beyond worlds and accessibility - uc berkeley · beyond worlds and accessibility part 1 wesley...

Beyond Worlds and Accessibility

Part 1

Wesley HollidayDepartment of Philosophy &

Group in Logic and the Methodology of ScienceUniversity of California, Berkeley

EASSLC 2014, July 2-8

Wesley Holliday: Beyond Worlds and Accessibility 1

Course Introduction

Course Description

The starting point of this course is standard possible-world semantics forepistemic, doxastic, and other modal logics, based on models with“possible worlds and “accessibility relations.”

We will explore a number of ways of enriching or departing from thesemodels, motivated by philosophical considerations.

Topics will include:

I models that add more structure to capture substantive theories ofknowledge (in the style of Dretske, Nozick, and others);

I models that replace total possible worlds with partial “possibilities”(in the style of Humberstone);

I models that replace total possible worlds with partial “situations”(in the style of Barwise and Perry).

In each case, we will study the logical properties of these classes ofmodels, illuminating the philosophical issues at stake.

Course Introduction

Course Description

Course Introduction

Course Description

Course Introduction

Course Description

Course Introduction

Course Description

Course Introduction

Course Description

Course Introduction

Course Description

Course Introduction

Outline of the Course

I Day 1 - Introduction and Background

I Day 1-3 - RA and Subjunctivist Semantics

I Day 3 - Set-Selection Function Semantics

I Day 4 - Possibility Semantics

I Day 5 - Situation Semantics

Course Introduction

For the slides and references for this course, go to

http://philosophy.berkeley.edu/holliday

then go to Short Courses – EASLLC 2014.

The slides are hyperlinked, so click on references to find them on the web.

The logical material today is primarily based on the following works:

I “Epistemic Closure and Epistemic Logic I: Relevant Alternatives andSubjunctivism,” Journal of Philosophical Logic, 2014.

I “Epistemic Logic and Epistemology,” Handbook of FormalPhilosophy, forthcoming.

Recent work on related topics comes from researchers in China! See, e.g.,papers by Zhaoqing Xu and Chenwei Shi on the webpage given above.

Course Introduction

From my “Summary of “Epistemic Closure and Epistemic Logic I:Relevant Alternatives and Subjunctivism”,” Logic Across the University :

An application of formal methods in philosophy gains value insofar as:

(1) it is faithful to the philosophical views being formalized;

(2) it can handle concrete examples discussed in the philosophicalliterature;

(3) it goes beyond particular examples to provide a systematic andgeneral view of the topic;

(4) it leads to philosophically-relevant discoveries that would be di�cultto make by non-formal methods alone;

(5) it leads to the development of new views that solve previousphilosophical problems.

(1) and (2) address the worry that by formalizing we may “change thesubject.” (3) - (5) address the question, “What do we get out of this?”Of course, there are other features that contribute to the value of anapplication of formal methods, as well as features that detract from it.

Course Introduction

(1) and (2) address the worry that by formalizing we may “change thesubject.”

(3) - (5) address the question, “What do we get out of this?”Of course, there are other features that contribute to the value of anapplication of formal methods, as well as features that detract from it.

Course Introduction

(1) and (2) address the worry that by formalizing we may “change thesubject.” (3) - (5) address the question, “What do we get out of this?”

Of course, there are other features that contribute to the value of anapplication of formal methods, as well as features that detract from it.

Course Introduction

Motivation for Today

Basic Epistemic Logic & What One Knows

Model what an agent a knows by what is true throughout a set Ra

(w) ofpossibilities that are compatible with her knowledge in world w :

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

Q: what exactly does it mean for v to be “compatible”?

A: v 2 Ra

(w) means that everything a knows in w is true in v .

Q: So you’re explaining when ‘Ka

j’ is true in terms of knowledge itself?

A: Yes.

Q: Isn’t that a problem? Somehow cheating or conceptually circular?

A: No! The point of basic epistemic models is to represent the content ofan agent’s knowledge—what the agent knows—not to give a substantiveaccount of what it takes to know something in other terms.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

A: No!

The point of basic epistemic models is to represent the content ofan agent’s knowledge—what the agent knows—not to give a substantiveaccount of what it takes to know something in other terms.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

Model what an agent knows by what is true throughout a set R(w) ofpossibilities that are “compatible” with her knowledge in world w :

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: The point of basic epistemic models is to represent the content of anagent’s knowledge—what the agent knows—not to give a substantiveaccount of what it takes to know something in other terms.

Qualification: two aspects of these models are not neutral with respectto di↵erent substantive accounts of what it takes to know something.

Most obviously, if we impose conditions on Ra

beyond reflexivity[w 2 R

(w)] such as symmetry [v 2 Ra

(w) ) w 2 Ra

(v)], we build insubstantive views (in this case, that ¬j ! K

j should be valid)that conflict with some accounts of knowledge (ditto for ‘information’).

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

(w) ) w 2 Ra

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

(w) ) w 2 Ra

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

Less obviously, while the ) direction of (1) follows immediately from thestipulation that v 2 R

(w) means everything a knows in w is true in v ,the ( direction of (1) does not follow just from that stipulation.

In fact, to add the ( direction is just to assume the principle thatknowledge is closed under known implication:

(j ! y)) ! Ka

If M,w ✏ Ka

j and M,w ✏ Ka

(j ! y), then by the ) direction of(1), 8v 2 R

(w) : M, v ✏ j and M, v ✏ j ! y, so M, v ✏ y.The ( direction then yields the conclusion that M,w ✏ K

The ( direction conflicts with some accounts of knowledge I will discuss.

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

(j ! y)) ! Ka

If M,w ✏ Ka

j and M,w ✏ Ka

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

(j ! y)) ! Ka

If M,w ✏ Ka

j and M,w ✏ Ka

(w) : M, v ✏ j and M, v ✏ j ! y, so M, v ✏ y.

The ( direction then yields the conclusion that M,w ✏ Ka

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

(j ! y)) ! Ka

If M,w ✏ Ka

j and M,w ✏ Ka

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

(j ! y)) ! Ka

If M,w ✏ Ka

j and M,w ✏ Ka

Epistemic Logic & What One Knows

A: The point of basic epistemic models is to represent the content ofan agent’s knowledge—what the agent knows—not to give a substantiveaccount of what it takes to know something in other terms.

Those who criticize basic epistemic models for not giving more substantiveinsight into what it takes to know something about the world (or what ittakes to get information from the world) are missing the point.

The beauty of it is that basic epistemic logic can nevertheless give illu-minating analyses of a wide range of epistemic phenomena, especially inmulti-agent settings with information change (think Muddy Children).

For many examples, see:

R. Fagin, J.Y. Halpern, Y. Moses, and M.Y. Vardi. 1995. Reasoning about Knowledge.

J.J.C. Meyer and W. van der Hoek. 1995. Epistemic Logic for AI and Computer Science.

H. van Ditmarsch, W. van der Hoek, and B. Kooi. 2008. Dynamic Epistemic Logic.

J. van Benthem. 2011. Logical Dynamics of Information and Interaction.

Epistemic Accessibility vs. Indistinguishability

Distinction: accessibility vs. other notions of indistinguishability.

Suppose that we replace R by a binary relation E on W , where ourintuitive interpretation is that wEv holds “i↵ the subject’s perceptualexperience and memory” in scenario v “exactly match his perceptualexperience and memory” in scenario w (Lewis, “Elusive Knowledge”).

M,w ✏ Ka

j , 8v 2 Ea

(w) : M, v ✏ j. (2)

Hence the agent knows j in w just in case j is true in all scenarios thatare experientially indistinguishable from w for the agent.

M,w ✏ Ka

j , 8v 2 Ea

(w) : M, v ✏ j. (2)

M,w ✏ Ka

j , 8v 2 Ea

(w) : M, v ✏ j. (2)

Epistemic Accessibility & Indistinguishability

Important di↵erences between the picture with E and the one with R :

I The epistemic model with E does not simply represent the contentof one’s knowledge; rather, it commits us to a particular view of theconditions under which an agent has knowledge, specified in termsof perceptual experience and memory.

I It is plausible that E has certain properties, such as symmetry (wEvi↵ vEw), which are questionable as properties of R .

Since the properties of the relation will determine valid principles forthe knowledge operator K , we must be clear about whichinterpretation of the relation we adopt.

“Considering Possible”

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

While one may read wRv as

I “for all the agent knows in w , scenario v is the scenario she is in,”

one should not read wRv as

I “in w , the agent considers scenario v possible,”

where the latter suggest a subjective psychological notion.

An agent may not subjectively consider it possible that his friend, whomhe has regarded for years as his most trusted ally, has betrayed him. Itobviously does not follow that he knows that his friend has not betrayedhim, as it would according to the subjective reading of R with (1).

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

An agent may not subjectively consider it possible that his friend, whomhe has regarded for years as his most trusted ally, has betrayed him.

Itobviously does not follow that he knows that his friend has not betrayedhim, as it would according to the subjective reading of R with (1).

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

An agent may not subjectively consider it possible that his friend, whomhe has regarded for years as his most trusted ally, has betrayed him. Itobviously does not follow that he knows that his friend has not betrayedhim, as it would according to the subjective reading of R with (1).

M,w ✏ Ka

j , 8v 2 Ra

(w) : M, v ✏ j. (1)

A: v 2 Ra

A: Yes.

Epistemology & What It Takes to Know

Notice that the way of modeling the content of an agent’s knowledge inbasic epistemic-logical models gives no instructions re the question:

In general, if one wants to know that something is true—not justbelieve or wish it to be true—what does it take to achieve that?

Philosophers have long tried to provide some guidance here. (Let usconsider empirical knowledge.) Here are two unpromising answers:

I The agent should keep making observations of the world until shecan eliminate every possible way in which p could be false, which isto say: until her experience is so extensive that it is impossible forthings to appear to her as they do in a world where p is false.

I She should keep investigating the world until her belief about pbecomes so strongly correlated with the truth of p that it would beimpossible for her to believe p as she does in a world where p is false.

Problem: according to these answers, nobody ever knows anything.

Philosophers have long tried to provide some guidance here.

(Let usconsider empirical knowledge.) Here are two unpromising answers:

Philosophers have long tried to provide some guidance here. (Let usconsider empirical knowledge.)

Here are two unpromising answers:

Many philosophers have realized this, e.g.:

J.L. Austin. 1946. “Other Minds.”

Alvin Goldman. 1976. “Discrimination and Perceptual Knowledge.”

Fred Dretske. 1981. “The Pragmatic Dimension of Knowledge.”

David Lewis. 1996. “Elusive Knowledge.”

“If knowledge required the elimination of all logically possible alternatives,there would be no knowledge (at least of contingent truths).”

(Goldman 1976, 775)

Many philosophers have realized this, e.g.:

J.L. Austin. 1946. “Other Minds.”

Alvin Goldman. 1976. “Discrimination and Perceptual Knowledge.”

Fred Dretske. 1981. “The Pragmatic Dimension of Knowledge.”

“There are always, it seems, possibilities that our evidence is powerlessto eliminate... If knowledge...requires the elimination of all competingpossibilities (possibilities that contrast with what is known), then, clearlywe seldom, if ever, satisfy the conditions for applying the concept.”

(Dretske 1981, 365)

Relevant Alternatives and Subjunctivism

Austin, Goldman, Dretske, Lewis, and others propose a di↵erent idea:

According to relevant alternatives theories of knowledge, the agentshould keep making observations of the world until she eliminates everyrelevant ¬p-possibility; an RA theorist then gives an account of whatfactors determine which possibilities are relevant and which are not.

Robert Nozick, Dretske, Ernest Sosa, and others propose related ideas:

According to subjunctivist theories of knowledge, the agent should keepinvestigating the world until her belief about p becomes su�cientlycorrelated with the truth of p that, e.g., the following counterfactual istrue: if p were false (in the “closest worlds”), she wouldn’t believe it.

Robert Nozick. 1981. Philosophical Explanations.

Fred Dretske. 1971. “Conclusive Reasons.”

Ernest Sosa. 1999. “How to Defeat Opposition to Moore.”

According to relevant alternatives theories of knowledge, the agentshould keep making observations of the world until she eliminates everyrelevant ¬p-possibility; an RA theorist then gives a story about whatfactors determine which possibilities are relevant and which are not.

According to subjunctivist theories of knowledge, the agent should keepinvestigating the world until her belief about p becomes su�cientlycorrelated with the truth of p that, e.g, the following counterfactual istrue: if p were false (in the “closest worlds”), she wouldn’t believe it.

—————————————————————————————–

Claim: these ideas are not just the work of philosophers’ imaginations,detached from real life. To the contrary, arguably considerations ofrelevance and counterfactuals can help to explain why people are actuallywilling or unwilling to attribute knowledge to other people in practice.

—————————————————————————————–

What do these theories have to do with epistemic logic?

Answer:they bear on the correctness of core principles of epistemic logic.

—————————————————————————————–

What do these theories have to do with epistemic logic? Answer:they bear on the correctness of core principles of epistemic logic.

Logical Omniscience vs. Epistemic Closure

Let’s distinguish two very di↵erent objections to

K (K j ^K (j ! y)) ! Ky.

First, the problem of logical omniscience: real (finite, boundedlyrational) agents don’t always “put two and two together.”

(Aside: one way to bypass the logical omniscience problem is to read K jas “j is entailed by what the agent knows.” Then K is fine.)

Second, the denial of epistemic closure: even if agents did always puttwo and two together, K should still be rejected in its fullest generalityfor knowledge (though for circumscribed applications it might be fine).

K (K j ^K (j ! y)) ! Ky.

Fallibilism, Closure, and Contradiction

According to Goldman, Dretske, and many others, the infallibilist viewthat for every proposition p, one must eliminate all ¬p-possibilities inorder to know p entails skepticism: we don’t know anything.

Rejecting infallibilism, they hold the fallibilist view that for at least somep, there are some ¬p-possibilities that one does not have to eliminate inorder to know p. Let s pick out such a set of possibilities, so knowing pdoes not require the agent to eliminate s-possibilities. But by K:

(Kp ^K (p ! ¬s)) ! K¬s,

which says that knowing p does require knowing ¬s, which requireseliminating s-possibilities (assuming that knowing something requireseliminating at least some counter-possibilities). Contradiction?

Rejecting infallibilism, they hold the fallibilist view that for at least somep, there are some ¬p-possibilities that one does not have to eliminate inorder to know p.

Let s pick out such a set of possibilities, so knowing pdoes not require the agent to eliminate s-possibilities. But by K:

(Kp ^K (p ! ¬s)) ! K¬s,

Rejecting infallibilism, they hold the fallibilist view that for at least somep, there are some ¬p-possibilities that one does not have to eliminate inorder to know p. Let s pick out such a set of possibilities, so knowing pdoes not require the agent to eliminate s-possibilities.

But by K:

(Kp ^K (p ! ¬s)) ! K¬s,

which says that knowing p does require knowing ¬s, which requireseliminating s-possibilities (assuming that knowing something requireseliminating at least some counter-possibilities).

Contradiction?

(Kp ^K (p ! ¬s)) ! K¬s,

Two di↵erences between these philosophical views of what it takes toknow and the basic epistemic-logical models of what an agent knows:

I The philosophical theories appeal to additional structure forrelevance or for counterfactuals, not in basic epistemic models;

I The philosophical theories are presented informally. They are notinvestigated mathematically as models for epistemic logic are.

But we can do something about this: add the extra structure to basicepistemic models and then investigate the theories mathematically.

Formalization

A Family of Theories

I will begin by providing model-theoretic, logical formalizations of a familyof relevant alternatives (RA) and subjunctivist theories of knowledge:

L the RA theory of David Lewis (1996);

D one way of interpreting the RA theory of Fred Dretske (1981);

H the RA theory of Mark Heller (1989, 1999);

N the basic tracking theory of Robert Nozick (1981);

S and the safety theory of Ernest Sosa (1999).

Formalization

Example (Medical Diagnosis)Two medical students, A and B, are subjected to a test. Their professorintroduces them to the same patient, who presents various symptoms,and the students are to make a diagnosis of the patient’s condition. Aftersome independent investigation, both students conclude that the patienthas a common condition c . In fact, they are both correct.

Yet only student A passes the test.

The professor wished to see if the students would check for anothercommon condition c 0, which causes the same visible symptoms as c .While student A ran laboratory tests to rule out c 0 before making thediagnosis of c , the student B made the diagnosis of c after only aphysical exam, having never considered the possibility of c 0.

Formalization

RA-style analysis:

I Although both students gave the correct diagnosis of c , student Bdid not know that the patient’s condition was c , since he did notrule out the alternative of c 0.

I Student B got lucky. Had the patient’s condition been c 0, student Bmight still have made the diagnosis of c , since the physical examwould not have revealed a di↵erence.

I Student A secured against this possibility of error by running thelaboratory tests.

Of course, student A did not secure against every possibility of error...

Formalization

RA-style analysis:

Formalization

RA-style analysis:

Formalization

RA-style analysis:

Formalization

Example (Medical Diagnosis continued)Suppose there is an extremely rare disease x1 such that people withdisease x appear to have c on many laboratory tests, even though peoplewith x are completely immune to c , and only extensive further testingcan detect the presence or absence of x in its early stages.

Should we say that student A did not know that the patient’s conditionwas c after all, since she did not rule out the possibility of x?

1Maybe never-before-documented, but a possibility dreamed up by ahypochondriac.

Formalization

Example (Medical Diagnosis continued)Suppose there is an extremely rare disease x1 such that people withdisease x appear to have c on many laboratory tests, even though peoplewith x are completely immune to c , and only extensive further testingcan detect the presence or absence of x in its early stages.

Should we say that student A did not know that the patient’s conditionwas c after all, since she did not rule out the possibility of x?

1Maybe never-before-documented, but a possibility dreamed up by ahypochondriac.

Formalization

RA-style analysis cont.:

I The requirement that one rule out all possibilities of error makesknowledge impossible, since there are always some possibilities oferror—however remote and far-fetched—that are not eliminated byone’s evidence and experience.

I If the student has no reason to think that the patient may have therare disease x , then it should not be necessary to rule out such aremote possibility in order to know that the patient has somecommon condition.

Formalization

RA-style analysis cont.:

I The requirement that one rule out all possibilities of error makesknowledge impossible, since there are always some possibilities oferror—however remote and far-fetched—that are not eliminated byone’s evidence and experience.

I If the student has no reason to think that the patient may have therare disease x , then it should not be necessary to rule out such aremote possibility in order to know that the patient has somecommon condition.

Formalization

I Suppose that all medical students know that people with x havecomplete immunity to c , as assumed above, so

KA (c ! ¬x) .

I Since the student did not run any tests that could possibly detectthe presence or absence of x , it would be unreasonable to claim thatshe knows that the patient does not have x , so

¬KA¬x .

I Then together with our judgment that the student knows that thepatient has condition c ,

we have a clear violation of closure under known implication,

(KAc ^KA(c ! ¬x)) ! KA¬x .

Formalization

KA (c ! ¬x) .

¬KA¬x .

(KAc ^KA(c ! ¬x)) ! KA¬x .

Formalization

KA (c ! ¬x) .

¬KA¬x .

(KAc ^KA(c ! ¬x)) ! KA¬x .

Formalization

According to Dretske, the RA theory explains why closure fails:

I In order to know the premises c and c ! ¬x , the agent must ruleout certain relevant alternatives. In order to know the conclusion¬x , the agent must also rule out certain relevant alternatives.

I But the sets of relevant alternatives for the premises and for theconclusion are not the same.

We have already argued that x is not a relevant alternative that mustbe ruled out in order for KAc to hold. But x certainly is a relevantalternative that must be ruled out in order for KA¬x to hold.

I It is because the relevant alternatives may be di↵erent for thepremises and the conclusion that closure under known implicationdoes not hold in general.

Question: which other closure principles fail (or hold) on the RA theory?

Formalization

Validity Omniscience and Doxastic ClosureDretske and Nozick argue that closure under known implication wouldfail even for agents with ideal rationality.

According to Dretske, “Were we all ideally astute logicians, were we allfully appraised of all the necessary consequences...of every proposition,perhaps then the epistemic operators would . . . [satisfy K]. It is this. . . claim that I mean to reject” (“Epistemic Operators,” 1010).

I Validity omniscience: one knows all valid logical principles;

I Full doxastic closure: one believes all the logical consequences ofthe (set of) propositions one believes.

One may satisfy the conditions for knowledge (ruling out the relevantalternatives, tracking the truth, etc.) with respect to some propositionsand yet not with respect to all logical consequences of the set of thosepropositions, even if one has explicitly deduced all of these consequences.

Formalization

Formalization: Relevant Alternatives

Relevant Alternatives Models

w1 c 0

w1 c , x

A (single-agent) RA model is a tuple M = hW ,_,�,V i, where:

w1 c 0

w1 c , x

I W = {w , v , u, . . . } is a non-empty set of “worlds” or “possibilities.”

w1 c 0

w1 c , x

I V : At ! P(W ), where At = {p, q, r , . . . } is a fixed set of atomicsentence symbols, the same for all models.

w1 c 0

w1 c , x

I _ is a reflexive relation on W (note omitted loops in diagrams).

w _ v means possibility v is uneliminated for the agent in w .

w1 c 0

w1 c , x

Lewis (1996) proposes that “a possibility . . . [v ] . . . is uneliminatedi↵ the subject’s perceptual experience and memory in . . . [v ]. . . exactly match his perceptual experience and memory inactuality” (553). Thus, for Lewis _ is an equivalence relation.

w1 c 0

w1 c , x

w _ v means possibility v is uneliminated for the agent in w .

Whether we assume transitivity, symmetry, Euclideaness, etc., inaddition to reflexivity will not a↵ect our main results.

w1 c 0

w1 c , x

I� assigns to each w 2 W a binary relation �

on some Ww

✓ W ;

1. �

is reflexive and transitive (preorder);

2. w 2 W

, and for all v 2 W

, w �

v (weak centering).

v means u is at least as relevant (at w) as v is;

w1 c 0

w1 c , x

on some Ww

✓ W ;

1. �

2. w 2 W

, and for all v 2 W

, w �

v (weak centering).

v means u is at least as relevant (at w) as v is;u �

v i↵ u �

v and not v �

w1 c 0

w1 c , x

on some Ww

✓ W ;

1. �

2. w 2 W

, and for all v 2 W

, w �

v (weak centering).

v means u is at least as relevant (at w) as v is;u �

v i↵ u �

v and not v �

v i↵ u �

v and v �

w1 c 0

w1 c , x

A (single-agent) RA model is a tuple M = hW ,_,�,V i ...

Many epistemologists state conditions for knowledge in terms of some kindof (possibly context-dependent) ordering on worlds: e.g., Heller’s (1989)RA picture of “worlds surrounding the actual world ordered according tohow realistic they are, so that those worlds that are more realistic are closerto the actual world than the less realistic ones” (25).

Mark Heller. 1989. “Relevant Alternatives.”

w1 c 0

w1 c , x

on some Ww

✓ W ;

1. �

2. w 2 W

, and for all v 2 W

, w �

v (weak centering).

The second condition corresponds to Lewis’s (1996) Rule ofActuality, that “actuality is always a relevant alternative” (554).

w1 c 0

w1 c , x

on some Ww

✓ W ;

1. �

2. w 2 W

, and for all v 2 W

, w �

v (weak centering);

3. if ∆ 6= S ✓ W

, then Min�

(S) 6= ∆ (well-founded).

For S ✓ W , Min�

(S) = {v 2 S \Ww

| ¬9u 2 S : u �

Definition (RA Model)A relevant alternatives model is a tuple M =

W ,_,�,Vi

where:

1. W is a non-empty set;

2. _ is a reflexive binary relation on W ;

3. � assigns to each w 2 W a binary relation �

on some Ww

✓ W ;

3.1 �

3.2 w 2 W

, and for all v 2 W

, w �

v (weak centering);

3.3 if ∆ 6= S ✓ W

, then Min�

(S) 6= ∆ (well-founded);

4. V : At ! P(W ).

We will also consider total RA models in which for all w 2 W ,�

is a total preorder, i.e., 8u, v 2 Ww

, u �

v or v �

These models represent a fixed context. For context change, see:

W. Holliday. 2012.“Epistemic Logic, Relevant Alternatives, and the Dynamics of Context.”

New Directions in Logic, Language, and Computation, LNCS.

W ,_,�,Vi

where:

on some Ww

✓ W ;

3.1 �

3.2 w 2 W

, and for all v 2 W

, w �

v (weak centering);

3.3 if ∆ 6= S ✓ W

, then Min�

4. V : At ! P(W ).

, u �

v or v �

W ,_,�,Vi

where:

on some Ww

✓ W ;

3.1 �

3.2 w 2 W

, and for all v 2 W

, w �

v (weak centering);

3.3 if ∆ 6= S ✓ W

, then Min�

4. V : At ! P(W ).

, u �

v or v �

Language

We could use our RA models to give semantics for a variety of di↵erentformal languages, but we’ll start with the basic modal language:

Definition (Epistemic Language)The (single-agent) epistemic language L is defined inductively by

j ::= p | ¬j | (j ^ j) | K j,

where p 2 At.

For simplicity in this talk, sometimes I will state results for the flatfragment in which only propositional formulas occur in the scope of K .

Semantics

We will consider three semantics for the K operator in RA models:

Semantics

We will consider three semantics for the K operator in RA models:

I C-semantics for Cartesian. C-semantics is not intended to captureDescartes’ view of knowledge. Rather, it is supposed to reflect ahigh standard for the truth of knowledge claims—knowledge requiresruling out all possibilities of error, however remote—in the spirit ofDescartes’ worries about error in the First Meditation.

Semantics

I C-semantics for Cartesian.

I D-semantics is one way (but not the only way or even the best way)of understanding Dretske’s (1981) RA theory, using Heller’s (1989,1999) picture of relevance orderings over possibilities.

Semantics

I C-semantics for Cartesian.

I D-semantics is one way (but not the only way or even the best way)of understanding Dretske’s (1981) RA theory, using Heller’s (1989,1999) picture of relevance orderings over possibilities.

I L-semantics is for Lewis’s (1999) RA theory (for a fixed context).

Semantics

Definition (Truth in an RA Model)Given an RA model M =

W ,_,�,Vi

, world w 2 W , formula j inthe epistemic language, and x 2 {c , d , l}, we define M,w ✏

j andJjKM

v 2 W | M, v ✏x

as follows:

M,w ✏x

p i↵ w 2 V (p);M,w ✏

¬j i↵ M,w 2x

j;M,w ✏

(j ^ y) i↵ M,w ✏x

j and M,w ✏X

M,w ✏c

K j i↵ 8v 2 JjKc

: w 6_ v ;M,w ✏

K j i↵ 8v 2 Min�

�JjK

�: w 6_ v ;

M,w ✏l

(W ) \ JjKl

: w 6_ v ,

where for S ✓ W , S = {v 2 W | v 62 S}.

C-semantics and the Problem of Skepticism

M,w ✏c

K j i↵ 8v 2 JjKc

: w 6_ v .

w1 c 0

w1 c , x

M,w1 2c

C-semantics looks like the standard epistemic semantics in the tradition ofHintikka (1962); but now that we are explicitly representing relevance, itformalizes the infallibilist idea that all possibilities of error, however remoteand far-fetched, must be eliminated—a view that leads to skepticism.

L-semantics and Non-skepticism

M,w ✏l

(W ) \ JjKl

: w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏l

L-semantics avoids skepticism by not requiring the elimination of everyerror possibility, but only those in the set Min

(W ) of relevant worlds.

L-semantics and Non-skepticism

M,w ✏l

(W ) \ JjKl

: w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏l

L-semantics avoids skepticism by not requiring the elimination of everyerror possibility, but only those in the set Min

(W ) of relevant worlds.

L-semantics and the Problem of Vacuous Knowledge

M,w ✏l

(W ) \ JjKl

: w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏l

With L-semantics, one can know ¬x even if x is possible and one has noteliminated any x-possibilities—even if one has not eliminated any possibil-ities at all. This would be knowledge of contingent empirical truths despitenot having empirically eliminated any possibilities—vacuous knowledge.

D-semantics and Closure

M,w ✏d

K j i↵ for all v 2 Min�

), w 6_ v .

w1 c 0

w1 c , x

D-semantics avoids the skepticism of C-semantics, since knowing j re-quires only that the closest ¬j-worlds be eliminated, and avoids the vac-uous knowledge of L-semantics, since if ¬j is possible there will be some¬j-world(s) that must be eliminated. It also leads to denial of closure.

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏d

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏d

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏d

Kc M,w1 ✏d

K (c ! ¬x)

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 ✏d

Kc M,w1 ✏d

K (c ! ¬x) M,w1 2d

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

Kc ^K (c ! ¬x) ! K¬x

D-Semantics vs. L-Semantics

M,w ✏d

): w 6_ v .

M,w ✏l

(W ) \ JjKl

: w 6_ v .

w1 c 0

w1 c , x

Fact (Known Implication)K j ^K (j ! y) ! Ky is C/L-valid but not D-valid.

In the terminology of Dretske (1970), the knowledge operator inD-semantics is not a fully penetrating sentential operator.

Fred Dretske. 1970. “Epistemic Operators.”

Yet Dretske wants knowledge to be semi-penetrating: “it seems to mefairly obvious that if someone knows that P and Q, he thereby knowsthat Q” and “If he knows that P is the case, he knows that P or Q is thecase” (1009). This is the “trivial side” of Dretske’s thesis.

However, if we understand the RA theory according to D-semantics,based on Heller’s picture, then even these closure principles fail.

D-Semantics and the Problem of Containment

While some welcome the failure of closure under known implication,D-semantics faces a problem of containment: closure failures spread,and they spread to where no one wants them.

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

K (c ^ ¬x) ! K¬x

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

K (c ^ ¬x) ! K¬xM,w1 ✏

K (c ^ ¬x)

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

K (c ^ ¬x) ! K¬xM,w1 ✏

K (c ^ ¬x)M,w1 2

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

Kc ! K (c _ ¬x)

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

Kc ! K (c _ ¬x)M,w1 ✏

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

M,w1 2d

Kc ! K (c _ ¬x)M,w1 ✏

KcM,w1 2

K (c _ ¬x)

M,w ✏d

): w 6_ v .

w1 c 0

w1 c , x

Fact (Distribution and Addition)K (j ^ y) ! K j and K j ! K (j _ y) are C/L-valid, but not D-valid.

A Dretskean Dilemma

This is only the tip of the iceberg . . . Yet it already points to a dilemma:

I If we understand RA theory according to D-semantics, then K is notsemi-penetrating, contrary to the “trivial side” of Dretske’s thesis.

I If we understand the theory according to L-semantics, then K isfully-penetrating, contrary to the non-trivial side of Dretske’s thesis;and we have the problem of vacuous knowledge.

It is di�cult to escape this dilemma while retaining the picture “of worldsarranged around the actual world in order of similarity, with those that aretoo far away from the actual world being irrelevant” (Heller 1999, 119).

A Dretskean Dilemma

I If we understand the theory according to L-semantics, then K isfully-penetrating, contrary to the non-trivial side of Dretske’s thesis;

and we have the problem of vacuous knowledge.

A Dretskean Dilemma

Comparison with Basic Relational Semantics

Recall that in basic epistemic models M = hW ,R ,V i, R is a binaryrelation on W , required to be at least reflexive. We take wRv to meanthat v is epistemically accessible from w , in the sense that everything theagent knows in w is true in v , which is clearly a reflexive relation.

The truth clause for K j is then given by:

M,w ✏ K j i↵ 8v 2 R(w) : M, v ✏ j,

where R(w) = {v 2 W | wRv}.

Let’s compare basic epistemic semantics with RA semantics (fromKnowing What Follows: Epistemic Closure and Epistemic Logic, §2.A). . .

M,w ✏ K j i↵ 8v 2 R(w) : M, v ✏ j,

Defining Accessibility

To do so, we’ll define epistemic accessibility relations within RA models.

Given an RA model M =h

W ,_,�,Vi

, we define a derived epistemicaccessibility relation R

on W as follows:

v i↵ 8j : if M,w ✏x

K j, then M, v ✏x

Two important observations to be made about this definition.

1. As we’ll see, the Rd

and Rl

relations may be di↵erent than the _relation; hence the set of epistemically accessible worlds and theset of uneliminated worlds may be distinct;

2. The basic relational semantics applied to Rd

M,w ✏ K j i↵ 8v 2 Rd

(w) : M, v ✏ j,

may give di↵erence truth values than D-semantics,

M,w ✏d

�JjK

�: w 6_ v .

W ,_,�,Vi

on W as follows:

K j, then M, v ✏x

and Rl

(w) : M, v ✏ j,

M,w ✏d

�JjK

�: w 6_ v .

W ,_,�,Vi

on W as follows:

K j, then M, v ✏x

and Rl

(w) : M, v ✏ j,

M,w ✏d

�JjK

�: w 6_ v .

W ,_,�,Vi

on W as follows:

K j, then M, v ✏x

and Rl

(w) : M, v ✏ j,

M,w ✏d

�JjK

�: w 6_ v .

For example, in the model M below, we have w1 _ w3 but not w1Ed

w3or w1E

w3, since M,w1 ✏d ,l Kc but M,w3 2

d ,l c .

Hence for the RA theory, there can be uneliminated possibilities that arenot epistemically accessible. One can also check that for all v 2 W suchthat w1E

v , M, v ✏ ¬x . Hence M,w1 ✏ K¬x according to basicrelational semantics. Yet as we have seen, M,w1 2

K¬x .

w1 c 0

w1 c , x

For example, in the model M below, we have w1 _ w3 but not w1Ed

w3or w1E

w3, since M,w1 ✏d ,l Kc but M,w3 2

d ,l c .

Hence for the RA theory, there can be uneliminated possibilities that arenot epistemically accessible. One can also check that for all v 2 W suchthat w1E

v , M, v ✏ ¬x . Hence M,w1 ✏ K¬x according to basicrelational semantics. Yet as we have seen, M,w1 2

K¬x .

w1 c 0

w1 c , x

Live Possibilities

Say that v is epistemically live for the agent in w i↵ she does not know inw that possibility v does not obtain, where this is understood as follows.

Let’s assume that we are dealing with RA models in which each world uis uniquely definable by a formula j

of the epistemic language.

W ,_,�,Vi

, we can define an epistemicliveness relation L

on W as follows:

v i↵ M,w 2x

It may seem that the live worlds should be exactly the accessible worlds orperhaps the uneliminated worlds. In fact, things are more interesting. . .

Live Possibilities

W ,_,�,Vi

on W as follows:

v i↵ M,w 2x

Live Possibilities

W ,_,�,Vi

on W as follows:

v i↵ M,w 2x

Proposition (Comparing Accessible, Live, and Uneliminated)

1. For all RA models, Ed

✓ Ld

= _, but there are some with Ed

2. For all RA models, El

✓ _, but there are some with Ll

Thus, if we go beyond basic epistemic logic as we have, then we mustkeep apart three di↵erent notions that are easily conflated.

That Ld

= _ shows that a D-semantical theorist can take eliminating apossibility to be (extensionally) equivalent to knowing that the possibilitydoes not obtain. However, allowing E

means allowing that in wan agent can know something that is false in v (so v is not accessible)without knowing that v does not obtain (so v is live/uneliminated).

That Ed

is another way of seeing closure failure. . .

✓ Ld

That Ld

That Ed

✓ Ld

That Ld

That Ed

Another View on Closure Failure

w1 c 0

w1 c , x

That Ed

is another way of seeing closure failure: at the pointedmodel M,w1 above, according to D-semantics student A knows c , whichis false at w3, but she does not know that w3 does not obtain, becauseshe does not know ¬(x ^ ¬c), and (x ^ ¬x) uniquely defines w3.

✓ Ld

is another expression of closure holding in L-semantics.

However, since the semantics allows Ll

( _, an L-semantical theoristcannot take eliminating a possibility to be equivalent to knowing that thepossibility does not obtain. Pryor (“Highlights of Recent Epistemology”)also observes that such an equivalence is not available to RA theoristswho wish to maintain closure. They require an independent notion ofeliminating possibility, such as Lewis’s notion involving perceptualexperience and memory or an alternative notion that Pryor suggests.

✓ Ld

is another expression of closure holding in L-semantics.

However, since the semantics allows Ll

( _, an L-semantical theoristcannot take eliminating a possibility to be equivalent to knowing that thepossibility does not obtain. Pryor (“Highlights of Recent Epistemology”)also observes that such an equivalence is not available to RA theoristswho wish to maintain closure. They require an independent notion ofeliminating possibility, such as Lewis’s notion involving perceptualexperience and memory or an alternative notion that Pryor suggests.

The Dynamics of Context

Context Change

Although we won’t go into the topic here, one can use tools fromdynamic epistemic logic to analyze the e↵ects of context changes, whichare modeled by changes in the relevance orderings �

See “Epistemic Logic, Relevant Alternatives, and the Dynamics of Context.”

w1 c 0

w1 c , x

w1 c 0

w1 c , x

Figure : result of context change by raising the possibility of disease x

Context Change

Although we won’t go into the topic here, one can use tools fromdynamic epistemic logic to analyze the e↵ects of context changes, whichare modeled by changes in the relevance orderings �

See “Epistemic Logic, Relevant Alternatives, and the Dynamics of Context.”

w1 c 0

w1 c , x

w1 c 0

w1 c , x

Figure : result of context change by raising the possibility of disease x

Context Change

w1 c 0

w1 c , x

w1 c 0

w1 c , x

w1 c 0

w1 c , x

Figure : di↵erent results of context change by " x and & x

Formalization: Counterfactuals and Beliefs

Subjunctivist Theories

Models similar to RA models can be used to formalize another veryinfluential family of “subjunctivist” theories of knowledge. . .

The reason they’re called “subjunctivist” is that their key conditions canbe expressed with subjunctive conditionals (or counterfactuals) of theform “if it were that A, then it would be that B” or something similar.

Subjunctivist Theories

Models similar to RA models can be used to formalize another veryinfluential family of “subjunctivist” theories of knowledge. . .

The reason they’re called “subjunctivist” is that their key conditions canbe expressed with subjunctive conditionals (or counterfactuals) of theform “if it were that A, then it would be that B” or something similar.

Counterfactual Belief Models

w1 c 0

<w1 c , x

A (single-agent) CB model is a tuple M = hW ,D,6,V i, where:

I W , 6, and V are defined as W , �, and V are in RA models.

w1 c 0

<w1 c , x

I W , 6, and V are defined as W , �, and V are in RA models.

We can think of 6w

either as a relation of comparative relevance(for RA theories) or of comparative similarity for counterfactuals.

w1 c 0

<w1 c , x

I D is a serial binary (“doxastic accessibility”) relation on W .

wDv means that everything the agent believes in w is true in v .

Subjunctivist Semantics

Now we’ll define three more semantics for K :

I H-semantics for a simple version of Heller’s RA theory.I N-semantics for a simple version of Nozick’s tracking theory.I S-semantics for a simple version of Sosa’s safety theory.

These simplify the full philosophical theories in various ways. See“Epistemic Closure and Epistemic Logic I,” JPL, for discussion.

Subjunctivist Conditions

Let Bj stand for “the agent believes that j.”

Here are three key subjunctivist conditions:

I sensitivity: closest ¬j-worlds are ¬Bj-worlds (i.e., if j were false,the agent would not believe it—this is a subjunctive conditional);

I adherence: among close worlds, all j-worlds are Bj-worlds;

I safety: among close worlds, all ¬j-worlds are ¬Bj-worlds.

Semantics

To succinctly state the truth clauses for K , we extend our language witha belief operator B , but our results will be for the epistemic language.

Definition (Truth in a CB Model)Given a well-founded CB model M =

W ,D,6,Vi

with w 2 W and jin the epistemic-doxastic language, define M,w ✏

j as follows:

M,w ✏x

Bj i↵ 8v 2 W : if wDv then M, v ✏x

M,w ✏h

K j i↵ M,w ✏h

Bj and(sensitivity) 8v 2 Min6

�JjK

�: M, v 2

M,w ✏n

K j i↵ M,w ✏n

�JjK

�: M, v 2

Bj,(adherence) 8v 2 Min6

�JjK

�: M, v ✏

M,w ✏s

K j i↵ M,w ✏s

Bj and(safety) 8v 2 Min6

�JBjK

�: M, v ✏

Semantics

W ,D,6,Vi

j as follows:

M,w ✏x

M,w ✏h

K j i↵ M,w ✏h

�JjK

�: M, v 2

M,w ✏n

K j i↵ M,w ✏n

�JjK

�: M, v 2

�JjK

�: M, v ✏

M,w ✏s

K j i↵ M,w ✏s

�JBjK

�: M, v ✏

Semantics

W ,D,6,Vi

j as follows:

M,w ✏x

M,w ✏h

K j i↵ M,w ✏h

�JjK

�: M, v 2

M,w ✏n

K j i↵ M,w ✏n

�JjK

�: M, v 2

�JjK

�: M, v ✏

M,w ✏s

K j i↵ M,w ✏s

�JBjK

�: M, v ✏

Semantics

W ,D,6,Vi

j as follows:

M,w ✏x

M,w ✏h

K j i↵ M,w ✏h

�JjK

�: M, v 2

M,w ✏n

K j i↵ M,w ✏n

�JjK

�: M, v 2

�JjK

�: M, v ✏

M,w ✏s

K j i↵ M,w ✏s

�JBjK

�: M, v ✏

Semantics

W ,D,6,Vi

j as follows:

M,w ✏x

M,w ✏h

K j i↵ M,w ✏h

�JjK

�: M, v 2

M,w ✏n

K j i↵ M,w ✏n

�JjK

�: M, v 2

�JjK

�: M, v ✏

M,w ✏s

K j i↵ M,w ✏s

�JBjK

�: M, v ✏

The model below is for our medical diagnosis example:

w1 c 0

<w1 c , x

w1 c 0

<w1 c , x

That the only arrow from w1 goes to itself indicates that in w1, studentA believes that the actual world is w1, where the patient has c and not x .(Note that we don’t require that D be functional, although it happens tobe in this model.) Hence M,w1 ✏ B(c ^ ¬x).

w1 c 0

<w1 c , x

That the only arrow from w3 goes to w1 indicates that in w3, A believesthat w1 is actual; since w3 is the closest (to w1) x-world, we take this tomean that if the patient’s condition were x , A would still believe it was cand not x (because A did not run any of the tests necessary to detect x).

w1 c 0

<w1 c , x

That the only arrow from w3 goes to w1 indicates that in w3, A believesthat w1 is actual; since w3 is the closest (to w1) x-world, we take this tomean that if the patient’s condition were x , A would still believe it was cand not x (because A did not run any of the tests necessary to detect x).

Hence M,w1 2h,n K¬x , because sensitivity is violated.

The CB model below is for our medical diagnosis example:

w1 c 0

<w1 c , x

However, one can check that M,w1 ✏h,n Kc , because sensitivity is sat-

isfied (in the closest world where c is false, w2, the agent does not believec) for H-semantics

w1 c 0

<w1 c , x

However, one can check that M,w1 ✏h,n Kc , because sensitivity is satis-

fied (in the closest world where c is false, w2, the agent does not believe c)for H-semantics and adherence is also satisfied (in all of the closest worldswhere c is true, which is just w1, the agent believes c) for N-semantics.

w1 c 0

<w1 c , x

If we draw the model for student B, we replace the arrow from w2 to w2by one from w2 to w1, reflecting that if the patient’s condition were c 0, Bwould still believe it was c (because B made the diagnosis of c after onlya physical exam, and c and c 0 have the same visible symptoms).

w1 c 0

<w1 c , x

w1 c 0

<w1 c , x

Hence M

0,w1 2h,n Kc , where M

0 has w2Dw1 instead of w2Dw2.

Semantics

W ,D,6,Vi

j as follows:

M,w ✏x

M,w ✏h

K j i↵ M,w ✏h

�JjK

�: M, v 2

M,w ✏n

K j i↵ M,w ✏n

�JjK

�: M, v 2

�JjK

�: M, v ✏

M,w ✏s

K j i↵ M,w ✏s

�JBjK

�: M, v ✏

w1 c 0

<w1 c , x

Finally, let’s consider the model from the perspective of S-semantics.

w1 c 0

<w1 c , x

Observe that M,w1✏s

K¬x , because student A believes ¬x in w1 and atthe closest worlds to w1, namely w1 and w2, ¬x is true. Therefore, Asafely believes ¬x in w1.

w1 c 0

<w1 c , x

Observe that M,w1 ✏s

K¬x , because student A believes ¬x in w1 andat the closest worlds to w1, namely w1 and w2, ¬x is true. Therefore, Asafely believes ¬x in w1.

Similarly M,w1 ✏s

Kc , because A safely believes c in w1.

w1 c 0

<w1 c , x

Observe that M,w1 ✏s

K¬x , because student A believes ¬x in w1 andat the closest worlds to w1, namely w1 and w2, ¬x is true. Therefore, Asafely believes ¬x in w1.

Similarly M,w1 ✏s

Kc , because A safely believes c in w1.

Yet if we add the arrow from w2 to w1 for B, one can check that B doesnot safely believe c at w1, so M

0,w1 2s

A story similar to that told for the RA theories applies to thesubjunctivist theories:

I S-semantics avoids the skepticism of C-semantics, but allowsvacuous knowledge.

I H/N-semantics avoid the skepticism of C-semantics and the vacuousknowledge of S-semantics.

I But all of the closure principles shown to fail for D-semantics alsofail for H/N/S-semantics, so they face the containment problem.

It’s now time to go beyond case-by-case assessment of closure principlesto results of a more general nature.

A story similar to that told for the RA theories applies to thesubjunctivist theories:

I S-semantics avoids the skepticism of C-semantics, but allowsvacuous knowledge.

I H/N-semantics avoid the skepticism of C-semantics and the vacuousknowledge of S-semantics.

I But all of the closure principles shown to fail for D-semantics alsofail for H/N/S-semantics, so they face the containment problem.

It’s now time to go beyond case-by-case assessment of closure principlesto results of a more general nature.

Formalization: Results

Toward the Closure Theorem

Definition (Closure Principle)Given (possibly empty) sequences of propositional formulas j1, . . . , j

and y1, . . . ,ym

and a propositional conjunction j0, we use the notation

cn,m := j0 ^K j1 ^ · · · ^K j

! Ky1 _ · · · _Kym

Call such a cn,m a (flat) closure principle.

A closure principle states: if the agent knows each of j1 through jn

(andthe world satisfies j0), then she knows at least one of y1 through y

and y1, . . . ,ym

cn,m := j0 ^K j1 ^ · · · ^K j

! Ky1 _ · · · _Kym

A closure principle states: if the agent knows each of j1 through jn

(andthe world satisfies j0), then she knows at least one of y1 through y

By propositional logic, any flat formula is equivalent to a conjunction ofclosure principles.

and y1, . . . ,ym

cn,m := j0 ^K j1 ^ · · · ^K j

! Ky1 _ · · · _Kym

Our question is: which closure principles are valid according to the philo-sophical theories of knowledge we have formalized?

and y1, . . . ,ym

cn,m := j0 ^K j1 ^ · · · ^K j

! Ky1 _ · · · _Kym

Our question is: which closure principles are valid according to the philo-sophical theories of knowledge we have formalized?

The answer is provided by the “Closure Theorem” of

Epistemic Closure and Epistemic Logic I: Relevant Alternatives and Subjunctivism.

Journal of Philosophical Logic, 2014.

and y1, . . . ,ym

cn,m := j0 ^K j1 ^ · · · ^K j

! Ky1 _ · · · _Kym

Despite the di↵erences between the relevant alternatives and subjunctivisttheories of knowledge as formalized by D/H/N/S-semantics, shortly theClosure Theorem will provide a unifying perspective:

I the valid closure principles are essentially the same for these theories.

Theorem (Closure Theorem - Simplified Version)For a flat c

n,m := j0 ^K j1 ^ · · · ^K jn

! Ky1 _ · · · _Kym

1. cn,m is C/L-valid over RA models i↵

(a) j0 ^ · · · ^ jn

! ? is valid or

(b) for some y 2 {y1, . . . ,ym

j1 ^ · · · ^ jn

! y is valid.

2. cn,m is D/H/N/S-valid over RA/CB models i↵ (a) or

(c) for some F ✓ {j1, . . . , jn

} and y 2 {y1, . . . ,ym

j2Fj $ y is valid.

n,m := j0 ^K j1 ^ · · · ^K jn

! Ky1 _ · · · _Kym

(a) j0 ^ · · · ^ jn

! ? is valid or

(b) for some y 2 {y1, . . . ,ym

j1 ^ · · · ^ jn

! y is valid.

2. cn,m is D/H/N/S-valid over RA/CB models i↵ (a) or

(c) for some F ✓ {j1, . . . , jn

} and y 2 {y1, . . . ,ym

j2Fj $ y is valid.

n,m := j0 ^K j1 ^ · · · ^K jn

! Ky1 _ · · · _Kym

(a) j0 ^ · · · ^ jn

! ? is valid or

(b) for some y 2 {y1, . . . ,ym

j1 ^ · · · ^ jn

! y is valid.

3. cn,m is D-valid over total RA models i↵ (a) or

(d) for some F ✓ {j1, . . . , jn

}, nonempty Y ✓ {y1, . . . ,ym

j2Fj $

y2Yy is valid;

beyond worlds and accessibility - uc berkeley · beyond worlds and accessibility part 1 wesley...

Documents