Truth and Transcendence Turning the Tables on the Liar Paradox GILA SHER 11.1. AMethodological Turnaround Confronting the liar paradox is commonly viewed as a prerequisite for developing a theory of truth. As soon as the truth theorist accepts one of the most minimal principles of truth, the equivalence principle, which in one of its forms is often formulated (schematically) by (E) <P> is true iff (if and only if) P, where "P" stands for any sentence and" <P>" stands for a name of that sentence, the liar paradox arises. And it is only after the truth theorist incorporates some device Oike a Tarskian hierarchy or a K.ripkean grounding process) for blocking the paradox that he is entitled to continue developing his theory. In this chapter I would like to turn the tables on this traditional conception of the relation between the liar paradox and the theory of truth. I would like to show that the theorist of truth need not worry about the liar paradox in developing her theory, that if she focuses deeply enough on the "material" adequacy of her theory (i.e., whether it adequately accounts for the nature of truth) as distinct from its "formal" adequacy (i.e., whether it is a consistent theory), the liar challenge is unlikely to arise for her theory at all.1 In approaching the liar challenge in this way, we are treating the theory of truth like most other theories. Consider a physical, psychological, geometrical, or moral theory. Normally, the theorist in any of these fields is aware that his theory must sati�fy certain norms of formal adequacy. But his focus in construct- ing his theory is not on this matter. His focus is on the material adequacy of his theory, on its success in giving a correct, comprehensive, and explanatory account of the physical structure of the world, human psychology, the geometry of space, the grounds of human morality, etc. Once his theory is completed (or reaches a 281 282 REFLECTIONS ON THE LIAR temporary state of completion), the theorist has to verify, to the best of his abil- ity, its formal consistency. Of course, if it turns out that his theory is inconsistent, this is a serious matter and the theorist must re-examine the theory: its mate- rial principles, its logical-linguistic framework, its background assumptions, etc. But under normal circumstances, the need to guard against inconsistency does not dominate, or even noticeably affect, the construction of our theories. Our theories are constructed based on material considerations, and the consistency check is just that, a check. In contrast, the theorist of truth, on the approach I am challenging, cannot go about his business-the construction of a correct, comprehensive, and informative theory of truth-in this way. Here, the issue of consistency is a major concern right from the beginning, and he is not entitled to proceed in developing the material content of his theory before taking care of its formal adequacy, or so the common wisdom says. My claim in this chapter is that the task of constructing a theory of truth is methodologically more similar to that of constructing a physical, psychological, geometrical, or moral theory than it is commonly thought to be. More specifically, if the theorist of truth builds a materially good theory (choosing some-any- reasonable logical-linguistic framework for working in), then the liar challenge is unlikely to arise at all for her theory. The key to this turnaround is methodological, and one way to explain it is by saying that according to the prevalent conception of the theory of truth, constructing a theory is like constructing a logical inference. But in fact con- structing a theory is very different from constructing a logical inference. Let me explain. If you think of your theory of truth as a logical inference, where each of its principles is an independent premise and the question is whether these premises entail a contradiction, then you conclude that if the equivalence prin- ciple, E, leads to a paradox, then the equivalence principle together with any other material principle of truth, M, will form a conjunction that also leads to that paradox: (1) [E => Paradox] => [E & M => Paradox].2 If this were correct, it would justify the requirement that we take care of the liar paradox as soon as we accept the equivalence principle. But this view of a theory is wrong. A theory-say, a philosophical theory-is not like a set of premises in a logical inference, and the process of building a philosophical theory is not that of creating a conjunction of independent principles. When you build a philosophical theory, each added principle constrains, and is constrained by, the other prin- ciples. That is, it affects, or can in principle affect, their scope and content, hence their consequences, and is similarly affected by them. As a result, if *�" is the operation of placing two principles, P1 and P2 , together in a theory, then: (2) [P1 =>Paradox]#- [P1 � P2 :;;, Paradox]. Truth and Transcendence 283 The difference between (2) and (1) is due to the difference between [81 and &. The process of constructing a theory is such that the addition of a new principle to the theory generally involves the updating of all its principles, old and new. We may say that P 1[81P 2 is more like ( (P 1 1 P 2) & (P 2 1 P 1)], where "X I Y" stands for "X as updated by Y", than like P1&P2• And the fact that for an arbitrary consequence C explains why My suggestion is that there is some material principle of truth, M, a principle arrived at by investigating the nature of truth itself, such that while (5) E :.;, Liar Paradox, (6) E[81M '1=> Liar Paradox. Which material principle of truth is M? There could, theoretically, be multiple such principles. In this chapter I will focus on one such principle, "IMMANENCE", introduced in Sher (2004). More specifically, I will focus on two subprinciples of IMMANENCE, "immanence" Oower case) and "transcendence", which, with the help of a third subprindple, "normativity", do the active work of preventing the liar paradox from arising. This pair of material principles says that truth is inher- ently hierarchical, and as such it is not susceptible to the liar paradox. We might say that the addition of IMMANENCE to equivalence updates the latter in a way that blocks the liar paradox. This turnaround in our understanding of the relation between the liar paradox and the theory of truth is not just significant by itself, but it has significant con- sequences for issues widely discussed in the literature: 1. It undermines a major criticism of Tarski's theory of truth, namely, that the hierarchical structure it attributes to truth is ad hoc. More precisely: 2. It shows that the hierarchical element in Tarski's and others' theories of truth can be justified on material and not just on formal grounds, Le., that it has deep roots in the material nature of truth, and is not just a technical device for dealing with a formal problem. 3. It further shows that many anti-Tarskian theories of truth share a hierarchical element with the theory they reject. 4. It disconnects both the liar paradox and the hierarchical conception of truth from the question of which logic is the right "logic of truth", bivalent logic or nonbivalent logic. This it does by showing that the hierarchical element in truth is independent of bivalence and compatible with nonbivalence. 284 REFLECTIONS ON THE LIAR 5. It also disconnects the liar paradox and the hierarchical conception of truth from the question of natural language versus artificial language. 6. It draws new lessons from the existence of non-Tarskian solutions to the liar paradox. 7. It points to an additional advantage of a substantivist over a deflationist theory of truth besides the straightforward advantages of greater depth, informa- tiveness, and explanatory power. Investigating the nature of truth beyond the minimalist equivalence principle, it shows, might save us from problems- including formal problems-that arise for deflationist or, more generally, "bare-bones" theories of truth. I should emphasize, however, that saying we do not need to consider the liar para- dox prior to the construction of a material theory of truth does not minimize the value of facing the liar paradox on its own. On the contrary, confronting the liar paradox on its own, i.e., independently of the development of a materially adequate theory of truth that might block it, has led to immense progress in our understanding of such central topics of philosophical inquiry as language, logic, semantics, definition, circularity, self-reference, vagueness, contextuality, revi- sion, truth itself, and more. 11.2. A Substantivist Theory of Truth Our main task is to present a material principle of truth that blocks the liar para- dox. But this principle does not exist in a vacuum. It belongs to a certain theory, with its underlying methodology, goals, and perspective.3 Among the underlying features of this theory are (i) a substantivist orientation, (ii) a holistic, or more precisely, a "foundational-holistic" methodology, and (iii) a focus on the cognitive-- epistemic role of truth. Let me briefly explain the nature of and motivation for these underpinnings: (i) Substantivist Orientation. The motivation for pursuing a substantive theory of truth in contrast to a trivial, thin, or a bare-bones deflationist theory of truth is straightforward and commonsensical. If the subject matter of truth is thin and trivial, there is no need to develop a theory of this subject matter. If, on the other hand, the subject matter of truth is rich and philosophically salient, it requires a substantive theory. In this respect, too, I regard the field of truth as similar to all other fields of knowledge, both within and outside philosophy. If the physi- cal structure of the world is a rich and important subject matter, it requires a substantive-deep, thorough, rich, informative, explanatory-theory. And the same holds for knowledge, logical reasoning, morality, and other philosophical subject matters, including truth. But the development of substantive theories of many philosophical sub- ject matters faces special challenges, both challenges due to the nature of these Truth and Transcendence 285 subject matters and challenges due to philosophers' conception of the structure of philosophical theories of such subject matters. Subject matters like truth and knowledge are characterized by extraordinary breadth, diversity, and complex- ity; yet one influential conception of a philosophical theory of these subject mat- ters is that of a single and simple definition or necessary and sufficient condition. In the case of knowledge such a "theory" used to take the form of a definition like "Knowledge =ot True, justified belief" or a necessary and sufficient condition like "x knows that P iff x has a true, justified belief that P". In the case of truth, it often takes the form of the equivalence biconditional, under one formulation or another. The combination of these two circumstances-the breadth of many philosophical subject matters and philosophers' narrow conception of a theory of these subject matters-introduces a nontrivial challenge to substantivist philosophers. In the case of truth, this challenge may appear insurmountable. Truth is a rich and philosophically significant subject matter, interwoven in many areas of our life, from the epistemic and theoretical to the moral and practical, and this creates a serious methodological problem for at least one central branch of the theory of truth, that dealing with truth conditions. The truth conditions of sentences (thoughts, beliefs, propositions, theories, bodies of knowledge, etc.)4 are tied up with their content, and the enormous diversity of types of content creates a plural- ity of types of truth conditions. (For example, the truth conditions of a discourse about causally accessible everyday physical objects are likely to be quite different in kind from those of a highly abstract discourse-discourse about causally inac- cessible physical phenomena, mathematical laws, moral principles, etc.) A major challenge for the theorist of truth is dealing with this diversity. The traditional conception of the theory of truth as comprised of a single and simple definition or necessary and sufficient condition, one that captures the one and only common denominator of all truths, or the one and only essence of truth, or the one and only necessary and sufficient condition for any sentence to be true, makes this challenge impossible to meet. If truth is too diverse, too complex, and too multidimensional to be exhausted by a single common denominator, then a substantive theory of truth is unfeasible. But this conclusion, I have shown in Sher (1999a, 2004), is unwarranted. If the conception of a theory of truth as a single and simple definition is untenable, it can and should be replaced by that of a cluster of interconnected prindples of truth (of various degrees of complexity and generality), a conception widely accepted by theorists in most fields of knowledge. Within philosophy, such a conversion has been successfully accomplished in some fields, notably epistemology, where during a certain period in the twentieth century many theorists viewed their task as producing a definition of, or a necessary and sufficient condition for, "x knows that P," and today only few follow in their footsteps. Indeed, the diversity challenge is not unique to philosophy. A similar challenge arises for natural science as a whole, that is, as our theory of nature. The diver- sity, complexity, and multidimensionality of nature pose a serious challenge to 286 REFLECTIONS ON THE LIAR the theorist of nature. This challenge has been widely discussed by philosophers and scientists under the rubric of "the disunity of science" and, not surprisingly, many of its lessons are independent of the empirical nature of natural science. In Sher (2004) I suggested that philosophers ought to adopt the same common- sense guidelines for dealing with this challenge as those devised by scientists. The gist of this idea is concisely captured by Dyson when he says that "every theory needs for its healthy growth a creative balance between unifiers and disunifiers" (1988, p. 4 7). In the case of truth, my suggestion is that we think of a substantive theory as a body of substantive principles-some more general, others more particular, some simpler, others more complex, some manifesting greater inner unity, others greater diversity, but all interconnected. This strategy frees us to search for, rather than legislate in advance (e.g., in the form of a definition by fiat), the material principles of truth, including principles that involve a significant adjustment, or updating, of other principles. (ii) Foundational-Holistic Methodology. Another challenge facing the theo- rist of truth, along with theorists of other philosophical subject matters, is the "foundational" challenge. Philosophical theories often deal with very basic subject matters that call for a foundationgl treatment. But foundational studies in phi- losophy have come upon serious difficulties and many philosophers view them as doomed to failure. In Sher (2010 and 2013a) I have suggested that this is due to the fact that the foundational project is commonly associated with a self-defeating traditional methodology, the so-called foundationalist methodology. The founda- tionalist methodology sets strict, unsatisfiable requirements on the foundational project, including a strict ordering of all areas of knowledge. To conduct a foun- dational study of an area of knowledge X we must limit ourselves to cognitive resources generated in areas lower than X. This means that it is impossible to gen- erate tools for a foundational study of the most basic areas of knowledge, those lying at the base of the ordering. The foundational study of such areas is bound to involve either circularity or resources generated higher-up in the ordering, and as such it violates a fundamental principle of the foundationalist methodology. This has led many philosophers to shift from the foundationalist to the holistic meth- odology, but holism is commonly believed to involve a complete renouncement of the foundational (and not just the foundationalist) project or else acquiescence to a very limited (e.g., a coherentist or a narrowly naturalistic) foundational project. In the above-mentioned works I argue that holism is perfectly compatible with a robust, noncoherentist, non-narrowly-naturalistic foundational project. The foundational project, rightly conceived, is the project of substantively and critically studying the main principles of some basic subject matter, with special emphasis on explanation and justification. Now, if we think of this project as one that is pursued in ways appropriate for humans (rather than, say, for gods), then we are seeking not absolute certainty or instantaneous knowledge (achieved by a flash of intuition), but progress achieved by probing inquiry, carefully monitored Truth and Transcettdrnce 287 imagination and insight, smart decisions, experimentation, openness to criti- cisms, willingness to institute revisions, and so on. Thus understood, the founda- tional project does not require a strict or a rigid methodology. Rather, it favors a flexible and dynamic methodology, one that sanctions less-than-perfect and tem- porary tools, back-and-forth movement, shifts in perspectives, and so on-i.e., the kind of methodology that is manifested in the holistic metaphor of Neurath's boat (minus coherentism). On this interpretation, Neurath's boat is a boat on a mission, a mission to study the sea and its residents, and to achieve this goal its occupants are ready to use any tools available to them flexibly and constructively yet also thoughtfully and critically. It is this methodology, which I call "founda- tional holism", that makes a substantive, foundational study of basic philosophi- cal subject matters like truth possible. (iii) Cognitive-Epistemic Focus. Every philosophical theory approaches its sub- ject matter from some perspective. The theory of truth considered in the present chapter approaches truth from a cognitive-epistemic perspective.5 This does not mean that it identifies truth conditions with epistemic conditions (e.g., justifica- tion conditions). On the contrary; it regards the task of assigning distinctly truth conditions to sentences and theories as integral to the epistemic project. Truth is fundamental to the human cognitive-epistemic project due to a combination of circumstances characterizing the human condition: strong cognitive inter- ests (both practical and intellectual), complex world, and a mixed assortment of cognitive resources (limited in some respects, rich and intricate in others). This complex situation creates an abiding need for reality checks by human cognizers, and that, in turn, requires a standard of truth for human thoughts (beliefs, theo- ries). In other words: It is due to the ever-present threat of a gap between our thought on the one hand and reality on the other that the question of truth is so crucial for us.6 Had life been so hospitable that we needed no information about the world, had we no intellectual interest in the world (no desire to have a theo- retical understanding of the world), were we incapable of cognitively diverging from or going beyond what is actually the case, had we no imagination, no drives, interests, or motives that blocked or distorted information about the world, then a standard, a concept, and a theory of truth would have been of little cognitive use for us. But since none of this is the case, the question of truth, the question whether our thoughts7 measure up to reality, always arises for us and is of great importance to us. 11.3. The IMMANENCE Thesis and Its Three Constituting Principles: Immanence, Transcendence, and Normativity The IMMANENCE thesis addresses a semi-Kantian question about truth, namely: Under what cognitive conditions does truth emerge as a fundamental standard of correctness for human thought? Its answer is that for truth to arise, .,.,,,,- . ., .... ,. " Mf""'- �-.... , • ..-1 .. ,.,,.,... .:� - .... ':i:;:r.;: 288 REFLECTIONS ON THE LIAR three conditions on human cognition have to be satisfied: immanence, transcendence, and normativity. Put differently: Truth emerges in the intersection of three basic modes of human cognition: immanence, transcendence, and normativity. Immanence. The first condition for the emergence of truth as a fundamental standard for human thought is the ability, and practice, of directing our gaze at the world, or at some things in the world, and saying something about them, or attributing some property, relation, or state to them. The mode of thought we use in satisfying this condition I call the "immanent mode" and thoughts exhibiting the character- istics of this mode-immanent thoughts. Immanence, thus, is both a mode and a property of thought: a mode of thought as an act, and a property of thought as an object. My use of "immanence" for this mode/property is influenced by Quine. In some of his writings (e.g., 1981, pp. 21-22) Quine says that to speak imma- nently is to speak from within a theory. But speaking from within a theory is, typically, saying something about things outside the theory, things in the world. In my own use, speaking immanently is speaking in the way one typically speaks when one speaks from within a theory, namely, speaking about some subject matter, attributing properties/relations to some objects, or saying how the world is. Immanence, thus, exhibits a basic dialectic of human theories: theories are human creations, yet they are focused on something external to them-the world, in a broad sense of the word. Accordingly, "immanent" connotes "being internal to a theory", but "being internal to a theory" connotes "being directed at something external to the theory".8 Our conception of immanence is also related to ideas by other philosophers: it is related to some philosophers' idea of intentional thought (see, e.g., Siewert 2016), to Frege's view that "in every judgment ... a step is made from the level of propositions to the level of the nominata (the objective facts)" (1892, p. 91).' to James's statement that "human thought appears to deal with objects inde- pendent of itself" (1890, p. 271), to Wittgenstein's claim that "(t]he general form of propositions is: This is how things are" (1921, 4.5), etc. My use of "immanence", however, is also different from many uses of this term in the philosophical literature, including some aspects of Quine's use of this terlll- For example, Quine (1970, 1986, 1995) restricts immanent statements to state- ments belonging to our mother tongue, to a given object language, to scientific discourse, or to naturalistic discourse. My own conception of immanence does not impose any of these restrictions. Immanent thought, on my conception, is com- monly translinguistic. The principles of general relativity, for example, are imma- nent in my sense, yet they do not belong to a specific language. Similarly, Kant's conditions for the possibility of knowledge are immanent, yet they are not part of a scientific or a naturalistic discourse. The category of immanent thought determines the domain of truth-bearers, i.e., the range of thoughts for which truth serves as a standard and to which truth- properties (truth, falsehood, truth-indeterminacy, etc.) apply. Immanence sets no Truth and Transcendence 289 limit on the complexity of thoughts, on whether they address their subject matter literally or nonliterally, directly or indirectly. Given our broad conception of world, thoughts themselves are part of the world, and therefore the category of imma- nent thoughts includes thoughts directed at thoughts. The category of immanent thought is very broad, but not all thoughts are immanent. In particular, thoughts that do not intend, or do not succeed, in saying something about the world or in attributing a property (relation) to some things in the world are not immanent. To be immanent is to genuinely attribute a property to something in the world, something of the kind that the property in question can be attributed to, and this is not a trivial thing. Normally, the category of immanence encompasses all statements and theories of all genuine fields of knowledge as well as large parts of everyday discourse. But depending on the type of thought involved (and/or the type of objects and properties involved) immanence might pose certain specific requirements, as we will see in the case of truth-thoughts-thoughts that attribute truth-properties to thoughts-below. Desirably, the boundaries of immanence are delineated in a systematic manner. But even before an adequate systemati- zation is available, we can use our judgment to decide, in many particular cases, whether a given thought is immanent. 10 A nontrivial presystematic candidate for a nonimmanent thought is "The number 2 is laughing".11 While immanence is a basic condition for truth, immanence by itself does not suffice to yield truth. To focus on the world, to say something about it, is not yet to approach it through the prism of truth. Transcendence. A second condition for the emergence of truth as a fundamental standard for human cognition is transcendence. By "transcendence" I mean the ability, and actual practice, of moving outside a given thought in order to reflect upon it, examine it, say something about it, ask and answer questions about it, set norms or standards for it, challenge it, attribute properties to it, and so on. The transcendence required for truth is transcendence of immanent thoughts. Henceforth, we will understand by "transcendence" transcendence of this kind. Transcendence has fallen into disrepute lately. To say that truth is transcen- dent, it is claimed, is tantamount to saying that we have a "God's-eye view" on language and the world. But "transcendence" does not need to have this connota- tion, and in our own use of this word it does not. Transcendence, as we under- stand it here, is not something mysterious or superhuman; rather, it is something quite simple and commonplace. Transcending an immanent thought, or a region of immanent thoughts, is casting a reflective look at it from a standpoint that holds it in view yet is located within the purview of human thought. In accordance with our foundational-holistic methodology, our conception of transcendence is holistic. Transcending a thought is finding a standpoint anywhere on Neurath's boat from which we can see it. Transcendence of this kind is also dynamic. We transcend sociology in order to view it from a philosophical standpoint, and we transcend philosophy in order to view it from a sociological standpoint. What counts as appropriate transcendence varies according to task and circumstance. 290 REFLECTIONS ON THE LIAR If our task is to describe the syntax of a given theory, then, as Godel (1931) has shown, if the theory is sufficiently rich, we can view its syntax from within it. I.e., a theory itself can provide an adequate standpoint for examining some of its features. But if our task is different, we may need a standpoint outside it. While the principle of transcendence, like the principle of immanence, calls for a sys- tematization, here too we are able to judge in many particular cases whether an appropriate transcendence is achieved prior to a full systematization. Although transcending an immanent thought or a domain of immanent thoughts is merely a human act, it is a cognitively powerful act. Godel's complete- ness and incompleteness theorems, Church's thesis, Turing's proof of the unsolv- ability of the halting problem, the Lowenheim-Skolem theorem, Lindstrom's theorems, all testify to power of the transcendent mode in one of its forms- ascent to a metalanguage. Transcendence is central to numerous fields of knowledge: psychology, soci- ology, many areas of philosophy, metalogic, metamathematics, etc. On the one hand, the very idea of metalogic or of philosophy (sociology, psychology) of, say, knowledge, requires transcendence; on the other, transcendence enables us to develop tools that are needed in these disciplines. It is important to recognize that transcendence does not conflict with imma- nence. On the contrary, most transcendent thoughts are immanent. Their tar- get is something in the world-human thoughts; they attribute properties to their target thoughts, relate them to other thoughts as well as to things other than thoughts, and so on. As such, they are genuinely immanent. There are many types of transcendent thoughts, and one of these is the truth-thought-a thought that attributes a truth-property to some thought(s). To be an admissible truth- thought, i.e., a truth-thought that is admissible as a truth-bearer, a truth-thought must be both immanent and transcendent, and its object(s)-the thought(s) to which it attributes a truth property-must also be immanent. I will call this interplay between immanence and transcendence the 'immanence-transcendence complementarity'. Immanence and transcendence by themselves, however, are still not sufficient for truth. By transcending an immanent discourse either to a higher level of dis- course or to a standpoint on Neurath's boat that has it in view, we can do many things that are not related to truth: we can ask questions of a variety of kinds about these thoughts (e.g., are they mathematical thoughts), attribute properties and relations of multiple types to them (e.g., being thoughts about something funny), set standards (norms) of different sorts for them (e.g., clarity), doubt, challenge, justify, refute them on diverse grounds, enjoy them or be disgusted by them, etc. Truth arises when we ask questions of a special kind about imma- nent thoughts and set standards (norms) of a special kind for such thoughts. This requires a third mode of thought. Normativity. A third condition for the emergence of truth as a fundamental standard for human cognition is the ability to engage in normative activities and Truth and Transcendence 291 the actual engagement in such activities. The mode of thought characteristic of such engagement I call the "normative mode of thought". The normative mode of thought is a mode of questioning, evaluating, setting standards for, sanctioning, etc., our thoughts, decisions, and actions in light of what we value, positively or negatively. As such it requires a transcendent standpoint. Normative thoughts are often associated with critical questions, and truth emerges as a standard for a positive answer to one, especially fundamental, normative-transcendent question: the question of correctness (of a given imma- nent thought). Roughly, and informally, the question of truth, or correctness, as it applies to a given immanent thought X can be expressed by: (QT) Is the world the way the immanent thought X says it is? Do the objects that X talks about have the properties, or stand in the relations, that it attributes to them? At issue is whether X is connected to reality in a way that warrants a positive answer to this question.12 Due to the complexity of the human cognitive situation noted above, the question of truth is one of the main engines of the human cognitive-epistemic project (the project of cognizing or knowing the world), and truth is a standard (or a norm) for a positive answer to this question. When a given immanent thought satisfies this standard we say that it is true, or that it has the property of truth. When it does not, we say that it is not true or, in many cases, false .13 We may never be able to give a final answer to the question of truth concerning a given immanent thought, an answer that could not be questioned or challenged in return. But it is just for this reason that the question of truth is a central driving force in the search for knowledge rather than a superfluous scholastic question. I have called immanent thoughts that attribute truth-properties to imma- nent thoughts "truth-thoughts". Such thoughts, on my account, are both nor- mative and descriptive. Inasmuch as they are concerned with the satisfaction of the truth standard, they are normative; inasmuch as they are concerned with the possession of the truth property, they are descriptive. The fact that truth is a transcendent standard of correctness sets special requirements on the transcendence of truth-thoughts. To attribute a truth- property to a given immanent thought, to determine whether a given immanent thought is true or false, we need to transcend it not just to any standpoint from which we can see it, but to a standpoint from which we can see both it and the world-specifically, that part, facet, or aspect of the world (objects in the world) that it is directed at, says something about, attributes properties to. It is at this point, a point from which we measure a given immanent thought, as a content- conveying thought (rather than as a syntactic object), in relation to its subject matter in the world, that truth arises in our cognitive life. 292 REFLECTIONS ON THE LIAR Such a standpoint is required to be transcendent in a strong sense. It is required to have in view both the target immanent thought and that part of the world that this thought is directed at, and it must afford us a critical view of the relation between them. For that reason, in the case of truth-thoughts we are required to step outside the target thought and cannot stay inside it (as we can do-using Godel numbering-if our interest is the syntax rather than the truth value of the target thought). This need to "step outside" means that in a sentence of the form "Xis true" ("Xis false", "Xis not true", etc.), the sentence named by "X" has to stand on its own as an immanent sentence. I.e., if we take a token of "X is true" and remove "is true" from it, we are left with a name of a token imma- nent sentence. ("Xis true" minus "is true" names a bona fide immanent sentence.) Furthermore, the transcendence inherent in truth-thoughts is antisymmetric: if a given truth-thought, t1 , transcends another truth-thought, t2 (which, as a truth- thought, is itself transcendent), the thought transcended by t2 is not t1, and the truth predicate belonging to t2 does not transcend the truth predicate belonging to t1• 14 Now, all these constraints on the transcendence of truth-thoughts are also constraints on their immanence: For a truth-thought to be appropriately immanent its truth predicate must be appropriately transcendent to some appropriate imma- nent thought(s) in the sense indicated above. The immanence-transcendence complementarity is required to satisfy this condition I call the combination of principles discussed in this section the "IMMANENCE thesis". The IMMANENCE thesis, like its constituting principles, is a material the- sis. It says that truth arises in the intersection of the immanent, transcendent, and normative modes of thought in the way just described (be it only briefly and infor- mally). We can sum it up by saying: A. The question of truth arises for all and only immanent thoughts. B. Truth is a normative-transcendent standard for immanent thoughts. (Truth and falsehood (nontruth) are normative-transcendent properties of immanent thoughts.) C. Truth-thoughts are immanent, transcendent, and normative. The IMMANENCE thesis has significant ramifications for other principles of truth, including the equivalence principle and, through it, the liar paradox. Since the equivalence principle and the liar paradox are commonly viewed as pertaining to small units of thought-sentence-like units rather than whole-theory-like units-from now on I will identify "thought" with "sentence-like thought" or, for short, "sentence". 11.4. Ramifications for the Equivalence Principle The IMMANENCE thesis says that given an immanent sentence P, we transcend it to make the truth-statement that P is true, and our standard of truth says that Truth and Transcendence 293 P is true iff the world is as P says it is, i.e., on one formulation, "< P > is true iff P". In this way, the IMMANENCE thesis supports, or gives rise to, the equivalence principle, E. But the IMMANENCE thesis, and in particular two of its subprinciples, immanence and transcendence (with significant help from normativity), also restrict E by setting certain constraints on it. Two such constraints are: (Im) For E to hold for any given sentence P, P must be a bona fide immanent sentence. (Trans) If Pis a truth-sentence, then for E to hold for P, P must be appropriately transcendent. (The truth predicate of P must be appropriately transcendent.) Accordingly, our theory of truth sanctions only those instances of E that satisfy the two material conditions, Im and Trans. Three instances of E that are sanc- tioned by these conditions are: (E1) "Snow is white" is true iff snow is white, (E2) "'Snow is white' is true" is true iff "snow is white" is true, 15 (E 3 ) "'Snow is white' is false" is true iff "snow is white" is false. E1 says that the immanent sentence "Snow is white" satisfies our (transcendent) standard of truth, or has the (normative-transcendent) property of being true, iff the world is as the sentence "Snow is white" says it is, i.e., under one formula- tion, iff snow is white. E2 says that the immanent truth-sentence "'Snow is white' is true" satisfies our (transcendent) standard of truth, or has the (normative- transcendent) property of being true, iff the world is as the immanent truth- sentence "'Snow is white' is true" says it is, i.e., under one formulation, iff "Snow is white" is true. And E3 says that the immanent truth-sentence '"Snow is white' is false" satisfies our (transcendent) standard of truth, or has the (normative- transcendent) property of being true, iff the world is as the truth-sentence" 'Snow is white' is false" says it is, i.e., under one formulation, iff "Snow is white" is false. All these instances of the equivalence principle satisfy the requirements Im and Trans: "Snow is white", "'Snow is white' is true", and" 'Snow is white' is false" are appropriately immanent, and the last two are appropriately transcendent. Now, it follows from the material constraints Im and Trans that Ep E as applied to P is an inadmissible instance of E in at least two circumstances: (a) Pis not a bona fide immanent sentence. (b) Pis a truth-sentence and Pis not appropriately transcendent (P's truth predi- cate is not appropriately transcendent). I.e., all instances of E that satisfy either one of the material conditions (a) or (b) are blocked by our substantivist theory of truth. 294 REFLECTIONS ON THE LIAR Another way of expressing these constraints is by saying that E can be instantiated only by admissible truth-bearers, where Im and Trans are reformulated as conditions on admissible truth-bearers. Thus formulated, Im says that for any sentence to be an admissible truth-bearer it must be immanent, and Trans says that for any truth- sentence to be an admissible truth-bearer it must be appropriately transcendent (or that its truth predicate must be appropriately transcendent to it). We may say that IMMANENCE induces an update of E, or that E � IMMANENCE involves an update of E which is expressed by constraints placed on the scope of E, so that now we have: (E) < P> is true iff P, where "P" stands for any sentence satisfying Im and (where applicable) Trans, and "< P>" stands for a name of this sentence. 11.S. Blocking the Liar Paradox The liar paradox arises from a particular instance, or cluster of instances, of the equivalence schema. One classical representative of these instances is (EL) "Lis false" is true iff Lis false, where "L" abbreviates "'L is false'". We arrive at the liar paradox by replacing "'L is false'"with "L", getting: (LP) L is true iff L is false. Our theory, however, blocks LP by ruling, based on the material conditions Im and Trans, that EL is an inadmissible instance of E or, alternatively, that the liar sentence, (L) Lis false. is an inadmissible truth-bearer. Before explaining why L/EL fails to satisfy Im and Trans, let us set the context for our explanation by a few preparatory and background comments: (i) In discussing whether L satisfies the Im and Trans conditions we will not appeal to any mechanical (or semimechanical) test. This is connected with the fact that at this stage we are not committed to any specific systematization of the immanence and transcendence principles. Systematization has many benefits and is an important goal of theorizing; but it often comes with a price: an ad hoc treatment of outliers.16 Truth and Transcendence 295 (ii) Given our cognitive-epistemic perspective, our goal is not to rule out only (or specifically) paradoxical sentences as admissible truth-bearers; our goal is to rule out any sentence that from a cognitive perspective is not a genuine truth-bearer. This distinguishes us from theorists like Kripke who aim at rul- ing out only paradoxical sentences. For example, for us it might be just as important to reject (T) T is true as to reject L as an admissible truth-bearer.17 (iii) Since the liar paradox is commonly thought to enter theories of truth through some instance of E, in this chapter I focus on E as a "gate" to the Liar. But it is worthwhile to note that from a cognitivist-epistemic perspective E is less central to theories of truth than it is often taken to be. The connection of immanent sentences to the world (through their truth conditions) is very central, but E itself offers just one particular way of representing this connec- tion and there could in principle be alternative representations. For example, in the case of the immanent sentence "Snow is white" it is possible to repre- sent its connection to the world (through its truth conditions) by a non-E- biconditional such as "Snow is white" is true iff snow reflects light of all huescompletely and diffusely. And for some purposes (e.g., informativeness) this representation might be superior to the more common E-representation: "Snow is white" is true iff snow is white. Thus one possible way of blocking the paradox could be to remove E from the theory altogether and introduce a different equivalence principle in its place. Rejecting E altogether, however, would be ad hoc, since most instances of E are true. For that reason I prefer to approach the task by limiting the scope of E. (iv) Given that L is a self-referring sentence and that this trait is sometimes associated with its paradoxicality, the question naturally arises whether immanence and transcendence ban (either individually or together) all self- reference. The answer is no. For example, the sentence (S) S is short, is self-referring, yet in spite of this it is not banned by either Im or Trans. Since S is not a truth-sentence, it is not subject to Trans. It is subject to Im but it has no difficulty satisfying Im, since it attributes a property applicable to syntactic entities (being short) to a bona fide syntactic entity in the world, namely, to itself as a syntactic entity, or to the syntactic facet of itself (as a sentence). 296 REFLECTIONS ON THE LIAR (v) Another feature that is often attributed to L is circularity. Do Im and/or Trans ban all forms of circularity? The answer, here too, is no. This can be seen both by considering the holistic nature of our theory, which sanctions nonvicious circularity (e.g., the kind of circularity represented by back-and- forth moves within Neurath's boat), and by considering particular sen- tences. Consider the sentence (A) The property of being abstract is an abstract property. This sentence is not blocked by Trans since it is not a truth-sentence, and it is not blocked by Im, since it is an immanent sentence: it attributes a property X to something that this property is applicable to, namely, to the property Y of being abstract. (The identity of X and Y does not change this fact.) As such it falls under (E).18 (vi) To say that a given sentence is not immanent is not to reject it as a sen- tence or even as a meaningful sentence. Immanence identifies a subclass of sentences, namely, those that potentially partake in the cognitive proj- ect, a project that gives rise to a notion of truth (or calls for a standard of truth). But language, as I will note in section 10.6 below, has multiple functions, and therefore a sentence that fails to contribute to the cogni- tive project might yet fulfill another function that could render it mean- ingful (for example, the sentence 'I do' uttered by a court witness while being sworn in). (vii) To say that a given sentence is transcendent is not necessarily to say that it belongs to a metalanguage. The object-language-metalanguage comple- mentarity is one model (systematization) of the immanence-transcendence complementarity, but it is not its only model, as we will see in discussing Kripke below. We are now ready to turn to the liar sentence, L, and the corresponding E- instance, EL. L says of itself that it is false, i .e., it attributes a truth-property to itself. As such it is a truth-sentence and therefore subject to both Im and Trans. The question is whether L satisfies these requirements. To satisfy Trans, the operation of truth predication in L must be appropriately transcendent to the object it is predicated of, namely L itself. I.e., in light of the conditions set on transcendence by the normativity of truth (see last paragraph of subsection of normativity) L minus "is false" must be immanent. But once we remove "is false" from L, no immanent sentence is left. So L fails to satisfy Trans: there is no appropriately immanent object for L to transcend.19 It follows that L is an inadmissible truth-bearer according to our theory' hence EL is an inadmissible instance of E and the liar paradox (in its classical form) is blocked. We have shown that the liar paradox can be blocked by material principles arrived at in the course of developing a substantive theory of truth. To do that we have focused on one, albeit classical, form of the paradox. This is sufficient to Truth and Transcendence 297 make our main point, but the question naturally arises whether the same material principles are capable of blocking other forms of the paradox as well. Let us examine two other liar sentences, leading to other forms of the liar para- dox. First, let us consider a type of Liar exemplified by (7) (8) is false, where the sentence referred to in (7) is (8) (7) is true. To show that (7) is not an admissible truth-bearer we note that for (7) to be an admissible truth-bearer, it must satisfy Trans, and this requires both that (8) be an admissible truth-bearer and that (7) be appropriately transcendent to (8). But for (8) to be an admissible truth-bearer, it has to be appropriately transcendent to (7), and this is impossible. Transcendence is an antisymmetric relation, hence (7) and (8) cannot mutually transcend each other. As in the case of L, (7) fails to satisfy Im. The next "liar sentence" we consider is only contingently a liar sentence. Let me explain. So far we have considered sentences that give rise to the liar paradox due to their own semantic structure and/or the semantic structure of other sen- tences they are relevantly connected to. Kripke (1975) shows that the liar paradox can arise due to circumstances that hold in the world, so that a sentence with a given semantic structure can be paradoxical under some circumstances and non-paradoxical under others. Are liar sentences of this kind ruled out by Im and Trans? And if they are, are they ruled out in all circumstances or only in problem- atic circumstances? The answer is that they are ruled out in problematic circum- stances (of the kind considered by Kripke) but not in all circumstances. Kripke's example of a contingently liar sentence is: (9) Most (i.e., a majority) of Nixon's assertions about Watergate are false. (1975, p. 691)20 While under most circumstances this sentence is nonparadoxical, under some circumstances it is. Consider the following two constellations of circumstances: Constellation of Circumstances #1: (i) One of Nixon's assertions is (10) Everything Jones says about Watergate is true (Kripke 1975, p. 691); (ii) Aside from this sentence, 'Nixon's assertions about Watergate are evenly bal- anced between the true and the false' (Kripke 1975, p. 691), and (iii) Jones made exactly one assertion about Watergate (9). 298 REFLECTIONS ON THE LIAR It is easy to see that under this constellation of circumstances (9) is paradoxical, and it is also easy to see that Im and Trans rule out (9) as an admissible truth- bearer under this constellation. Given (i)-(iii), both transcendence and imma- nence break down, or malfunction, in (9), as they do in (7). Constellation of Circumstances #2: (i) Nixon made only one assertion about Watergate. (ii) Nixon's assertion about Watergate is: (11) I have nothing to do with the Watergate break-in. Clearly, under this constellation of circumstances (9) does not yield a paradox, nor is it rejected by Im or Trans. With this we have completed the task of showing that the liar paradox can be blocked by substantivist theories of truth due to their material principles. It fol- lows that philosophers need not impose liar-motivated constraints on their theo- ries of truth in advance, constraints that might unnecessarily limit their options in constructing their theories. They may wait until a later stage in the construction of their theory to decide which constraints (among all the alternatives offered by different solutions to the paradox) are most suitable for their theory, if the need for such constraints arises at all. Our next task is to explore how changing our attitude toward the liar paradox in this way affects our perspective on existent solutions to the paradox and on some of the factors commonly associated with these solutions. 11.6. Tarski, Kripke, Natural Language1 and Bivalence It is quite common to view Tarski's (1933) hierarchical solution to the liar paradox as an unnecessarily radical, ad hoc solution and contrast it with other, less radi- cal and less ad hoc solutions like Kripke's (1975). Our discussion of truth, tran- scendence, and the liar puts the two types of solution in a new perspective: both solutions are based on a largely non-ad hoc systematization of the immanence- transcendence complementarity, Tarski's on an external-hierarchy systematization and Kripke's on an internal-hierarchy systematization, and each systematization has its own advantages and disadvantages. Tarski's External Hierarchy as a Systematization of the Immanence-Transcendence Complementarity. In the Tarski systematization the immanence-transcendence complementarity is represented by an object-language-metalanguage hierarchy. We start with a so-called object language, L0, which is a purely immanent lan- guage. Sentences of this language are directed at something external to them- selves, something in the world (broadly construed). The language has no resources for transcendence, e.g., no direct resources for naming its own sentences and no semantic predicates (or tools for creating them). This object language serves as a Truth and Transcendence 299 basis for the construction of a hierarchy of languages and is the lowest element in this hierarchy. The next language is a so-called metalanguage, ML0 or L1, and we arrive at it by transcending L0 • is fully equipped for (truth-)transcendence: it has resources for talking about L0, resources for talking about that part or aspect of the world that L0 is directed at, and resources for setting a standard of truth for sentences of L0. Furthermore, L1 is itself an immanent language, and as such has its own metalanguage, (= ML1 MML0) to which it stands as an object language. Each metalanguage has a truth predicate applicable to its predecessor in the hierarchy, and as a result, truth itself is represented by a hierarchy of predi- cates: Ti, T2, T3, • • • , which apply to sentences of 10, L11 L2, • • • , respectively. As a systematization of the immanence-transcendence complementarity Tarski's hierarchy has both advantages and disadvantages. Among its advantages are a clear, sharp, unambiguous, and formally rigorous structure, as well as one that has proved extremely fruitful in multiple fields of knowledge. Let me briefly reflect on its fruitfulness. It is a remarkable fact that while Tarski used the object- language-metalanguage duality as a formal solution to a formal problem concern- ing the largely material notion of truth, this duality, partly under his influence, has been thoroughly integrated into our conception of knowledge and has proved extremely profitable in a number of fields, in particular metalogic and metamath- ematics. My present suggestion is that one of the reasons this duality has been a major force in advancing human knowledge is that it captures the cognitively fun- damental and highly fruitful complementarity of immanence and transcendence. Among the weaknesses of the Tarskian hierarchy, from our perspective, is a rigid and inflexible structure, as reflected in, e.g., the exclusion of many self-referring sentences satisfying Im and (if applicable) Trans. Its rigidity is also reflected in some artificial aspects of its treatment of truth, for example, the existence of multiple truth predicates, each limited to a particular language in the hierarchy, rather than a single predicate, common to the entire hierarchy. Another weakness is the narrow range of languages that Tarski's hierarchy encompasses, namely, its limitation to so-called "formalized languages of the deductive sciences". These languages by no means exhaust the full range of cognitively efficacious languages licensed by the (presystematic) principles of immanence and transcendence. A different model of the immanence-transcendence complementarity is sug- gested by Kripke's solution to the liar paradox. Kripke's Internal Hierarchy as a Systematization of the Immanence-Transcendence Complementarity. In Kripke's model we have only one language and only one truth predicate. The idea of immanence is represented by the concept of groundedness and the idea of transcendence by a stage by stage determination of the extension (and counterextension) of the truth predicate. Kripke informally describes the intuition underlying his concept of groundedness as follows: It has long been recognized that some of the intuitive trouble with Liar sentences is shared with such sentences as [T21] which, though 300 REFLECTIONS ON THE LIAR not paradoxical, yield no determinate truth conditions .... In general, if a sentence . .. asserts that (all, some, most, etc.) of the sentences of a certain class C are true, its truth value can be ascertained if the truth values of the sentences in the class C are ascertained. If some of these sentences themselves involve the notion of truth, their truth value in turn must be ascertained by looking at other sentences, and so on. If ultimately this process terminates in sentences not mentioning the concept of truth, so that the truth value of the original statement can be ascertained, we call the original sentence grounded; otherwise, ungrounded .. .. Sentences such as [T], though not paradoxical, are ungrounded. (Kripke 1975, pp. 693-694) The grounded sentences in Kripke's system are all immanent in our sense, and most immanent sentences in our sense are included in the class of Kripke's grounded sentences.22 The idea of transcendence is represented by Kripke's conception of the (single) truth predicate as defined in stages, or its extension as determined in stages. In Stage 1 the truth predicate, Tr, is assigned an extension (and a counterextension)23 in the set of all grounded/immanent sentences that contain no truth predicate (or semantic predicates more generally). In Stage 2 the extension (and counterexten- sion) of Tr is extended to all grounded/immanent sentences whose truth value is determined in Stage 1. And so on. In this way, the assignment of a truth value to each immanent sentence in Kripke's system is made from a standpoint transcen- dent to it, albeit internal to the (single) Kripkean language, and Kripke's system can be viewed as representing the immanence-transcendence complementarity by an internal hierarchy. Things change, however, once we get beyond the "'minimal' or 'smallest' fixed point" (Kripke 1975, p. 705), i.e., beyond the point in which the hierarchy is limited to grounded/immanent sentences. Beyond this point Kripke deals with sentences like T, which are neither grounded nor paradoxical, and shows how a conventional assignment of truth values to such sentences can proceed. Since these sentences are nonimmanent (ungrounded), they do not fall under our immanence and transcendence principles, and as such are not admis- sible truth-bearers according to our theory.24 One important feature of Kripke's system is its allowing a sentence to "seek its own level'' (1975, p. 696) in the internal hierarchy. Whether or not a given sentence has a place in the internal hierarchy, and if it does, what this place is, Kripke points out, may be determined by empirical circumstances surrounding the sentence. It is in this context that he introduces the sentence numbered (9) in the present chapter, which, he rightly says, is an admissible truth-bearer in some circumstances, inadmissible (and paradoxical) in others. Among the advantages of Kripke's hierarchy over Tarski's from our perspective are its considerable flexibility and highly dynamic nature. These are reflected in its ability to recognize the admissibility of sentences that are excluded from Tarski's Truth and Transcendence 301 hierarchy as truth-bearers (e.g., some self-referring sentences satisfying Im and (where applicable) Trans that are banned from Tarski's hierarchy) as well as in its ability to account for the role of empirical circumstances in the admissibility and place in the hierarchy of sentences like (9). Importantly, its flexibility saves it from the need to artificially multiply truth predicates as in Tarski's hierarchy.25 But Kripke's system has significant disadvantages as well. In particular, its internal hierarchy is too confined to encompass the whole Tarskian hierarchy, leaving many metalinguistic truth-bearers unaccounted for. Kripke himself is fully aware of the indispensability of an (external) hierarchy like Tarski's: It seems likely that many who have worked on ... [non-Tarskian] approach[es] to the semantic paradoxes have hoped for a universal language, one in which everything that can be stated at all can be expressed .... Now the languages of the present approach contain their own truth predicates and even their own satisfaction predicates, and thus to this extent the hope has been realized. Nevertheless the pres- ent approach certainly does not claim to give a universal language, and I doubt that such a goal can be achieved. First, the induction defining the minimal fixed point is carried out in a set-theoretic metalanguage, not in the object language itself. Second, there are assertions we can make about the object language which we cannot make in the object language. For example, Liar sentences are not true in the object lan- guage, in the sense that the inductive process never makes them true; but we are precluded from saying this in the object language by our interpretation of negation and the truth predicate. If we think of the minimal fixed point ... as giving a model of natural language, then the sense in which we can say, in natural language, that a Liar sentence is not true must be thought of as associated with some later stage in the development of natural language, one in which speakers reflect on the generation process leading to the minimal fixed point. It is not itself a part of that process. The necessity to ascend to a metalanguage may be one of the weaknesses of the present theory. The ghost of the Tarski hierarchy is still with us. (1975, p. 714) Natural Langu.age and the Cognitive Perspective on Truth. Many of the objections to Tarski's hierarchy in the philosophical literature have to do with natural lan- guage. In a way, Tarski himself invited these objections. In setting the ground for his theory of truth he described his goal as constructing a philosophical definition of the concept of truth, and his explanation suggested that it is "the meaning of the term 'true sentence' in colloquial language" (Tarski 1933, p. 152) that such a definition aims at capturing. Yet Tarski emphasized that a "thorough analysis of the meaning current in everyday life of the term 'true' is not intended here" (1933, p. 153), and he proceeded to question the very consistency of natural 302 REFLECTIONS ON THE LIAR language on the ground that it leads to paradox: Due to the occurrence of the liar paradox in natural language, "the very possibility of a consistent use of the expression 'true sentence' which is in harmony with the laws oflogic and the spirit of everyday lan- guage seems very questionable, and consequently the same doubt attaches to the pos- sibility of constructing a correct defmition of this expression » (1933, p. 165). Finally, he formulated his goal as that of defining truth for a narrow class of highly spe- cialized artificial languages: the "formalized languages" of "the deductive sciences» (1933, p. 166). It is not surprising, therefore, that many philosophers questioned Tarski's approach to the construction of a theory of truth, starting with his sweep- ing conclusion about the inconsistency of natural language and ending with his narrow focus on artificial logical languages. 26 Leaving the controversy over Tarski's approach to truth aside, let me briefly explain how natural language is viewed from the perspective of the theory pre- sented here-both from its broader substantivist perspective and from its nar- rower, though still very broad, cognitivist perspective. From a substantivist perspective the diverse, multifaceted, and sometimes conflicting uses of "truth" and its cognates in natural language(s) reflect the extraordinarily breadth and diversity of truth, and an account of the breadth and diversity of truth and their manifestations in natural language is one of the tasks of a substantive theory (family of theories) of truth. But it is not the only task, or necessarily the most important philosophical task, of the theory of truth. Indeed, there are several complicating factors in focusing the philosophical study of truth on natural language. Among these are (i) the fact that natural lan- guage is a natural phenomenon, and (ii) the fact that natural language is a mul- tipurpose tool. These facts suggest that the use of "truth" in natural language is shaped by multiple, sometimes conflicting and often accidental determinants, and that as a result natural language is less than an optimal source of understanding the role of truth in specific human endeavors, including cognition or knowledge. This is the point at which we, given our interests, distance ourselves from natural language; but it is also the point at which we distance ourselves from Tarski's for- malized languages. We are investigating truth from a specific, yet relatively broad perspective, the cognitivist-epistemic perspective, and this means that on the one hand we cannot limit ourselves to the behavior of truth in formalized languages, but on the other hand we are not concerned with all its manifestations in natural language. Moreover, our interest in truth is partly normative. This means that as far as language is concerned we are interested in a model of language that is recognizably a model of human language, but one that is geared toward effective cognition of the world rather than toward other tasks. Thus, from our perspective, a study of the behavior of the term 'true' in natural language is valuable in under- standing some aspects of truth, but for understanding the role of truth in human knowledge and cognition natural language is secondary. The Immanence-Transcendence Complementarity and Bivalence. It is common to contrast Tarski's and Kripke's hierarchies not just along the formal-language Truth and Transcendence 303 versus natural-language dimension, but also along the bivalence versus triva- lence (or more generally nonbivalence) dimension. From my perspective, these contrasts are only partly warranted. In particular, the viability of Tarski's and Kripke's hierarchies as models of the cognitive immanence-transcendence com- plementarity has little to do with these contrasts. Focusing on the bivalence- trivalence dichotomy, let me emphasize, first, that the immanence-transcendence complementarity is perfectly compatible both with classical logic and with non- classical logic, both with classical set theory and with nonclassical set theory. There is nothing in my account of immanence or transcendence to favor one logic (or set theory) over the other. If the world, or some aspects of the world, exhibit a tripartite rather than a bipartite property structures, then immanence will be better represented by a nondassical logic (and/or a nonclassical set theory) than by a classical one. The account of the formal structure of properties will have a trivalent rather than a bivalent operation of complementation, so that given a domain D and a property P, complementation will partition D into three rather than two regions. Transcendence would then require a standpoint from which we can view the world in its tripartite structure, and the determination of truth values would be made based on this view of the world, hence involve three rather than two truth values. There is thus no intrinsic sense in which Tarski's notions of object language and metalanguage, or my notions of immanence and tran- scendence, or my material conditions of Im and Trans, are tied up with, or limited to, classical logic or bivalence. Significant structural differences will still exist between Kripke-like and Tarski-like models of the immanence-transcendence complementarity, but the differences between formalized and natural language on the one hand and bivalence versus trivalence on the other are not central to them. Notes 1. The contrast between "material" and "formal" is intended to invoke the similar contrast in Tarski (1933), but my notion of "material adequacy" is not limited to "extensional correct- ness," as Tarski's notion is sometimes thought to be. "Material; in the sense intended here, has the connotation of "internal to the subject matter of the theory," whereas "formal" has the connotation of "having to do with formal matters that are extraneous to the subject matter of the theory." (When the subject matter of a theory is itself formal in the sense of dealing with, say, logical or set theoretical issues, the contrast between "material" and "for- mal" in the present sense is reduced or altogether disappears.) 2. Here and below the arrows represent metalogical consequence relations, or more broadly, monotonic consequence relations. 3. For various aspects of this theory, including further discussion of issues that I will only briefly gloss over below, see Sher (1999a, 2004, 2012, 2013b, and 2015). 4. I intentionally construe potential truth-bearers very broadly. 5. "Cognitive" and "epistemic; or "cognition" and "knowledge," are used here as cognate notions, with "cognition" being the broader, weaker, and less specific of the two. While "knowledge" connotes success in cognition, "cognition" connotes mainly an attempt at acquiring knowledge (information). 304 REFLECTIONS ON THE LIAR 6. I use "reality" and "world" as synonyms. 7. Here and in the remainder of this chapter I use "thought" as a general term that covers sentence, statement, belief, judgment, cognition, theory. body of knowledge, thought proper, etc. 8. We could replace "immanence" with the Quinean expression "world oriented", but to emphasize both the contrast, or complementarity, to transcendence and the inherent dia- lectic of theories we will stay with "immanence". 9. This is especially clearly expressed in the Feig! translation. In the original: "(I]n jedem Urteil .. . der Schritt von der Stufe der Dedanken zur Stufe der Bedeutungen (des Objectiven) geschehen ist." 10. Indeed, working outside a full systematization is sometimes advantageous, since the act of systematizing a theory may force us to introduce some ad hoc adjustments. 11. But see n. 16 below. 12. Elsewhere (Sher 2013b and 2015) I discuss the diversity of ways in which an immanent thought can, or need, be connected to reality in order for the answer to this question to be positive. 13. See discussion of bivalence in section 11.6 below. 14. Note that this does not conflict with the possibility, mentioned above, of transcending philosophy in order to study it sociologically and vice versa. In the course of studying phi- losophy by sociology, the latter is transcendent to the former, and not vice versa, while in the course of studying sociology by philosophy the situation is reversed. So in the course of each study the transcendence relation is antisymmetric. Of course, we can transcend (the relevant parts of) both sociology and philosophy to talk about their possible inter- relationships (as we are presently doing). But this view, too, involves an antisymmetric transcendence relation. 15. l use the convention that embedded quotation marks follow the pattern<",',",', ... >. 16. I should note, however, that sometimes this price is worth paying. In those cases pragmatic considerations guide us in making choices. For example, we may choose, for the sake of sys- tematicity, to render every first-level predicate applicable to every singular term. This might lead us to say. on pragmatic grounds, that "The number two is laughing" is meaningful or even immanent. Due to transcendence, however, we can always move to a standpoint from which we can distinguish (and explain the differences between) genuine and conventional (or ad hoc) meaningfulness and immanence as well as genuine and conventional truth. 17. Theorists of truth vary in their attitude to (T): Tarski regards it as not falling under (E); Kripke regards it as falling under (E). From our perspective, it does not fall under (E) on material grounds, namely: it does not attribute a property to a genuine object. More spe- cifically, it does not satisfy the (material) test for truth-sentences formulated in section 11.3: "Xis true" minus "is true" names a bona fide immanent sentence. Although treating (T) as a truth-bearer does not lead to a paradox and although technically it is possible to reformulate the theory in a way that will leave open the possibility that (T) satisfies (E). I prefer not to do so here. (I would like, however, to thank an anonymous reviewer of this chapter for suggesting how such a reformulation could be done.) My goal is a theory of truth that captures the material content of truth in a substantive manner, and sanctioning (T) does not contribute to this goal. 18. Of course, if and when we set out to systematize the theory, we might be led to exclude this sentence from falling under (E). Under one scenario this would be a compromise we would be willing to make in order to arrive at an overall (materially) better systematization. Under another scenario, further investigation will lead us to update the considerations that lead us to sanction this sentence now so we end up rejecting it on (direct) material grounds. (See discussion of updating in section 11.1.) 19. If we did not include immanence as a condition of appropriate transcendence (i.e .• if we did not say that for a truth-sentence to be transcendent it must be transcendent to an appropri- ate immanent sentence), then we could say that to satisfy Im L must fail to satisfy Trans and to satisfy Trans L must fail to satisfy Im. Truth and Transcendence 305 Another way to pinpoint the blocking of L is to say that to be appropriately immanent, L must attribute its truth-property to itself as a semantic, content conveying sentence. But once its truth-property is removed from its content (by transcendence) it is no longer a content conveying, hence appropriately immanent, sentence. 20. "(9)" is my own number for this sentence. 21. I.e., the sentence we called "T" in section 10.5 above. 22. For important exceptions, see below. I should note that Kripke, like me, distinguishes between meaningful and immanent (for him, grounded) sentences. (See 1975, 699-700.) 23. The counterextension of Tr is the set of all false sentences, leaving truth-wise indeterminate sentences outside both extensions. (Kripke's system is based on a trivalent logic. See discus- sion of bivalence below.) 24. The sentences "T is not paradoxical" and "Technically, T can be assigned a truth value in a consistent system", however, are immanent according to our theory, and therefore the upper echelon of Kripke's system does make a significant contribution to under- standing truth, according to our theory, in spite of dealing with empty (inadmissible) truth-bearers. 25. Another purported advantage of Kripke's system is its ability to account for the behavior of the truth predicate of natural language. I will briefly discuss the relevance of natural lan- guage to the theory of truth below. 26. My own view, which I have expressed elsewhere (Sher 1999b, 2004), is that what Tarski's theory actually does is provide a model of the logical factor in truth, and in particular, the role logical structure plays in the truth conditions of sentences. This justifies Tarski's choice of logically formalized languages to develop his definition, and it explains why his definition of truth had such fruitful applications in metalogic. References Dyson, F. 1988. Infinite in All Directions. New York: Harper & Row. Frege, G. 1892. "On Sense & Nominatum." Translated by H. Feig! in Readings in Philosophical Analysis, ed. H. Feig! and W. Sellars, 85-102. New York: Appleton-Century-Crofts, 1949. Godel, K. 1931. UOn Formally Undecidable Propositions of Principia Mathematica and Related Systems." In Collected Works, Volume 1. Eds. S. Feferman et al., 145-195. New York: Oxford University Press, 1986. 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