first draft: April 1998 / last revision: August, 21, 1998

Philosophy of language course 1998-9 / Teaching material
General assessment and suggestion for further readings on Kripke's puzzle [see Bibliography]

Carlo Penco

Kripke's puzzle about belief

A traditional argument is often used against Mill's theory of names (the meaning of a name is exhausted by its referent). Mill's theory implies transparency of proper names (coreferring proper names are substitutable salva veritate); but examples like Frege's and Quine's show that proper names are not transparent in belief contexts. This could be thought to be a reductio ad absurdum of Mill's theory.
In " A puzzle about Belief" (1979; 1988) Kripke builds up an argument which aims to show that the same problems, given by the principle of transparency of proper names, can also be generated without the use of that principle, but with some weaker and more general principles, which seem to be difficult to reject. (see Donellan)
Therefore, the traditional argument against Mill's theory does not work. If you want to reject Mill's theory with some reductio ad absurdum, you should reject two very intuitive and apparently valid principles.
The well known puzzle is based on the assumption that our speaker is normal non omniscient, sincere, reflective and not conceptually confused. The two principles used are the Disquotational Principle (DP) and the Translation Principle (TP):

If a speaker of a language L assents to p and "p" is a sentence of L, then he believes that p.

If a sentence of one language expresses a truth in that language,
then any translation of it into another language also expresses a truth in that other language

Pierre, a Frenchman, heard in Paris about London's beuty. He therefore assents to the sentence:

(1) Londres est jolie.

Emigrated in England he learns English by exposure, takes up residence in London and, after observing the surroundings, he assents to:

(2) London is not pretty

He does not realize that the town where he lives is the town depicted in the nice pictures he saw in Paris; he has not updated his earlier belief expressed once as "Londres est jolie". Then, given that "Londres" and "London" (just as the old-fashioned "Hesperus" and "Phosphorus") have the same reference - or the same semantic value (the object referred to by the names which are rigid designators), it follows that Pierre believes that London is pretty and he believes that London is not pretty.
A similar puzzle may arise also with the homophonic case, when Pierre meets on two occasions Paderewski, once in a music hall and another time at a political conference. He does not realize that he met the same person and he assents to two different sentence:
Paderewski has musical talent
Paderewski has no musical talent

In both cases, we are compelled to admit that this supposed rational person holds contradictory beliefs, therefore this person is not as rational as supposed. Is it a real puzzle? If it is, either we have to reject the causal theory of reference, or we have to find an answer to the puzzle. Some answers could say that if the puzzle works, then it is worse for the causal theory of reference. A more precise answer could be that, as in the case of the reductio ad absurdum of Mill's theory via the traditional argument, we may have a reductio ad absurdum criticizing the validity of the disquotational principle. Beyond the difficulty of abandoning an apparent acceptable principle, it has been suggested (by Sosa) that even that principle may be dispensable in building up the puzzle. Before rejecting such a principle, or rejecting the causal theory of reference, it should be shown that other theories can solve the puzzle.
But it is not so clear that a descriptive theory of reference can do better. A Fregean could say that the contents of Pierre's beliefs (thoughts) are senses: the way in which London is presented to Pierre the first time fits with the way in which the concept "pretty" is given; the way in which London is presented to Pierre the second time does not fit. This could be correct. The problem is that we have no idea of what these "modes of presentations" are. We could try something like this: " Pierre believes that the town depicted in a nice picture he heard about in Paris is pretty" and "Pierre believes that the town he lives in is not pretty". Even given these expressions, we cannot avoid the fact that in both cases Pierre believes of London that it is pretty, and he believes of the same town that it is not pretty.

Different solutions or discussions of the puzzle are connected with different authors who dealt directly or indirectly with it; among them I give some hint about the ideas of
Ruth Barcan Marcus
Robert Brandom
Michael Dummett
Hans Kamp
David Sosa
Epistemic Logics

Ruth Barcan Marcus
Robert Brandom makes a distinction on two aspects of the disquotational principle: a) linguistic avowals of belief vs attributions of belief
b) expressions used by the believer vs. expressions used by the reporter
Given this distinction we may say that while Brandom accepts a) and rejects b), Ruth Barcan Marcus actually would reject a) while retaining b). In fact she rejects the idea that an avowal should be identified with a belief, but only with a "claim to believe".
In order to maintain the expression used by the speaker we need to use a more sophisticated form of the disquotational principle, inserting the additional clause: "'p' describes a possible state of affairs".
Does Barcan Marcus forgets that both "London is pretty" and "Londres c'est jolie" describe two possible state of affairs? Not really because her discussion is framed inside a theory of rationality where we have to distinguish between asserting, believing and claiming to believe:
- asserting is "a relation between a language user and a linguistic utterance"
- believing which is "a relation between an epistemological agent and a state, or state of affairs".
- claiming to believe, which is a weaker relation than believing and somehow evaporates in asserting.
Therefore Pierre may and has to assent to "London is different from Londres"; this sentence follows from his previous avowals. But he does not believe it; he just claims to believe it, because he does not know that it amounts to a contradiction.
Once an impossibility or a contradiction (such as London is different from London) has been disclosed, the speaker would recognize that he himself (Peter) had only "claimed to believe that London was different from Londres"; being the same city, he should have believed that an object is not identical with itself, and that is impossible.

Robert Brandom

Michael Dummett
Dummett 1973 maintains that we may have contradictory beliefs. We may have contradictory beliefs if they are plugged into different cognitive contexts such that the contradiction does not invade the entire system of our beliefs. Should we then plug in some concept of "viewpoint"? In a cognitive context of a French viewpoint, London is pretty; in a cognitive context of an English viewpoint, London is not pretty. But what are we speaking about? We are still speaking about the same town. Okay. We need to find a way to attribute to Pierre contradictory beliefs, without having to attribute to him a direct contradiction. We have to avoid what Brandom calls the "weak" form of the puzzle (Pierre believes that p and not p) and adhere to the stronger version: Bel (Pierre p) and not Bel (Pierre, p). We have the key of the contradiction: we know that the speaker is speaking of the same town; therefore he is involved in a contradiction. Are we involved in a contradiction in expressing his contradictory beliefs? We are expressing a contradiction, reporting two contradictory beliefs. They are stacked inside the cognitive context of our representation of Pierre's set of beliefs (which is, by the way, very different from his representation of his beliefs). If we have to solve the problem, we have to give Pierre the possibility to update his beliefs and change his mind on London. [Or - maybe - to reach a more complex belief such as: London sometimes looks pretty and sometimes it does not.] Kripke says that "we regard individuals who contradict themselves as subject to greater censure than those who merely have false beliefs" (1988,p.122) However it is possible to have some false belief which produce a contradiction after some logical passage of which we are not aware of. If we keep our beliefs closed inside some cognitive context we may have unnoticed contradictions through different contexts.

Hans Kamp
In his "Prolegomena to a Structural Account of Belief and Other Attitudes" Kamp 1990 introduces a distinction between formal anchors and external anchors: external anchors are introduced to make space for the impact direct reference theory has on the philosophical and linguistic community. Discourse Representations Structures contains purely descriptive information; a "direct link with the world" is given plugging a sign for an external anchor into the representation ; this anchor would be "a pair consisting of a discourse referent and an object". Therefore anchored DRSs express Russellian singular propositions . On the other hand, Kamp suggests that there are also internal anchors, that is anchors which purport to represent an object, but do not make any existential assumption on the object (they are similar in that respect to "concept discourse referents" introduced by Asher 1989, p.357?). We may have then two possibilities: just formally anchored DRS and externally anchored DRS (the first being without external anchors and only with internal ones; the second having both internal and external anchors). Is this distinction enough to treat Kripke's puzzle?
An interesting point is given by the idea of "internal links" as a mean of making a formal anchor of one person available to another: we might use formal links to make available to us two different internal anchors in Pierre's DRS (one for "London" and the other for "Londres"), which we may link to the same external anchor in our DRS. But in our representation of Pierre's beliefs, his two utterances are linked to two different internal anchors and that gives expression to the possibility of his currently holding the two beliefs.
It this worked, it would be a nice solution to Kripke's puzzle. But in our representation of Pierre's beliefs we have to make the connection with our external anchor and we get again into contradictory belief (in contradictory attributions of belief, not in contradictory beliefs which are avoided by the game of internal anchors). [we should represent Pierre as believing that London is different from Londres, and at the same time asserting that London is identical with Londres].

David Sosa ("Import of the Puzzle About Belief", Philosophical Review, 1996) argues against the idea that Kripke's puzzle can be considered as a reduction of the disquotational principle; eventually he says that Kripke-type cases can be created even without any principle of disquotation. The moral is that "it is useless to try to stop Kripke's examples before they reach the step in which the belief that N is F and the belief that N is not F are attributed to the agent" (p.384-5). The point made by Sosa is that another hidden principle as at work in Kripke-type cases, the Hermeneutic principle:

Hermeneutic principle [H] : "If a name in ordinary language has a single referent then it may correctly be represented logically by a single constant"

[H] seems to represent in an explicit way the idea of an "unambiguous" name as a name which has a single referent and can be represented by a single logical constant (the example by Kripke being of an ambiguous name "Cicero" used to refer to the orator and to a pilot in World War II).
Kripek's arguments are valid only if we assume "that an agent has contradictory beliefs if and only if the agent has beliefs whose contents can be represented formally as Fa and - Fa", an assumption which clearly goes on well with the principle [H]. (388). A theory which allows a proper name to have a meaning which is not exhausted by the reference will deny the validity of [H]; on the other hand the only justification for [H] is the Mill's theory. Therefore Kripke's puzzle can be seen as a reductio ad absurdum of [H] and therefore of Mill's theory of proper names.
The presupposition Sosa rejects is that "ambiguity is a matter of multiple reference". "If we were to require, for a term to be ambiguous, that it have more than one referent, then, it seems to me, we would presuppose Millianism; such a requirement excludes a Fregean position in which a name with a single referent is ambiguous in virtue of having more than one sense, and Kripke's project is precisely to recreate a difficulty without presupposing Millinanism." (391).

Doxastic Logics
Kripke's puzzle is also a puzzle about doxastic logic and their ability to correctly represent our attributions of belief and our intuitions about belief. Doxastic logics attempt to represent the logic of belief (Greek: doxa) in counterposition to the logic of knowledge (Greek: episteme, from which also "epistemic logics"). Often people speak of "Epistemic logics" for logics for Knowledge and Belief (given that the traditional definition of knowledge is "justified true belief"). The first to discuss deeply the problem of a logic for belief-contexts after Frege was Carnap in Meaning and Necessity: here Carnap calls the belief contexts (expression such as "s believes that p") as "hyperintensional". They cannot be treated within the framework of an intensional logic, because the failure of substitution of equi-intensional expressions. We need a relation which is more powerful than equi-intensionality: intensional isomorphism or sameness of intensional structure (this idea has been developed by Cresswell 1985 with the idea of "structured meanings"). A major divide in the history of formal attempts to treat belief context is given by the results in the semantic of modal logic given by Kripke 1963, with the standard definition of a possible world semantics. After that, different attempts have been given for a logical treatment of belief contexts. One of the best treatments of the subject is Frixione 1994, chapters 8-11 (in Italian) . One of the main problems posed in such logics is the distinction between explicit and implicit belief, which permits the differentiation between the point of view of the believer (Pierre) and the reporter. A study in this field is needed, and Kripke's puzzle looks like a good test to verify the suitability of logic to our intuitions.


A traditional argument against Mill's theory of proper names and the intuitive criterion of difference of thought
(substitute "a" and "b" with "Hesperus" and "Phosphorus" or with "Cicero" and "Tully"; substitute "P" with "is a planet" or with "is a great orator" and you have Frege's and Quine's examples).

While in normal context we may have the inference:

(Pa, a = b) |- Pb

we cannot make the following:

(s believes that Pa, a = b) |- s believes that Pb

It is possible that, given the ignorance of the speaker, the speaker believes that Pa and does not believe that Pb. Therefore, if the meaning of a singular term is exhausted by its referent, we are compelled to fall into a contradiction; we have to assert that s believes that Pb and at the same time we have to assert that s does not believe that Pb

How to respond to this result?
Frege derived the following conclusion: if we want to preserve the rationality of the speaker and ours, we have to admit that she had two different thoughts, and therefore gives two different meanings to the singular terms, against the principle of transparency of proper names. Therefore, names must have meanings which are not exhausted by their reference, and in indirect contexts (as belief contexts) the reference of an expression is what in normal contexts is its sense.
Carnap develops the idea that expressions have not only reference or extension; they have also intension (propositions, properties and individual concepts). He derives however a different answer than Frege; the extensions and intensions of an expression do not change depending on the contexts; we have to distinguish extensional and intensional contexts as contexts where two different principles of substitutability hold: substitutability of extensions and of intensions. As far as belief contexts, we need something more in order to make a truth-preserving substitution: identity of intensional structures
Quine derives a further conclusion: if we want to maintain something like the Mill's theory of proper names and therefore the transparency principle, we have to abandon belief contexts, and any modal context where these principle are not valid.

Kripke wants to hold both modal (and belief) contexts and the rationality of the speaker. But he wants also to keep the Millian framework of direct reference. Therefore he cannot accept neither the Fregean nor the Quinean answer. He never hints at the problem of intensional structure as an answer (actually Carnap's answer has been considered not clear enough, and bound to be defeated by Mate's puzzle)

Donnellan 1990 has given this point in interpreting Kripke's paper: "...the same paradox upon which the direct reference theory is supposed to founder, can be derived from just those principles alone, without any theory of reference being assumed at all. In other words, quite independently of any theory of reference we might adopt, there is a paradox which can be derived from principles we use in ascribing beliefs to people. And those who pose the objection to the direct reference theory must implicitly rely upon those principles....The principles, of course, result in paradox only in certain cases - the very cases which are used to attack the direct reference view."(p.207)

Frege assumed since Die Grundlagen der Arithmetik, and later on in "Uber Sinn und Bedeutung", Leibniz' substitution principle: if you substitute a name in a sentence with another name without changing the truth value of the sentence, the two names have the same reference (eadem sunt quae uni substitui alteri salva veritate). This certainly works normally, but it does not work in the case of indirect speech. The point is made by Frege in a famous passage in Uber Sinn und Bedeutung :

"...the thought in the sentence 'the morning star is illuminated by the Sun' differs from that in the sentence 'the Evening Star is a body illuminated by the Sun'. Anybody who did not know that the evening star is the morning star might hold the one thought to be true and the other to be false."

This passage is quoted often as an expression of the "intuitive criterion of difference of thought" (the first to use the expression was Evans 1982, p.18). The criterion relates the sense of a sentence to the cognitive state of the speaker, and it is used as a key in the analysis of philosophers who work in the Fregean tradition; we might express the principle as:

"If it is possible to understand two sentences and coherently believe what one expresses while not believing what the other express, then those sentences express different senses (different thoughts)"

That the two sentences express different thoughts can be derived by the fact that the substitutional law does not apply to belief contexts; in fact, given that the morning star=the evening star, you cannot substitute one for the other, leaving unchanged the truth value of the sentence.

(a) "s believes that the evening star is a planet"
(b) "s believes that the morning star is a planet"

it may happen, given the ignorance of the speaker, that (a) is true and (b) is false. Therefore we cannot accept that "evening star" and "morning star" be two terms identical in all respect; they must have some difference which explains the fact that they cannot be substituted salva veritate; Frege will say that they are different in sense. Therefore the senses of the sentences to which they belong (the thoughts the sentences express) must be different.

Carnap ...


Quine defines a distinction between opaque contexts and transparent ones: a context is transparent when you can substitute any coreferential term preserving truth value; a context is opaque when you cannot. A typical example of opaque context is the use in formal mode

"giorgione" was so called for he was big
"barbarelli" was so called for he was big

We find another example of opaque contexts when we deal with belief contexts:

John believes that Cicero is an orator
John believes that Tully is an orator

As it is easy to see this example maps the one given by Frege with Hesperus/Phosporus. If John does not know that Cicero=Tully, then he does not believe that Tully is an orator. A development of this discussion has been done in a later paper:

Quine 1956 defined a difference between a relational or "de re" and a notational or "de dicto" sense of "belief"; Quine will call them "belief1" and "belief2":

(1) (Ex)(Ralph believes that x is a spy)
(2) Ralph believes that (Ex)(x is a spy)

Think of the difference: most of us are like Ralph in (2); that is we believe that there are spies, but few of us believe of somebody that he is a spy. However Quine tends to support the use of (2) against (1) which is a clear case of opaque context. But he also suggests that (1) involves "quantifying into a propositional idiom from outside", and this is "a dubious business". Why? Take the example by Quine: and Ralph has seen Ortcutt, the spy with a brown hat, in questionable circumstances and believes he is a spy. But the same man, Ortcutt, without hat and laying at the beach seems to Ralph a perfectly good man. How to interpret then (1)? Is it compatible with the two following?:

(3) Ralph believes that the man in the brown hat is a spy
(4) Ralph does not believe that the man at the beach is a spy

(3) and (4) are intuitively true; but on the other hand we know that:
Ortcutt = the man with the hat = the man at the beach
If we made substitutions with coreferential expressions we had a clear case of contradiction, as in
(5). Ralph believes that the man at the beech is a spy.
But "believes that" is "referentially opaque"; we cannot make substitutions of coreferential expressions. We do not accept the contradiction, but we cannot make sense of (1).
However, if we want to develop the relational sense of belief, we have to accept: (5)
Therefore we have to "acquiesce in the conclusion that Ralph believes1 that the man at the beach is a spy even though he also believes2 (and believes1) that the man at the beach is not a spy".
It is in this paper that Quine speaks of intentions [propositions, attributes] as "creatures of darkness", posing "most of the problems that a logic of belief has to deal with" Here also Quine distinguishes belief as a dyadic relation (between a speaker and a proposition) and as a triadic relation (between a speaker am object and an attribute):
(5) Ralph believes that Ortcutt is a spy
(6) Ralph believes x (x is a spy) of Ortcutt
(where the notation x(x is a spy) stands for the attribute "spyhood")
The conclusion of the paper is that any quantification you try to make will lead you unavoidably into some kind of contradiction, as the one given before between (4) and (5). Propositional attitudes are referentially opaque and their formal treatment leads to contradictions. Clarity is not restored re-inserting intensions.
Kaplan 1968 discusses abundantly the matter. The solution to distinguish between a strong sense of se re belief (where a demonstration is needed) and a weak sense of de re belief (where a denotation is enough) is developed by Brandom 1994 , ch.8,IV-V. For the relation with Kaplan's analysis see p.548.