1 Frege, Gottlob, “Der Gedanke: Eine Iogische Untersuchung,” in: Beiträge zur Philosophie des deutschen Idealismus 2, 1918–19,Google Scholar 58-77, here 74.

2 Kant, Immanuel, Logik, §1.Google Scholar

3 See Ludwig Wittgenstein, Philosophische Untersuchungen, §§198-202. In the following as PU.

4 See, for instance, Dummett, Michael, “Frege's Myth of the Third Realm,” in his Frege and Other Philosophers (Oxford: Oxford University Press, 1991), 249–62.Google Scholar For a critique of this reading and a proper development of the alternative on which I rely in the following, see Ricketts, Thomas, “Objectivity and Objecthood: Frege's Metaphysics of judgment,” in Frege Synthesized, ed. Haarparanta, L. and Hintikka, J., 65–95 (Dordrecht: Reidel, 1986).CrossRefGoogle Scholar

5 Frege, , “Der Gedanke,” 69.Google Scholar

6 Frege, “Der Gedanke,” 58-59.

7 Frege, Gottlob, Grundgeselze der Arithmetik, I/II (Hildesheim: Olms, 1998),Google Scholar Introduction, XV.

8 Ibid., XV.

9 In the opening paragraph of “Der Gedanke,” the question arises: “But may not logical laws also have played a part in this mental process [of judging]?” The answer is that he doesn't want to “dispute” this, but that this question doesn't belong to a logical investigation.

10 This sets Frege”s conception of logic apart from the standard contemporary view of logic. See Smith, Nicholas J.J., “Frege” s Judgrnent Stroke and the Conception of Logic as the Study of Inference not Consequence,” Philosophy Compass 4 (2009): 639–65.CrossRefGoogle Scholar

11 See Frege, Gottlob, “Logik” (1897), in Schriften zur Logik und Sprachphilosophie (Hamburg: Meiner, 2001), 35–73,CrossRefGoogle Scholar here 63-64.

12 Frege, , “Der Gedanke,” 77.Google Scholar

13 See Frege, Gottlob, “Über Begriff und Gegenstand,” Vierteljahresschrift für wissenschaftliche Philosophie 16 (1892): 192–205.Google Scholar

14 Frege, , Grundgesetze der Arithmetik, XVII.Google Scholar

15 The argument focuses on a scenario that the psychologist logician would have to regard as possible, namely that there could be creatures who think according to laws of logic that are fundamentally different from ours. The official aim is show that on closer inspection this scenario crumbles in one's hands: it is not clear what we are supposed to imagine. See James Conant, “The Search for Logical Alien Thought: Descartes, Kant, Frege, and the Tractatus,” Philosophical Topics 20, no. 1 (1991): 115-80, esp. 142-50.

16 See Burge, Tyler, “Frege on Knowing the Third Realm,” in his Truth, Thought, Reason: Essays on Frege (Oxford: Oxford University Press, 2005), 315.CrossRefGoogle Scholar

17 This is precisely what Łukasiewicz argues in his critique of Aristotle. See Łukasiewicz, Jan, “On the Principle of Contradiction in Aristotle,” Review of Meta-physics 24 (1971): 485–509.Google Scholar For a discussion see Irad Kimhi, Thinking and Being: The Two Way Capacity (unpublished manuscript).

18 Frege, , “Logik,” 65.Google Scholar

19 See PU §107. See also PU §81. Positively speaking this is the point Wittgenstein insists on in passages like this: “The laws of logic are indeed the expression of ‘thinking habits” but also of the habit of thinking. That is to say, they can be said to show: how human beings think, and also what human beings call ‘thinking.“” Ludwig Wittgenstein, Bemerkungen über die Grundlagen der Mathematik, I, §131. In the following cited as BGM.

20 That was, of course, Wittgenstein's strategy in the Tractatus. See Diamond, Cora, “Throwing Away the Ladder,” Philosophy 62 (1998): 5–27;Google ScholarConant, James, “The Method of the Tractatus,” in From Frege to Wittgenstein: Perspectives on Early Analytical Philosophy, ed. Reck, Erich, 374–462 (Oxford: Oxford University Press, 2002).CrossRefGoogle Scholar

21 See Ludwig Wittgenstein, PU, §§143-55 and §§185-242. It has been suggested, for instance, that the problem is about mathematics or, alternatively, that the problem arises due to unwarranted application of a mathematical model of determinacy to our ordinary concept use. (For the latter see, for instance, Cavell, Stanley, “The Argument of the Ordinary: Scenes of Instruction in Wittgenstein and Kripke,” in his Conditions Handsome and Unhandsome, 64–100 [Chicago: Chicago University Press, 1990].Google Scholar) Focusing on the use of signs in the school arithmetical illustration, other readers have suggested that the problem is a problem that only arises for the constitution of linguistic meaning and not for the constitution of conceptual content: (See, for instance, McGinn, Colin, Wittgenstein on Meaning [Oxford: Basil Blackwell, 1984]Google Scholar, 144ff.) Yet other readers suggest that it is about the epistemological problem how I can come know the meaning of what another means with his words. (See Davidson, Donald, “The Second Person,” in his Subjective, Intersubjective, Objective, 107–21, esp. 110-11 [Oxford: Oxford University Press, 2001].CrossRefGoogle Scholar) I don't want to deny that all of these questions are addressed in the Philosophische Untersuchungen. But I think that they present a distraction when it comes to the task of getting the rule-following problem into view. As I see it, the latter problem arises regardless of the preferred view on the relation between conceptual thinking and language and it is independent of any specific position on the difference between our use of mathematical functions and the deployment of ordinary concepts.

22 There is a further complication since one and the same thought can be divided in different ways. (See Frege, “Über Begriff und Gegenstand.“) But we can leave that aside for present purposes.

23 As I understand it, this is the point of PU §242, the culminating section of the rule-following discussion. The target of the remark is the Tractatus conception of what it is to understand a language or possess a system of concepts. The problem is that this conception leaves it open whether anyone's judgments are true. In the following I will leave the discussion of the relation between the Tractatus and the rule-following considerations aside, since it would require its own paper.

24 See BGM, I, §§102—4. The terminology of “internal relations” is, of course, from the Tractatus. (See Wittgenstein, Ludwig, Tractatus Logico-Philosophicus, 4.123.Google Scholar) It therefore peculiar when readers of the later Wittgenstein invoke that notion in order to articulate what they regard as the solution to the rule-following paradox. (See, for instance, Baker, Gordon and Hacker, Peter, Skepticism, Rules and Language [Oxford: Basil Blackwell, 1984], 95–96.Google Scholar) I sidestep the discussion of the Tractatus here because I think the difficulty about the relation between concepts and acts of deploying them arises already in Frege's work. It would be a different paper to show why Wittgenstein thought that his early self didn't do justice to this problem. In a nutshell, one might say that the problem about the way in which “internal relations” are conceived in the Tractatus is that it makes it impossible to see how the mastery of a language (or a system of concepts) can be explanaton; of any true judgment. The impossibility of that idea is addressed in the rulefollowing considerations culminating in PU, §242, that discards Wittgenstein's early attempt to ensure that “logic can take care of itself.“

25 Some philosophers simply deny that this phrase characterizes a necessary feature of thinking at all. (See, for instance, Davidson, “The Second Person,” 112-13.) This dismissal of the topic often goes together with linking the word “rule” right at the outset with Wittgenstein's talk about “practices” and “customs.” In this way of proceeding, the kind of” rule” at issue appears to be social, perhaps something in the order of a” convention.” And then it looks like, properly defined, the topic is the task is of specifying how the words of a natural or ‘shared” language have the meaning its speakers take them to have. In consequence, it seems that the task can be dismissed in one of the following two ways: by denying that thinking is language-dependent or by arguing that the prior notion of language is that of an idiolect rather than a shared language. As a result, the question that seems to have elevated the later Wittgenstein appears to be one that, strictly speaking, doesn't belong to philosophy, but rather to linguistics. To make it a legitimate object for the philosopher's worries, one first would have to establish the language-dependence of thinking and then the social character of language. I want to avoid these debates here. And I think we can do so by strictly distinguishing the basic sense of “rule” that is required to get the difficulty going from the richer sense this term acquires when Wittgenstein working towards a solution appeals to the idea of practices and social institutions. The problem can be presented as a puzzle about the metaphysics of our acts of conceptual thinking. Accordingly, claims such as that thinking is “language-dependent” or “essentially social” can only enter the dialectic if they contribute to the solution. Since I will focus on the difficulty, I won't say much about language and meaning. When language comes up, then, it is mainly for purposes of elucidation and mostly as a “vehicle of thought” and not as “means communication.” In the specification of an aspect of the generality of concepts, the “intersubjective” dimension of thinking will come up. But as far as setting up the problem is concerned, this involves just the idea of a possible second person and the imagined communication might as well operate through telepathy.

26 Aristotle, De Interpretatione, 7; Immanuel Kant, Kritik der reinen Vernunft, A25/ B40.

27 Frege, Gottlob, “17 Kernsätze zur Logik,” in: Nachgelassene Schriften (1983), 189, Kernsatz 4.Google Scholar

28 See Evans, Gareth, The Varieties of Reference (Oxford: Oxford University Press, 1982), 100–105.Google Scholar There are different versions of the Generality Constraint in the literature. This is how P.F. Strawson formulates it: “The idea of a predicate is correlative with that of range of distinguishable individuals of which the predicate can be significantly, though not necessarily truly, affirmed” (son, P.F. Straw, Individuals [London: Methuen, 1959]Google Scholar, 99﹜. By contrast to Kant's formulation quoted above, this implies that what is subsumed under the concept are singular representations. Another point discussed in the literature concerns the question whether the constraint demands that a subject must be able to understand any combination between any concept and any object terms she possesses. Evans introduces a “proviso about the categorical appropriateness of the predicates to the subjects” and thus rules out combinations like ‘Caesar is a prime number.” (See The Varieties of Reference, 101.) Similarly, Christopher Peacocke's version of the requirement reads like this: “If a thinker can entertain the thought Fa and also possesses the singular mode of presentation b, which refers to something in the range of objects of which the concept F is true or false, then the thinker has the conceptual capacity for propositional attitudes containing the content Fb” (Christopher Peacocke, A Study of Concepts [Cambridge, MA: MIT Press, 1992], 42). Elisabeth Camp has recently denied categorical restrictions on the subject's power of combination. (See her “The Generality Constraint and Categorial Restrictions,” Philosophical Quarterly 54, 215 [2004]: 209-31.) We can sidestep both issues here. All that matters for the purpose of setting up the rule-following problem is that the subject must be able to understand other sentences in which the predicative element figures.

29 The deployment of empirical concepts like green, square or fluffy to objects encountered in experience like this stolle, that table, or this dog is, of course, more basic than the extending of a number series. But the former cases introduce the vexed question of how the ‘concept” is joined with the matter ‘given” through the senses. The arithmetical example enables us to set these complications aside and focus on the joining of the ‘concept” and with the very act of judgment in which it ‘figures.'

30 On the grammar of this form of reaching ahead, see Thompson, Michael, “Naive Action Theory,” in his Life and Action (Cambridge, MA: Harvard University Press, 2008), 85–146.CrossRefGoogle Scholar

31 See BGM, VI, §15: “[W]hy was not this a genuine prediction: ‘If you follow the rule, you will produce this'? Whereas the following is certainly a genuine prediction: ‘If you follow the rule in all conscience, you will.…” The answer is: the first is not a prediction because I might also have said: ‘If you follow the rule, you must produce this.” It is not a prediction if the concept of following the rule is so determined, that the result is the criterion for whether the rule was followed.“

32 On the distinction between these two kinds of “rules,” see Rawls, John “Two Concepts of Rules,” in his Collected Papers (Cambridge, MA: Harvard University Press, 1999), 20–46.Google Scholar Note that nothing I have said so far requires that the rule is actually “shared.” For all that is relevant up to this point, the pupil might have figured out adding all by himself.

33 It seems that it would require a further clause in the formula expressing the rule for it to include the demand that one wears the helmet because of this rule- say, as a sign of respect for the company. Controlling whether your manners satisfy this rule would obviously be a much more complicated affair – so would describing what it is that you are doing when you act in accordance with it.

34 PU§185.

35 See Sellars, Wilfrid, “Some Reflections on Language Games,” in his Science, Perception and Reality, 321–58, esp. 322ff. (Atascadero, CA: Ridgeview, 1963 and 1991).Google Scholar See also John, Haugland's distinction between “rule-exhibiting” and “rule governed behaviour” in his “Truth and Rule-Following,” in his Having Thought, 305–62, esp. 305ff. (Cambridge, MA: Harvard University Press, 1998).Google Scholar

36 This formulation departs from Evans, who puts it in terms of a “common explanation,” not of the subject's judgments that Fa and that Fb, but of her understanding of ‘Fa” and ‘Fb'. (See Evans, The Varieties of Reference, 101.) The simplicity of the example of extending a number series by adding 2 is, I think, designed to allow us to put the complications aside that call for such qualifications. In the case of perceptual judgments, it can seem that what explains the judgment, if it is to be knowledge, can only be the individual thing and its properties. These, it seems, are the elements in actual reality that can appear in an explanatory role: the thinker judges that the thing is green because the thing is green. It is therefore harder to make out the other explanatory factor at work: the concept that figures as common explanation of a whole range of judgments. In the simple arithmetical case, there is only the latter explanatory aspect. In consequence, it becomes clear that what it explains is judgment about which number comes next.

37 Michael Thompson's view on the concept of life, for instance, entails that this formula is applicable throughout animate nature. See Thompson, “The Representation of Life,” in his Life and Action, 25-82.

38 See Dummett, Michael, Truth and Other Enigmas (London: Duckworth, 1978), 131Google Scholar. See also Sellars” distinction between “patterned governed” and “rule obeying behaviour” in his “Some Reflections on Language Games,” 324.

39 See PU §188.

40 This first horn of the dilemma has been articulated in PU §139-41 in relation to mental images. In PU §§198-201, the point is generalized to any kind of mediating mental item.

41 The “sceptical solution” that Saul Kripke famously considers is maybe the most radical in that it rejects the very idea that there are facts of meaning and understanding and tries to retain an independently intelligible notion of the ‘normative.” (See Kripke, Saul, Wittgenstein on Rules and Private Language [Oxford: Basil Blackwell, 1982].Google Scholar) But each one of our six constraints has been implicitly or explicitly rejected in some proposed solution to the paradox. Some reject the Transparency Constraint (See, for instance, Ruth Garrett, Millikan, “Truth Rules, Hoverflies, and the Kripke-Wittgenstein Paradox,” in Rule-Following and Meaning, ed. Miller, Alexander and Wright, Crispin, 209–34 [Montreal: MeGill-Queen's University Press]Google Scholar), others also the Normativity Constraint (See Paul Horwich, “Meaning, Use and Truth,” in Rule-Following an Meaning, ed. Alexander Miller and Crispin Wright, 260-73.) Even rejecting the Generality Constraint itself has been presented as a solution.ln all these cases, one has to ask whether the words expressing the constraints that are supposed to be retained can have their intended sense independently of the rejected constraints.

42 See, for instance, Goldfarb, Warren, “Kripke on Wittgenstein on Rules,” Journal of Philosophy 82, no. 9 (1985): 471–88;CrossRefGoogle ScholarMcDowell, John, “Wittgenstein on Following a Rule,” in his Mind, Value, and Reality, 221–62 (Cambridge, MA: Harvard University Press, 1998).Google Scholar

43 See John McDowell, “Meaning and Intentionality in Wittgenstein's Later Philosophy,” in his Mind, Value and Reality, 263-78.

44 See PU §197.

45 See Kant, Kritik der reinen Vernunft, B 93. See Frege, Gottlob, Grundlagen der Arithmetik (Stuttgart: Reclam 1987), X.Google Scholar This is not to deny that there are important differences here. But the relevant differences cannot come into view unless one first acknowledges that Wittgenstein” s celebrated slogan “meaning is use” is, to begin with, nothing but a fancy way of putting something Frege already knew, and the early Wittgenstein never got tired of emphasizing. (See Conant, James, “Meaning and Use,” Philosophical Investigations 21 no. 3 [1998]: 222–50.CrossRefGoogle Scholar) Charles Travis argues that the later Wittgenstein “radicalizes” Frege's Context Principle in such a way that this amounts to a rejection of the Generality Constraint. (See Travis, Charles, “On Constraints of Generality,” Proceedings of the Aristotelian Society 94 [1994]: 165–88.CrossRefGoogle Scholar) According to Travis, where Frege's Context Principle teaches that words only have meaning in the context of a sentence, Wittgenstein's radicalization of that principle teaches that sentences only have meaning in the context of a situation. But the Generality Constraint follows from the Context Principle, namely the idea that the basic units of semantic content are structured and that to understand this structure is to conceive of the components as something that could also occur in connection with other elements. To tie an utterance to the concrete situation in which it occurs is to deny the subject an understanding of the situation as structured. If there is to be radicalization of the Context Principle, it can't be a rejection of the Generality Constraint, but only a different articulation of the relevant generality.

46 See PU, §§ 209-10.

47 See PU §219.

48 For present purposes, it doesn't matter which vocabulary exactly is chosen for the retreat. All that matters here is the thesis that the explanatory aspect of concept possession – that is, the way in which that” something about the subject” provides a “common explanation” of her multiple acts of deploying the predicative element-must be specifiable in terms that are logically independent from the notion of “being logically determined“: whether it is in the language of fundamental physics or biology or in terms of the merely psychological.

49 See McDowell, “Meaning and Intentionality in Wittgenstein”s Later Philosophy,” 264, 270, 271-72.

50 The thesis that the relation between the “general pattern” as it is grasped by the subject and her application of it must be mediated by an interpretation was the consequence of one attempt to specify what the subject ‘grasps” or ‘represents” when she understands a concept. The proposal was that she represents the ‘unapplied rule.” This thesis in turn, was motivated by the realization that such understanding surely can't consist in the subject having implicitly already acknowledged all the true thoughts in which the concept figures.

51 See PU §§198–202.

52 See PU, §79 and §§146-51.

53 See Evans, The Varieties of Reference, 101-2.

54 See Evans, Gareth, “Semantic Theory and Tacit Knowledge,” in his Collected Papers (Oxford: Oxford University Press, 2000), 322–42,Google Scholar here 324, 328, and 331.

55 Dummett, Michael, “What Do I Know When I Know a Language?,” in his The Seas of Language (Oxford: Oxford University Press, 1993), 94–105.Google Scholar

56 Dummett, Michael, “Mood, Force and Convention,” “Mood, Force and Convention,” in his The Seas of Language, 202–23, here 221.Google Scholar

57 Michael, Dummett, “What is a Theory of Meaning (II),” in his The Seas of Language, 34–91, 54.Google Scholar

58 The counterfactual rendering leads us right back into the paradox: either it won't be informative about the particular step at hand or there would be an infinite set of counterfactuals representing each step. An analogous difficulty arises for the recent attempt to present the relevant kind of knowing how as a species of knowing that. Jason Stanley and Timothy Williamson have suggested that ‘I knows how to fʹ can be translated into a statement of the form ‘I know that this is a way for me to fʹ (See Stanley, Jason and Williamson, Timothy, “Knowing How,” Journal of Philosophy 98, no. 8 [2001]: 411–44.CrossRefGoogle Scholar) The problem is not whether such knowledge is ‘explicit” or ‘implicit'; the difficulty is rather that the phrase ‘way to fʹ is ambiguous between the general and the particular that is at the center of our problem. Applied to our scenario the phrase can be read in two ways: either it picks out a particular segment in the number series or it stands for the general way of continuing. In the first case it remains unexplained how representing.the particular step can connect the subject to the infinitely many other steps. In the second case the question arises how that general knowledge is to be applied to the segment at hand. This is, of course, no argument against Stanley and Williamson's analysis; it just shows that if their analysis is correct, it is useless to appeal to the notion of knowing how in the context of the rulefollowing paradox.

59 See, for instance, Goldfarb, Warren, “Wittgenstein on Understanding,” in The Wittgenstein Legacy, Midwest Studies in Philosophy, Vol. XVII (Notre Dame: University of Notre Dame Press, 1992), 109–22;Google Scholar and McDowell, John, “Are Meaning, Understanding, etc. Definite States?,” in his The Engaged Intellect (Cambridge, MA: Harvard University Press, 2009), 79–95.Google Scholar

60 PU, §199.

61 See, for instance, Wright, Crispin, Wittgenstein on the Foundations of Mathematics (London: Duckworth, 1980), 220;Google ScholarMalcolm, Norman, “Wittgenstein and Idealism,” in his Wittgensteinian Themes (Ithaca, NY: Cornell University Press, 1995), 87–108;Google ScholarChristopher, Peacocke, “Rule-Following: the Nature of Wittgenstein's Arguments” in Wittgenstein: To Follow a Rule, ed. Holtzman, S. H. and Leich, C. M., 72–98 (London: Routledge, 1981).Google Scholar

62 Wright, Crispin, Realism, Meaning, and Truth (Oxford: Blackwell 1987), 28.Google Scholar

63 Brandom, Robert, Making It Explicit (Cambridge, MA: Harvard University Press, 1994), 626.Google Scholar

64 The invocation of the perspectival differences arising within the social setting of two subject's interpreting each other doesn't supply any new resources that for describing the explanatory aspect of the nexus that links a person's judgment to her other acts in the relevant series of “going on in the same way.” For, in the proposed theoretical framework, the only available notion for talking about the ‘something about the judging subject” that links her present act with other judgments is the notion of a ‘disposition” understood either in counterfactual terms or by appeal to an underlying ‘categorical basis” (probably a neurological one). Davidson is good example. He famously rejects the demand for an account of linguistic competence in non-semantic terms. (See Davidson, Donald, “Reply to Foster,” in his Inquiries into Truth and Interpretation, 171–79 [Oxford: Oxford University Press, 1984].Google Scholar) But in answering the question of what explains the speaker continuing in the same way, Davidson appeals to the “speaker's dispositions to go on in a certain way” conceived as “real features of brains and muscles.” (See Donald Davidson, “The Second Person,” 111.)

65 For an investigation of the related concepts ‘disposition” and ‘practice” in their role moral philosophy, see Michael Thompson, “Practical Generality,” in his Life and Action, 149-210.

66 See, for instance, Bohossian, Paul, “The Rule-Following Considerations,” in Rule-Following an Meaning, ed. Miller, Alexander and Wright, Crispin (Montreal: MeGill-Queen's University Press, 2002), 141-87;Google ScholarEbbs, Gary, Rule-Following and Realism (Cambridge, MA: Harvard University Press, 1997), 279ff.Google Scholar

67 Goldfarb, , “Kripke on Wittgenstein on Rules,” 475.Google Scholar

68 Warren Goldfarb, “Rule-Following Revisited,” unpublished manuscript.

69 See Diamond, Cora, Realism and the Realistic Spirit (Cambridge, MA: MIT Press, 1995), 69–70.Google Scholar Similarly, Cavell, Stanley, The Claim of Reason (Oxford: Oxford University Press, 1979)Google Scholar. Finkelstein, David, “Wittgenstein on Rules and Platonism,” in Tile New Wittgenstein, ed. Crary, Alice and Read, Rupert, 53–73 (London: Routledge, 2000);Google ScholarMcDowell, John, “Wittgenstein on Following a Rule“; Ed Minar, “Paradox and Privacy: On §§201-202 of Wittgenstein's Philosophical Investigations,” Philosophy and Phenomenological Research, 54, no. 1 (1994): 43–75;Google ScholarStroud, Barry, “Mind, Meaning and Practice,” in his Meaning, Understanding, and Practice (Oxford: Oxford University Press, 2000): 170–92.Google Scholar

70 For an incredibly helpful discussion I thank the participants of the Wittgenstein Workshop, University of Chicago, 2011. For comments on earlier drafts of this material, I thank Ian Blecher, Matthew Boyle, James Conant, Anton Ford, Wolfram Gobsch, David Hunter, Andrea Kern, Ben Laurence, Douglas Lavin, Eric Marcus, John McDowell, Christoph Menke, Richard Moran, Thomas Ricketts, Sebastian Rüdl, and Michael Thompson.