drive the choice: the Common Ground and the circumstances of utterance. It seems clear that available knowledge is sensitive to what one is trying to do–prove a theorem, recover a sunken ship, convict a thief, deposit a check, figure out what we can conclude in a worst-case-scenario concerning belief . . .

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Stalnaker: available knowledge

It also seems clear that both available knowledge and desire are reflected in intensional/opaque attitudes toward their objects. Whether a given piece of information is available depends on its guise, on how the information is presented. Two sentences can be logically equivalent, with

• ne available and one not.

For a relevant example, under common background assumptions, “n is an even number greater than 4 and less than 1017” is equivalent to “n is the sum of two odd primes and less than 1017”. This suggests that we must relativize knowledge to the guises under which the information is available to the agent. It seems that the conditions relevant for specific actions include those guises. This is closely related to the problem of logical omniscience.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Lewis strategy

A preliminary strategy due to David Lewis: If you want to get clear on a certain notion, ask what that notion is supposed to do for us, and then figure out what can do those things. Our notion here is (available) knowledge. What is its job description?

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Lewis strategy

What does knowledge do? 1. represents aspects of the way-things-are 2. feeds practical and theoretical inference: what is known can serve as premises 3. distinguishes between logically equivalent propositions (so some kind of fine-grained representation is required–hyperintensionality, structured propositions, or Discourse Representation Structures, . . . )

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Lewis strategy

What does knowledge do? (continued) 4. comes in two flavors: available knowledge and unavailable knowledge, a different distinction from that between explicit and tacit knowledge. What it is to be available is not an inherent property of the object of knowledge, but is itself a function of both the purposes to which the knowledge would be put and the guise under which it’s represented. One guise of proposition P may be available in a given circumstance, while another is not. 5. when available, serves as a crucial foundation for action. Hence, only available knowledge is of practical or even theoretical interest in most uses.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Lewis strategy

What does knowledge do? (continued) 6. has a shiftable boundary with belief, e.g. what counts as available knowledge, as opposed to mere belief, is a function of what’s at stake, with standards adjusted accordingly 7. interacts with desire in the formulation of (a) plans for action, and (b) intentions to carry out those plans 8. may be iterated: knowledge of knowledge of other agents and in particular of their (available) knowledge and beliefs

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

Stalnaker [1999, 265]: An agent’s knowledge that ϕ is available under condition ψ, provided that the agent has the capacity to make its action depend on ϕ under that condition, and is disposed, under that condition, to make its action depend on whether ϕ whenever it wants its action to depend on whether ϕ.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

So one knows that P (in this sense) just in case decisions about what to do can reasonably depend on whether-P under the appropriate conditions, including the action to be performed: to be available or usable, the information we store in our representation of the world must be appropriately calibrated to the desires or other motivational states that determine how the information is to be used.” (Stalnaker [1999,270])

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

The above condition ψ indicates that whether-P is relevant to what one chooses to do, i.e. to address the Question-under-discussion. It depends

• n the kind of action/doing in question.

There may be several facets of this dependence, as reflected in:

• Stalnaker’s example about relativization of knowledge to

dependent action (concretely instantiated): A baseball outfielder might know where the ball is going to land [P = ‘the ball is going to land at spot x’] for the purpose of catching it, but not for the purpose of precisely describing that location to an onlooker.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

• Pierre knows that London is pretty for the purposes of

expressing that proposition in French, but not in English. I might know that Sam Clemens lived in the South for the purposes of describing his history, but not for the purposes of describing the author of Mark Twain.

• Stanley’s practical interests, and their relationship to

standards: I might know the bank will be open tomorrow for the purposes of making a non-vital deposit, but not for the purpose

• f making one where a lot of money depends on getting it

deposited by tomorrow.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

Part of what one needs to do, from a practical point of view, is access the information in question in a reasonable time frame. How reasonable a time frame depends on the task at hand.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

Recall the connection between (i) desire/motivation and (ii) available knowledge. Desire and the intention it fosters jointly direct and constrain attention For example, salience is more a function of intentions and attention that it is of recency.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Available knowledge

Attention facilitates the increased availability of general information pertaining to the object of attention, in accordance with information about that object in the common ground. What facilitates availability of information:

• attention (and hence, the intentions that facilitate that

attention, above)

• guises: evident features highlighted in the representations
• f the information, and the relations between those features

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Parting thoughts (in lieu of a conclusion)

Stalnaker [1999, 270]: People, animals, “robots who act, but do not speak” . . . may have different information-bearing states that tend to carry the same info, but that play a different role in guiding rational

• behavior. We need a finer-grained notion of content to give

perspicuous descriptions of what they know, and what knowledge they have available, but it is not clear that the semantic structure of sentences that might be used to express their knowledge will necessarily provide the relevant distinctions.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Parting thoughts (in lieu of a conclusion)

We cannot understand knowledge and belief as capacities and dispositions without “bringing the uses to which knowledge and belief are put into the concepts of knowledge and belief themselves” (Stalnaker [1991, 251). And again (254): I know whether P if I have the capacity to make my actions depend on whether P. But I may have this capacity for some actions, and not for others . . . the accessibility of knowledge and belief can be understood only relative to the actions they are being used to guide. In other words, we can know P under some guise only relative to a representation suitable for certain purposes.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro

Parting thoughts (in lieu of a conclusion)

However (253),

• n the face of it, it does not seem that when we attribute

knowledge or belief to someone we are making any claims about what the agent plans to do with that information. This reflects a tension between: (1) the assumption that it’s available knowledge/belief that matters (not explicit or implicit), and (2) the common assumption that knowledge and belief are independent of what we want/are inclined to try to get.

Epistemic modals and mathematics Craige Roberts and Stewart Shapiro