Truth, Assertion, and the Horizontal: Frege on The Essence of Logic William W. Taschek

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1 Truth, Assertion, and the Horizontal: Frege on The Essence of Logic William W. Taschek In the opening to his late essay, Der Gedanke, Frege asserts without qualification that the word true points the way for logic. But in a short piece from his Nachlass entitled My Basic Logical Insights, Frege writes that the word true makes an unsuccessful attempt to point to the essence of logic, asserting instead that what really pertains to logic lies not in the word true but in the assertoric force with which the sentence is uttered. Properly understanding what Frege takes to be at issue here is crucial for understanding his conception of logic and, in particular, what he takes to be its normative status vis-à-vis judgement, assertion, and inference. In this paper, I focus my attention on clarifying the latter claim and Frege s motivations for making it, exposing what I take to be a fundamental tension in Frege s conception of logic. Finally, I discuss whether Frege s deployment of the horizontal in his mature Begriffsschrift helps to resolve this tension. 1. Logic: truth and assertoric force Anyone having more than a superficial acquaintance with the writings of Gottlob Frege will be familiar with a number of passages in which he identifies logic as a science essentially and distinctively concerned with truth. They range from fairly early in his career to quite late. A passage from the earlier of two posthumously published manuscripts that Frege labelled Logic is already characteristic. 1 Frege begins by stating that the task of logic is to establish laws of valid inference not a surprising claim for the author of the recently published Begriffsschrift (Frege 1879). This claim immediately leads him to contrast the subject matter of logic with that of psychology, at which point he writes: there is a sharp divide between these disciplines, and it is marked by the word true. Psychology is concerned with truth in the way every science is, in that its goal is to extend the domain of truths; but in the field it investigates 1 Frege [between 1879 and 1891]. But also compare Frege 1893, p. 202; 1897, pp. 126, 227 8; 1918, p ; and 1919, pp All translations of Frege s work are from Beaney 1997, except where stated otherwise. Page numbers for quotations are in parentheses, following the quotation, and refer to the translated source. Dates appearing in square brackets are dates assigned by the editors of Frege s Nachlass. Mind, Vol April 2008 doi: /mind/fzn039 Taschek 2008

2 376 William W. Taschek it does not study the property true as, in its field, physics focuses on the properties heavy, warm, etc. This is what logic does. It would not perhaps be besides the mark to say that the laws of logic are nothing other than an unfolding of the content of the word true. Anyone who failed to grasp the meaning of this word what marks it off from others cannot attain to a clear idea of what the task of logic is. (p. 3) Given the steady supply of passages similar to this one, it would hardly seem risky to conclude that for Frege the notion of truth plays an essential role in characterizing what is distinctive about logic. But as safe as this claim in all its schematic vagueness might at first appear, Frege s Nachlass contains an important set of remarks that casts serious doubt on whether, for Frege, the notion of truth does or, indeed, can play such a central role in characterizing the essence of logic. The remarks occur in the set of jottings to which the editors of the Nachlass assign the date Frege himself entitled them, My Basic Logical Insights, and beneath this title he wrote, The following may be of some use as a key to the understanding of my results (Frege 1915, p. 322). The manuscript is striking as a whole, but for the time being consider the following central passage. After arguing that attaching the words is true to a sentential clause as predicate adds nothing to the thought expressed by the original sentence, Frege writes: This may lead us to think that the word true has no sense at all. But in that case a sentence in which true occurred as a predicate would have no sense either. All one can say is: the word true has a sense that contributes nothing to the sense of the whole sentence in which it occurs as predicate. But it is precisely for this reason that this word seems fitted to indicate the essence of logic. Because of the particular sense that it carried, any other adjective would be less suitable for this purpose. So the word true seems to make the impossible possible: namely, to allow what corresponds to the assertoric force to assume the form of a contribution to the thought. And although this attempt miscarries, or rather through the very fact that it miscarries, it indicates what is characteristic of logic. And this, from what we have said, seems something essentially different from what is characteristic of aesthetics and ethics. For there is no doubt that the word beautiful actually does indicate the essence of aesthetics, as does good that of ethics, whereas true only makes an abortive attempt to indicate the essence of logic, since what logic is really concerned with is not contained in the word true at all but in the assertoric force with which a sentence is uttered. (p. 323) If we are to take these remarks seriously, then contrary to what we were led to believe by the various passages alluded to earlier it would seem that logic is not in its essence concerned with what is contained in

3 Truth, Assertion, and the Horizontal 377 the word true, but with the assertoric force with which a sentence is uttered. If these remarks do not explicitly contradict Frege s earlier (and later) claims, then, at the very least, they raise serious interpretive difficulties about how we are to reconcile them with the various truth-orientated pronouncements concerning the goal, the task, the aim, and essence of logic that Frege makes both before and after these remarks were written. How, for example, are we to understand Frege s repeated assertion that the laws of logic are the laws of truth? Moreover, what sense are we to make of his repeated analogy between the way in which logic is distinctively concerned with truth and the way in which physics is distinctively concerned with such properties as heat and weight? Finally, and perhaps most importantly, if the word true does not in fact indicate the essence of logic, then how are we now to assess Frege s oft-repeated charge that psychologism s failure lies precisely in its inability to respect the central and distinctive role that truth plays in logic? Indeed, if we locate the essence of logic in assertoric force, will we not thereby have smoothed the way for a psychologistic understanding of logic? Of course, we always have the option of dismissing Frege s remarks in My Basic Logical Insights as an aberration. Frege was always struggling against what he viewed as the logical imperfections of ordinary language to find a way to articulate his views about logic. And perhaps, in the end, Frege found the strategy he adopted here misguided. After all, we do not find Frege, in the later Der Gedanke, insisting that it is not truth but assertoric force that best indicates the essence of logic. There, rather, he appears to revert to his truth-orientated way of characterizing that essence. 2 Nevertheless, I think it would be a serious mistake to dismiss these important remarks. First of all, as Frege goes on to say a bit further on in the My Basic Logical Insights manuscript, it is precisely due to the logical imperfection of ordinary language that we seem to find ourselves obliged to deploy the word true. While it may be that its use cannot precisely capture the distinctive concern of logic, it would seem that it is the best tool we currently have to gesture toward what really does indicate the essence of logic the assertoric force with which a sentence is uttered. Frege writes: How is it then that this word true, though it seems devoid of content, cannot be dispensed with? Would it not be possible, at least in laying the foundations of logic, to avoid this word altogether, when it can only create 2 Frege 1918, see especially pp And the same goes for his even later remarks in the Notes for Ludwig Darmstaedter, Frege 1919, especially pp

4 378 William W. Taschek confusion? That we cannot do so is due to the imperfection of language. If our language were logically more perfect, we would perhaps have no further need for logic, or we might read it off from the language. But we are far from being in such a position. Work in logic just is, to a large extent, a struggle against the logical defects of language, and yet language remains for us an indispensable tool. Only after our logical work has been completed shall we possess a more perfect instrument. (p. 323) It would seem, then, that Frege felt that he had little choice but to use and to emphasize the centrality of the word true for logic. And so it is no surprise that he does so again in Der Gedanke even if the remarks in My Basic Logical Insights express what Frege took to be a deep insight about the nature of logic. But why did he there not just talk about assertoric force as he did in My Basic Logical Insights? My own surmise is that Frege felt that he had not yet found a way to say that logic s principal concern is with assertoric force that he felt would not be misunderstood as introducing an element of psychologism. He found the rhetoric of truth in this connection more compelling, perhaps essential especially in the context of his anti-psychologistic polemics, which, it is worth noting, provide the context for nearly all of the remarks in which he writes that the essence of logic is bound up with the notion of truth. A second reason for not dismissing Frege s remarks in My Basic Logical Insights is that this is not in fact the only place in which he suggests that the essence of logic is centrally bound up with assertoric force. With considerably less fanfare, he comes very close to making the same claim in the later Logic manuscript, where he writes: If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, if I assert that it agrees with reality. Therefore, it is really by using the form of an assertoric sentence that we assert the truth, and to do this we do not need the word true. Indeed, we can say that even where we use the form of expression it is true that the essential thing is really the assertoric form of the sentence. (pp ) Essential, that is, so far as logic is concerned. The move here from reflection on the redundancy property of the truth predicate to a focus on the importance and fundamental status of assertion clearly anticipates the later discussion. 3 3 While it is true that in this passage Frege mentions the assertoric form of the sentence and not the assertoric force with which it is uttered, and while it is also true that in his later writings Frege emphasized that assertoric form does not suffice for assertion, he nevertheless did often write as if the assertoric form of an uttered sentence contained assertoric force unless this was somehow cancelled. Cf. for example Frege 1892, p. 158, and 1918, p This was more common earlier in his career.

5 Truth, Assertion, and the Horizontal 379 Thirdly, anyone even superficially acquainted with Frege s reflections on logic and its axiomatic formulation will have been struck by his unwavering insistence from the Begriffsschrift on that in so far as it is the goal of logic to articulate the laws of valid inference, logic must concern itself essentially with judged or asserted contents. 4 To be a wellformed Begriffsschriftsatz and therefore a potential premiss or conclusion in any inference requires being prefaced with a judgement stroke. 5 This suggests that from the very outset of his career Frege took the twin notions of judgement and assertion to play a central role in his understanding of that with which logic is essentially concerned. It is also noteworthy that in the Begriffsschrift, and the several early essays in which he explains and defends the aims and significance of that work, the notion of truth is barely mentioned. 6 And finally, despite Frege s repeated appeal to the notion of truth in characterizing the essence of logic, it is noteworthy that from 1897 on, but probably as early as 1891, Frege began to view the predicate is true as in various ways anomalous, if not downright problematic. Indeed, eventually, Frege came to think of the truth values the True and the False as objects that are the Bedeutungen of true and false sentences, respectively. In any case, given the problematic status of the word true for Frege, we need to be extremely cautious in assessing the import of the various truth-orientated characterizations of the essence of logic that Frege offers. 7 4 Even as late as 1923, two years before he died, in the Nachlass fragment Logical Generality Frege wrote emphatically: only a thought acknowledged as true can be made the premiss of an inference (Frege, [not before 1923], p. 261). 5 Cf. Frege 1893, pp Of course logic is also obliged to recognize and concern itself with unasserted contents, when they occur as parts of compound thoughts. But this is only because this is required to do justice to the logical assessments we make of our judgements, assertions, and inferences. 6 For a nice overview of the development of Frege s views on logic, the shifting vocabulary that he uses to present those views, and the central role that assertion plays throughout, see Sluga In addition to arguing (i) that the relation of truth to a thought is that of a Bedeutung to a sense (cf. Frege 1891, 1892, 1893, 1902, 1904, 1906c, 1914, 1919), we find him arguing (ii) that attaching the words is true to a sentential clause as predicate as, for example, in It is true that seawater is salty adds nothing to the thought expressed by the sentence Seawater is salty alone (cf. Frege 1893, 1906c, 1914, 1915, 1918), (iii) that the word true is sui generis and indefinable (cf. Frege 1897, 1918), and indeed, (iv) that it is a mistake to think that truth is a property of sentences or thoughts at all (cf. Frege 1918, but especially 1914, pp. 234). Each of these claims raises delicate issues, and the story about how they are all supposed to fit together is quite complicated. In what follows, I will have the opportunity to touch upon some of these claims, though, not in as much detail as they deserve. For more extended discussions of these issues, see Burge 1986 and 2005b, and Ricketts 1996.

6 380 William W. Taschek All in all, then, it seems to me that we have compelling reasons to take the remarks in My Basic Logical Insights very seriously. Indeed, I believe that they present a unique insight into Frege s distinctive understanding of the nature and status of logic. What remains to be seen is whether, or to what extent, we can reconcile these remarks with his other, truth-orientated pronouncements about the essence of logic. My own view, as will become evident, is that while we can go quite a ways toward reconciling them, in the end these prima facie conflicting pronouncements are symptomatic of a deep tension in Frege s overall conception of logic. 2. The status of logic: descriptive and normative To see what I have in mind, we need to remind ourselves of another prima facie dichotomy in the way that Frege characterizes the nature and status of logic. On the one hand, as is now well known, Frege views logic as a descriptive science, though, to be sure the maximally general science. As Frege says in the opening paragraph of Der Gedanke, The word law is used in two senses. When we speak of moral or civil laws we mean prescriptions, which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature are general features of what happens in nature, and occurrences in nature are always in accordance with them. It is rather in this sense that I speak of laws of truth. Here of course, it is not a matter of what happens but of what is. From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. (p. 325) This passage suggests that, as with any science, the aim of logic is to identify a finite set of basic laws substantive generalizations appropriate to its subject matter, and to derive from these any other laws applying to its domain. The axioms of Frege s Begriffsschrift are these basic laws. 8 What distinguishes logic from other sciences is not, of course, that it has as its aim the establishment of such general truths for that is the aim of any science whatsoever. Nor are the laws of logic distinguished from other laws by containing some specialized or proprietary vocabulary that segregates its subject matter from that of other sciences; for the only (ineliminable) vocabulary needed to express logical laws is a vocabulary that will be required to express the laws of any 8 I use the italicized Begriffsschrift to denote Frege s early work. I use the unitalicized Begriffsschrift to refer to Frege s formal system(s) distinguishing the early Begriffsschrift, as presented in the Begriffsschrift from the mature Begriffsschrift introduced in Function and Concept and fully articulated in the Grundgesetze.

7 Truth, Assertion, and the Horizontal 381 science whatever. 9 Rather, what distinguishes logic from other sciences is the fact that its laws are in no significant way restricted with respect to subject matter. The laws of logic are laws that hold of all reality, all of what is. There is no subject matter there is, so to speak, no truth that is not subsumed under the laws of logic; and it is in this sense that they count, for Frege, as the most general laws of truth (Wahrseins). The finer details of and motivations behind this universalist conception of logic have been well described and documented by others; so I will not pursue this further here. 10 It is also noteworthy, however, that in virtually every context in which Frege lays out this picture of logic as the maximally general descriptive science, we find him also insisting on the fundamentally normative character of logic. The last sentence of the passage from Der Gedanke quoted immediately above is characteristic: From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. Here is how Frege puts it in the later Logic manuscript: Like ethics, logic can also be characterized as a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand that it should go into what is particular to each branch of knowledge and its subject matter. On the contrary, the task we assign logic is only that of saying what holds with the utmost generality for all thinking, whatever its subject matter. We must assume the rules for our thinking and for our holding something true are prescribed by the laws of truth. The former are given along with the latter. (p. 228) Of course, to claim that the laws of logic articulate what holds with the utmost generality for all thinking, whatever its subject matter (my emphasis) is, as Frege fully realizes, potentially quite misleading. Unless we keep squarely in mind that what is at issue here is what holds prescriptively for all thinking, we are likely to take the laws of logic to be descriptive laws of thinking and, thereby, fall into the morass of psychologism. So, as Frege points out later in this same text, if we call [the laws of logic] laws of thought, or, better, laws of judgement, we must not forget we are concerned here with laws which, like principles of morals or laws of the state, prescribe how we are to act, and do not, like laws of nature, define the actual course of events. Thinking, as it actually takes place, is not always in agreement with the laws of logic any more than people s actual behaviour is always in agreement with the moral law. (pp ) 9 There is, possibly, a noteworthy exception here. It is far from clear that every science requires names of courses-of-values. 10 Cf. especially van Heijenoort 1967, Ricketts 1986 and 1996, and Goldfarb 2001.

8 382 William W. Taschek Notice first the interesting shift in emphasis here from characterizing the laws of logic as prescriptive laws of thinking to prescriptive laws of judgement. This is noteworthy because it suggests a connection a connection I will be emphasizing later between Frege s commitment to viewing logic as essentially normative and his claim in My Basic Logical Insights that the essence of logic is best indicated not by the word true but by reflecting on the assertoric force with which sentences are uttered. Since assertion just is, for Frege, the public manifestation of judgement, and since logic, understood now in its normative guise, articulates prescriptive laws governing judgement and, so, assertion Frege s claim that the essence of logic is located in assertoric force can be understood as, in effect, suggesting that the essence of logic is to be found in its normative role vis-à-vis judgement and assertion and not, or at least not exclusively, in its descriptive role. But I am getting ahead of myself here. If we are properly to appreciate Frege s understanding of the normative status of logic, there are two intimately related issues about which we need to get clearer. On the one hand, we need to understand the sense in which Frege thought that the relevant prescriptions follow from, as he sometimes puts it, the laws of logic qua descriptive generalizations. On the other hand, we need a clearer sense of the distinctive kind of normative force that Frege understood the laws of logic have over judgement and inference. At first glance, it can look as if Frege thought that there is nothing distinctively prescriptive about logical laws as such, as opposed, say, to the laws of any other science. For example, Frege says in a passage from the later Logic manuscript, almost immediately following the last quote, that We could with equal justice think of laws of geometry and laws of physics as laws of thought or laws of judgement, namely as prescriptions to which our judgements must conform in a different domain if they are to remain in agreement with the truth. (p. 247) This suggests that Frege held that from any law, any general truth, there follows a prescription that one ought to think in accordance with it. In this case, logical laws are not distinctive in virtue of their issuing in prescriptions about how one ought to judge, but are distinctive only in virtue of the maximal generality of their subject matter and, so, only in the maximal scope of their prescriptive influence. A similar suggestion seems to be implicit in the following important passage from the Preface to the Grundgesetze: That the logical laws should be guiding principles for thought in the attainment of truth is generally admitted at the outset; but it is easily forgotten.

9 Truth, Assertion, and the Horizontal 383 The ambiguity of the word law is fatal here. In one sense it states what is, in the other it prescribes what should be. Only in the latter sense can the logical laws be called laws of thought, in laying down how one should think. Any law that states what is can be conceived as prescribing that one should think in accordance with it, and is therefore in that sense a law of thought. This holds for geometrical and physical laws no less than for logical ones. The latter then only deserve the name laws of thought with more right if it should be meant by this that they are the most general laws, which prescribe universally how one should think if one is to think at all. (p. 202) A bit later he adds: Anyone who has once recognized a law of truth has thereby also recognized a law that prescribes how judgements should be made, wherever, whenever, and by whomever they may be. (p. 204) From these passages, it would seem that Frege is suggesting that the normative status of the laws of logic, their normative authority over our thinking, is immediately consequent upon their status for us as general truths. In this respect, the normative authority that the laws of logic have over our thinking is not different in kind from that which derives from our acceptance of any law, whether in physics, geometry, or psychology. To be sure, there is a difference in scope, but not a difference in kind. If this is right, then the laws of logic cannot be distinguished from the laws of physics or, for that matter, any other special science on the grounds that the former are, apart from scope, in some distinctive sense prescriptive vis-à-vis judgement while the latter are not. 3. The distinctive normativity of logic Perhaps this is indeed Frege s view; there is certainly textual support for supposing so. However, I am inclined to think that this cannot be quite right. As the concluding remarks in the first of the two passages from the Grundgesetze just quoted reveal, Frege clearly thinks that the laws of logic do have a proprietary claim to the title laws of thought, conceived of normatively. The question, then, is whether Frege believed that this proprietary status is to be or even can be accounted for simply on the basis of what we might call logic s maximal descriptive scope. What the concluding remark of that passage says, recall, is that the laws of logic deserve to be called laws of thought (in the normative sense) with more right than, say, the laws of physics, only if they are taken to prescribe universally how one should think if one is to think at all (my emphasis). The implication here, I suggest, is that what is distinctive about logical laws in virtue of which they deserve to be called the laws

10 384 William W. Taschek of thought with more right than the laws of physics is not merely that they are maximally general in scope, but that in an important sense they issue in, while the laws of physics do not, constitutive norms of thinking as such. In suggesting that for Frege the norms of logic are constitutive norms of thinking as such, I am not suggesting that he held the absurd view that one cannot be counted as a thinker unless one s thoughts more importantly, one s judgements, assertions, and inferences accord with these norms. The second to last quotation above from the later Logic manuscript clearly rules this out. Rather, I am suggesting that, for Frege, if one is to count as a thinker at all, then one s judgements, assertions, and inferences must at the very least be subject to evaluation by the norms of logic. Judgements and, so also, assertions and inference are by their nature subject to evaluation by logical norms. Indeed, I would argue that the constitutive status that logical norms have for Frege vis-à-vis thinking as such has an even more profound dimension: to count as a thinker at all one must acknowledge the categorical normative authority that logic has over one s practice of judging, asserting, and inferring. For Frege, then, one s status as a thinker would be seriously compromised by a failure to acknowledge the categorical normative claim that logical principles have over one s thinking. To explicate with any precision what exactly this acknowledgement consists in would be a very tricky matter. For present purposes, however, the following clarificatory remarks should suffice to indicate what I have in mind. First of all, to acknowledge the categorical normative authority of logic over one s practice of judging, asserting, and inferring cannot consist in a judgement to the effect that such-and-such is a law of logic, or even in the judgement that the law is true, on pain of the sort of regress exposed by Lewis Carroll s (1895) parable of Achilles and the Tortoise. As we shall see, an issue closely related to this will come up again in what follows. In any case, it would be absurd to suppose that one s status as a thinker, in Frege s sense, required that one have already judged of the relevant logical norms that they are such. Rather, to acknowledge the categorical authority of logic will involve one s possessing a capacity to recognize when being sincere and reflective, and possibly with appropriate prompting logical mistakes both in one s own judgemental and inferential practice and that of others. Moreover, upon recognizing such a mistake in one s own case, one will feel an unconditional obligation to correct it. Thus, if one were to recognize or take oneself to recognize such a mistake in the case of another person, and if that person could not, even after appropriate prompting, be per-

11 Truth, Assertion, and the Horizontal 385 suaded to acknowledge it as a mistake a mistake with respect to which she felt an unconditional obligation to correct herself that would provide one with grounds to question that individual s status as a thinker. In the Preface to the Grundgesetze, for example, Frege proposes that if we were to confront anyone who appeared not to acknowledge the categorical normative claim on our thinking that issued from a law of logic, we should suppose that here we have a hitherto unknown kind of madness (Frege 1893, p. 203). 11 There is, however, another passage in the Preface to the Grundgesetze occurring toward the end of his principal argument against psychologism with respect to logic that might appear to cast doubt on my claim that Frege took logic to issue in constitutive norms vis-àvis judgement, assertion, and inference of the sort at issue here. He writes: Leaving aside logic, one can say: we are forced to make judgements by our nature and external circumstances, and if we make judgements, we cannot reject this law of identity, for example; we must recognize it if we are not to throw our thought into confusion and in the end renounce judgement altogether. I do not wish to either dispute or endorse this view and only remark that what we have here is not a logical implication. What is given is not a ground of being true, but of our holding as true. And furthermore, this impossibility of our rejecting the law does not prevent us from supposing that there are beings who do reject it; but it does prevent us from supposing that these being are right in doing so; it also prevents us from doubting whether we or they are right. (p. 204) The suggestion here that we cannot reject a logical principle without throwing thought into confusion and in the end renouncing judgement altogether appears to give expression to the very thesis about the constitutive nature of logic for thinking as such to which I have claimed Frege commits himself. But what is noteworthy about this passage is that Frege, rather than endorse it, remains explicitly agnostic about this sug- 11 An important issue in recent Frege scholarship concerns how we are best to understand Frege s claim that the basic laws of logic are self-evident and how we are to understand this selfevidence in such a way that we can see it as warranting our acceptance of these laws. At stake, of course, is the proper understanding of what Frege might have had in mind by his various and nonsystematic appeals to self-evidence here. See, for example, the discussions by Burge 1998, Jeshion 2001, Weiner 2004 in this journal. The proposal that, for Frege, to count as a thinker requires that one acknowledge the categorical normative authority of logic vis-à-vis judgement, assertion, and inference suggests that a proper account of the self-evidence to which Frege appeals will be one in which self-evidence is itself grounded in or explicated in terms of the constitutive requirement that one acknowledge the categorical normative authority of logic on one s own thinking. A proper explication of what this acknowledgement comes to would, I believe, help us to see how it grounds our commitment to the truth of the basic laws of logic. Unfortunately, developing this idea is not something I can pursue here.

12 386 William W. Taschek gestion, declining to endorse it. Moreover, he seems to suggest that there is nothing problematic in our imagining creatures that do reject logical principles. These two features of this passage might seem to suggest that we ought to be dubious about committing Frege to the constitutivity thesis. 12 A closer reading of the passage makes it clear, however, that the suggestion as issue here was introduced by Frege as a purely descriptive psychological hypothesis about our thinking and, as such, cannot plausibly be taken to give expression to the constitutivity thesis,. After all, the passage explicitly begins, Leaving logic aside. It is with respect to this suggestion qua descriptive claim about current human psychology that Frege declines to commit himself. In effect, Frege is asking: If we were to suppose it to be a psychological fact about our thinking that we cannot now fail to accept logical laws, then what would follow? What Frege says is that, even if this suggestion, as a descriptive psychological claim, were true for us, nothing would follow from it about whether or not there might be other thinking creatures who reject logical principles that we ourselves cannot. And this is certainly right. But surely, he insists, any such creature would at the very least nevertheless be mistaken. At which point he wryly remarks, If others dare to recognize and doubt a law [of truth] in the same breath, then it seems to me like trying to jump out of one s own skin, against which I can only urgently warn. Anyone who has once recognized a law of truth has thereby also recognized a law that prescribes how judgements should be made, wherever, whenever, and by whomever they may be made. (p. 204) 13 All of this shows that the passage in question cannot very plausibly be understood as one in which we find Frege declining to endorse the constitutivity claim. Indeed, Frege s final remark in the last quotation comes as close to expressing his commitment to the constitutivity claim as we are likely to want. For in it he seems precisely to be saying that to acknowledge a logical law is thereby to acknowledge the categorical normative authority of that law over judgement as such. If this is right, however, serious difficulties arise for the suggestion that the normative authority that the laws of logic have on our thinking cannot be distinguished in kind from the laws of physics or, for that matter, any other special science. First of all, it is implausible to view any prescriptions that issue from the laws of physics or geometry as constitutive of thinking in any analogous sense. Specifically, one s status 12 I want to thank Gary Ebbs for suggesting that I attend more carefully to this passage. 13 Cf. the discussion in Frege 1897, pp

13 Truth, Assertion, and the Horizontal 387 as a thinker is in no way compromised by failing to acknowledge the truth of a law of physics or even geometry. If we were to confront someone who failed to acknowledge the truth of some law of physics that we ourselves accepted, there is no temptation nor is it plausible to suppose Frege would be tempted to suggest that here we have a hitherto unknown type of madness. To be sure, if I do acknowledge the truth of a law of physics or, for that matter, of any other special science I do thereby find myself under an obligation to think in accordance with it. But what can this mean except that I am under a logical obligation to have my other judgements be consistent with this one and, in particular, to be willing to acknowledge as true all instances of the general claim? This, though, is an essentially logical obligation; it does not in any sense derive from the law of physics itself, independently of the normative demands of logic. The normative demands of logic must, so to speak, already be in place for there to be any obligation, once one has committed oneself to the truth of a laws of physics, to think in accordance with that law. The latter obligation presupposes wholly general logical obligations. I suggest, then, that Frege did take the normative status of the laws of logic to be distinct in kind from the normative status of laws in the specialized sciences and not merely distinct in scope. These last reflections show that there is, in fact, something quite peculiar, if not incoherent, in the idea that a constitutive prescription about how one ought to judge can be immediately grounded in our acceptance of any true generalization independently, that is, of our already acknowledging the normative demands logic places on our thinking. The problem for Frege is that this would seem to be just as true for those generalizations that he took to express logical laws, as it is for the laws of physics. On Frege s universalist conception of logic, when I accept a logical law, I acknowledge as true a substantive generalization. To be sure, it is maximally general with respect to subject matter. But so what? Here, as before, by judging the law as true, I find myself under an obligation to think in accordance with it. But again, there seems to be no way to understand this obligation except as grounded in a prior logical obligation to have all my other judgements be consistent with this one and, in particular, to accept all instances of this general truth. What, if anything, grounds this prior obligation? Here again, we cannot allow that it is itself grounded in my acceptance of some other substantive logical law on pain of the sort of regress exposed by Carroll s parable of Achilles and the Tortoise to which we have already alluded. Eventually, we must find ourselves acknowledging

14 388 William W. Taschek a commitment to a logical norm concerning judgements in effect, a rule of inference 14 where this commitment cannot itself consist in (or be grounded upon) our accepting as true some further substantive law. If this is right, then the prescriptive force of any substantive logical law must itself be viewed as grounded in our prior commitment to some fundamentally non-descriptive normative logical principles, which themselves are not grounded in our acceptance of any more fundamental substantive claim. Is this a problem for Frege? Yes and no. On the one hand, Frege was quite clear to distinguish rules of inference from logical axioms in his formal system. And it appears that he was also fairly clear about the ineliminability of rules of inference though, so far as I can tell, there is no place where he explicitly addresses this issue. 15 To this extent, then, he seems to have acknowledged and taken into account even if only inchoately an irreducible normative element in logic. Indeed, there are places in Frege s published work where he appears to commit himself more straightforwardly to the idea that the essence of logic is revealed, in the first instance, by reflecting on its constitutive normative status vis-à-vis our practice of judgement assertion, and inference. This is, perhaps, most readily seen in the context of the specific critique of psychologism that Frege mounts in the Preface to the Grundgesetze. In general, Frege s criticisms of psychologism consist in his pointing out how, in one way or another, a psychologisitic approach to logic fails to capture something that belongs to the essence of logic. In the Grundgesetze, he clearly takes himself to have shown that psychologism is committed to a conception of logical laws that is incapable of doing justice to the constitutive prescriptive role that the laws of logic play vis-à-vis thinking as such. In particular, psychologism cannot support a conception of judgement or assertion according to which they are constitutively subject to normative governance by the laws of logic in the way Frege conceived them to be. Rather, the psychologistic conception of logical law forces an assimilation of judgement and assertion to non-cognitive attitudes and their expression of the sort exemplified by matters of subjective taste and their expression in yuks and yums. 16 A psychologistic logician is committed to supposing that failure to acknowledge the categorical normative claim of a logical principle on 14 It is noteworthy that Frege s rules of inference are always couched in terms of assertions. The permissions they encode are explicitly restricted to expressions prefaced by the judgement stroke. 15 See for example Frege 1914, p This theme is also developed at length in Frege 1897, pp. 230ff. The discussion in Ricketts 1986 is especially helpful as well.

15 Truth, Assertion, and the Horizontal 389 one s judgement merely exposes a difference in psychological laws governing one s practice of judging those laws are valid for them, these for us. But as we have noted already, Frege takes it to be obvious that such a failure would manifest a hitherto unknown type of madness. It is precisely in this connection that the claim Frege makes about assertoric force in My Basic Logical Insights comes into its own. If the laws of logic just are those laws that issue in constitutive norms governing judgement and assertion, then if we are to understand their distinctive status, their essence, we need to appreciate fully the aim that is constitutive of our practice of assertion. And it is just this that is revealed in the assertoric force with which a sentence is uttered. To utter a sentence with assertoric force is just to express a judgement. For Frege, the notion of judgement is fundamental. Throughout his career, Frege consistently explicates the notion of judgement as the acknowledgment [Anerkennung] of the truth of a thought. 17 Frege insists, however, that this explication is not any sort of definition, and that acknowledgement is not itself a species of judgement. It would be a mistake precisely the sort of mistake that Frege s insistence that this is not a definition was intended to forestall if, for example, we supposed that understanding the nature of judgement required a prior and independent grasp of the notion of truth or, for that matter, of thought. And in particular, it would be a mistake if we were to allow this explication to seduce us into understanding truth as if it were a substantive property of thoughts. For this might in turn lead us to assimilate judgement to predicating truth of a thought. But Frege repeatedly inveighs against making this mistake. That this is a mistake, Frege took to be clearly exposed by reflecting on the redundancy feature of the ordinary truth predicate when applied to thoughts. The following passage from the 1914 Logic in Mathematics manuscript is characteristic. He begins by allowing that, Of course treating truth as a property of sentences or of thoughts is in accordance with linguistic usage. If we say The sentence 3 > 2 is true the corresponding thing holds of the thought. Still the predicate true is quite different from other predicates such as green, salty, rational, for what we mean by the sentence, The thought that 3 > 2 is true can more simply be said by the sentence 3 is greater than 2. Thus we do not need the word true at all 17 Though it has become standard to translate Anerkennen (and its cognates) in this context using recognize (and its cognates), I prefer to translate it using acknowledge (and its cognates). To my ear, recognize is most naturally heard as a factive, while acknowledge is not. In any case, Frege clearly allowed that one could make mistakes in judgement judging a thought to be true when in fact it is not. See, for example, Frege [between 1879 and 1891], p. 2; 1924/25, pp ; 1914, p. 234; 1918, p For a comprehensive discussion of these and related issues, see Kremer 2000.

16 390 William W. Taschek to say this. And we see that really nothing at all is added to the sense by this predicate. In order to put something forward as true, we do not need a special predicate: we only need the assertoric force with which the sentence is uttered. (p. 233) He then adds, If a man says something with assertoric force that he knows to be false, then he is lying. This is not so with an actor on the stage, when he says something false. He is not lying, because assertoric force is lacking. And if an actor on the stage says it is true that 3 is greater than 2 he is no more making an assertion than if he says 3 is greater than 2. Whether an assertion is being made, therefore, has nothing at all to do with the world true ; it is solely a matter of the assertoric force with which the sentence is uttered. (p. 234) The word true is neither necessary to express a thought, nor is it sufficient, when attached as a predicate to a thought or sentence, to effect an assertion. But now it will hardly be surprising to find that it is this very line of reasoning that leads Frege to his conclusion, in My Basic Logical Insights, that it is not the word true but the assertoric force with which a sentence is uttered that correctly indicates the essence of logic. After offering his standard elucidatory explication of judgement as the acknowledgement of a thought as true, Frege proceeds to point out the redundancy feature of the predicate true. This then leads him to conclude: If I assert it is true that seawater is salty, I assert the same thing as if I assert seawater is salty. This enables us to recognize that the assertion is not to be found in the word true, but in the assertoric force with which the sentence is uttered. (pp ) It is precisely this insight, he goes on to claim, that leads us to see that it is not the word true that indicates the essence of logic but the assertoric force with which a sentence is uttered. To be sure as Frege s elucidatory explication of judgement reveals it is fundamental to our understanding of judgement and assertion that we appreciate that their aim is truth. Judgements, and so assertions, are by their nature subject to assessment as correct or incorrect. So when, in My Basic Logical Insights, Frege declares that the essence of logic is to be found in the assertoric force with which thoughts can be expressed, it is all but inevitable that he should take the essence of logic to be fundamentally normative. Of course, the notion of correctness constitutive of judgement is inextricably bound up with the notion of truth. But, for Frege, we have no independent grip on the notion of truth apart from our appreciation of it as the constitutive aim of the norm of correctness for judgement. And, for Frege, our

17 Truth, Assertion, and the Horizontal 391 appreciation of truth as a norm of correctness for judgement is ultimately bound up with our appreciation of judgements as constitutively subject to normative governance by the laws of logic. It is in just this sense, then, that for Frege as he puts it in Der Gedanke The meaning of the word true is spelled out in the laws of truth (p. 326). Or, as we saw Frege write already in the earlier of the two Logic manuscripts, It would not perhaps be besides the mark to say that the laws of logic are nothing other than an unfolding of the content of the word true (p. 3). 4. Final reflections: Frege s horizontal It cannot be denied that Frege recognized an ineliminable normative element in logic. He sought to accommodate this in his formal system through his inference rules together, of course, with his insistence that logic is concerned, in the first instance, only with judged (and not merely judgeable) contents. But it also cannot be denied that Frege felt committed to an ineliminable descriptive element in logic as well. This shows itself in his unwavering insistence that an ideal formulation of logic will be axiomatic and include, as substantive general axioms, logical laws precisely those laws he calls laws of truth. But how, given the ways in which Frege has exposed the problematic nature of the ordinary language predicate true, and given his suggestion that a more adequate conception of the notion of truth involves appreciating its status as a constitutive norm of judgement, are we to understand Frege s repeated characterization of these laws of logic as substantive laws of truth? In particular, how, if at all, can we rescue Frege s repeated analogy between the way in which the laws of physics concern themselves with such fundamental properties as heat and weight and the way in which the laws of logic concern themselves with truth? The laws of physics will clearly contain predicates whose sense will oblige us to recognize such fundamental properties as heat and mass. But in what sense, if any, do the laws of logic contain predicates that oblige us to recognize a property of truth? If truth were a property of thoughts, we might expect the laws of logic to be laws concerning when this property holds of, or what follows from its holding of, various thoughts. But we know already that Frege was keen to deny that the word true picked out any such property and that language is misleading precisely to the extent that it seduces us into understanding truth in this way. Moreover, it is plain from Frege s mature Begriffsschrift, as presented and deployed in the

18 392 William W. Taschek Grundgesetze, that Frege s logic at no point finds itself obliged to deploy a predicate whose extension is the set of all true thoughts or all true sentences. In this sense, then, it is quite clear that Frege does not deploy a truth predicate in anything like the modern semantic sense. So how does Frege want us to think of the laws of logic, in their descriptive guise, as substantive laws of truth? While it certainly is true that Frege s axiomatic formulation of logic contains no truth predicate in the modern sense (nor, in its presentation, is he obliged to appeal to any), is it right to conclude that his logic is not concerned in a substantive way with truth? Or, to put this another way, is it right to conclude that his logic contains nothing at all worth calling a truth predicate? What about the expression in Frege s mature Begriffsschrift that he called the horizontal? 18 Frege s horizontal, recall, is the descendent of what, in the Begriffsschrift, Frege called the content stroke (Inhaltsstrich). According to the rules of formation set forth in the Begriffsschrift, the content stroke (a longish dash ) can prefix an expression A only if A expresses a judgeable content. And in the Begriffsschrift it is only to an expression prefaced with a content stroke (e.g. A ) that one can legitimately attach the vertical stroke that Frege (both there and later) calls the judgement stroke (Urteilsstrich) thus issuing in an expression like A, which expresses the judgement that A (Frege 1879, pp. 52 3). But once Frege introduced the distinction between Sinn and Bedeutung and more specifically identified the Bedeutung of a sentence with its truth value either the True or the False he reinterpreted the content stroke (which he now simply calls the horizontal ) as a first-level function name that issues in the value the True only when completed by an argument expression that names the True (e.g. any true sentence), and has as its value the False when completed by the name of any object except the True (which would include, of course, any false sentence). From a logical point of view, then, the horizontal of Frege s mature Begriffsschrift is a predicate expression a first-level concept name. Moreover, the only object in its extension is the True that is, the object Frege identifies with the Bedeutung of true sentences. In this quite specific sense, then, the horizontal is a kind of truth predicate: it is a predicate true of the True and of no other thing. Moreover, the horizontal is an absolutely essential feature of Frege s Begriffsschrift. Though rarely remarked upon, the negation stroke, the 18 The transformation of the Begriffsschrift s content stroke into the functor of his mature Begriffsschrift that Frege came to call the horizontal is first explained in Frege 1891, pp. 141ff., and gets fully elaborated in Frege 1893, pp. 215ff.