Kripke's Puzzle and Semantic Granularity

(Added October 2016: my most up-to-date treatment of granularity can be found in Chapter 6 of my PhD thesis. This is an early, undeveloped attempt.)

This is the post where I first introduced my doctrine of semantic granularity. Follow-ups so far:

Facts and Granularity
Granularity and Quine
Metaphysical Realism and Conceptual Relativity: An Application of Granularity
Granularity and Relativism about Truth
Granularity and the Paradox of Analysis
The Principle of Compositionality and Semantic Granularity
Two Opposite Types of Granularity Difference

Meanings of expressions and belief-contents can be carved up at different granularities. That is, it can sometimes be the case that, when operating at one granularity it is correct to bundle two expressions or beliefs together as having the same meaning or content, while at another granularity it is correct to put them in separate bundles. I think this must be acknowledged in order to fully solve Kripke's puzzle about belief.

There are aspects of the puzzle which do not require this - i.e. the puzzle has some morals which do not involve this. But until semantic granularity is recognized there will be a remainder.

One aspect of Kripke's puzzle is like Frege's puzzle: we need difference-makers for 'Londres' and 'London' and the propositions they appear in, so that we can avoid the conclusion that Pierre here believes some proposition as well as the negation of that very proposition (i.e. we need to differentiate his 'Londres'-mediated beliefs from his 'London' ones). I do this with my accounts of names and propositions.

Another, closely related, aspect of the puzzle is that, once we have the required difference-makers, we need to put them to work somehow in distinguishing the sense in which Pierre has inconsistent beliefs from the sense in which he does not have inconsistent beliefs. Accordingly I distinguish internal and external inconsistency. Two beliefs are internally inconsistent iff no two beliefs with the same internal meaning could both be true. Two beliefs are externally inconsistent iff those very two beliefs, with their actual external projective relations to reality, could not both be true. Internal inconsistency implies external, but not the other way around. And one of the morals of Kripke's puzzle is that merely internal inconsistency does not constitute irrationality.

But there is a further aspect that remains puzzling even with the required difference-makers, and the distinction between internal and external consistency with its associated moral about rationality. And this comes out in Kripke's summing up of the puzzle: does Pierre, or does he not, believe that London is pretty?

And here, to see that this is still puzzling, it is very important that we take to heart Kripke's stipulation that he is using the language of belief reports not in a de re sense, but in a de dicto sense; that he is using forms like 'S believes that a is F' not in the sense of 'S believes, of a, that it is F' or 'S has a belief concerning a to the effect that it is F', but to actually specify belief-contents. Kripke gives a supplementary explanation of his meaning by saying that we could emphasize it by putting a colon in place of the that-clause: 'S believes: a is F'.

It is important to take this to heart because, if we stick to a de re sense, we can give an answer with what we already have; we can say, in answer to Kripke's puzzle question reproduced at the end of the second last paragraph, 'He does; Pierre believes, of London, via his "Londres" concept (or via his symbol "Londres" with its attendant use or internal meaning) that it is pretty. But he also believes, of London, that it is not pretty, but in that case his belief goes via his "London" concept (or via his symbol "London")'.

And this is just using the stuff we needed anyway to solve Frege's puzzle. And furthermore we can add that there is no irrationality on Pierre's part here, since his two conflicting beliefs, while externally inconsistent, are internally consistent, and he doesn't know that they concern the same object.

But this doesn't enable us to answer Kripke's puzzle-question as he intended it, namely in a belief-content specifying sense. Indeed, it can seem to be part of the problem. We wanted to allow that Pierre is not being irrational, and distinguish his 'London' concept from his 'Londres' concept. But then what was going on when, in the first part of the story, we felt the pull of saying that Pierre believes that London is pretty – i.e. that Pierre believes the same thing that we mean when we say 'London is pretty'?

The solution is to see that a shift in granularity has taken place, and that the answer to Kripke's question - indeed, the meaning of that question - depends on what granularity one is operating at. In the first part of the story, we naturally go for a granularity coarser than the one we will end up at, in order to capture in an efficient way what Pierre's and our contents have in common. Then, when the special “splitting” (mistaking one for two) situation arises, it becomes much more convenient to describe the situation using the same device of belief reports, but at a finer granularity. 

Kripke's puzzle is puzzling because one part of the story induces one granularity, and another part induces another. With granularity kept in the background as an unarticulated and untheorized contextually variable aspect of the sense of belief reports, the results seem to contradict each other. Once we realize what is going on, the results can be seen to be no more contradictory than 'All the beer is in the fridge over there', under a certain natural contextual restriction of quantifiers, is of 'There is beer at the pub'.

Philosophers already talk about different granularities, but generally the distinction is made between two quite different notions: for example, the set-of-worlds conception of propositions is said to be more coarse grained than the Russellian. Here, I am keeping to one conception (which, in comparison with those just mentioned, is left more intuitive), but saying that we can operate at different granularities in individuating meanings, roles in language systems, and the contents of beliefs. The idea is that semantic notions such as that of synonymy and belief content are flexible devices, in that they can be used to bundle expressions and representations together in multiple ways.

The underlying idea here, analogues of which appear in connection with other things besides linguistic meaning and the content of belief, is quite commonsensical. For example, consider someone who takes a board game and alters some rules, inaugurating a social institution of playing to the altered rules which goes on along side the practice of playing the original game. Are we to say there are two different games here, or two different versions of the one game? It seems like common sense to say that one can say either. It's not as though there's some answer here which we haven't yet managed to find out. So, we individuate games at different granularities. And this is part and parcel of the usefulness and flexibility of our concept of a game. I think the same holds for the concept of meaning.

While this idea is quite commonsensical, the idea that it should be taken seriously in analytic philosophy of language appears quite radical. (It is as though, without really thinking it over, people have rejected any such move as inherently inimical to analytic conceptions and methods. A bit like vagueness before analytic philosophers began taking that seriously.)

Interestingly, after I had independently started applying the terminology of granularity and bundling to the matters of Kripke's puzzle and internal meaning, I found that AI researchers working on word sense disambiguation have been talking the same way (without apparently reflecting much on it philosophically, let alone from the point of view of the problems of analytic philosophy of language).

In this post I have tried to introduce the doctrine of semantic granularity, a doctrine which has come to assume an important role in my thinking. I have motivated it in the first instance using Kripke's famous puzzle about belief, which is also how I arrived at it. In subsequent posts I will develop the idea further and outline further applications of it.

[Here is one further application. - TH 25/8/14.]

Reference

Kripke, Saul A. (1979). A puzzle about belief. In A. Margalit (ed.), Meaning and Use. Reidel. 239--83. [Online here and here as of 4/3/14.]

Propositions: A Neo-Wittgensteinian Approach

(Added October 2016: my most up-to-date treatment of propositions can be found in Chapter 6 of my PhD thesis. This is an early, undeveloped attempt.)

As I use the term 'proposition', propositions are propositional signs (e.g. declarative sentences) together with their internal uses or meanings, plus any external projective relations borne to reality by the component signs.

This is partly inspired by the conception of propositions espoused in the Tractatus, but also the conception of meaning espoused in Part I of Philosophical Grammar, 'The Proposition and its Sense' and various contemporaneous documents of Wittgenstein's thought– that is, the middle Wittgenstein. Roughly speaking, the 'projective relations' component comes from the Tractarian conception, and the internal meaning component comes from the later conception of meaning.

Before explaining, in my own terms, these conceptions as I have them, I will briefly quote these sources. Here is entry 3.12 of the Tractatus:

I call the sign with which we express a thought a propositional sign.—And a proposition is a propositional sign in its projective relation to the world.

And here is an excerpt from remark 27 of Philosophical Grammar:

[…]

A name has meaning, a proposition has sense in the calculus to which it belongs. The calculus is as it were autonomous. - Language must speak for itself.

I might say: the only thing that is of interest to me is the content of a proposition and the content of a proposition is something internal to it. A proposition has its content as part of a calculus.

The meaning is the role of the word in the calculus.

The meaning of a name is not the thing we point to when we give an ostensive definition of the name; that is, it is not the bearer of the name.

I do not, however, want to give the impression that I agree with everything in Part I of Philosophical Grammar. Indeed, immediately after the passage quoted above, Wittgenstein asserts that the name 'N' is synonymous with the definite description 'The bearer of “N”'. I think this is a sheer mistake, for Kripkean reasons which were not highly visible when Wittgenstein was writing – these expressions exhibit, for example, different behaviour across counterfactual scenarios. But this doesn't mean I don't agree with Wittgenstein's general way of thinking about (internal) meanings as roles in language-systems – I just don't agree about this case, i.e. don't agree that 'N' and 'The bearer of “N”' play the same role.

And remark 36:

If we look at the actual use of a word, what we see is something constantly fluctuating. In our investigations we set over against this fluctuation something more fixed, just as one paints a stationary picture of the constantly altering face of the landscape.

When we study language we envisage it as a game with fixed rules. We compare it with, and measure it against, a game of that kind.

And from the earlier work in Philosophical Remarks:

3. […] The words 'Colour', 'Sound', 'Number' etc. could appear in the chapter headings of our grammar.

7. Grammar is a 'theory of logical types'.

15. What does it mean, to understand a proposition as a member of a system of propositions? (It's as if I were to say: the use of a word isn't over in an instant, any more than that of a lever.)

I will now briefly explain what I mean by 'projective relations' (or 'external meaning') and 'internal meaning'. As a preliminary, I should say that I make no presumption that I mean what Wittgenstein meant in the Tractatus by 'projective relations'. The ideas are certainly related, but I think not likely to be the same, especially since I have two components side by side here – internal and external meaning – the first of which is inspired by later work of Wittgenstein's. The second component I have been inspired by the Tractatus to gloss as 'projective relations', but since in the Tractatus it was the only component besides the sign, it seems reasonable to suppose that the Tractarian notion may have covered more, so to speak, than mine. Furthermore, this way I adopt of distinguishing between internal and external aspects of meaning – although it owes a lot to Frege's distinction between sense and reference and similar earlier distinctions, and does some of the same work, is not quite there even in the later Wittgenstein, although his work no doubt helped us along the way to it. Rather, it begins in a clearly recognizable form with Putnam's Twin Earth thought experiment and subsequent discussions of semantic externalism.

I will try to explain what I mean by 'projective relations' by explaining how, and why, this notion differs from that of 'referential relations': we can distinguish between my name 'O' for some object in my environment, and a name 'O' used in exactly the same way (internally speaking), but for another object, e.g. by a Twin Earth counterpart of me. For this, the notion of referential relations does the job. But there is an analogous distinction we can make for terms which fail of reference. For example, 'John' when I falsely believe that a man called John came to my friend's house, when my friend has in fact fabricated the whole thing, and 'John' used in the same way on Twin Earth. Both terms fail to hit a mark, so to speak, but their trajectories are in different places. Or: both fail to catch anything in their net, but the nets (which are the same) are cast in different places. So we can say that 'John' and Twin Earth 'John', as well as 'O' and Twin Earth 'O', bear different projective relations to the environment. As for referential relations on the other hand, both 'John' and Twin Earth 'John' bear none, and so cannot be differentiated on that score.

To be clearer about what I mean by '(internal) uses or meanings': by 'internal use' I do not mean to include all the historical facts about how the symbol gets used – rather, I use it similarly to 'internal meaning', namely to mean something like: an expression's role in the language-system it belongs to. Uses or meanings can be individuated in different ways – at different granularities, and factoring in different sorts of features..

So, we may say that propositions are propositional signs together with their internal meanings and their external meanings. And internal meanings can be carved up at different granularities – what you may at one granularity call two instances of the same proposition may at another granularity count as (instances of) different propositions.

As with my account of names, this account of propositions can only be properly understood once the distinction I make between internal and external meaning, and my doctrine of semantic granularity, are properly understood. I will try to explain these things in forthcoming posts.

There seems to me to be an important methodological difference between, on the one side, this conception of propositions which I advocate, and on the other side, certain conceptions prominent in contemporary analytic philosophy. The conceptions I have in mind could be called more technical – we may roughly characterize them by saying that they perform theoretical identifications; they conceive objects, often using formal methods such as basic set theory, and then identify propositions with these. Two basic examples are:

(1) Sets of possible worlds conceptions, on which, for example, the proposition that snow is white is identified with the set of possible worlds in which snow is white.

(2) Russellian conceptions, on which, for example, the proposition that Socrates is mortal is identified with the n-tuple whose first member is Socrates, and whose second member is mortality.

And things get more sophisticated from there – to take three examples: Russellian annotated matrices, two-dimensional semantic values, and objects comprising, among other things, sentences of formal languages.

It is important to realize that there is a methodological difference here, lest it appear that my account is so to speak on the same level, playing the same game, only less precise and less technically developed than, e.g., those mentioned above.

To my way of thinking, those objects which are sometimes identified with propositions are models of, so to speak, real propositions. It may in some cases of other views be indeterminate, or in determinate cases may vary from case to case, whether there is substantive disagreement here or only terminological difference. I am not trying to police other people's use of words, and if they want to call technical constructions like those mentioned above 'propositions', I am not going to object (although I prefer to talk differently). But when these things are said to be – without the words I am about to use being given special technical meanings – objects of belief, or meanings of sentences, or the things we convey in communication, or work with in deductive arguments, I become uneasy at the very least, and in many cases (e.g. set-of-worlds conceptions) disagree confidently. Such ideas seem like category-mistakes to me, or perhaps more true to the case, so impossibly revisionary that they constitute methodologically misguided philosophy (I am thinking of David Lewis here).

My method, then, is to work with semantic concepts more as they are, treating them as more sui generis. I think this is what Wittgenstein did, and what Moore did (I have not gotten a lot out of Moore directly), and I think it is also the way Kripke works in Naming and Necessity and auxilliary works like 'A Puzzle about Belief' (but this is not to deny that much of that work was informed and inspired by Kripke's formal work). I am trying to refine special versions of these notions for philosophical purposes – I am not doing ordinary language philosophy, for example, or concerning myself directly with how “the folk” think about meaning. Rather, I am using my natural lights as one of those folk, together with what I've learned from philosophers, and trying to do philosophy of language in a way which stays close to the phenomena and works with intuitively compelling considerations.

One reason this sort of methodology has fallen out of favour in some circles, I think, is Quinean skepticism about semantic concepts, and hangovers therefrom. Few still go as far as Quine, but the confusing individuation-behaviour he cited as his main reason for abandoning serious use of semantic concepts - the phenomenon I explain in terms of semantic granularity-shifts – seems now to militate, not toward self-conscious abandonment of semantic concepts, but supposed “reconstructions” or cleaned-up versions of them which are so far from the real thing, that they are better thought of as models – models which involve serious idealizations and often great limitations.

So far, I have said in broad outline what propositions are on my account: propositional signs together with their internal meanings (i.e. systematic roles) and their external meanings. I have said a little bit about internal/external meaning, deferring further explanation to other posts. And I have made some methodological remarks about what sort of account this is.

A word on my concept of 'propositional sign': this is no purely syntactic category (whatever that means exactly). Its objects – the things which are propositional signs – are indeed to be regarded as syntactic items, not intrinsically carrying meaning, but the concept which picks them out should probably be thought of as doing so via the broadly semantic notion of a proposition. That is, propositional signs do not count as such in virtue of their intrinsic syntactic features, but because they get used propositionally. This of course means that we haven't got any kind of reductive analysis here of 'proposition', or a 'propositional sign', or 'used propositionally' for that matter, but that was never the intention. (I am sympathetic to Wittgensteinian views on this matter. For example, I think it can be said that 'proposition' is not a sharply bounded concept. Perhaps it can be said to be a family resemblance concept. Perhaps it can be said to function by means of paradigms.)

A word on type-token ambiguity: I have deliberately left my account type-token ambiguous. Just as with sentences, I think we should have a type-token ambiguity which can be resolved when need be, when talking about propositions on my conception. So that we can say (in “token” mode) things like 'Regarding that proposition written on that piece of paper ...' and 'Regarding the proposition that just came out of your mouth ...' without any inaccuracy (e.g. without having to maintain that, strictly speaking, we were talking about instances of propositions, so that to make what we said both explicit and literally correct we should have to add 'instance of a' before 'proposition' in the exampled phrases), but also, for example, 'He uttered the same proposition as I did'.

A word on usage: 'proposition' is often used in such a way that no particular signs – whether tokens or types – are part of the entity in question. For example, when it is said that 'Snow is white' and 'Schnee ist weiss' 'express the same proposition'. I have no special problem with talking like that, and am open to doing it myself in informal contexts or the context of someone else's philosophy, but on my account, and the way I like to use words in conjunction with it, what we are talking about in such cases is not a proposition, but a proposition-meaning. (We may think of proposition-meanings as comprising both internal and external aspects, as propositions minus the sign, so to speak. And of course we may also pick out just one component.)

I will conclude by mentioning some applications to, or connections with, other topics. One application of the account is in solving Frege's Puzzle, in the way indicated in the post on names. Another, where granularity comes into its own, is with Kripke's Puzzle (about belief), which I will discuss in a future post. Among other applications which will be dealt with in future posts are accounts of three major notions in propositional typology: the a priori/empirical distinction, the analytic/synthetic distinction, and the distinction between necessity and contingency in the metaphysical or subjunctive sense.

References

Wittgenstein:

Wittgenstein, Ludwig (1974). Philosophical Grammar. Blackwell.

Wittgenstein, Ludwig (1975). Philosophical Remarks. University of Chicago Press.

Wittgenstein, Ludwig (1922). Tractatus Logico-Philosophicus. Dover Publications.

Other:

Saul A. Kripke (1980). Naming and Necessity. Harvard University Press.

Saul A. Kripke (1979). A puzzle about belief. In A. Margalit (ed.), Meaning and Use. Reidel.

Sidelle and the Contingency of Conventions: An Objection Regained

This is a draft of a paper.

Alan Sidelle's conventionalism about modality is well-known, and only a bit less of a whipping-boy than the earlier positivistic views of Ayer (1936) and Carnap (1947). Published in his 1989 book Necessity, Essence and Individuation: A Defense of Conventionalism, it focuses on the problem of explaining the existence of the necessary a posteriori from a metaphysical standpoint according to which (in Sidelle's phrase) we, and not the world, are the source of modality. (I think this isn't very clear, but I don't want to press that here.)

I will not give a proper exposition of Sidelle's account here. I will just say that the basic idea is that a modal claim such as 'Necessarily, water is H20' follows from the a posteriori claim 'Water is H20' together with an a priori claim, such as 'If water is H20, then necessarily, water is H20'. The idea is that the a priori claim is somehow a matter of convention.

There are many problems with such a view – see Yablo's incisive (1992) review for a start. For example, can the a priori claim really be said to be a matter of convention? How doesn't this fall prey to the following argument?: it may be that what sentences mean is conventional, but we can't make the propositions they mean true by convention, except for the special case of propositions about conventions. (Yablo calls this the Lewy point, citing Lewy (1976). See also Quine (1936).) And even if this is somehow surmounted, aren't we still in the dark about what necessity is? (While Sidelle's aims, when stated carefully, do not seem to include saying what necessity is, some of his more impressionistic rhetoric does seem to try to say something about that. In any case, his account leaving us in the dark about the nature of necessity, if it does, is something worth taking due note of, since it is commonly taken to be addressing that issue.)

Here I will discuss another central objection (or type of objection) – that from the contingency of conventions. Or rather, Sidelle's recent response to it; in a 2009 paper called 'Conventionalism and the Contingency of Conventions', Sidelle defends his conventionlism about modality from this sort of objection. He carefully distinguishes two objections here, one focusing on truth-making, the other on necessity-making:

1. Truth-making version. If conventions were different, certain necessary truths would not be true. This seems to follow from conventionalism, catching it in a contradiction – since what it is to be a necessary truth is not failing to be true in any circumstances.
2. Necessity-making version. If conventions were different, certain necessary truths may have been contingent. This seems to follow from conventionalism, but seems wrong.

Sidelle argues (convincingly, in my view) that (1) is wrong – the conventionalist isn't committed to that. (I refer readers to his paper for this.)

Sidelle acknowledges (2) to be more serious, and devotes his paper to responding to it. Here, I will argue that his response to (2) fails at an early step, for use-mention reasons.

Sidelle considers but rejects one possible avenue of response, a partly bullet-biting response which says: OK, so this shows that, at least sometimes, what is necessarily so may not have been necessarily so (and also that, at least sometimes, what is contingently so may not have been contingently so). Such truths, then, are contingently necessary and contingently contingent, respectively. This is tantamount to rejecting the characteristic axiom of S4 – that what is necessary is necessarily necessary.

Sidelle will not have this. It is simply too implausible that the S4 axiom fails for metaphysical modality. Indeed, there is reason to think that the appropriate system is S5 (since an unrestricted accessibility relation seems appropriate), which is stronger than S4. Furthermore, he says, conventionalists, in his opinion, ought to try to “save the modal phenomena” and not be highly revisionary.

He also has an argument to the effect that even biting this bullet wouldn't suffice, but I do not understand that argument (I think because it involves certain confusions bound up with Sidelle's form of conventionalism, but I won't try to go into that here).

Sidelle's strategy with (2) is to consider an example – that of 'bachelor', and what would be the case if our conventions governing it were different – and try to show that, if we are careful to stick to the proper mode of evaluating counterfactuals, namely where we keep our conventions, and the meanings of our terms, intact, we can see that the relevant (2)-like counterfactuals are not true.

Sidelle supposes for the sake of argument that our conventions make it that 'bachelor' applies to unmarried but eligible men, and not women, and then considers an alternative situation in which the conventions differed so that unmarried, eligible women fell in the extension of 'bachelor':

With such a convention, we would call unmarried Linda ‘a bachelor’, and so, ‘necessarily, bachelors are male’ would be false. However, how should we describe this situation? Is Linda a female bachelor? Of course not—someone counts as a bachelor only if they are male. Our rules for applying ‘bachelor’ tell us that one must be (give or take) ‘a never-been-married, but eligible male’ [footnote 14]—so ipso facto, the rules tell us that what rules the speakers in that world use is quite irrelevant to whether or not someone is a bachelor. They are no more relevant than the rules of Spanish if we are, in English, describing a situation in Mexico. And of course, this is perfectly general. Notice that this has nothing at all to do with Conventionalism—it is what anyone should believe about evaluating counterfactuals, when those counterfactuals contain words governed by certain semantic conventions—and of course, one doesn’t need to be a Conventionalist to believe there are at least some, or even many, such conventions. [footnote 15] And as the conventions in that situation are irrelevant to the truth of ‘Linda is a female bachelor’, so are they to the question of the necessity of bachelors’ being male there, and so, to whether our necessary truth is itself necessarily so (i.e. to whether or not it is necessarily necessary that bachelors are male). Thus, if the conventionalist story is correct, it will not be true that ‘had our conventions been different, what is necessary would (could) have been false’, or not necessary.

The first part of this quote is an unexceptionable rehearsal of how to evaluate counterfactuals dealing with situations where the meanings of words differ: don't get confused into using the words with those different meanings in describing the situation: it isn't the case that, if 'tail' meant 'leg', dogs would have four tails – although 'Dogs have four tails' would be true in such a situation, ceteris paribus. So while 'Dogs have four tails' would, in that situation, say something true, it does not actually say something true of that situation, i.e. about what happens in that situation.

Similarly with the sentence 'Linda is a female bachelor' – it would say something true in the situation in question, but it isn't – given what it actually means – actually true of that situation.

So far, so good. The trouble is in the last two sentences, when Sidelle tries to conclude from his unexceptionable rehearsal that it's not the case that, if our conventions were different, which propositions are necessary might be different. The last sentence just gives the conclusion. The whole argument, really, is in the second last sentence, so we will concentrate on that. Here it is again, with two words capitalized by me: 

And as the conventions in that situation are irrelevant to the truth of ‘Linda is a female bachelor’, so are they to the question of the necessity of bachelors’ being male there, AND SO, to whether our necessary truth is itself necessarily so (i.e. to whether or not it is necessarily necessary that bachelors are male).

Firstly, there is an ambiguity in Sidelle's phrase 'the truth of “Linda is a female bachelor”'. The conventions in that situation are obviously not irrelevant to the truth, in that situation, of the sentence 'Linda is a female bachelor'. But it is true that they are irrelevant to whether or not that sentence is actually true of the situation: it isn't of course, because there can't be female bachelors in any possible situation. So we can accept this and move on to see what Sidelle is likening it to.

The way Sidelle has put the point, it is not easy to see what the similarity is. The conventions in that situation are irrelevant to the truth of some sentence here, and similarly, to the necessity of bachelors being male there? The points would seem more similar if Sidelle semantically descended for the first bit: just as the conventions in that situation are irrelevant to whether Linda is a female bachelor in that situation, so too are they irrelevant to whether the bachelors are necessarily male there.

In any case, the point can be accepted: bachelors are necessarily male, in all situations. So in a situation where the conventions were different, any bachelors would still need to be male.

But, and this is the crucial point, in saying this, we are using our language, with our conventions, and describing a counterfactual scenario. Our proposition 'Necessarily, all the bachelors are male' is true of that situation. Call the situation S – our more explicit proposition 'Necessarily, all the bachelors are male in S' is true. And you can substitute for 'S' the name of any possible situation.

To ask of that situation, of S, whether the bachelors are necessarily male there, is palpably not to ask whether the proposition that bachelors are male – the proposition now, or whatever thing bears modal statuses, not just the sentence – is necessarily true in that situation. That question just hasn't been raised.

And this is why 'AND SO' is capitalized – it is spurious. It just doesn't follow from all the bachelors in situation S necessarily being male – that's us describing the scenario from here, remember – that the proposition that bachelors are male is necessarily true in that situation – and so you can't conclude from it that some proposition of ours which is necessarily true is necessarily necessarily true. Of course, such a conclusion is itself plausible, but that doesn't mean Sidelle – a conventionalist about the modal statuses of propositions – is entitled to it! And his argument only gets there by means of a subtle, illicit use-mention shift.

Having established to his satisfaction that he is not committed to what is necessary varying with convention, Sidelle then faces the task of explaining why the following plausible constraint on explanation fails in this instance: if A explains B, it can't be that no change in B would ever come about if A changed.

I think there are serious problems with his attempt, and I hope to make this clear in future. My purpose here has just been to show that the previous step, which led Sidelle to having to face this question about explanation, is fallacious. Sidelle has slid from mention to use in the consequents of the counter-conventional, counterfactual conditionals at issue: he can agree with everyone else that, if conventions had been different, any bachelors would still necessarily be male, but this is not the same as being able to agree that, if conventions had been different, the proposition that any bachelors are male would still be necessary. His argument from common knowledge about how to evaluate counterfactuals does not succeed in earning him the right to the latter, only the former. We can conclude from this alone that Sidelle has not adequately responded to the (necessity-making focused) objection from the contingency of conventions. 

References 

Ayer, A.J. (1936). Language, Truth and Logic. London, V. Gollancz, Ltd.

Carnap, Rudolf (1947). Meaning and Necessity. University of Chicago Press.

Lewy, Casimir (1976). Meaning and Modality. Cambridge University Press.

Quine, W.V. (1936). Truth by Convention. In The Ways of Paradox and Other Essays.

Sidelle, Alan (2009). Conventionalism and the contingency of conventions. Noûs 43 (2):224-241.

Sidelle, Alan (1989). Necessity, Essence, and Individuation: A Defense of Conventionalism. Cornell University Press.

 Yablo, Stephen (1992). "Review of Alan Sidelle, Necessity, Essence and Individuation." Philosophical Review 101: 878-81.

Names: Between Mill and Frege

Followup posts: 

Empty Names and Negative Existentials

Names and the Articulation Assumption

Names and the Publicity of Meaning

(Added October 2016: my most up-to-date treatment of names can be found in Chapter 6 of my PhD thesis. This is an early, undeveloped attempt.)

Kripke at the beginning of 'Vacuous Names and Fictional Entities':

One of the main concerns of my previous work (Kripke 1980) [Naming and Necessity] is the semantics of proper names and natural kind terms. A classical view which Putnam mentioned, advocated by Mill, states that proper names have as their function simply to refer; they have denotation but not connotation. The alternative view, which until fairly recently has dominated the field, has been that of Frege and Russell. They hold that ordinary names have connotation in a very strong sense: a proper name such as ‘Napoleon’ simply means the man having most of the properties we commonly attribute to Napoleon, such as being Emperor of the French, losing at Waterloo, and the like. Of course, intermediate views might be suggested, and perhaps have been suggested.

My aim here is to propose just such an intermediate view. In future posts I will flesh out the proposal and offer some speculations about why it has not been generally adopted already.

As is well known, the cardinal problem with Millianism about names is Frege's Puzzle, given in his famous article 'On Sense and Reference': Millianism leaves us unable to semantically distinguish, in a systematic compositional way, 'Hesperus is Phosphorus' from 'Hesperus is Hesperus'; since Hesperus is Phosphorus, both names involved in those propositions have the same referent, and thus, on the Millian view, the same meaning. But the two propositions do not seem to have the same meaning - the first is an empirical scientific discovery, and the second is not. Further problems arise with singular negative existentials like 'Santa does not exist' - here, in addition to Frege's-Puzzle-type problems of differentiation ('Santa doesn't exist' and 'Noddy doesn't exist' mean different things, even though both names are the same as regards their referents; neither has one), we have the problem of seeing how any of them could be true, or even mean anything.

As is also well known, the cardinal problems for descriptivism - what Kripke above calls the view of Frege and Russell - are given by Kripke in Naming and Necessity. I will not try to summarize Kripke's whole case properly, but it is often divided into three prongs: the semantic objection, the epistemological objection, and the modal objection. Very roughly, the semantic objection is 'Which description or descriptions constitute the meaning of some given name? Isn't any answer bound to be arbitrary? And since different people might associate different descriptions with the same name and the same object, how aren't they just talking past each other when they use the name?'. The epistemological objection is 'Take any plausible meaning-giving description or cluster thereof. I can surely use the name in question correctly without knowing all this - I don't have to know much of anything about someone in order to pick them out with a name'. And the modal objection is 'Suppose "Aristotle" means "the teacher of Alexander". How does it come about, then, that "Had things gone differently, Aristotle might not have been the teacher of Alexander" seems true, while "Had things gone differently, Aristotle might not have been Aristotle" seems false?'.

I am not concerned here to argue that no versions of Millianism or descriptivism have legs. I will not, for example, argue against Millianisms which try to fill the apparent semantic gap in their accounts by means of linguistic pragmatics, nor will I argue against "wide-scope" or "actualizing" versions of descriptivism. I want to propose what I think is an elegant and natural view in between Millianism and descriptivism which avoids the problems of both.

I have two ways of expressing what I take to be essentially the same view, but others may prefer to think of me as offering a disjunction of two structurally similar views. The first is in terms of the use of a name, or the role it plays in the system of language to which it belongs. The second is in terms of individual concepts - the ideas of particular objects which we (sometimes) tie names to. I more-or-less identify these things in my own thinking, but since the concepts 'use' and 'role' on the one hand, and 'concept' and 'idea' on the other are quite different (as are the uses or roles of these terms!), it is worth giving both formulations, in case someone prefers one over the other. The use-conception is inspired, at least in part, by Wittgenstein (particularly the middle period, for example Philosophical Grammar). The concept/idea-conception is more traditional.

The view, then, is that names have uses, or are tied to individual concepts, and that these are partly constitutive of their semantics. (We may say that uses, or individual conepts, constitute the internal meanings of names.)

Individual concepts or name-uses do some of the work Frege wanted to do with his senses, but there are important differences. One important difference is that Frege held, of his senses, that they determine reference, whereas individual concepts or name-uses avowedly do not do this in general; someone on Twin Earth can use a name in the same pattern (i.e. with the same concept or use), but with a different referent - in a word, semantic externalism is true of individual concepts or name-uses. (We can of course have a notion which adds an extensional component to the individuation of these things, so that two concepts or uses are distinct if they have different objects, or different projective relations to reality, but we can also isolate the internal component.)

Another important difference is that, while Frege indicates that the sense of a name is that of, or can be given by means of, a definite description, I hold no such thing with respect to individual concepts or name-uses. Names are in an important sense indefinable, as Wittgenstein held in the Tractatus. But that does not mean their referents are all there is to (what you might, though possibly misleadingly, call) their meanings, i.e. Millianism doesn't follow. (Wittgenstein expresses Millianism too in the Tractatus, when speaking of 'names', although there is an exegetical question whether this term is meant to cover ordinary proper names, which is what I am talking about, or names in some philosophically idealized sense, e.g. names in an "ideal logical language".)

Individual concepts or name-uses combine, in a very simple way, the difference-making power of Frege's senses with invulnerability to Kripke's arguments against descriptivism. 

Frege's Puzzle is solved, much in the same way as Frege did with his senses: 'Hesperus is Phosphorus' and 'Hesperus is Hesperus' are different propositions with different meanings, since 'Hesperus' and 'Phosphorus' are tied to different individual concepts and have different uses. (Likewise with 'Clark Kent' and 'Superman'.)

But unlike with Frege's senses, this conception of names is not only compatible with Kripke's rigid designation thesis, but predicts it, at least when formulated in terms of individual concepts: if names are associated with individual concepts - concepts of particular objects - then it is immediate that they will designate the same object in all possible worlds where that object exists; designating another object is out of the question, since we are holding fixed the meaning of the proper name - the associated individual concept.

Individual concepts or name-uses also allow for an important kind of flexibility, which we must recognize in order to solve Kripke's puzzle about belief. We can individuate name-uses and individual concepts - as well as the uses or meanings of other terms, and other concepts, and larger units such as propositions - at different granularities, so that what at one granularity might count as instances of different uses/concepts may count on another (coarser) granularity as instances of the same. This is an important ingredient of my view, and will be discussed in a future post.

I have now at least mentioned all the main ingredients of the view of names I want to propose. Further posts - on semantic granularity, on internal and external meaning, and on why the view I propose hasn't already been generally adopted - will fill out the picture. I will conclude this post by trying to avert a couple of possible misunderstandings:

(1) The term 'individual concept' is sometimes used in philosophical logic and technical philosophy of language to refer to functions from possible worlds or state-descriptions to individuals (or similar constructions). I am not using it that way. For one thing, that way of going would make the problem of empty names harder - i.e., it would reduce the utility of my approach with respect to the problem of empty names - since one needs individuals for the functions to map to. For me the notion of an individual concept is more basic - it is just a refined version of the ordinary idea of an idea of an object.

(2) I am not saying that any sort of theory which associates names not with name-uses or individual concepts (in my sense), but with something else, and calls the associates 'the semantic values' of the names, is wrong. My attitude here is that expressed by Chalmers in this passage from 'The Foundations of Two-Dimensional Semantics':

A methodological note: in this paper I will adopt the approach of semantic pluralism, according to which expressions can be associated with semantic values in many different ways. Expression types and expression tokens can be associated (via different semantic relations) with extensions, various different sorts of intensions, and with many other entities (structured propositions, conventionally implied contents, and so on). On this approach, there is no claim that any given semantic value exhausts the meaning of an expression, and I will not claim that the semantic values that I focus on are exhaustive. (I think that such claims are almost always implausible.)

References

Chalmers, David J. (2006). The foundations of two-dimensional semantics. In Manuel Garcia-Carpintero & Josep Macia (eds.), Two-Dimensional Semantics: Foundations and Applications. Oxford University Press.

Frege, Gottlob. (1952). first pub. 1893. ‘On Sense and Reference’, in P. Geach and M. Black (eds.) Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell.

Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.

Kripke, Saul A. (2011). Vacuous Names and Fictional Entities. In Saul A. Kripke (ed.), Philosophical Troubles. Collected Papers Vol I. Oxford University Press.

Wittgenstein, Ludwig. (1922). Tractatus Logico-Philosophicus. Dover Publications.

Wittgenstein, Ludwig. (1974). Philosophical Grammar. Blackwell.

Formal Logic Isn't a General Science of Validity

'Tell me how you are searching, and I will tell you what you are searching for.' - Wittgenstein.

Introduction 

A curious feature of the philosophical landscape is that the most fundamental and influential discussions of several key issues in the philosophy of formal logic occur, in large part, in introductory logic textbooks. There's something Wild-West-ish and exciting about this, even if it stems partly from regrettable neglect of these issues. While proficient philosophically-trained logic teachers all impart a certain core of technical understanding, they differ greatly on what they tell their students formal logic is all about.

Here I want to give an argument for a thesis in the philosophy of formal logic:

(NGV) Formal logic is not aptly regarded as having, among its aims, that of giving a general account of validity.

This argument, if any good, should be of considerable interest, since the view which (NGV) denies – call it (GV) – is widely held, and plays a large role in our collective understanding of the nature of formal logic. I will use as a foil N.J.J. Smith's 2012 textbook Logic: The Laws of Truth, which endorses (GV).

My argument in compressed form is as follows: there are glaring lacunae in formal logic construed as a general account of validity; if that were one of its aims, it would be blatantly neglecting certain basic phenomena which it is meant to account for. But no one with any close relationship to formal logic cares, and we cannot put this down to laziness or stupidity. Therefore, it is inappropriate to construe formal logic as having a general account of validity among its aims. 

What I Mean by 'Formal Logic' and 'Validity' 

Before explaining the argument further with an example, a brief word about what I mean by 'formal logic' and 'validity'. 'Formal logic' means the formal, mathematized discipline of logic – syntax, formal semantics, proof-theory. Observations about the validity of natural language arguments, even if validity can be aptly conceived as turning on the form of propositions, are not themselves bits of formal logic, on this usage. By 'validity' I mean the thing which gets explicated sometimes as 'necessary truth preservation in virtue of form', or 'necessary truth-preservation in virtue of the meanings of logical (or subject-matter neutral) vocabulary', but also in other ways. A key point is that the valid inferences are a subset of the deductive (in a common sense of 'deductive'). We may deduce 'There is a vehicle' from 'There is a car', but this is not a valid inference in the relevant sense, since it turns on 'vehicle' and 'car'. 'There is a car, therefore there is a car or a vehicle', on the other hand, is valid. We say that the conclusion of a valid argument is a logical consequence of the premises. (Often what I am calling 'validity' here gets called 'logical validity', or even 'narrowly logical validity'. This may not be a bad practise, but here I follow the logicians who use 'validity' in this already-narrow way.)

There are different ways of spelling out what validity consists in. 'Valid' may also be a vague or indeterminate predicate (if, for example, the form/content distinction or the distinction between logical and non-logical vocabulary is vague or indeterminate), but that doesn't matter.

This should suffice to identify the relevant basic intuitive idea of validity, an idea which plays a major role in thinking about logic. It is not our purpose here to investigate the notion of validity, or the nature of validity, any further. We just needed to get it clear enough for the purpose of giving the argument for (NGV).

(GV)

Here is a clear expression of a conception involving (GV) - the view that formal logic is aptly regarded as having, among its aims, that of giving a general account of validity - in The Laws of Truth (pp.20-21, Chapter 1: Propositions and Arguments):

When it comes to validity, then, we now have two goals on the table. One is to find a precise analysis of validity. (Thus far we have given only a rough, guiding idea of what validity is: NTP [necessary truth-preservation - TH] guaranteed by form. As we noted, this does not amount to a precise analysis.) The other is to find a method of assessing arguments for validity that is both

1. foolproof: it can be followed in a straightforward, routine way, without recourse to intuition or imagination—and it always gives the right answer;
and
2. general: it can be applied to any argument.

Note that there will be an intimate connection between the role of form in the definition of validity (an argument is valid if it is NTP by virtue of its form) and the goal of finding a method of assessing arguments for validity that can be applied to any argument, no matter what its subject matter. It is the fact that validity can be assessed on the basis of form, in abstraction from the specific content of the propositions involved in an argument (i.e., the specific claims made about the world—what ways, exactly, the propositions that make up the argument are representing the world to be), that will bring this goal within reach.

One of the philosophically most important sections of the book is 14.4, 'Expressive Power', where four kinds of propositions are discussed which may seem to create difficulties for first-order logic with identity (FOL=) construed as a general account of validity (what follows is summary and paraphrase, not quotation):

(1) 2 + 2 = 4. (Translatable into FOL=, but the result is considerably more complex, and this may make us hope for something more elegant.)
 
(2) She will see the doctor but she hasn't yet. (Quantification over times is required to translate this adequately into FOL=, and, like with (1), we might hope for something else. And there is something else: tense logic.)

(3) Propositions involving vague predicates. (FOL= requires predicates to have definite extensions. But there is fuzzy set theory and other formal tools which could deal with these.)

(4) 'There are finitely many Fs'. (Can't be expressed in FOL=, but there are extensions in which it can be.)
 
Note carefully that Smith is not claiming that classical first-order logic is enough for a general account of validity. He admits, for example, that it cannot adequately represent propositions of type (4). But none of this casts any real doubt on (GV).

I want to bring a different kind of example into the mix. It will perhaps seem a bit boring or basic, but that's actually what makes it philosophically important.

Example: From 'Only' to 'All' 

Consider this argument:

Only horses gallop, therefore all gallopers are horses.

This is at least as non-trivial as 'A and B, therefore A', which formal logic deigns to capture.
 
I think this is, in a way, a more philosophically instructive sort of "problem for classical logic", in that it doesn't seem like much of a problem, and this gives us a way of seeing that (NGV). The argument isn't "fancy", it doesn't seem to make us want new systems, and the problem-propositions involved are translatable in a loose sense with no worries at all. But there is no way of capturing the inference in first-order logic. You either wind up with the premise the same as the conclusion, or a premise which differs from the conclusion (e.g. by using existential quantification), but no more similar to the natural language premise than the formal conclusion is, and so not capturing the original inference at all. 

This translatability in a loose sense is important. Since we can quite easily see that only Xs Y if and only if all Y'ers are X's, we just translate 'only' propositions using ordinary first-order quantifiers. But the logical insight we needed for that was at least as real and non-trivial as that which we need to see that 'A and B' implies 'A'.
 
We must ask the logician: if you're bothering to codify things like 'A and B, therefore A', and you're quite generally interested in the validity of arguments in natural and other languages, why are you so content to leave 'Only horses gallop, therefore all gallopers are horses' unaccounted for by your science?

The fact that we are so content shows (GV) to be misleading, indeed false. In doing formal logic, we are actually doing something quite different from: trying to construct a general account of validity.

It seems like what we are really doing is something more like: developing artificial languages fulfilling certain desiderata, arguments in which can be easily inspected for validity. (Quine, regimentation.) Or perhaps, getting a sense of what validity turns on, without trying to give a general account. (Realistically, it's probably both of these plus many others, right down to aesthetic or even social motivations.) At least, if we were doing either of those things, our formal neglect of the Only-to-All argument would be quite intelligible.

Again: if we're so keen on a general accoubt of validity, how can we be so ready to not care about having no formal account of 'Only Xs Y, therefore all Y'ers are Xs'? Because it is so trivial? That can't be right, since things we do try to capture are no less trivial. No, there is no way out - the fact that we don't care shows that we really aren't so keen on a general formal theory of validity. That idea doesn't capture the real life of formal logic.

We might tell ourselves we want a general account of validity, but then when we're underway, we think 'Why bother formalizing "only", when we've already formalized "all" and can use that?' - the fact that we take that attitude gives the lie to the idea that we're really after a general account of validity.

We sense that it would be fairly uninstructive to formalize 'only' once we've got 'all' etc. - we should let that help guide our thinking about what we're really doing when we do formal logic (hence the Wittgenstein quote at the beginning).

That is my argument for (NGV) and against (GV).

Other Kinds of Examples
 
The above example may be resisted on the grounds that 'only' has been studied from within Montague semantics, and perhaps from other perspectives. My reply to that, in the first instance, is that these contributions were not generally thought of as part of formal logic, nor have they come to be incorporated into it. I think that's the right reply, but it isn't entirely satisfying - it might seem overly legalistic and trusting in the status quo. Fortunately, other examples are to hand.

What other kinds of examples are there, besides the Only-to-All argument, which show the same thing? A couple more I have thought of are:

- 'All men are mortal, therefore everything is such that its being a man materially implies its being mortal', or 'There are men, therefore there is something which is a man'. That is, arguments which take us from basic natural quantifier-constructions to something of the quantifier-variable form. These arguments embody logical insights due to the inventors of quantification theory (such as Frege and Peirce), and yet it seems we don't care about giving formal accounts of the arguments themselves.

- Arguments involving truth-functional compounds where the premise is written in some non-standard notation (such as: Venn diagrams, Gardner-Landini shuttle diagrams, Wittgensteinian ab-notation, truth-tables construed as propositional signs), and the conclusion is written with regular connectives (or vice versa).

These two kinds of examples differ from the Only-to-All argument in involving technically-formed propositions, whereas 'All gallopers are horses' and 'Only horses gallop' are forms that come naturally to us.

Philosophers' Carnival #156

Welcome to the 156th edition of the Philosophers' Carnival! It's exactly one year since I took over organizing duties from Richard Chappell, so I have chosen to host this month.

Below the Carnival proper you will find a Metacarnival covering the last twelve months' editions (procrastinators beware: a large amount of absorbing content lies only 1 to 2 clicks away).

- Are basic moral principles empirically falsifiable? - by David Sobel at PEA Soup. 

- The Nature of Desire: A Liberal, Dispositional Approach - by Eric Schwitzgebel at The Splintered Mind.

- The Space of Languages - by Jeffrey Ketland at M-Phi. 

- A Problem for Hume's Problem of Induction - by Jason Zarri at Philosophical Pontifications.

- Respect for Truth in Science and the Humanities - by David Maier at 3QuarksDaily.

- Pretend Numbers - by Richard Brown at One More Brown.

- Alief and Knowledge from Fiction - by Allan Hazlett guesting at Aesthetics for Birds. 

- Seven Puzzles of Pictorial Content - by Gabriel Greenberg guesting at Aesthetics for Birds.

- Theistic frequentism and evolution - by Alexander Pruss at Prosblogion.

- Testimony and Moral Understanding - by Richard Chappell at Philosophy et cetera.

- Pets, Livestock, and Narrative Value - by David Killoren guesting at Philosophy et cetera. 

- Against countable additivity - by Wolfgang Schwarz at wo's weblog.

- The Libet experiment as a refutation of dualism - by Bill Skaggs, a neuroscientist guesting at Brains. 

- Relativism about Epistemic Modals: Some Experimental Data - by Joshua Knobe and Seth Yalcin at Certain Doubts.

- A voting puzzle, some political science, and a nerd failure mode - by Chris Hallquist at Lesswrong.

- On the Limitations of Philosophical Writing, or What is Wisdom? - by Eric Schliesser at NewAPPS. 

- Why is it easier to get crap published? The Stanley Hypothesis - by Jon Cogburn at NewAPPS.

- Skolemizing Laws - by Robbie at Metaphysical Values.

- The Truth-Tracking Account of Knowledge: Two New Counterexamples - by Tristan Haze (me) here.

The next edition will be hosted next month at FsOpHo, a blog about logic and epistemology. Submit here.


Metacarnival 

#155: Blogging the End - Septermber 10, 2013.

#154: Nick Byrd's Blog - August 10, 2013

#153: Philosophy on Philosophy - July 10, 2013

#152: Siris - June 10, 2013

#151: Camels With Hammers - May 10, 2013             

#150: The Splintered Mind - April 10, 2013
#149: kennypearce.net - March 10, 2013
#148: In Search of Logic - February 10, 2013
#147: Philosophy and Polity - January 10, 2013
#146: Talking Philosophy - December 10, 2012
#145: Philosophical Pontifications - November 10, 2012
#144: Sprachlogik - October 10, 2012

The Truth-Tracking Account of Knowledge: Two New Counterexamples

In recent years Nozick's notion of knowledge as tracking truth has witnessed a revival. - Horacio Arló-Costa, 2006.

[This is a draft of a paper.] [Added 3/9/15: The paper is forthcoming in Logos & Episteme.]

Here I present two counterexamples to the truth-tracking account of knowledge. As far as I have been able to tell, they are new.

The simple version of Nozick's famous (1981) truth-tracking account runs as follows:

S knows that p iff
1. p is true
2. S believes that p
3. If p weren’t true, S wouldn’t believe that p
4. If p were true, S would believe that p

Counterexample 1: I have a deep-seated, counterfactually robust delusional belief that my neighbour is a divine oracle. He is actually a very reliable and truthful tax-lawyer. There is a point about tax law he has always wanted to tell me, p. One day, he tells me that p, and I believe him, because I believe he is a divine oracle. I would never believe him if I knew he was a lawyer, being very distrustful of lawyers.

In this case, it seems to me, I do not know that p: my belief rests on a delusion, albeit a counterfactually robust one. But it is true, I believe it, and my belief tracks the truth: if it were true, I would have believed it, and if it were false, I would not have believed it. (The lawyer, being reliable and truthful about tax law, would not have told me that p if p were not the case.)

Counterexample 2: My neighbour is a tax lawyer. Here, unlike in the previous counterexample, I have no delusional belief. It is my neighbour who is the strange one: for years, he has intently nurtured an eccentric plan to get me to believe the truth about whether p, where p is a true proposition of tax law, along with five false propositions about tax law. His intention to do this is very counterfactually robust. He moves in next door and slowly wins my trust. One day, he begins to regale me with points of tax law. He asserts six propositions: p and five false ones. I believe them all.

It seems to me that I do not know that p in this case either. But I believe it, it is true, and my belief tracks the truth: if p were the case, I would have believed it, and if p were not the case, I would not have believed it (remember, the tax lawyer has long been anxious that I believe the truth about whether p).

These counterexamples carry over to Nozick's more complicated method-relativized version of the account (since there is only one method in question in each case). That version runs as follows:

S knows, via method (or way of knowing) M, that p iff
1. p is true
2. S believes, via method M, that p
3. If p weren’t true, and S were to use M to arrive at a belief whether (or not) p, then S wouldn’t believe, via M, that p
4. If p were true, and S were to use M to arrive at a belief whether (or not) p, S
would believe, via M, that p.

The final account of knowing is then: 

S knows that p iff there is a method M such that (a) she knows that p via M, her belief via M that p satisfies conditions 1 – 4, and (b) all other methods via which she believes that p which do not satisfy 1 – 4 are outweighed by M.

(Formulation taken from Matthew Nudds, 'Truth Tracking' (handout).)

They also carry over to the recent account of Briggs and Nolan (2012), which replaces counterfactuals with dispositions. (Their account was designed to deal with cases where the truth-tracking account undergenerates. Here, it overgenerates.)

Furthermore, they are unaffected by a recent defence of the truth-tracking account, due to Adams and Clarke (2005), against already-known putative counterexamples; these ones seem importantly different, and nothing Adams and Clarke say carries over to them, at least in any way I have been able to discern.

Thanks to John Turri, Fred Adams and Murray Clarke for helpful correspondence.

References

Adams, F. & Clarke, M. (2005). Resurrecting the tracking theories. Australasian Journal of Philosophy. 83 (2):207 – 221.

Briggs, R. & Nolan, D. (2012). Mad, bad and dangerous to know. Analysis. 72 (2):314-316.
 
Nozick, R. (1981). Philosophical Explanations. Harvard University Press.