Contextualism, Revision Theory of Truth, and the Liar Paradox

Transcription

1 Contextualism, Revision Theory of Truth, and the Liar Paradox by Vanessa Eaken, A.B. A Thesis In Philosophy Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of Master of Art Approved by: Dustin Tucker Chair of Committee Christopher Hom Dominick Casadonte Interim Dean of the Graduate School May, 2013

2 Copyright 2013, Vanessa Eaken

3 TABLE OF CONTENTS Abstract... iii I. Introduction... 1 Some History... 2 Classical Logic... 4 Families of Solutions... 9 II. More on RTT and Contextualism Some Superficial Similarities A More-Complete Picture Revision Theory of Truth Contextualism Some Differences III. My Own Theory A Verbal Dispute The Takeaway A Simpler Theory Fitting into the Big Picture Conclusion Bibliography ii

4 ABSTRACT Most philosophers agree that Contextualism and Revision Theory of Truth are incompatible. My suggestion, however, is that what many people take to be important differences between these theories are merely the result of a verbal, and non-substantial, disagreement between writers like C. Parsons and M. Glanzberg, on the side of Contextualism, and A. Gupta and N. Belnap, on the side of Revision Theory of Truth. By carefully analyzing these theories, we will come to understand three things: (1) Though Contextualists openly reject the Tarski-biconditional and Revision Theorists claim to accept all the principles of classical logic, careful analysis shows that both manipulate the Tarski-biconditional so as to demonstrate the same feature of ideal reasoning a recognition of the fact that the Liar sentence is true at some steps in reasoning, or hierarchical levels, but false on others. (2) The Contextualist solution involves a step that the Revision Theorist of Truth solution does not a step which is the result of an assumption Contextualists make about the semantic status of L when it is analyzed from the same hierarchical level that it is interpreted in. And, (3) Contextualists suggest that shifts from one hierarchical level to another are the result of a change in the domain of the truth predicate, while Revision Theorists of Truth suggest that this change is the result of our changing our interpretation of the truth predicate a difference that is merely the result of Revision Theorists of Truth assuming that the ideal reasoner will begin with hypotheses regarding each and every sentence in his language. The only substantive differences between Contextualism and Revision Theory of Truth, then, are these two background assumptions regarding the semantic status of L and iii

5 what things the ideal person makes hypotheses about. But, even with these differences in mind, we are able to conceive of a theory that both Contextualists and Revision Theorists of Truth would approve of one that brings to light the theory of the Liar Paradox buried within both theories. iv

6 CHAPTER I INTRODUCTION A quick analysis of Contextualism and Revision Theory of Truth reveals that their solutions to the Liar Paradox are similar in several superficial ways. But, upon closer inspection, we see that there are also important differences between the two theories. With these apparent differences in mind, most philosophers take Contextualism and Revision Theory of Truth to be incompatible. The point of this essay, however, is to show that these apparent differences are not real differences at all. Instead, they are merely a misunderstanding that results from the authors of these theories C. Parsons and M. Glanzberg, on the side of Contextualism, and A. Gupta and N. Belnap, on the side of Revision Theory of Truth talking past one another with regard to a variety of topics. Consequently, we can discuss a more abstract theory (detailed in Ch.3, Sect. 3 of this essay) which represents more generally the solution to the Liar Paradox that both theories endorse. In what follows, then, I will discuss the Liar Paradox and why it has been so problematic, historically, for classical logicians. In Chapter 2 I will go on to discuss the general structures of Revision Theory of Truth and Contextualism, as well as the superficial similarities and differences between those structures. The main butt of my argument, however, will come in Chapter 3, where I will show how these apparent 1

7 differences can be explained away so as to reveal the underlying theory of the Liar Paradox that is buried inside each of these theories. Some History To begin, then: As many philosophers have discussed throughout the ages, there is something very odd about Epimenedes of Crete asserting that all of the assertions Cretans make are falsehoods, or, in other words, they are always liars. First, Epimenedes statement is strange insofar as the pragmatic result is Epimenedes dubbing every Cretan, including himself, a liar. But, it is strange for a different reason, too. When we analyze his statement at a semantic level, it seems impossible to judge whether it is a truth or a falsehood. The nature of his claim is such that if we take what he says to be true (and make various other assumptions about the falsity of all of the claims that Cretans have and will make), then he and every other Cretan must always speak falsehoods. But, basic reasoning tells us that if he s always speaking falsehoods, then his assertion that all Cretans always lie must be false. Similarly, however, if we take what he says to be false (while still holding our beliefs about him and every other Cretan always and forever being a liar), then Epimenedes assertion is not a falsehood and he does not always lie. So, by basic reasoning, we can conclude that his assertion must be true. It seems impossible, then, to determine whether Epimenedes assertion is true or false using our basic reasoning skills. Whenever we find that his assertion is true, basic reasoning leads us to the conclusion that it is false, and whenever we find that his 2

8 assertion is false, basic reasoning leads us to the conclusion that it is true. And, yet, we know by basic reasoning that his assertion must be either true or false either Cretans always speak falsehoods or they don t. Consequently, Epimenedes remark and others like it draw our attention to one of the oddities of classical logic paradoxicality. What is particularly interesting about this sort of paradox commonly referred to as the Liar paradox is that it cuts to the heart of where our methods of basic reasoning fall short. What s more, it does so without bringing on extra baggage, by which I mean that one might try to dismiss the paradoxicality of Liar sentences, like Epimenedes, by claiming that they are the result of a sort of vagueness involved in the term liar or that they are the result of self-reference, there are always ways we can shape our Liar sentences so as to avoid these pitfalls and still show that there is something paradoxical involved in asserting them. 1 This teaches us that the various versions of the Liar sentence are paradoxical at their core. They re not paradoxical because of vagueness or self-reference. Instead, their paradoxical nature stems from the truth predicate itself. That is, there s something about the way that we reason about truth and falsity that makes Liar sentences paradoxical. 1 The paradigm example of this, I think, is Saul Kripke s thought-experiment from his Outline of a Theory of Truth. Here he discusses two men, Nixon and Jones, and their views on the Watergate scandal. Jones says of Nixon that he often lies about the Watergate scandal that a majority of the statements he s made in the past and will make in the future about the Watergate scandal are false. But, Nixon says of Jones that every one of the statements he s made in the past and will make in the future about the Watergate scandal is false. There is no self-reference involved each of the political candidates has asserted something about his opponent, not himself. Furthermore, there is no vagueness we ve cashed out liar as someone who has always said, and always will say, (a majority of) false things about this particular subject. But, we still run into the Liar paradox. 3

9 In what follows, then, I will focus my attention on one version of the Liar sentence, what I will refer to as L from here on out, known as the simple-untruth 2 : L: L is untrue. But, with the above thought in mind, it should be the case that what I say about the simple-untruth will apply equally as well to the other members of the Liar family, for each member is paradoxical in the same way. Classical Logic Classical logic offers us a formalized version of our basic methods of reasoning. It does so by representing the heuristics we generally use in everyday thought as laws that an ideal reasoner ought to follow. And each of these laws, or principles, intuitively seems to consistently lead us to true conclusions. Unfortunately, the combination of these principles leads us into a conundrum when it comes to Liar sentences paradoxicality. Before I show how this conundrum can be formally represented in classical logic, I think it would be useful to lay out each of the principles involved. These principles are 2 I choose to discuss this particular version of the Liar because, as far as I can tell, it is less-susceptible to revenge paradoxes paradoxes which reappear after a possible solution has been offered. That is, some versions of the Liar sentence are misleading in such a way that when a solution is offered, the paradox reappears in a slightly-different form. For example, the simple-falsity, which says This sentence is false, ends up returning in a different form after we clarify how one ought to reason about falsity; the simple-falsity reappears as the simple-untruth. But, the simple-untruth does not have this same weakness; it is not as susceptible to a revenge paradox as other versions of the Liar. 4

10 Excluded Middle, Tarski-biconditional, Adjunction, and Disjunction. 3 The principle of Excluded Middle asserts that for every proposition, either it or its negation is true. More formally: Excluded Middle: ϕ v ~ϕ (where ϕ represents some proposition) Next is the Tarski-biconditional, or T-biconditional. This principle asserts that ϕ in English is true if and only if ϕ holds in the real world. More formally: Tarski-Biconditoinal: T( ϕ ) ϕ (where ϕ represents the Gödel number of ϕ, some sentence) This principle can be broken up into two parts, the capture and release aspects of the Tarski-biconditional. The capture aspect tells us that if ϕ holds in the real world, then we know that the truth predicate captures ϕ and includes that sentence in its domain. The release aspect captures the Tarski-biconditional in the other direction. It says that when ϕ is within the domain of the truth predicate, that predicate releases ϕ and tells us that ϕ must hold in the real world. These capture and release aspects can be represented more formally: Capture: ϕ T( ϕ ) Release: T( ϕ ) ϕ 3 There are, of course, other principles of classical logic that do not appear in the proof below. But, because of the purposes of this essay, it does not seem important to lay them out explicitly. Consequently, I will only be discussing those principles explicitly cited in the classical proof for the paradoxicality of L. 5

11 One should keep in mind that these features of the Tarski-biconditional make it rather obvious that the biconditional is meant to be a material one for example, the sentence grass is green is only true if and only if grass is actually green. This is a fact that will become more important later on in this essay. But, back to the discussion at hand. The third principle used in this proof will be the Principle of Adjunction. It claims merely that when you have two premises α and β, you are allowed to conjoin those two sentences so as to form the sentence α ^ β. More formally: Adjunction: α, β α ^ β Last is the Principle of Disjunction. This principle asserts that if you know that either α or β is true, and you also know that both α and β imply γ individually, then you are allowed to conclude that γ. More formally: Disjunction: (α v β, α γ, β γ) γ With these principles, and the formal language they make use of, it is easy to see the conundrum that that comes from the Liar sentence, L, according to classical logicians. We can formally represent our reasoning about L in the following manner: 1. T( L ) v ~T( L ) (by Excluded Middle). 2. Suppose: ~T( L ). 3. L (by definition of L ). 4. T( L ) (from the capture aspect of T-biconditional). 6

12 5. So, ~T( L ) ^ T( L ) (from 2, 4, and Adjunction). 6. Or, similarly, we might suppose: T( L ). 7. L (from the release aspect of T-biconditional). 8. ~T( L ) (by definition of L). 9. So, T( L ) ^ ~T( L ) (from 6, 8, and Adjunction). 10. Consequently, T( L ) ^ ~T( L ) (from 1, 2-5, 6-9, and Disjunction). This proof alone shows merely that when we reason about the truth-value of L, we run into a contradiction. But, in classical logic, contradictions, like the one found in (10), are especially problematic because of the Principle of Explosion. According to Explosion, it is possible to derive anything and everything from a contradiction a conclusion which follows our intuitions insofar as it seems to be the case that, if we are in such a world where contradictions exist, then we must be in a world where anything holds. This means that sentences like L and the reasoning above lead us to incoherent conclusions: 11. Therefore, finally, X (where X represents anything we want) (by Explosion). But, (11.) is a disagreeable consequence and, so, we d like to reject the reasoning that led us to it if at all possible. As I mentioned before, however, each of these steps seems acceptable on an individual basis. That is, each principle of classical logic that we used to get to (11.) 7

13 appears to be acceptable in its own right. It is thus difficult at first glance to decide where our reasoning has gone wrong in the proof. Consider, for example, premise (1) and the principle of Excluded Middle. It seems relatively clear that any sentence P is either true or false. That is, when a person utters P, he is either speaking a truth or a falsehood there seems to be no middle ground. Consequently, it would be difficult to reject premise (1), for if we do so we seem to be giving up on the claim that every sentence is either true or false, a move which is unintuitive to most philosophers and falls prey to revenge paradoxes. Perhaps, then, it is one of the other principles which has led us astray. But consider premises (4) and (7), which stem from our acceptance of the Tarski-biconditional. Once again, rejecting these seems to be misguided, for, our rejecting the Tarski-biconditional results in a disconnect between the real world and the realm of truth and falsity. That is, if we reject the Tarski-biconditional, then we seem to be open to the idea that the truth or falsity of a sentence does not relate to that sentence s position in the real world (or the position in the real world of the proposition expressed by that sentence) in any important way an unintuitive result, which seems to be missing crucial facts, either about what a truth predicate does or about what sentences do. But, the other three principles of classical logic that come into play in this proof are even more difficult to deny than the two we have just discussed. It s clearly the case that the principle of adjunction is correct: when a person has both α and β, the deduction of α 8

14 ^ β by means of Adjunction is merely a way for him to make explicit what he already knows. Similarly, the principle of Disjunction seems undeniable: when one has evidence that either α or β is true and knows that both it is possible to deduce γ from α and from β, it s obvious that γ holds whether it turns out that α is what makes the sentence α v β true, or β does, or both do. What s more, it s obvious that, if the above reasoning is entirely correct, then we do indeed have a contradiction on our hands. And, with a contradiction it seems difficult to deny that Explosion occurs 4 : our living in a world that materially realizes contradictions means that we live in a world where anything is possible. We are left with five principles to reject to avoid the Liar Paradox Excluded Middle, Tarski-biconditional, Adjunction, Disjunction, and Explosion each of which is extremely plausible intuitively. However, no matter how unappetizing it may seem to give up any one of these principles, it seems that an explosion of inappropriate conclusions is an even harder pill to swallow. Consequently, we must do away with at least one of them. The question is which principle is least important to ideal reasoning. Families of Solutions There are various families of solutions that we can discuss each of which focuses on the insufficiencies of at least one classical principle. Though I will briefly 4 Graham Priest, in his essay, What is so bad about contradictions?, denies that Explosion occurs. But, his views are highly controversial for, they seem to ignore crucial facts about negation. 9

15 mention a few of these solutions, I will focus my attention on two resolutions in particular, which I take to be members of the same family: Revision Theory of Truth and Contextualism. Paracomplete theories belong to the family of solutions that denies the principle of Excluded Middle is correct. By rejecting that sentences can only have two truthvalues, true or false, this leaves room for alternative types of truth-value. Paracomplete theorists opt for the possibility of truth-value gaps the possibility that some sentences are neither true nor false and so they are able to avoid the Liar paradox by simply rejecting (1), shutting down the classical proof before it begins. As another family of solutions, the Paraconsistent theories argue that there are truth-value gluts that some sentences are both true and false. In doing so, they deny that contradictions are problematic, and consequently they are able to avoid the negative consequences of the classical proof by rejecting (11). These types of solution are, of course, available to us. But, each is problematic for a number of reasons that are not relevant to this essay. What is important here is simply that we have good reason reason based on intuition to think that these theories, and their abrupt stops to the classical proof, are excessive. Our intuitions tell us that there is something right about what goes on in the classical proof and that we embrace much of what it tells us about the pathological nature of the Liar sentence. 10

16 Revision Theorists of Truth and Contextualists agree with this sentiment. They reject the Tarski-biconditional in its classical form and aim to avoid contradiction and explosion by demonstrating why it is wrong to think of the Tarski-biconditional as material in nature. And, as I have suggested, I will argue that the Revision Theorist of Truth and the Contextualist have much more in common than merely rejecting the same classical principle in order to avoid contradiction and explosion. In the next chapter I will explain these similarities in full. 11

17 CHAPTER II MORE ON RTT AND CONTEXTUALISM Some Superficial Similarities There are a number of similarities between Revision Theory of Truth and Contextualism, some of which the authors admit and others that they do not. The most obvious of these similarities (beyond those that have already been mentioned) is that both Revision Theorists of Truth and Contextualists base their theories in natural language. That is, they both observe the way that we discuss the Liar sentence in English (or whatever other natural language we might be speaking) and try to mimic our success in doing so using their more formal theories. What s more, each theory has the same goal in mimicking our use of the Liar sentence in natural language: each intends to describe how an ideal reasoner reasons about pathological sentences like the Liar. 5 One might be wondering at this point, What is so successful about the way that we discuss L in natural language? What does this success have to do with ideal reasoners? But, the answer is actually quite simple. When we talk about the Liar sentence in an everyday discussion we all understand that the Liar sentence is such that when we take it to be true, it turns out to be false, and when we take it to be false, it turns out to be true. This, of course, is exactly what we understand a paradox to be. And, yet, 5 One might notice that it is unclear from what I have said here whether these theories are descriptive or normative. I have intentionally avoided taking a position on this subject. The distinction seems vague to me, but, also, unimportant to my theory. 12

18 in our everyday conversations, the fact that L appears to be true through some methods of basic reasoning and false through others doesn t lead to an explosion of inappropriate conclusions. Instead, we simply recognize that there is something pathological about the Liar and other sentences like it. Or, at least, we do if we re ideal reasoners. Both Revision Theorists of Truth and Contextualists try to mimic this pattern of thought in their formal theories. They attempt to show that in some way the Liar sentence is both false and true, preserving our intuitions about the Liar being pathological. But, they also show how this fact does not lead us directly to an explicit contradiction, thereby avoiding explosion. Generally speaking, both Revision Theory of Truth and Contextualism attribute the paradoxical nature of the Liar sentence its being both true and false in some sense to the fact that its truth-value is dependent on the stage of reasoning you ve reached put another way, the hierarchical level you analyze it from. That is, they claim that on some hierarchical levels the Liar is true and on others it s false. In this way they are able to avoid contradiction and explosion. What follows from this appeal to the stages of reasoning, or hierarchical levels, is that whatever goes on when we reason about the truth or falsity of the Liar flips us from one hierarchical level to another, viewing the sentence as true, then false, then true again, ad infinitum. Each theory differs, however, with regard to its technical explanation of why this is the case. The differences in these explanations are exactly the ones that I 13

19 want to focus on in the remainder of this essay the ones that I want to explain away in order to show the deeper similarities between these theories. A More- Complete Picture Now, obviously it s the case that Revision Theorists of Truth and Contextualists take themselves to be describing unique theories which are very different from one another. As I said before, they would probably admit to the similarities discussed in the section above. However, they would stress that there are important differences between their theories as well. In what follows, I will discuss each of these theories in detail. First, I will start with a discussion of Revision Theory of Truth, or RTT. This may seem counterintuitive, considering that RTT is a more recently established theory, but, RTT is simpler and is, therefore, a better starting point for the discussion. The next step will be to go through the details of the Contextualist solution to the Liar Paradox. After that, I will discuss the deeper differences between the two. Finally, this will lead us to Ch. 3, a discussion of how these differences can be explained away and what we can take away from the similarities between these two theories. Revision Theory of Truth The Revision Theory of Truth is a relatively new theory, first published on by Hans Herzberger, Anil Gupta, and Nuel Belnap, in Because of its recent 6 Credit is given to both Herzberger and Gupta for independently conceiving of the theory. See Philip Kremer s Revision Theory of Truth, SEP. 14

20 establishment, the theory hasn t gained much popularity outside of discussions about the Liar Paradox. Nonetheless, I suspect that, in time, the theory will gain popularity in the fields of logic and philosophy of language more broadly. RTT is, after all, able to preserve all of the principles of classical logic, and therefore our intuitions about how basic reasoning ought to go, while also explaining how classical logicians have failed to understand what s actually going on with pathological sentences like the Liar. They do so by specifying more carefully how the Tarskibiconditional works and, therefore, how we ought to understand claims about truth and falsity. When we consider how we reason about the Liar sentence using natural language, we see that each step in our reasoning involves us supposing, or hypothesizing, that it is true or false. When we suppose the Liar is true, we go through some basic reasoning about the definition of the Liar and that reasoning leads us to believe that our supposition was wrong and that the Liar is actually false (and vice versa). Our reasoning about the definition of the Liar sentence, then, indicates that we ought to update, or revise, our hypothesis about the Liar being true (or false). Revision Theorists of Truth focus in on this idea of hypothesizing to show that the classical Tarski-biconditional is flawed insofar as it is based on a mistaken conception of what it is to be true or false. More specifically, Revision Theorists of Truth say of the 15

21 Tarski-biconditional that it should not be considered a material biconditional. Instead, we should view it as a definitional one: T( ϕ ) = (def) ϕ where = (def) is the if and only if of definitional concept introduction. 7 8 To make this clearer, let us consider the following example: Grass is green. The assertion that Grass is green is true, according to Revision Theorists of Truth, if the definitions of grass and green are intertwined in such a way that it is true that what we have dubbed grass actually has whatever property we have dubbed green. If we speak about the revision of hypotheses on a larger scale, then: When we reason about the hypothetical truth-values that we ve assigned to sentences of interest, we often appeal to the definitions of those sentences. Sometimes, in doing so, we come to realize that one, or more, of the truth-values we ve assigned needs to be adjusted. When this happens, we adjust the interpretation of the truth-predicate so as to incorporate this new information (thereby, adjusting our model). Under this new interpretation, we start with a new set of hypotheses, reason some more about how our hypotheses fit with the definitions of the sentences we are interested in, and change our interpretation, again, if 7 See Philip Kremer s Revision Theory of Truth, Sect. 4.2, SEP. 8 With this new conception of the Tarski-biconditional, we are able to define a previously undefined concept by reference other concepts that we have already defined (like the components of ϕ ). The circularity involved in such a project might be alarming to some. But, these worries are unwarranted, I think. For, it seems to be the case that this is perfectly in line with Kurt Gödel s conception of the Gödel number in his work on incompleteness. Gödel numbers are, after all, supposed to be a number assignment to each and every mathematical symbol or string of mathematical symbols. See Kurt Gödel s Über Formal Unentscheidbare Sätze Der Principia Mathematica Und Verwandter Systeme I,

22 need be. As we switch from one set of hypotheses to another we make use of what is often referred to as the Revision Rule. For some sentences (those which behave regularly), this process of switching from one hypothesis to another eventually ends think, for example, of tautologies. So, with these sentences, we come to a point where, every time you reason about the truth values you ve assigned to the sentence, you maintain that your previous hypothesis regarding its truth-value is still correct. In other words, for some sentences, the truthvalue settles and you can confidently assert that it has one truth-value or another. Consider, for example, a sentence like, ϕ is either true or false. Because it is a tautology, as we reason about its truth-value, we will eventually determine that it must always be true even if we start with hypotheses regarding the truth-value of ϕ and ~ϕ that make our sentence, ϕ v ~ϕ. When we reason about the definitions of the sentences involved and the hypotheses that we ve made regarding their truth-values, we will always come to the realization that ϕ can only be true when ~ϕ is false. Consequently, after that point, at every hierarchical level the two sentences will always have opposite truth-values that is, it will always be the case that one is true and the other is false. This fixes the truth-value of our sentence, ϕ v ~ϕ, in such a way that it will be stably-true. For, its truthvalue is simply dependent on at least one of its disjuncts, ϕ or ~ϕ, being true. But, other sentences (like the Liar) will never stabilize. When it comes to these sentences, every time you reason about the truth-value you ve assigned to it, you realize 17

23 that a correction must be made. This becomes an endless process of hypothesizing and revising. The Liar sentence is pathological, or unstable, in this way because of a combination of the circularity of truth and the definition of L. Recall from Chapter 1, Section 1, above, that: L: L is not true. When we consider the meaning of the terms involved here, we see that L is true whenever L is not within the domain of the truth predicate. Consequently, anytime see that L is outside of the truth-predicates domain we adjust our hypotheses so as to put L inside of the domain. But, when it is inside of the domain, we see that that makes L untrue, and, so, we update our hypotheses by removing L from the domain of the truthpredicate. This becomes an endless process of hypothesizing and updating; the truthvalue of L never stabilizes. With this in mind, we can see that, according to Revision Theory of Truth, the classical proof breaks as is indicated below: 1. T( L ) v ~T( L ) (by Excluded Middle). 2. Suppose: ~T( L ). 3. L (by definition of L ). 4. T( L ) (from the capture aspect of T-biconditional). 18

24 5. So, ~T( L ) ^ T( L ) (from 2, 4, and Adjunction) Consequently, T( L ) ^ ~T( L ) (from 1, 2-5, 6-9, and Disjunction). 11. Therefore, finally, X (where X represents anything we want) (by Explosion). Because the Tarski-biconditional is definitional (and the definition of L is such that it is true whenever it is not in the domain of the truth-predicate), Revision Theorists of Truth suggest that we should not assert the conjunction of (2) and (4) (or, (6) and (8)). Or, in other words, they suggest that a contradiction like (5) can never be materially realized. In doing so, they are able to avoid explicit contradiction and explosion. Our reasoning about the Liar sentence, instead, indicates that, because of the definition of L, we ought to update our hypothesis regarding whether the Liar sentence is true or false. The Revision Theorists of Truth, instead, want to represent our basic reasoning about L in the following manner: 1. T( L ) v ~T( L ) (by Excluded Middle) 2. Suppose: ~T( L ). 3. L (by definition of L ). 4. T( L ) (from the capture aspect of T-biconditional). 5. So, we should change our hypothesis to T( L ) (by the Revision Rule). 9 The premises (6)-(9) are structured similarly to (2)-(5), in the sense that (5) and (9) are rejected for the same reasons. Consequently, it is not problematic if we ignore the middle-part of the proof. 19

25 6. Suppose, then: T( L ) 7. L (by the release aspect of T-biconditional). 8. ~T( L ) (by the definition of L ). 9. So, again, we should change our hypothesis (by the Revision Rule). We will come to see, in the section that follows, that this proof appears to be quite different from that of the Contextualist. Contextualism Unlike Revision Theory of Truth, Contextualism is quite popular amongst philosophers of language and logicians. Consequently, there is an abundance of literature on the subject. But, for the purposes of this essay I am going to focus my attention on two works, written by Charles Parsons and Michael Glanzberg, which focus on how Contextualism can serve as a solution to the Liar paradox. Both Parsons and Glanzberg agree that the key step in solving the Liar paradox is to acknowledge that the Tarski-biconditional of classical logic is deeply flawed. Only sentences that express propositions can be true or false. But, this fact is not acknowledged by: ϕ is true ϕ Consequently, Contextualists argue that the Tarski-biconditional should be adjusted to read as follows: 20

26 (x) (x is a proposition ϕ expresses x.. x is true ϕ) That is, for all sentences ϕ, where ϕ expresses a proposition x, x is true if and only if ϕ holds. (Notice that Contextualists are comfortable with the biconditional being a material one, unlike Revision Theorists of Truth.) This distinction is important because Contextualists take the Liar sentence to be syntactically structured such that L should not be read as the classical logicians suggest: L: L is untrue. Instead, we should understand that there are unarticulated constituents in L such that it should read: L: L expresses a proposition that is untrue. (where a is an existential quantifier that refers to the domain of the truth predicate). Because we run into a contradiction using the classical proof to reason about L, Contextualists deny that the Liar sentence as it is interpreted in some particular context, n, can be captured by the Tarski-biconditional in that same context. If it were the case that L n could be captured so as to conclude T n ( L n ), then we would run into a contradiction and explosion would occur (as is demonstrated by the classical proof). Consequently, the only time that the Tarski-biconditional captures the Liar sentence is when it is capturing the Liar sentence interpreted in a lower-level context (and vice versa for the release-aspect): 21

27 Capture: L n T n+1 ( L n ) Release: T n+1 ( L n ) L n This move is what allows Contextualists to claim that the truth or falsity of the Liar sentence is dependent on the hierarchical level that we analyze it from. To make this thought perfectly explicit: The truth-value of L n is dependent on whether you analyze it from the level of context n or from the level of n+1. We know from the classical proof that L n is false in n. But, as we switch from context n to context n+1, we acknowledge this fact. Consequently, during this hierarchical shift we become aware of the fact that L n s being not true makes it true. Unfortunately, we are also well aware that L n+1 cannot be true in context n+1, otherwise we run into contradiction and an explosion occurs. Over and over again, ad infinitum, as we use our basic reasoning skills to determine the truth-value of our sentence L and add that information to our context, we are forced to revaluate the truth-value we ve assigned to it. With this in mind, Contextualists are able to show that the classical proof is flawed in the ways indicated below: 1. T( L ) v ~T( L ) (by Excluded Middle). 2. Suppose: ~T( L ) L (by definition of L ). 4. T( L ) (from the capture aspect of T-biconditional). 10 Saying that we suppose L is untrue is misleading from a Contextualist point-of-view. Instead of supposing that (2) is the case, we come to make this claim because of the semantic status of L n, when it is analyzed from n (where n just is some arbitrary context). But, more on this in a bit. 22

28 5. So, ~T( L ) ^ T( L ) (from 2, 4, and Adjunction) Consequently, T( L ) ^ ~T( L ) (from 1, 2-5, 6-9, and Disjunction). 11. Therefore, finally, X (where X represents anything we want) (by Explosion). Contextualists deny (5) and (9). Because of their claim that the Liar sentence s truthvalue is dependent on the hierarchical level that you analyze it from, L will never be true on the same level that it is false. In this way, then, Contextualists are able to avoid contradiction and explosion. 12 Using a formal language very similar to that used in the section on classical logic, except that it uses this new version of the Tarski-biconditional, the Contextualist s method of reasoning about the Liar sentence can be depicted in the following way: 1. The proposition expressed by L n is not in the domain of propositional quantification in context n (because if it did express a proposition, then we would be in trouble it would lead to a contradiction and explosion). 2. ~T n ( L n ). (because, if it doesn t express a proposition at all, then it can t express a true one). 3. L n (in context n+1) (by the definition of L). 11 Note that for Contextualists the classical proof breaks in the same place as it does for Revision Theorists. This goes back to both theories belonging to the same family of solutions to the Lliar paradox insofar as they each reject the Tarski-biconditional in its simple, material form. 12 As one might expect, Contextualists and Revision Theorists of Truth, then, agree on why the classical proof fails. Because both theories reject the classical version of the Tarski-biconditional, this fact should be unsurprising. For, each takes it to be the case that the classical proof breaks for the same reason it fails to accommodate the subtle details of how the truth-predictae captures and releases propositions. But, it is important to note, at this point, that Revision Theorists of Truth and Contextualists give different reasons for their rejections of (5.) and (9.). 23

29 4. T n+1 ( L n ) (by the capture aspect of the T-biconditional). 5. But, ~T n+1 ( L n+1 ) (for the same reasons as (1.)). 6. So, T n+2 ( L n+1 ) (by definition of L). Notice how different this proof is from that of the Revision Theorist of Truth (which can be found on page 18 of this essay). Some Differences In the more detailed versions of these theories we see that, though each theory criticizes the classical version of the Tarski-biconditional for the same reasons, their suggestions for how one ought to remedy the problem are quite different. Revision Theorists of Truth argue that classical logicians are right so long as we view the Tarskibiconditional is definitional, and not material. But, Contextualists, on the other hand, argue that the Tarski-biconditional of classical logic ought to be rejected and replaced with a very similar principle one that acts as a partial definition of truth while also admitting that pathological sentences like the Liar express propositions that fail to be in the domain of propositional quantification in some contexts. This is the first of three major differences that I must explain away if I am to show that these two theories really are in agreement about how to solve the Liar paradox. The second of these major differences is that Contextualism is a more complex theory than Revision Theory of Truth insofar as their reasoning about when a hierarchical shift occurs includes an extra step. When explaining the work that the definition of L 24

30 does in our basic reasoning, Revision Theorists of Truth suggest that, so long as we know ~T( L ) we can assert L. But, the Contextualist story has an additional step built into each hierarchical shift. Contextualists don t simply assert: ~T n ( L n ) L n. In fact, they argue explicitly against this move. Instead, they suggest that the definition of L is such that: ~T n ( L n ) L n+1. Or, in other words: Contextualists suggest that, as we recognize that ~T n ( L n ) satisfies the definition of L, we shift from context n to context n+1, so that we end up asserting L n (or, L as it is understood in context n) in context n+1. Finally, the third major difference between these two theories is that each claims that the model we interpret the Liar sentence under changes due to different features of the model being updated. In Revision Theory of Truth, whenever a hierarchical shift occurs, reasoners simply change their interpretation so that it reflects the information they have just learned about the truth-value of L. Contextualists, on the other hand, argue that, during a hierarchical shift, the model changes insofar as the domain expands to include the newly learned information about L that L n is true in n+1. For example, when we shift from context n to n+1, the definition of L n tells us to expand the domain of the truth predicate so that L n is true in n+1. With these differences in mind, many philosophers, including the authors of these theories, conclude that Revision Theory of Truth and Contextualism are distinct members of the same family of solutions. They admit that there are prima facie similarities between these two theories, but deny that the theories are similar on a deeper level. But, in the section that follows, I will argue that this is an inappropriate way of looking at the 25

31 above differences between these two theories. For, the differences are a result of the authors talking past one another and are not evidence of any real disagreement. 26

32 CHAPTER III MY OWN THEORY A Verbal Dispute To see why these so-called differences are simply the result of a verbal dispute between Revision Theorists of Truth and Contextualists, we must take a step back from the specifics of each of these theories and consider, again, what these theories look like when they are presented in their abstract forms. Each theory agrees that on some hierarchical levels L is true and on others it s not true. If we say that L is not true, we run into problems because L says of itself that it is not true. In this way, when we say that the Liar sentence is untrue, it becomes true. But, it s not the case that it is true on the same hierarchical level as it was untrue. Contextualists suggest that every time we use our basic reasoning skills, the context changes and we shift to a new hierarchical level. This thought is what under rides their own conception of the Tarski-biconditional and, in particular, their conception of its capture and release aspects. They suggest that the truth predicate from a higher hierarchical level (i.e. n+1) captures a Gödel number representing a sentence from a lower level (n), and that it cannot capture a Gödel number from the same level (n+1) (because doing so runs the risk of contradiction). 27

33 Revision Theorists of Truth agree with this sentiment. Their Tarski-biconditional simply appears in a different form because they take no stand on the semantic status of L (as it is interpreted in and analyzed from some context n). They feel no need to specify why our basic reasoning in this instance causes a shift in hierarchical level. (This is, after all, why Contextualists think it is important to discuss the possibility of L failing to express a proposition in a context.) They simply assert that it does so. And, with this in mind, we can make sense of the apparent differences between the Revision Theorist of Truth s and the Contextualist s views on the Tarski-biconditional. This, too, explains the extra step in the Contextualist s method of reasoning about L. Because Contextualists believe it is important to stress the idea that L, as it is understood in context n, cannot express a proposition in n (while Revision Theorists of Truth feel no need to address this issue), they force this extra step into their method of reasoning. L n is untrue in context n, but true in n+1. But, it s also the case that L n+1 is untrue in n+1 while it is true in n+2. And, this thought must be accounted for in the Contextualist s formal reasoning. Finally, Contextualists and Revision Theorists of Truth do specify different causes of our model changing as we shift from one hierarchical level to another Contextualists cite a change in domain, and Revision Theorists of Truth, a change in interpretation. But, this is merely another result of authors from each side having a slightly different project than the other. Because Revision Theorists of Truth take 28

34 themselves to be describing, in a not-so-specific fashion, how basic reasoning ought to go, they do things a bit differently than Contextualists. The most important difference is that Revision Theorists of Truth propose that the ideal reasoner considers a jumble of hypotheses at once all of those hypotheses that are definitionally related to the sentence whose truth-value he is analyzing. More to the point, Revision Theorists of Truth claim that each time he reasons about a sentence like L he doesn t simply consider his current hypothesis about the Liar sentence s truth-value and how it fits with the definition of L. Instead, he considers all of his hypotheses, his hypotheses about each and every sentence in his language, and how they relate to the definition of L and his current hypothesis regarding its truth-value (a suggestion which is in accordance with Gödel s thought that all of our definitions for all of these sentences will be intertwined). Consequently, the domain of the truth predicate cannot be extended anymore to incorporate the newly-learned information (as is the case in the Contextualist picture). Instead, our interpretation of it has to be adjusted. The Contextualist picture is quite different from this. They argue that the hierarchical level changes when we add whatever new information we ve learned about L into the domain of the truth-predicate, extending its domain to include the Liar sentence as it is interpreted in earlier contextual levels. But, this is merely the result of Contextualists being narrow-minded about what is contextually relevant to the considerations at hand. Revision Theorists of Truth seem to be more liberal about what definitions ought to be taken into consideration during this period of basic reasoning. 29

35 It seems clear, then, that these apparent differences really are just the result of a verbal dispute between the authors of these two theories. The only substantial differences between Revision Theorists of Truth and Contextualist stem from their assumptions about how an ideal reasoner ought to think about L. Revision Theorists of Truth seem to assume that the ideal reasoner has hypotheses about the truth-value of each and every sentence in his language in mind when he considers what truth-value to assign to L. Meanwhile, Contextualists assume that the ideal reasoner will use the classical proof as evidence for denying that L will ever be true in the same context that it is interpreted in thereby recognizing that in those circumstance which he deems L to be true, he is well aware that it is true on a higher contextual level than the one he is interpreting the sentence on. Whether these differences are irreconcilable is something to consider in the last section of this essay. What is important here is simply that these assumptions, which appear to be benign at first, cause a chain of reactions in their resulting theories. The results of these reactions are perceived by the majority of philosophers (including the authors of these theories) as significant differences between Revision Theory of Truth and Contextualism. But, as long as we keep these two assumptions in the back of our minds, we can ignore the superficial differences between Revision Theory of Truth and Contextualism regarding their adjustments to the Tarski-biconditional, the extra step in Contextualism, and their conceptions of how the ideal reasoned changes his model as he moves from one step of reasoning, or hierarchical level, to another. 30

36 The Takeaway When we see that the superficial differences between Revision Theory of Truth and Contextualism are simply the result of a verbal dispute between the authors, it becomes easier to see the theory of the Liar Paradox that is buried inside of these Revision Theory of Truth and Contextualism. Each theory agrees that the ideal reasoner makes use of stages of reasoning, or hierarchical levels, in an attempt to show how the truth predicate operates on pathological sentences like the Liar. As the ideal reasoner considers what truth-value he ought to assign to the Liar sentence, he shifts from one level to another. More specifically, the ideal reasoner recognizes that when he begins with the hypothesis that L is not true, for example 13, and reasons about what the definition of L entails, he ll come to find that his hypothesis was incorrect, at least in some sense. For, from a higher hierarchical level one can see that when L is untrue, the definition of L is satisfied, which makes L true again. And, yet, from an even higher level, this makes L untrue again. This is just what happens when one reasons about the simple-untruth Liar sentence. The ideal reasoner will make claims about what the truth-value of L is on various hierarchical levels while being fully aware of the fact that, in a technical sense, his hypotheses can never be right. But, he will be unworried by this apparent 13 He could just as easily begin with the thought that L is true. The proofs that I ve given all begin with ~T( L ), but this need not be the case. We could adjust them, if we wanted to, to show that when assume that L is true, we run into contradictions a thought which gives us reason to believe that our hypothesis was incorrect. 31

37 contradiction for, the simple-untruth Liar is true in some sense, but false in another, on every hierarchical level. To make this buried theory of the Liar Paradox more explicit, in the section below I will attempt to lay out a more formal account of this ideal person s method of abstract reasoning. A Simpler Theory My suggestion is that we can represent the ideal reasoner s thoughts on the Liar sentence by using the Tarski-biconditional as it is presented by the Revision Theorists of Truth: T ϕ = (def) ϕ This is because, by referencing the definition of the sentence involved, we see that this conception of the Tarski-biconditional says of L: T n+1 ( L ) = (def) ~T n ( L ) (where n is some hierarchical level and n+1 is some level above n) Or, in other words, L is true in context n+1 if and only if it is not true in the context below that, n. The definition of L is such that it already has built into it that 'L is true just in those cases where it has already been deemed untrue. Consequently, for the purposes of finding a solution to the Liar paradox, we need to make explicit reference to the fact that 32