Boudewijn de Bruin - Epistemic Logic and Epistemology
Published as "Epistemic Logic and Epistemology." New Waves in Epistemology. Ed. Vincent F. Hendricks and Duncan Pritchard. Basingstoke: Palgrave Macmillan, 2008. 106-136.
This paper is about the fun and troubles of connecting epistemic logic and epistemology. Epistemic logic was developed by philosophers such as Jaakko Hintikka, Edward John Lemmon and Georg Henrik von Wright with rather firm convictions about its relevance for the theory of knowledge (Hintikka 1962; Lemmon 1967; Von Wright 1951). But it seems that the two have parted soon after they first met. Epistemologists are concerned with such issues as scepticism and the conceptual analysis of knowledge, whereas logicians tend to focus on such mathematical intricacies as the completeness of their proof systems. Recently, Johan van Benthem and Vincent Hendricks have argued independently for reconsidering the connection between epistemology and epistemic logic (Van Benthem 1991; 2004; 2006; Hendricks 2003; 2005). This paper, although critical at various places, is an attempt to strengthen this case.
A very brief introduction (which one may wish to skip) reviews the formalism. Then, I give a survey of the most important applications of epistemic logic in epistemology. I will show how it is used in the history of philosophy (Mark Steiner's reconstruction of Descartes' sceptical argument), in solutions to Moore's paradox (Hintikka), in discussions about the relation between knowledge and belief (Lenzen) and in an alleged refutation of verificationism (Fitch) and I will examine an early argument about the (im)possibility of epistemic logic (Max Hocutt). Subsequently, I deal with interpretive questions about epistemic logic that, although implicitly, already appeared in the first section. I contend that a conception of epistemic logic as a theory of knowledge assertions is incoherent and I will argue that it does not make sense to adopt a normative interpretation of epistemic logic. Finally, I show ways to extend epistemic logic with other branches of philosophical logic so as to make it useful for some epistemological questions. Conditional logics and logics of public announcement are used to understand causal theories of knowledge and versions of reliabilism. Temporal logic helps understand some dynamic aspects of knowledge as well as the verificationist thesis.
The main motivation behind this paper is to emphasize connections between the theory of knowledge and epistemic logic. In addition, however, it is, to some extent, meant to show ways to follow Dana Scott's paper from 1970 on 'Advice on Modal Logic':
'Here is what I consider one of the biggest mistakes of all in modal logic: concentration on a system with just one modal operator. The only way to have any philosophically significant results in deontic or epistemic logic is to combine these operators with: Tense operators (otherwise how can you formulate principles of change?); the logical operators (otherwise how can you compare the relative with the absolute?); the operators like historical or physical necessity (otherwise how can you relate the agent to his environment?); and so on and so on.' (1970: 143)
It also follows Vincent Hendricks when he says:
'It seems that when determining the strength of epistemic operators in a way pertinent to both epistemic logic and epistemology alike---and thus construct a logical epistemology we have to consider at least 3 dimensions besides the agent involved . . .' (2003: 491)
Hendricks sets apart an axiomatic dimension of particular modal axioms (the K-axiom, the T-axiom and so on), a temporal dimension of synchronic and diachronic interpretations of the logical formalism and a linguistic dimension (first person, third person). My paper is an excursion through all dimensions. Sometimes I argue that a dimension collapses, sometimes I add precision to the definition of the dimension and sometimes I even add dimensions: the dimension of facts versus values (normative versus descriptive conceptions of epistemic logic), the dimension of epistemic states versus epistemic assertions and, to some extent, the dimension of explicit versus implicit knowledge.
The logical symbols of (propositional) epistemic logic (the logic of knowledge) include: ~ (negation), & (conjunction), v (disjunction), -> (implication) and <-> (equivalence). The epistemic operator Ka is used with intended interpretation 'a knows that . . .' The reference to the precise epistemic subject a will sometimes be left out. The non-logical symbols of some specific epistemic logic depend, of course, on the object of knowledge with which the logic is concerned. Proposition letters p, q and so on, may stand for all kinds of different statements. The axioms of epistemic logic are listed here:
[prop] all propositional tautologies
[K] Ka (&phi -> &psi) -> (Ka &phi -> Ka &psi)
[T] Ka &phi -> &phi
 Ka &phi -> Ka Ka &phi
 ~ Ka &phi -> Ka ~ Ka &phi.
The K-axiom, named after Saul Kripke, is the standard axiom for any normal modal logic and expresses a closure condition on knowledge often associated with 'deductive cogency'. The T-axiom stands for 'truth': what is known has to be true; that is, knowledge has to be 'veridical'. Axiom 4 (also called the 'KK-principle') and axiom 5 (also called the 'E-axiom', because of the corresponding 'Euclidean' frames) date back to C.I. Lewis's (1912) writings on modal logic The 4-axiom is associated with 'self-awareness' or 'positive introspection', and the 5-axiom with 'wisdom' or 'negative introspection'. The two proof rules of epistemic logic are modus ponens and (epistemic) necessitation:
[MP] if |- &phi -> &psi, |- &phi, then |- &psi
[Nec]if |- &phi, then |- Ka &phi.
There is, of course, some discussion in the literature about what is the true proof system (that is, set of axioms) for epistemic logic. All systems include the K-axiom. Some answers about the extra axioms are: the system T (add the truth axiom), the system S4 (system T plus the 4-axiom), the system S5 (the system S4 plus the 5-axiom). In this paper, all three systems will be used. It will be clear from the discussions about the axioms where each is being used.
Doxastic logic (the logic of belief as opposed to the logic of knowledge) has the same set of logical symbols as epistemic logic, except that the knowledge operator and its dual are replaced by their doxastic versions. That is, we have an operator Ba with intended interpretation 'a believes . . .' The non-logical symbols of doxastic logic are the same as those of epistemic logic. The axioms of doxastic logic are listed here:
[prop] all propositional tautologies
[K] Ba (&phi -> &psi) -> (Ba &phi -> Ba &psi)
[D] Ba &phi -> ~ Ba ~ &phi
 Ba &phi -> Ba Ba &phi
 ~ Ba &phi -> Ba ~ Ba &phi.
The T-axiom disappears and the D-axiom (for 'deontic', because it is an axiom prominent in deontic logic) replaces it. It is an axiom about 'consistency' of beliefs. The two proof rules of doxastic logic are modus ponens and (doxastic) necessitation:
[MP] if |- &phi -> &psi, |- &phi, then |- &psi
[Nec] if |- &phi, then |- Ba &phi.
There is, of course, some discussion in the literature about what is the true proof system for doxastic logic. All systems include the K-axiom. Some answers about the extra axioms are: the system K45 (add the 4-axiom and the 5-axiom), the system KD45 (the system K45 plus the D-axiom). In this paper, both systems will be used. It will be clear from the discussions about the axioms where each is being used.
Combining Epistemic and Doxastic Logic
To combine some logic for knowledge and some logic for belief, a language is used in which the Ka operator as well as the Ba operator occur. The epistemic axioms from the particular logic for knowledge and the doxastic axioms from the particular logic for belief are, of course, included and the rules of modus ponens, epistemic necessitation and doxastic necessitation are available, too. In addition, to make the resulting combination interesting, interrelation axioms appear. The following axioms are discussed in this paper:
Ka &phi -> Ba &phi
Ba &phi -> Ka Ba &phi
Ba &phi -> Ba Ka &phi.
Steiner, or What Cartesian Scepticism Presupposes
One of the first to put to work the formalism of epistemic logic in (the history of) epistemology is Mark Steiner (1979). In a short, but illuminating paper, Steiner examines Descartes' argument, from the Meditations, that, since he cannot be sure he is not dreaming, he does not know that he is seated by the fire. Let me review Steiner's analysis. What Descartes wishes to establish is:
I do not know that I am seated by the fire.
Or, in the formalism of epistemic logic where KI. . . stands for 'I know. . .' and p for 'I am seated by the fire':
Descartes, Steiner claims, arrives at this conclusion via the following two premises. First:
If I am dreaming, then I do not know that I am seated by the fire.
That is, with q for 'I am dreaming':
q -> ~ KI p. (2)
This is self-evident because of 'the very nature of dreaming' (1979: 39).
Were Descartes dreaming he was seated by the fire, he would still not know that to be the case (even if, in fact, he were seated by the fire). Or so Steiner claims for Descartes. Second:
I do not know that I am not dreaming.
Which in the formalism is:
~ KI ~ q. (3)
This is not self-evident, but for the sake of the argument we may take it as 'definitely proved' (1979: 39). For Descartes observes that there are no certain indications to distinguish the state of sleeping from the state of being awake.
It is plain that Descartes cannot prove 1 from 2 and 3 in ordinary propositional logic. The exegetical task Steiner sets himself is to find out what extra premises, or what kind of alternative logical system, Descartes could have had in mind when he wrote his Meditations.
(i) The first idea is simply to add the proposition q (that is, 'I am dreaming') to the set of premises. Modus ponens would indeed lead to the desired conclusion, without using 3 and without transcending the realm of propositional logic. But as a representation of Descartes this will not do, since he does not, of course, claim to be dreaming.
(ii) The second idea is to use the full force of epistemic logic. To anticipate, the main trick of the argument is to transform 2 in such a way as to make it fit for playing a role in a derivation of 1 from 3. To begin with, a proof rule specific to epistemic logic is needed: the rule of (epistemic) necessitation. This rule allows us to prove KI &phi, once we have proved &phi, in the same way as the rule of modus ponens guides the construction of a proof of &psi from proofs of &phi -> &psi and &phi. The intuition behind the rule of necessitation is that the epistemic subject knows all provable propositions. Of course, the plausibility of this rule may be questioned. But that is none of our business in this survey.
By contraposition the 'self-evident' 2 is equivalent to:
KI p -> ~ q. (4)
Applying the rule of necessitation to 4 yields:
KI (KI p -> ~ q). (5)
(The rule of necessitation can only be brought into play if 2 or 4 is an axiom or a provable sentence. So, in fact, the logic we are working with is an extension of epistemic logic with some axioms about dreaming and knowledge. In other words, this is not true for arbitrary proposition letters.) This helps quite a bit, but we do need one or two other principles from epistemic logic. First, we need to transform the knowledge of some implication as embodied in 5 into an implication of knowledge statements. To that end, the so-called K-axiom is put to work. It says, applied to this specific case:
KI (KI p -> ~ q) -> (KI KI p -> KI ~ q). (6)
Combining 5 and 6 with modus ponens makes:
KI KI p -> KI ~ q. (7)
That is, if one knows that one knows that one is seated by the fire, then one knows that one is not dreaming. The second principle of epistemic logic Steiner uses to represent Descartes' argument is the KK-principle. It says that something known is known to be known. Again and for all principles to be introduced, we have to set aside suspicions about the plausibility for a moment here. The special case of the axiom needed is:
KI p -> KI KI p. (8)
Combining 7 and 8 yields:
KI p -> KI ~ q. (9)
From which, by contraposition, we get:~ KI ~ q -> ~ KI p. (10)
It is now straightforward to derive the demonstrandum 1 from 3 and 10 via modus ponens.
(iii) Steiner has some worries about the plausibility of his reconstruction of Descartes' argument for those who believe in a causal theory of knowledge because the KK-principle would not convince its followers (Cf. Gordon 1979). (We will see Steiner's argument for that claim later.) For those who wish to avoid the KK-principle, Steiner gives an alternative based on a weaker principle that reads as follows, for any proposition &phi:
If I am committed to ~ KI &phi, then it is irrational for me to assert &phi.
(Some may find the reference to rationality in this principle and the next problematic: it may, for instance, be quite rational to lie in certain circumstances.) We resume the thread of the derivation at:
KI KI p -> KI ~ q. (7)
Instead of applying the KK-principle to obtain 9 and so on, we take the contraposition of 7, that is:
~ KI ~ q -> ~ KI KI p.
from which, with premise 3, we obtain:
~ KI KI p.
Steiner goes on to show that Descartes is committed to ~ KI KI p because of the very proof just given and that, on the basis of the earlier assertability principle, it is irrational for Descartes to assert KI p . However, it might be argued that it is not the assertability of KI p , but that of p , which is at stake here. Applying a stronger principle:
If it is irrational for me to assert KI &phi, then it is irrational for me to assert &phi
allows Steiner to block the assertion of p, too: it is irrational for Descartes to assert p.
(iv) George Schlesinger (1985) believes that Descartes had a different argument in mind (or, at least, he wishes to make clear that Steiner's reconstruction is not the only plausible one). Recall the premises:
q -> ~ KI p (2)
~ KI ~ q. (3)
While Steiner's idea was to transform 2to function in some derivation in which (besides 3) extra axioms and the rule of necessitation play a role, Schlesinger wishes to transform 3 to function in some derivation in which (besides 2) only one extra axiom and no rule of necessitation play a role. Actually, Schlesinger cannot really be said to 'transform' 3. What he does is to show that Descartes' argumentation behind 3 in fact establishes a different proposition, too, which, then, occurs in the extra axiom. In addition, Schlesinger does not provide us with a derivation of 1, but rather of 11. This means that, at the end, unless Schlesinger disagrees with Steiner about what is really the demonstrandum of Descartes' argument, he still needs an argument for ~ KI KI p -> ~ KI p . In other words, Schlesinger does not give us a way to circumvent the use of either the KK-principle or the principles about rationality of assertion.
Descartes, according to Schlesinger, puts forward proposition 3 on the basis of the fact that his, Descartes, experiences do not give him
'a rational basis for believing with enough confidence that. . . what goes on in his mind is generated by corresponding external factors any more than this is true in the case of a dream.' (1985: 8)
This, says Schlesinger, not only establishes that Descartes does not know he is not dreaming, premise 3, but also that objectively speaking it is not rational for Descartes to accept the proposition that he is not dreaming on the basis of the information he possesses. In Schlesinger's notation:
~ BjI_I ~ q. (12)
The mnemonic is 'justified belief'. An axiom involving this notion now finishes the argument in favour of the demonstrandum 11 almost immediately. Schlesinger defends:
((&phi -> &psi) & KI &phi) -> BjI_I &psi.
Which says that objectively speaking the fact that some proposition is entailed by a known proposition makes it rational for the knower to believe the former (although, in fact, the knower need not be aware of the entailment). With a little bit of logical rewriting we obtain as an instance of this axiom:
((KI p -> ~ q) & ~ BjI_I ~ q) -> ~ KI KI p.
Together with (the contraposition of) 2 and 12, modus ponens gets us at 11. I will make a sudden transition here, as elsewhere, to the next application.
Hintikka, or How to Solve Moore's Paradox
'Moore's problem' (Moore 1912) is the problem how to explain the strange nature of assertions such as:
p but I do not believe that p.
One of the first applications discussed, by Jaakko Hintikka, of the formalism of doxastic and epistemic logic he developed in Knowledge and Belief, is a solution to Moore's problem (Hintikka 1962). An observation preceding the solution is that the strangeness of the assertions disappears if we change it to:
p but he does not believe that p
p but I did not believe that p.
This may be fortunate in the sense that the problematic character of the first assertion does not depend on the logical form of the assertion as such, because in all three assertions the logical form is, one may say:
p & ~ \believe p (13)
Yet it also means that a solution on the basis of doxastic logic has to transcend the surface structure of the assertion. For, indeed, in doxastic (as in epistemic) logic no contradiction can be derived from 13. What Hintikka does, then, is to consider:
BI (p & ~ BI p). (14)
(It will soon become clear why.) This proposition is contradictory. To show this (in a proof-theoretic way quite different from Hintikka's original model-theoretic argument), first observe that from 14 it follows that:
BI p & BI ~ BI p (15)
via an easy derivation using an instance of:
BI (&phi & &psi) -> (BI &phi & BI &psi). (16)
From the first conjunct of 15 we derive, by means of the above mentioned KK-principle (but now, for belief rather than knowledge, more adequately called the BB-principle) BI BI p. Recombining with the untouched second conjunct and pulling back the conjunction under the scope of the doxastic operator, gives:
BI (BI p & ~ BI p). (17)
This, however, amounts to a belief in a contradiction and Hintikka's logic does not allow for such beliefs. The crucial axiom here is the so-called axiom of 'consistency':
BI &phi -> ~ BI ~ &phi.
According to which you do not believe the negation of something you believe. Since, by propositional logic, we can prove BI p v ~ BI p, by applying the rule of doxastic necessitation, we can prove BI (BI p v ~ BI p). Applying the axiom of consistency proves:
~ BI ~(BI p v ~ BI p).
Which (modulo some classical propositional logic) is the negation of 17.
That is all very fine, but how is it supposed to solve Moore's problem? To show why it is strange to assert:
p but I do not believe that p.
Hintikka introduces a principle according to which:
If I assert some proposition &phi, then I should believe, or at least conceivably believe, that &phi.
This makes it easy to finish Hintikka's solution to Moore's problem. If someone asserts the problematic sentence, he---or, of course, she---either does not make a truthful announcement (in which case the contradictory 14 would not hold), or violates this principle.
Lenzen, or Are Knowledge and Belief the Same?
In Steiner's reconstruction of Cartesian scepticism the epistemic operator was used; in Hintikka's treatment of Moore's paradox, the doxastic operator. The difference between the two operators is, of course, that they are guided by different axioms. Apart from the KK-principle discussed in the previous sections, knowledge has to be veridical, known propositions have to be true propositions, or formally, the T-axiom:
K &phi -> &phi. (18)
Moreover, one has to know what one does not know, the axioms of 'negative introspection':
~ K &phi -> K ~ K &phi. (19)
(The precise agent does not matter much here, so I leave out the index.) In contrast, for belief we do not assume veridicality (although, of course, beliefs may be true), but we assume the earlier consistency requirement:
B &phi -> ~ B ~ &phi.
It states that if one believes that some proposition is true one cannot simultaneously believe its negation to be true. In addition, we may assume analogues of the KK-principle and negative introspection for belief, too.
Up to now, we have only been working with epistemic or doxastic logics in isolation. What will happen if we combine the two? In order to make such a fusion interesting, we need interrelation principles. One of the most prominent is the principle according to which one believes what one knows:
K &phi -> B &phi. (20)
One step further, one may be taken to know to believe what one believes, or:
B &phi -> K B &phi. (21)
Stronger even, 'Moore's principle' says that if one believes something one believes one knows it:
B &phi -> B K &phi.
But lo and behold, Wolfgang Lenzen (1972) showed a peculiar consequence of this line of reasoning. For in the combined system we can derive the following four implications. Moore's principle gives us:
B &phi -> B K &phi.
B K &phi -> ~ B ~ K &phi.
~ B ~ K &phi -> ~ K ~ K &phi.
~ K ~ K &phi -> K &phi.
B &phi -> K &phi.
Recombining with 20:
K &phi <-> B &phi.
This means that knowledge and belief collapse. Or in grander terms, if we combine knowledge and belief, we cannot distinguish between them any longer. Etrange, n'est-ce pas?
Fitch, or Against Verificationism
A little adaptation of Hintikka's argument can be used to derive a reductio of verificationism. The argument is due to Frederic Fitch (1963). The verificationist thesis can be formalized as:
&phi -> <> K &phi. (23)
It says that every true proposition &phi can be known. The auxiliary (or in the formalism, the <>) is used in the sense of logical possibility; an alethic modality, so to speak. Fitch substitutes for &phi the sentence p & ~ K p arriving at the following instance of the verificationist thesis:
(p & ~ K p) -> <> K(p & ~ K p). (24)
We can rewrite the consequent in three steps to falsum so as to conclude (p & ~ K p) -> falsum. First, it is rephrased as:
<>(K p & K ~ K p)
using an analog of the earlier doxastic principle 16; subsequently, as:
<>(K p & ~ K p)
on a simple application of the principle that what is known is true, principle 18; finally, as:
by means of an argument using the necessitation rule for the alethic modal operator. And, as announced, this means that we have derived:
(p & ~ K p) -> falsum
which is equivalent to:
p -> K p.
Altogether, what we have shown is that in a proof system consisting of the axioms of epistemic logic and the axiom of verificationism (principle 23) we can derive p -> K p for any p; in other words, truth entails knowledge. Since knowledge entails truth by principle 18, at the end for all p it turns out to be true that:
K p <-> p.
Conclusion: knowledge and truth collapse. Or in grander terms, if we combine epistemic logic with verificationism, we cannot distinguish between the truth of a proposition and the fact that a proposition is known or not. Again, etrange, n'est-ce pas?
Hocutt, or Is Epistemic Logic Possible?
Max Hocutt's 1972 paper is an attempt to provide a negative answer to the question whether epistemic logic is possible (Cf. McLane 1979). Although the various critical observations may not all be entirely original (or even correct), the way he sums up and rearranges alleged problems of epistemic logic is quite useful. Hocutt distinguishes two questions: is epistemic logic really a 'logic', and is it really 'epistemic'?
To argue against the claim that epistemic logic is a logic, Hocutt goes back to the well-known problem of referential opacity. In epistemic logic, Hocutt claims, the following principle holds:
(&phi -> &psi) -> (K &phi -> K &psi). (25)
Of which he considers the instance obtained by substituting 'Bugs Bunny is a rabbit' for &phi and 'Bugs Bunny is an Oryctolagus cuniculus' for &psi. Obviously &phi -> &psi is true, Hocutt claims, but an actual epistemic subject 'who knows rabbits but no Latin' may make K &phi -> K &psi false (1972: 435) Hocutt concludes that this shows that the principles from epistemic logic are not true in all possible worlds and that hence they do not constitute a 'logic'.
Hocutt errs. In standard epistemic logic, principle 25 is not provable at all. As one may suspect, putting an epistemic operator in front of its antecedent is of no avail. It is indeed provable in epistemic logic that:
K(&phi -> &psi) -> (K &phi -> K &psi).
And due to the rule of epistemic necessitation any provable implication is provable to be known. So, if the implication about rabbits is provable then so is the implication about knowledge about rabbits. Hocutt's true problem then seems to be that he finds fault with the fact that for epistemic logic all tautologies are known (it is, I believe, doubtful whether the conventions of zoological nomenclature are tautologies at all) rather than with referential opacity.
Against the epithet 'epistemic', Hocutt criticizes the KK-principle. He observes that for an actual epistemic subject it may very well be the case that he knows some proposition without having 'realized self-consciously' that he knows it (1972: 446). And this, Hocutt concludes, shows that epistemic logic, if a logic at all, does not constitute an 'epistemic' logic because it is not about epistemic subjects in an ordinary sense.
In what is probably the most interesting part of the paper, Hocutt sets out to show that epistemic logic is best interpreted as 'a set of criteria for assessing acts of advancing first-person knowledge claims' (1972: 447). He observes that, indeed, there is something strange about assertions such as, say:
I know that p but I do not know that I know that p.
They go against, it may seem, the KK-principle. However, although epistemic logic could function as a tool to study knowledge assertions, we do not need epistemic logic, Hocutt claims, as a set of rules determining the unacceptability of such assertions. For it is already sufficient to have principles such as:
Know what you are talking about
Always tell the truth
to exclude such assertions (1972: 448).
No Logic of Assertions
In the first part of this paper, I presented two applications of epistemic logic showing the inadequacy or unacceptability of particular assertions involving the verb 'to know'. The Moorean assertion:
p but I do not believe that p
was seen to be unacceptable because, as Hintikka argued using doxastic logic, it is either untruthful or it conflicts with the maxim that one only asserts what one, as he describes it, 'at least conceivably believes'. And asserting:
I know that p but I do not know that I know that p
was, Hocutt argued, seen to conflict with the KK-principle of epistemic logic.
Something remarkable happens. Since the second assertion goes against the KK-principle, to show its unacceptability it is sufficient to point out that its direct translation into the language of epistemic logic (that is, the sentence KI p & ~ KI KI p) is inconsistent with epistemic logic. To exclude the first assertion, however, it will not help to reflect deeply on its translation (that is, p & ~ BI p) as it is not inconsistent with epistemic logic.
Let us distinguish two logics. A logic of epistemic assertions would deal with direct translations of knowledge assertions. Acceptability of the assertion would be tested by checking for consistency (as in the example 'I know that p but I do not know that I know that p'). A logic of epistemic states, in contrast, would be concerned with structural or conceptual properties of knowledge. Possible epistemic states would have to have consistent translations in that logic. The comparison of the two assertions from above shows that epistemic or doxastic logic (the kind of logic developed by Hintikka and others) is not a logic of assertions because it fails to exclude the Moorean sentence on the basis of consistency considerations. I will first argue that there is no need for a logic of assertions to develop a theory about the acceptability of doxastic or epistemic assertions. Subsequently, I will examine whether it is nonetheless possible.
No logic of assertions is needed. Doxastic and epistemic logic together with maxims of assertion do the job. For instance, the unacceptability of the statement:
I know that p but I do not know that I know that p
can be defended on the basis of a similar kind of maxim (be it a bit stronger and perhaps not plausible for everyone) as the one used to exclude the Moorean assertion:
If I assert a proposition &phi, I should know that &phi.
Applied to the earlier statement, I would be committed to:
KI (KI p & ~ KI KI p).
Which is equivalent to:
KI KI p & KI ~ KI KI p.
Which is contradictory (use the truth axiom). It is crucial to be aware of the fact that this argument should not be rephrased as showing that the unacceptability of the assertion is derived from the impossibility of the asserted statement to be true. Granted, in Hintikka's epistemic logic the asserted statement is false because it goes against the KK-principle. But even if we drop the KK-principle, the assertion remains unacceptable. Only the truth axiom is needed to see that.
This has important consequences. First, it is strongly suggested that a logic for first-person assertions is not needed and often not even desirable, for studying (knowledge) assertions. Second, a defence of the KK-principle by referring to the strangeness, or even unacceptability, of statements such as the one above, is misguided (as Hocutt suggests Hintikka chooses to argue at times). And further, the plausibility of not adopting the KK-principle in the argument contra the assertion is strengthened once we realize that, indeed, an epistemologist such as Timothy Williamson postulates exactly the kind of maxim above and rejects the KK-principle. Yet, Williamson rejects the assertion, too (1996).
Let us look briefly at some obstacles one might stumble upon when, notwithstanding the above conclusions, one tries to develop such a logic. First, the differences between the first person and the third person are not about different kinds of epistemic states but about different kinds of assertions. They have to do with the peculiar indexical function of the first-person pronoun, not with the differences in ways to access first-person and third-person mental states. A logic about first-person knowledge, then, would be a logic about assertions. Vincent Hendricks (2005) develops an alternative view according to which the differences between first person and second person are not linguistic but methodological.
Second, there is a problem about how to read the sentences that the logic would produce. Using such a logic we should be quite clear about which of the statements the logic gives us are important. At first sight, for instance, a statement such as:
KI p -> KI KI p (26)
could be considered as reflecting that if I assert that I know that p I should also be prepared to assert that I know that I know that p. And:
~ KI p v KI KI p
should be read as urging me either to assert that I do not know p, or that I know that I know it. Where the first statement did not sound problematic, the second already sounds odd: why can I not remain silent about p? Also, compositionality would be given up, for the rule of epistemic necessitation would, for instance, not lead to knowledge of some conditions on assertions.
Third, there is a more general problem about what to do with the provability symbol of some logic of assertion. Or more precisely, how, with the help of the provability symbol, to model formally 'it is acceptable to assert &phi'. The provability sign is used to express the fact that some proposition is derivable in the logic. But for the purposes of applying the logic, we can use it in different ways. One possibility to model the acceptability of asserting &phi is to take |- &phi to stand for such acceptability; another, ~ |- ~ &phi. The first is certainly wrong, because you would never be able to assert any contingencies. The latter, to exclude Moorean assertions, would need the principle &phi -> K &phi conflating truth and knowledge.
Is a logic of assertion impossible, then? At least the precise epistemic logic we are concerned with most does not form such logic. One way to overcome (some of) the above problems is to apply some kind of operator for assertions. In fact, the earlier statement 26 we read, for some agent a, as:
Aa Ka p -> Aa Ka Ka p.
It says that if a is willing to assert that a knows p, then a should be willing to assert that a knosw that a knows p, too. It is not the assertion:
If a knows that p, then a knows that a knows that p.
Now it does not seem impossible at all to develop a logic of epistemic assertions with the help of such an extra modality. Multi-modal logics are widely studied and the kind of logic needed is not extraordinarily complex. Apart from the regular epistemic axioms, a plausible connection principle would be:
Aa &phi -> Aa Ka &phi.
To avoid assertions about assertions, however, we would need to restrict the &phi to the vocabulary without Aa operators. And, to avoid knowledge about assertions, this should also be true of the epistemic axioms.
Another route takes logics of public announcement as it starting point (Van Benthem 2004; Van Ditmarsch 2003; and references therein). Such logics are dynamic epistemic logics where statements of the form [&phi] &psi have intended meaning 'after truthful announcement of &phi, it is the case that &psi holds'. Up to now, no variants of such logics have been studied in which all necessary distinctions between audience and speaker can be made. A desideratum of a logic to study epistemic assertions, especially one to exclude Moorean assertions, should make it provable that:
[p & ~ Ka p]a falsum.
It has intended meaning that after a's announcing p & ~ Ka p falsum holds. (So [&phi]a stands for the announcement of &phi made by a.) To exclude the Moorean assertion, Barteld Kooi (personal communication) has suggested to take the wonderfully natural:~ Ka &phi -> [&phi]a falsum.
On the other hand, what should hold is things such as:
[p & ~ Kb p]a Kb p.
That is, after a's announcement of p & ~ Kb p (to an audience a member of which is b) it holds that b knows p to be true.
No Normative Interpretation
Some claim that epistemic logic is a logic for ideal agents; and some claim that this is a weaker claim than its opposite claim about real agents. As in the case of assertions versus epistemic states, here, too, there are various possible ways to go about modelling. The easiest is probably to interpret the provability sign in Hintikka's epistemic logic (the clause |- &phi, or the clause ~ |- ~ φ it is, again, a bit of a problem, vide infra) as saying that any agent (who falls under the intended scope of the norm) has the obligation to make it the case that, or bring it about that, or see to it that &phi be true. Such an agent would have to make it the case that, for instance, if he believes something, then he believes he believes that and he would also have to make sure he believes all tautologies. A peculiar consequence, however, is that he should also see to it that, if he wishes his beliefs to be real knowledge, they be true. And that is unnatural. If one has the obligation to make one's belief in proposition &phi true, then in general this may involve an obligation to change the world in such respects that &phi ends up true (unless the belief is true). But such an obligation is not an obligation about epistemic or doxastic agents. It does not have anything to do with how ideal agents would arrange and organize their knowledge and beliefs. One cannot be held responsible for not making true the beliefs one has. In addition, one may even doubt whether one can have obligations to form beliefs at all, since, as one may argue, the formation of beliefs would be involuntary. Note that one cannot respond by saying that since knowledge implies truth, the requirement is empty. Because, then, all requirements a normative interpretation of epistemic logic yields would be empty if at the same time the axioms of epistemic logic were taken as correctly describing epistemic states.
To conclude, epistemic logic is not a logic specifying norms for ideal epistemic agents. At most it describes. Hocutt's criticism of Hintikka, to the extent that he wishes to see epistemic logic as a normative enterprise, is seen to be unconvincing. Doxastic logic, in contrast, does not run into the problem connected with the truth axiom. And to that extent it can be used as a normative theory for ideal doxastic agents. However, two problems show up.
First, what do the proven statements mean? Does the fact that the BB-principle is provable mean that if I believe &phi, I am obligated to believe that I believe it? That is, something like:
BI &phi -> OI BI BI &phi.
Which only gives me a conditional obligation. Or does it mean that I have the unconditional obligation to make the BB-principle true no matter what? That is:
OI (BI &phi -> BI BI &phi).
Second, what does the provability symbol mean? Am I forbidden to &phi if ~|- &phi, or if |- ~ &phi. The first is silly, since you would simply be forbidden to believe any (contingent) proposition p. The second is probably the more plausible reading.
What I will do now is to clarify some of these questions very tentatively by adding deontic logic to the system of doxastic logic. One way to present deontic logic (Hilpinen 2001) is to view it as an attempt to lay bare the logical structure and the logical properties of statements such as:
It is obligatory for a that &phi iff &phi is necessary for a to be a good person.
Obligation is, then, defined in terms of necessity and goodness and hence its logic is a straightforward version of alethic logic:
Oa &phi :<-> (Good(a) -> &phi).
For our purposes it is, of course, the notion of 'epistemic' (or rather, given what we said above, 'doxastic') goodness that should figure in the definition of obligation. And obligations should be obligations to believe something. One way of combining doxastic and deontic logic would be to add the rule:
If |- &phi, then |- Oa Ba &phi.
To ensure that an agent has the obligation to conform to doxastic logic, it should perhaps be restricted to &phi containing belief operators only. For instance:
Oa ((Ba (&phi -> &psi) & Ba &phi) -> Ba &psi) (27)
simply expresses the norm that one has to believe the believed consequences of what one believes.
A well-known problem of deontic logic concerns so-called 'contrary-to-duty' obligations (Hilpinen 2001). Certain formalizations of such duties carry inconsistencies. And it is not at all clear that they cannot arise here. Suppose an agent believes a propositional contradiction &phi. It is clear that he should not do that:
Oa ~ Ba &phi.
But as he does, he has the obligation to believe another contradiction &psi, because he has the obligation to believe all consequences of &phi and &phi -> &psi. But the first contradiction being a consequence of the second, too, (that is, &psi -> &phi being the case) he has, at the end, the obligation to believe the first contradiction &phi:
Oa Ba &phi.
For this contradiction to arise, the rule of deontic necessitation is essential. Otherwise, we would not be able to say that the agent is obligated to believe the consequences of what he believes. But we do not need that rule. We could add to doxastic logic some axioms involving the obligation, without an extra proof rule. And apart from barring the paradoxes of contrary-to-duty obligations, this move has the extra advantage that it allows us to make a sharper distinction between obligations to ensure certain structural or formal properties of doxastic states to hold and substantial requirements about the very content of someone's beliefs. This is so because, in particular, there would not be any obligation in this system to believe every propositional tautology. Along the lines sketched above, one might say this is a plausible view of what a normative theory of beliefs should do. Another advantage of this milder combination of deontic and doxastic logic is that, in the very axioms, we could decide to solve the problems about the scope of the deontic operator. That is, in less formal wording, we could decide only to include axioms that specify conditional obligations, such as Ba &phi -> Oa Ba Ba &phi.
Incidentally, Lennart Aqvist (1967) has shown that adding deontic logic to epistemic logic (as opposed to doxastic logic) may lead to problems that are structurally analogous to the contrary-to-duty problems if we accept the rule of deontic necessitation. Some have argued that this shows that the rule of deontic necessitation is not valid (Goble 1991). But I think such a diagnosis is unwarranted. As I showed, normative interpretations of epistemic logic do not make sense, because obligations to make beliefs true do not make sense. That is, Oa Ka &phi is senseless (a teacher's such admonition is in fact an expression of a norm to believe &phi) and that is why adding deontic logic to epistemic logic is a bad idea anyway.
To finish up, we have reached the conclusion that epistemic logic (not doxastic logic) fails to have a sensible normative interpretation. At most it describes. But if if describes, what does it describe? Some notes. Is it plausible to set apart, here, epistemic logic as a description language of epistemic states from a language to describe actual epistemic agents, or a formalism to describe the standard use of the verb 'to know'? If one is convinced that Hintikka's epistemic logic correctly describes some properties of epistemic states in, what some would call, the 'philosopher's sense' of the word, then why would one hesitate to reject assigning these properties to actual agents (of course, only in case they possess knowledge), or to talk about these states and agents by means of the verb 'to know'?
A mild view would hold that we may reject the rule of epistemic necessitation for actual epistemic agents and for the correct usage of the verb 'to know'. An epistemic subject knows that the earth is not flat, even if he does not know any propositional tautology. And this is not so because the requirement embodied by the rule of epistemic necessitation is too strong, but rather because it is of the wrong kind. Knowledge requirements, the argument would go, should be about structural or formal epistemic aspects, not about the content of knowledge. The KK-principle, under this mild view, would not necessarily be problematic since it is a kind of internal requirement on states, agents, or usage.
Causal Theories of Knowledge
An admittedly slightly recherche formalization of some kind of causal theory of knowledge uses again logics of public announcement. Constructs of the form [&phi] &psi, however, have crucially different intended interpretation: '&phi is a cause for &psi'. The motivation to use public announcement logics nonetheless is that, for the purposes of understanding causal theories of knowledge, one may think of something causing a belief about something as an announcement given by nature. (A well-known topos: physics as a game in which nature discloses its laws to shrewd scientists.) Of course, we are mainly interested here in propositions such as [&phi] Ba ψ that is, &phi causes a to believe that &psi. A definition of knowledge could run like this:
Ka &phi :<-> &phi & Ba &phi & [&phi] Ba &phi.
It says that, apart from the conditions on truth and belief, a proposition is known if it has caused the belief in it. (I will blithely ignore the subtle distinctions between facts, events and propositions.)
In the discussion about Mark Steiner's formalization of Descartes' sceptical argument, we noted that he found himself forced to revise the first version, with the KK-principle, so as to render the argument acceptable for the adherent of a causal theory of knowledge. But why would the causal theorist doubt the plausibility of the KK-principle? For him, the principle reads:
(&phi & B &phi & [&phi] B &phi) -> B B &phi & B [&phi] B &phi & [&phi & B &phi & [&phi] B &phi] & B (&phi & B &phi & [&phi] B &phi).
(Redundancies have been avoided by some logical rewriting.) First, assuming:
B &phi -> B B &phi
does not go against a causal theory of knowledge simply because no causality is involved. Second:
[&phi] B &phi -> B [&phi] B &phi
is more problematic. In general, the causes of your beliefs may be unknown to you. You may believe that Johannes Vermeer used sand in his View of Delft, but perhaps you do not recall whether this was your own observation, or not. However, causal theorists of knowledge would, perhaps, hesitate to go this way. Third, let us study the third conjunct of the consequent. To start with, we note that:
(&phi & B &phi & [&phi] B &phi) -> [B &phi] B B &phi
may be rejected, but not on the grounds of a causal theory of knowledge. It is a rather strong principle about particular forms of mental causation and one may well argue that it too strong, but it seems not to contradict a causal theory. But, in addition:
(&phi & B &phi & [&phi] B &phi) -> [[&phi] B &phi] B ([&phi] B &phi)
should hold. This principle is problematic and the reason is quite simple. The clause [&phi] B &phi expresses, not that some event happens or has happened, but that there is a relation between two events, namely, a causal relation. And it is highly dubious whether the fact that some causal relation obtains can itself be a cause of something again.
Steiner, following David Shatz (1977), thinks that what is wrong with the KK-principle for the causal theorist is that it would give rise to the dubious inference:
[K &phi] B K &phi -> [K K &phi] B K K &phi.
But the above analysis shows that the problem for the causal theorist is rather that one particular ingredient of the analysans of knowledge cannot be a cause of something. This, however, has quite a strong consequence for causal theories in general: not only is the KK-principle implausible, it simply does not make sense to speak about knowledge about knowledge. But that means that the entire argument by Steiner will fail to convince the causal theorist. This is because the causal theorist will have to say that he does not understand what the antecedent means at stage 7 of Steiner's reconstruction of Descartes' argument:
K K p -> K ~ q.
Which states that if I know that I know that I am seated by the fire, then I know that I am not dreaming.
Peter Unger (1967) has suggested the following definition of knowledge: a person a knows that &phi iff it is not accidental that he is right about &phi. A plausible formalization is:
Ka &phi :<-> &phi & Ba &phi & (&phi -> Ba &phi).
Here the  is the alethic modality of necessity. The notorious tripartite format of analyses of knowledge is clear: knowledge is true, the agent believes it to be true and it satisfies some extra justification condition. Logically speaking, what is interesting is that Unger makes use of one of the first attempts to formalize subjunctive conditionals, namely, as 'strict conditionals'. This is not generally considered an entirely successful approach because even under very weak assumptions about necessity (about the  operator) it gives rise to counterintuitive results. To see that, first observe that for the alethic modality the following kind of transitivity condition is true:
(&phi -> &psi) & (&psi -> &chi) -> (&phi -> &chi).
If, then, we formalize subjunctive conditionals as 'boxed' material implications, it would follow from:
If Carter had died in 1979, he would not have lost the election in 1980
If Carter had not lost the election in 1980, Reagan would not have been president in 1981
If Carter had died in 1979, Reagan would not have been president in 1981.
But, as Donald Nute and Charles Cross, from whom I borrow the example, note, it is 'far from clear' that this is a valid conclusion (Nute and Cross 2001: 8). Counterexamples like this are among the main reasons why an analysis of subjunctive conditionals as strict conditionals is rejected.
One way to represent Robert Nozick's (1981) tracking view of reliabilism, drawing heavily on the work of Johan van Benthem (2005), is to see it as Unger's approach with a more advanced formalization of subjunctive conditionals. (I abstract away form problems involving the non-trivial construction of a semantics for the combined system.) Using > for the more refined conditional implication, knowledge would then be defined as:
Ka &phi :<-> &phi & Ba &phi & (&phi > Ba &phi) & (~ &phi > ~ Ba &phi).
An interesting logical question is what, if any, axioms are needed to guarantee the traditional axioms from epistemic logic to hold. (It is clear that the truth axiom follows from the definition.) And this means that we have to look, first, at the logic of conditionals. One of the standard such logics has the following axioms:
[ID]&phi > &phi
[MP](&phi > &psi) -> (&phi -> &psi)
[MOD](~ &phi > &phi) -> (&psi > &phi)
[CSO]((&phi > &psi) & (&psi > &phi)) -> ((&phi > &chi) <-> (&psi > &chi))
[CV]((&phi > &psi) & ~(&phi > ~ &chi)) -> ((&phi & &chi) > &psi))
[CEM](&phi > &psi) v (&phi > ~ &psi).
And its proof rules are, apart from modus ponens:
[RCEC]if |- &phi <-> &psi, then |- (&chi > &phi) <-> (&chi > &psi)
[RCK]if |- (&phi_1 & . . . & &phi_n) -> &psi, then |- ((&chi > &phi_1) & . . . & (&chi > &phi_n)) -> (&chi > &psi).
A number of interesting logical observations may be made about (the formalization of) Nozickian tracking.
First, the third conjunct is redundant as long as we stay this particular version of conditional logic. For unless we drop axiom CEM, we can prove that:
(&phi & &psi) -> (&phi > &psi).
(To prove the contraposition, observe that the following implications hold: by CEM, ~(&phi > &psi) -> (&phi > ~ &psi); by MP, (&phi > ~ &psi) -> (&phi -> ~ &psi); and by propositional logic, (&phi -> ~ &psi) -> ~(&phi & &psi).)
Another immediate logical question is whether the principles from epistemic logic follow from the tracking definition of K &phi, or whether tracking versions of these axioms are needed. Of course, the truth axiom is captured by the definition. Interestingly, the KK-principle (which Nozick rejects) turns out to be provable if we add some axioms that, in some sense, are not as strong as the KK-principle itself. They involve, not knowledge about knowledge, but belief about knowledge. The axioms are:
K &phi -> (K &phi > B K &phi)
~ K &phi > ~ B K &phi.
For negative introspection, unsurprisingly we need a kind of dual axioms:
K &phi > ~ B ~ K &phi,
~ K &phi > B ~ K &phi.
(Of course, as in the analysis of causal theories of knowledge, the axioms can be spelled out in terms of doxastic operators only.)
Another illustration concerns Wolfgang Lenzen's equivocation of knowledge and belief from the perspective of Nozickian tracking. To understand how a logical approach, on the basis of Nozick's analysis of knowledge, may help, a bit of context may be needed. The traditional argument about Lenzen's collapse of knowledge and belief blames:
B &phi -> B K &phi.
But what would be the reasons? It is interesting to see what would be needed in a Nozickian reformulating of the argument. The crux would be an axiom such as:
B &phi -> B(~ &phi > ~ B &phi)
stating that one would only believe a proposition &phi if one would believe this belief to be robust. One would have to believe that if &phi were not true, but if other things remained more or less similar, that then one would not believe &phi. That is quite a strong axiom. It is, in addition, an axiom that is easier shown to be implausible in certain cases than that of B &phi -> B K &phi. Take the example, adapted from Lenzen, where someone claims to believe that the Netherlands will win the next World Cup. Does he really believe that in a possible world in which the Netherlands do not play all that well, but which in all other respects is as close to the actual world as possible, that in such a world he would not believe the Netherlands would win the World Cup? Probably not.
Static and Dynamic
Two properties of Hintikka's epistemic logic have received quite some attention in the formal as well as in the informal literature on knowledge: logical omniscience and the KK-principle. The literature on both is vast and I will be concerned with one approach to the KK-principle only. Vincent Hendricks (2003) suggests we should distinguish between two readings of the KK-principle. The axiom is 'synchronic' iff antecedent and consequent hold at the same time; it is 'diachronic' iff the antecedent precedes the consequent in time. In this section I will take up this suggestion and try to give it more precise content by using temporal logic.
Two observations. First, Hendricks's definition of synchronicity and diachronicity is sensitive to the precise form of the axiom. If we rewrite the KK-principle as:
~ K &phi v K K &phi
it remains unclear how to apply the two notions (for there is no antecedent and no consequent). And if we rewrite it as:
~ K K &phi -> ~ K &phi
it is easy to assign the temporal labels wrongly, so to speak. Second, it is desirable to phrase the notions of synchronicity and diachronicity in a subtler vocabulary than Hendricks's. What precise point in time counts for synchronicity? Should K &phi imply K K &phi at one specific moment, the present, for instance, or at all moments, past and future? And should diachronicity relate the past to the future, or the present to the future? And so on and so forth.
The use of some version of temporal logic helps to add more precision. The logical language which we use to phrase the various possible KK-principles will include the epistemic operator K, temporal operators G ('henceforth. . .' or 'it is going to. . .') and H ('hitherto. . .' or 'it has been. . .') and their dual operators F ('somewhere in the future. . .') and P ('somewhere in the past. . .'). Obvious axioms for temporal logic are principles such as:
[DB]G(&phi -> &psi) -> (G &phi -> G &psi)
H(&phi -> &psi) -> (H &phi -> H &psi)
[CV]&phi -> G P &phi
&phi -> H F &phi
G &phi -> G G &phi
H &phi -> H H &phi.
And as proof rules we have modus ponens and necessitation for G and H.
The most obvious candidate for a diachronic KK-principle in this language is:
K &phi -> G K K &phi.
But this principle is implausible since an instance of it is:
K ~ K p -> G K K ~ K p.
It states that if I (now) know that I do not know some proposition p, henceforth I will know that I know that I do not know it. But that would exclude that I come to learn p at some future moment.
Another problem with this rendering of the KK-principle is that it does not specify any reasons why the move from knowledge to knowledge about knowledge is acceptable. In other words, we might expect that a more plausible axiom enriches the antecedent with another conjunct. It seems natural to expect this extra condition to be a condition about knowledge; that is, it will start with an epistemic operator. Something like:
(K &phi & K H K &phi) -> G K K &phi.
Using temporal logic also makes it quite clear that at least these kinds of diachronic reading of the KK-principle are extraordinarily strong in that they do not allow epistemic subjects to forget what they know. A weaker version would keep antecedent and consequent in the present, only requiring some recall of previous knowledge:
(K &phi & K H K &phi) -> K K &phi.
Let us turn to Frederic Fitch's clever reductio of verificationism. What he proved was that if we assume:
&phi -> <> K &phi
truth and knowledge collapse in the sense that for all p it holds that:
p <-> K p.
A whole range of solutions have appeared in the literature (weakening the underlying epistemic or alethic logics or weakening the verification principle). As an application of a combination of epistemic and temporal logic, I suggest a different solution by temporalizing the verificationism thesis. If I do not know now some proposition that is nevertheless now true, then I may come to know later that this proposition was true and I may even come to know that I did not know that proposition at the moment it was true. That is:
&phi -> <> F K P &phi.
This is only an (admittedly very rough) suggestion of a solution and it may well be that different temporal operators add more and necessary, precision. A more detailed solution using logics of public announcement is given by Johan van Benthem (2004).
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Boudewijn de Bruin
Boudewijn de Bruin is assistant professor in the Faculty of Philosophy of the University of Groningen. De Bruin did his undergraduate work in musical composition at Enschede, and in mathematics and philosophy at Amsterdam, Berkeley, and Harvard. He obtained his Ph.D. in philosophy from The Institute for Logic, Language, and Computation in Amsterdam. His doctoral dissertation, under supervision of Johan van Benthem and Martin Stokhof, was on the logic of game theoretic explanations. read more