The Semantic Paradoxes: A Diagnostic Investigation Charles Chihara
The Philosophical Review, Vol. 88, No. 4 (Oct., 1979), 590-618. Stable URL: http://links.jstor.org/sici?sici=0031-8108%28197910%2988%3A4%3C590%3ATSPADI%3E2.O.CO%3B2-V The Philosophical Review is currently published by Cornell University.
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The Philosophzcal Reuiew, LXXXVIII, No. 4 (October 1979).
THE SEMANTIC PARADOXES: A Diagnostic Investigation Charles Chihara
I
n this paper, I shall give "diagnoses" of the principal semantic paradoxes (or antinomies) that have played such a significant role in developments in the foundations of logic and mathematics.' I use the term 'diagnoses' rather than the more standard 'solutions' for two principal reasons. First of all, I wish to indicate that I am concerned with only one of two closely related problems raised by the paradoxes. Alfred Tarski once remarked: "The appearance of an antinomy is for me a symptom of disease."2 But what disease? That is the diagnostic problem. We have an argument that begins with premises that appear to be clearly true, that proceeds according to inference rules that appear to be valid, but that ends in a contradiction. Evidently, something appears to be the case that isn't. The problem of pinpointing that which is deceiving us and, if possible, explaining how and why the deception was produced is what I wish to call 'the diagnostic problem of the paradox'. The related problem of de-
--
Many of the basic ideas of this paper are contained, in an obscure and inchoate form, in an article I wrote many years ago (1973) entitled "A Diagnosis of the Liar and Other Semantical Vicious-Circle Paradoxesn-hereafter, "A Diagnosis." In The Bertrand Russell Memorial Volumes, Vol. I, edited by George Roberts, Allen & Unwin (London, 1979), 52-80. Although the present paper is self-contained, the reader may wish to consult the earlier paper for additional references and for certain details that I have omitted in this paper. I have been discussing my diagnoses of the paradoxes in my philosophy of mathematics course for the past four years. In addition, versions of this paper were read at U.C.L.A., Reed College, and at Berkeley. Many who attended my talks or my course lectures have aided me in clarifying my thoughts on the subject, and for this I am most grateful. I would also like to thank Robert Martin, Brian Skyrms, Carol Chihara, a referee, and the editors of this journal for their useful criticisms and suggestions. This paper was written while I was on sabbatical leave with financial support from the University of California Humanities Research Fellowship Program: I wish to express my gratitude to the university for this opportunity. The conventions I use for "mentioning" linguistic entities are those of Benson Mates' Elementary Logic, 2nd ed., Oxford University Press (New York, 1972). ' "Truth and Proof,"Scientz/ic American 220 ('June, 1969), p. 66. I
SEMANTIC PARADOXES
vising languages or logical systems which capture certain essential or useful features of the relevant semantical concepts, but within which the paradox cannot arise, I shall call 'the preventative problem of the ~aradox'.~ Now, if an attractive solution to the preventative problem of a paradox is found, one cannot infer that the diagnostic problem of that paradox has been solved. Furthermore, it is clear that nothing should be called a solution (or resolution) of the paradox that does not solve its diagnostic problem. Confusion about these matters can be found, even in the writings of specialists. For example, Irving Copi has argued that Principia Mathematica resolves the semantic paradoxes, on the grounds that none of these paradoxes can be reproduced within its logical ~ y s t e m But . ~ since there are many systems within which the paradoxes cannot be constructed, additional arguments are needed to show that this particular solution to the preventative problem provides us with genuine insights into what is generating the paradoxical consequences. In this paper, I put forward solutions only to diagnostic problems; to mark this fact, I call the solutions 'diagnoses'. My second reason for using the term 'diagnoses' is connected with the tendency of many people to think that any acceptable solution to the paradoxes must have a certain quality-a quality which I am inclined to express by the words "Of course, that's it." For many paradoxes, it is true that when a correct solution is proposed, things click, and the fallacy or error stands out dist i n ~ t l yBut . ~ there is no good reason for supposing that all paradoxes should be solvable in this way, especially those that have resisted solution for over two thousand years. Hopefully, my use of the term 'diagnoses' will make it clear that the solutions I propose here are not intended to have the "Of course, that's it" quality; 3 variation of the preventative problem might be stated so as to require, roughly, the capturing of "all the really important features" of the relevant concepts (where what are really important features would be specified as the result ofconceptual analysis). Note: the characterization of the preventative problem presented here differs slightly from that given in "A Diagnosis," p. 53. See his Theory of Logical Types, Routledge & Kegan Paul (London, 1971), 88, 91. Cf. the Three Salesman paradox and its solution described in my Ontology and the Vicious-Circle Principle, Cornell University Press (Ithaca and London, 1973), pp. 2-3.
for a physician diagnosing a puzzling illness need not believe he (she) has found an irresistibly correct diagnosis in order to be convinced that some particular one is right. I intend to build a case for accepting my diagnoses, realizing that there is room for differences of judgment on these matters.
Suppose a person tells us that he is going to define 'glub'. He says: "An animal is a glub if, and only if, it is not a mouse; and it is not a glub if, and only if, it is neither a mouse nor different from itself." As a result, we then state: [*I For every animal x, [a.l] x is a glub iff x is not a mouse; and [a.2] x is not a glub iff x is neither a mouse nor different from x. Suppose, for some reason, we were blind to the defects of this definition. Then, the following might seem paradoxical. [I] Lassie is a glub (assumption) [2] Lassie is not a mouse ([I] and [a.l] of [*I) [3] Lassie is not different (identity theory) from itself [4] Lassie is not a glub ([21, [31, and [ a 4 of [*I) [5] Therefore, Lassie is not (reductio ad absurdum of [I]) a glub [6] Therefore, Lassie is a ([5] and [a.l] of [*I) mouse [7] Therefore, Lassie is a ([6] and [a.2] of [*I) glub A proper diagnosis of this "paradox" would point out that [*I, which expresses the definition of 'glub', is inconsistent. It would then be easy to see why we can derive a contradiction from it. After all, the dangers of introducing contradictions into one's theories by means of "creative definitions" are well-known.6 Why would the above reasoning seem paradoxical to those who fail to see the defects of the definition? Because it would appear that a contradiction has been derived from true premises using --
For more on "creative definitions," see Mates, Elementary L o g ~ ,2nd ed., pp. 197-203.
SEMANTIC PARADOXES
only valid rules of inference. Thus, pointing out that [*I is inconsistent makes it obvious that one of the premises used in the argument is not true-indeed could not be true since no inconsistent statement could be true.' Another example of the "illness" I have in mind is generated when two clubs, A and B, set down their rules. Among A's rules, one finds: "Any person is eligible to join this club if, and only if, he (she) is eligible to join club B." But B's rules state: "Any person is eligible to join this club if, and only if, he (she) is not eligible to join club A." By the usual sort of reasoning, one can construct a paradox. In other words, from [i] For every person x, x is eligible to join A iff x is eligible to join B. [ii] For every person x, x is eligible to join B iff x is not eligible to join A. we can derive a contradiction. But [i] and [ii] form an inconsistent pair of statements and hence cannot both be true. Again we have not inferred a contradiction from true premises. But why, it might be wondered, do people tend to think [i] and [ii] are true? Evidently, because they state the eligibility conditions given by the rules of the clubs: [i] and [ii] seem to be true byftat, as did [*I. Of course, no inconsistent pair of statements can be true; so a fortiori, no such pair can be true by fiat.
Consider now a full-fledged semantic paradox. Imagine a situation in which many clubs have hired secretaries but have es: It has been frequently objected that the above definition of 'glub' only results in the specification of a n empty extension for the predicate and is not significantly different from the following, which has been generally regarded as permissible by logicians a t least since the time of Frege: For every object x, x is a glob iff x is a mouse & x is not a mouse. This definition, I agree, is perfectly in order. But I d o not agree that it is not significantly different from that of 'glub'. For it is a simple matter to show (by the usual logical tests) that [*I is not satisfiable whereas a sentence expressing the latter definition is satisfiable. So unless one is willing to espouse the absurd view that there is no significant difference between a satisfiable and a n unsatisfiable statement, the above objection cannot be sustained.
C H A R L E S C H I H AR A
tablished rules excluding such secretaries from membership. Suppose that these secretaries form their own club, Secretary Liberation (or "Sec Lib" for short), the rules of which state: "A person is eligible to join this club if, and only if, he (she) is secretary of a club which he (she) is not eligible to join." All goes well for the club until it hires itself a secretary, a certain Ms. Fineline, who has the misfortune of being secretary of no other club. The paradox arises: Is she, or is she not, eligible to join Sec Lib? O n the assumption that she is, it follows that she is not; and if she is not, she The contradiction is derived from: [I] For every person x, x is eligible to join Sec Lib iff there is a club of which x is a secretary and which x is not eligible to join. [2] Ms. Fineline is secretary of Sec. Lib. [3] Ms. Fineline is not secretary of any other club. [2] and [3] are just given facts that are empirically determinable. So this suggests that [I] is false. But why is one inclined to think that [I] is true? As in the previous paradox, it would seem that [I] has been made true by fiat: after all, that is what the rules say. In the previous cases, it was thought that one could make an inconsistent pair of statements true by fiat. In this case, it is thought that one can make a statement that is inconsistent with statements of fact true by fiat. But one can no more do the latter than one can the former. Hence, we should reject [I]. But there are special reasons why most people do-not think of questioning [I]. It is hard to question the premise since the eligibility rules seem to be in order, as can be seen from the fact that they work in general: in most situations, Sec Lib's rules function W n e might argue that this paradox is not, strictly speaking, a semantic paradox, on the grounds that 'eligible to join' is not a semantic relation. But the expression 'semantic paradox' has come to denote any paradox of the sort Russell called 'vicious-circle paradox' that is not purely logical or mathematical in nature, and it is for this reason I call the Sec Lib a semantic paradox. Historically, F. P. Ramsey divided the vicious-circle paradoxes into two groups and attributed the contradictions of the second (which corresponds to the semantic paradoxes) to "epistemology" (in his "The Foundations of Mathematics," The Foundations of Mathematics and other Logical Essays, Routledge & Kegan Paul (London, 1954), p. 21). The Sec Lib is due to Frank Cioffi, who evidently got the idea for it from some science fiction story. A slightly different version of the paradox can be found in "A Diagnosis."
SEMANTIC PARADOXES
without difficulty, leaving no doubt as to whether or not a candidate is eligible. The idea that a general rule might be perfectly adequate in most situations and yet defective, or even inconsistent, when applied to some special case is not a familiar one. Yet it is easy to construct such rules once the possibility is r a i ~ e d . ~ The above diagnosis explains why the Sec Lib is more puzzling than the following version of Bertrand Russell's Barber paradox:'' The village council decrees that the village barber is to shave any inhabitant of the village if, and only if, that inhabitant does not shave himself. Since the village barber happens to be a n inhabitant of this village, one might argue to the paradoxical conclusion that this barber shaves himself if, and only if, he does not shave himself. The argument, however, is not very paradoxical. For the crucial premise states that the barber shaves any inhabitant of the village if, and only if, that inhabitant does not shave himself; and we haven't been given strong reasons for believing it. Of course, the village council did decree that the village barber is to d o just that. But since it is impossible for him to do it, there is little temptation to think that the decree made the premise true by fiat. It does not even seem to be the sort of statement that is made true by fiat." T h e Sec Lib is more puzzling because [ I ] does seem to be the sort of premise that can be made true by decree: the officials of the club d o seem to have the authority to make it true by simply laying down the rules of eligibility that way.
For an example, see "A Diagnosis," pp. 60-1. Although the Barber is generally attributed to Russell, the version I take u p here is due to Evert Beth, The Foundations of Mathematics, NorthHolland (Amsterdam, 1959), pp. 491-2. Cf. the version I give in "A Diagnosis," p. 55. I' However, there is the remarkable House Bill No. 246 of the Indiana State Legislature (1897), authored by a certain Edwin J. Goodwin, which introduced the new "mathematical truth" that the area of a circle is to the square of the quadrant of its circumference as the area of a square is to the square of one of its sides. (This was "offered as a contribution to education to be used only by the State of Indiana. . .") Perhaps some people believed that one could bring it about by legislation that a = 4! I should also mention that the Barber is sometimes classified as a "pseudo-paradox," presumably because its solution seems so obvious. lo
CHARLES CHIHA RA
In this section, I develop a test which can be used to confirm or disconfirm the sort of diagnoses I have given above. I wish to decrease the probability of making the sort of error, committed in the following, of attributing to a premise (supposedly true by fiat) the defects of something else, such as a general rule of logical inference. Imagine that the following rule of logical inference (affirming the consequent) has been accepted: [AC] From 'If p, then q1 and q, infer p. The predicate 'is a flub' is defined: [Dl For every object x, x is a flub iff x is a two-headed dog. And it is established empirically that [A.1] Lassie is a dog and [A.2] Lassie is not two-headed. A contradiction can now be derived from the above. Let us call such a derivation 'the Flub paradox'. A diagnosis that attributed the Flub to [Dl would be erroneous. But such an error can be avoided, generally, by applying a procedure (to be called 'the first-order test') that consists of translating the relevant premises (those thought to be true by fiat, in addition to the "axioms") into some standard first-order quantificational language and then testing the set of sentences for satisfiability. Thus, using Benson Mates' language L, [Dl, [A.l], and [A.2] of the Flub get translated into: (x) (Fx t, (Dx & Tx))
Da - Ta
and this set can be shown to be satisfiable. Hence, we have good reason for thinking that the definition of 'flub' is not incompatible with the other axioms and hence is not producing the contradiction. The other sort of result (a "positive result") from the first-order test would suggest that the premises thought to be true by fiat are actually inconsistent or, as the case may be, actually do conflict with the other premises. Thus, applying the test to the Sec Lib, we obtain confirmation of the diagnosis: the set of translations of [I], [2], and [3] (of section 2) consists of
S E M A N T I C PARADOXES
(Ex5 0 ( 3 Y) (SYJ& -&Y)) sjs (x) (sjx -+ X = s) and can easily be shown to be unsatisfiable.12
We begin with some definitions. A predicate c$ applies to predicate I/ if, and only if, the result of appending c$ to a quote name of I/ is a true sentence. And a predicate is heterological if, and only if, it does not apply to itself. As a result of these "defining speech acts," we are inclined to assert: [I] A predicate is heterological iff it does not apply to itself. [2] For every predicate x, 'is heterological' applies to x iff x is heterological. If it is asked why [I] and [2] are held to be true, there seems to be no answer possible but "That is how I understand the definitions given" or "That is what the definitions say." So [I] and [2] seem to be true by definition (or fiat). But from [I] and [2], a contradiction is easily derived. So we have a paradox. As in the previous cases, I propose to analyze the paradox as resting on the mistaken belief that certain crucial premises, namely [I] and [2], are true by fiat: Such a diagnosis may seem controversial since it differs significantly from the typical ones given. As James Thomson points out: [A111 the 'solutions' of this paradox which are usually discussed come to the same thing. . . . The essential thing in each case is that the word 'heterological' is so explained that for it itself to be heterological it is necessary and sufficient that it both be not heterological and also satisfy some other condition. Then we seek to avoid the by denying, with or without argument, that this further condition is satisfied.'" It needs to be emphasized that, in many cases, a positive result from the first-order test may not provide decisive confirmation. T h e test is confirming in so far as a certain sort of error is found to be less likely. One can always challenge the test itself by challenging the translations involved. Also I d o not wish to suggest that all semantic paradoxes can be confirmed by such a method: certain modal semantic paradoxes may not be appropriately tested in this way. Finally, another test called "the elimination test" is given in "A Diagnosis," p. 63. It can be verified that the elimination test can be applied to all the paradoxes discussed in this paper to yield additional confirmation. Thomson, "On Some Paradoxes," Analytic Philosophy, edited by R. J.
'v.
CHARLES CHIHARA
Thomson then goes on to point out that these "solutions" are generally felt to be unsatisfactory, partly because such declarations of what 'heterological' must mean have an air of dogmatism. Surprisingly, Thomson's own "solution" has such an "air of dogmatism" aboui it. For after presenting an analysis under which a predicate is taken to be a special kind of function, he suggests that the simplest way out of the difficulty is to say that the "function" denoted by ' x is heterological' is not defined for itself as an argument.14 According to Thomson, "we may have a wrong conception of what the word's meaning is." l5 But again, I would want to ask: "Why must the definition of'is heterological' be such as to preclude its being defined for itself as an argument?" Basically, Thomson's answer is that, otherwise the definition would contradict a certain law of first-order logic.16 But this comes down to little more than replying that, otherwise, a paradox will result. So far as I can see, Thomson's stipulation about the meaning of 'heterological' cannot be justified by semantic laws that are either intuitively correct or empirically justified. I prefer, therefore, to allow the possibility that the argument range of the definition of 'is heterological' does include 'is heterological'. After all, that is the natural way of taking the definition-indeed that is how we all understand the definition when we are first presented with the paradox. Another reason for not resorting to the definitional doctrines advocated by Thomson is this: even if he is right about the definition of 'heterological', [I] and [2] would still have to be rejected, and it would still be reasonable to hold that these premises are thought to be true because they are thought to express (part of) what has been laid down in the definitions. So the acceptance of Thomson's view of definitions would not change the basic diagnosis I have given. This shows that the resort to such a view is not needed after all to remove ourselves from the paradoxical situation in which a contradiction is apparently derived from true premises according to valid rules of inference. Since ThomButler, Basil Blackwell (Oxford, 1962), p. 113. The "usually discussed" solutions Thomson is referring to above include Russell's, Ryle's and the hierarchy of language solution based on Tarski's work. l 4 Ibid., pp. 110, 112, 114. l5 Ibid., p. 110. p. 110. '"bid.,
SEMANTIC PARADOXES
son's view of definitions is motivated primarily by the desire to find a way out of such paradoxical situations, the view loses much of its plausibility. Finally, there are definite advantages to allowing the sort of "inconsistency of definitions" that the definitional doctrines were brought in to preclude-advantages which will emerge later (in section 7) when a "strengthened" version of the Liar paradox is analyzed. I should mention here that although my diagnosis can be confirmed by the first-order test (as the reader can verify), such confirmation does little to decide the issue between Thomson and myself since both diagnoses imply that [ I ] and [2] are mistakenly thought to be true by fiat.
Unlike the preceding paradoxes, the Berry does not provide us with an explicit definition or rule for analysis. Russell presents the paradox in the following way: '[Tlhe least integer not nameable in fewer than nineteen syllables' is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction. "
There is a certain looseness in this statement of the paradox, which makes it difficult to diagnose. What is meant by 'nameable'? It is nowhere said. Clearly, whether the argument is valid or not depends on what is meant by this crucial term. It is not difficult to provide a reasonable definition of the term, the appeal to which will not engender the above contradiction. Because of " In his "Mathematical Logic as Based on a Theory of Types," Logic and Knowledce, edited by Robert Marsh, Allen & Unwin (London, 1956), p. 60. The Berry paradox played a role in the controversy between Poincart and Russell over whether the assumption of the actual infinite gave rise to the paradoxes. Since the set of phrases not nameable in fewer than nineteen syllables is finite, the Berry seems to be a paradox not requiring the actual existence of an infinite set. See my Onto10~pyand the Vicious-Circle Principle p. 140. For more on the Berry, its authorship and its origin, see I. GrattanGuinness, Dear Russell-Dear Jourdaz'n, Columbia University Press (New York, 1977), pp. 50-1. As the Richard and the Konig paradoxes are quite similar to the Berry and can be dealt with in the way the Berry is, I do not take them up in this paper. See "A Diagnosis," p. 79, fn. 21.
CHARLES CHIHA RA
this unclarity, it is possible that no definitive diagnosis of the paradox can be given. Perhaps one can construe the meaning of 'nameable' in a multitude of ways. For this reason, I propose to examine a more rigorous version of the paradox, which seems to capture all the essential features of the original without incorporating the vagueness. I first eliminate the unclarity about the meaning of 'nameable' by using a new expression, 'dginitely describe', which is defined in terms of a precise test: To obtain necessary and sufficient conditions for it to be the case that phrase a definitely describes natural number P, construct a sentence from the schema A def2nitely describes B iff B is C by replacing 'A' with the quote name of a , 'B' with the Arabic numeral denoting P, and 'C' with a token of the same tYPe as P. Thus, 'the first odd prime' definitely describes three. Secondly, instead of talking about the set of phrases consisting of fewer than nineteen syllables (a somewhat vaguely defined set), I shall discuss the set of L-phrases defined as follows: An L-phrase is either the phrase the least natural number not dejiinitely described by an L-phrase or is constructable from the schema the natural number greater by two than twice n by replacing 'n' with an Arabic numeral of a natural number less than a thousand and one. (Notice that the set of L-phrases has the finiteness property that the set of phrases consisting of fewer than nineteen syllables was supposed to have-a property that figured significantly in the early discussions of the paradox [see footnote 171.) A paradox can then be constructed in, essentially, the above way. We first note that not all natural numbers are definitely described by an L-phrase, since only finitely many natural numbers can be so definitely described. Hence, the set of natural numbers not definitely described by an L-phrase is nonempty. This set must have a least element, since every nonempty set of natural numbers has a least element. Let m be this least element. Then m is the least natural number not definitely described by an L-phrase. So, 'the
SEMANTIC PARADOXES
least natural number not definitely described by an L-phrase' definitely describes m. Thus, we have concluded that m both is and is not definitely described by an L-phrase. This version of the Berry can be given the sort of diagnosis I provided for the previous paradoxes. We can hypothesize that certain statements, which we accept as expressing at least part of what was laid down in the definitions, cannot be true by definition as they seem to be. However, the first-order test cannot be applied in this case in quite the direct manner it was previously, because of the way the defining conditions were laid down (that is, via a schema). So I propose that the test be applied to statements that express what is obtained by relatively straightforward applications of the defining rules. Thus, [a] 'the least natural number not definitely described by an L-phrase' definitely describes 1 iff 1 is the least natural number not definitely described by an L-phrase results from constructing a sentence according to the definition of 'definitely describe' and can reasonably be said to express for the case of a specific phrase and specific natural number what was laid down generally by the definition. Similarly, we obtain [b] 'the natural number greater by two than twice 1' definitely describes 1 iff 1 is the natural number greater by two than twice 1 and 'the natural number greater by two than twice 2' definitely describes 1 iff 1 is the natural number greater by two than twice 2 and 'the natural number greater by two than twice 1000' definitely describes 1 iff 1 is the natural number greater by two than twice 1000 by applying the definition repeatedly and forming a conjunction. In a similar way, [c] 'the least natural number not definitely described by an L-phrase' is an L-phrase and [dl Every L-phrase different from 'the least natural number not definitely described by an L-phrase' is either 'the natural number greater by two than twice 1' or 'the natural number
CHARLES CHIHARA
greater by two than twice 2' or . . . or 'the natural number greater by two than twice 1000' can be said to express what was laid down in the definition of 'L-phrase'. (Note: the ' . . . ' in [b] and [dl are the "dots of lazyness. ") Now it can easily be shown that this set of premises, when augmented by a few theorems from number theory, is unsatisfiable. The following will indicate how this is done. D: definitely describes Q L: Q i s an L-phrase A? @is a natural number e: 'the least natural number not definitely described by an L-phrase' a , : 'the natural number greater by two than twice 1' a,: 'the natural number greater by two than twice 2'
aloo0: 'the natural number greater by two than twice 1000' [I] eD1 t, ( (x) (Lx+-xD1) & (x) ( (Nx &-(3y) (Ly & yDx) ) + x A 1)) 2 . 1) & . . . & ( a l o 0 ~ l +=l2 + 2 . 1000) [2] (a,D1-+1 = 2 PI Le [4] (x) ((Lx & xf e) -+ ( x = a , v . . . v x = a l o o 0 ) ) [5] (x) (Nx-+x 2 1) [6] (1 # 2 + 2 - 1 ) & ( l f 2 2.2)& . . . & ( I 5 2 + 2.1000) Now [I] - [4] are translations of [a] - [dl respectively; and [5] and [6] are translations of theorems of number theory. The set consisting of [I] - [6] is unsatisfiable.
+
+
I now put forward considerations of a general nature that provide additional support for the above diagnoses. (1) Special Reasons. A good diagnosis should do more than merely provide a way of avoiding the contradictions; for the paradoxes can be blocked in a variety of ways, and one needs special reasons for picking any one of these alternatives as the
SEMANTIC PARADOXES
crucial one. l8 Recall, in this light, that I gave independent reasons for thinking both that a certain premise of the Sec Lib was false, and also that it played a crucial role in the paradox; I even provided explanations of why the paradox is puzzling and why the solution is so easily overlooked. (2) Reproducibility. One way of testing the claim that a particular virus causes cancer is to isolate the virus, apply it to appropriate test organisms, and then see if the disease is produced. One should test proposed solutions to the paradoxes in a similar manner, by seeing if paradoxes result when we reproduce the conditions that, according to the diagnosis, engender the paradoxes. For example, the claim that the paradoxes are due to self-reference does not pass this test, since many sentences involving self-reference do not generate paradoxes." My own diagnoses fare much better since, clearly, if one produces some premises, each of which appears to be true by fiat, but that, as a set, is inconsistent (or is incompatible with known facts), then one can generate a paradox. Indeed, once one sees how these semantic paradoxes are produced, not only is it quite easy to construct new ones, but we are given information as to how we can make them either more or less difficult to s01ve.'~ (3) Simplicity. Simplicity considerations also provide support, since an examination of the literature on the paradoxes provides striking evidence that these diagnoses have a kind of simplicity found in few other solutions: no appeal is made to complex logical or semantic theory. ( 4 ) Occam's Razor. In accordance with Occam's principle, In Although this minimum condition of adequacy may seem obvious once it is stated, it is surprising how many proposed "solutions" fail to meet it. For example, the only reason Frederic Fitch gives (in his "Comments and Suggestions," T h e Paradox of the Liar, edited by Robert Martin, Yale University Press (New Haven, 1970), p. 77) for adopting his "solution" to the Liar-a solution which consists in maintaining that the Liar sentences are not well-formed-is the fact that by doing so "there is no resulting wellformed sentence to cause trouble." It is clear that one ought not argue "This must be what is going wrong in the paradoxes, since otherwise one could get a contradiction." Practically any diagnosis could be "defended" in that way. For a sample of the many ways of blocking the paradoxes, see The Paradox of the Liar. "' or a specific example of a paradox constructed in the light of the sort of diagnoses given here, see "A Diagnosis," pp. 75-6.
C H A R L E S CHZHARA
scientists are generally reluctant to accept some new law or principle postulated to explain some phenomenon if there is an explanation of the data, already at hand, which is based on accepted theory. Now many philosophers have appealed to new and largely untested logical or semantic principles to solve the paradoxes.'O My own diagnoses have been made within the framework of classical logic: the essential principles and ideas used in this paper have been part of standard logical theory for years. (5) Conservation. One consideration that moves reasonable people to prefer a theory to a competitor is whether the acceptance of the one would require less revision of our presently accepted scientific theories than would acceptance of the other. The diagnoses presented in this paper do not require the rejection or revision of any of our well-established scientific theories, unlike, say, Russell's solutions of the paradoxes.'l
I have saved the most difficult of all these paradoxes for the last for several reasons: (1) I wish to make it clear that the acceptability of the preceding diagnoses does not rest upon the acceptability of my diagnosis of the Liar. (2) The issues surrounding the Liar are more complex and tangled than in the other cases; what has gone before will allow me to omit certain details and concentrate on issues that are not mere variations on previously discussed themes. Consider the following version: '' Let us use 'L' as the name of: 'O Cf. the various solutions proposed in The Paradox of the Lzar, edited by Martin. Note that many solutions have been proposed utilizing many-valued logic. The following advice of Dana Scott's is relevant to such proposals: ". . . I have yet to see a really workable three-valued logic. I know i t c a n be defined, and at least four times a year someone comes up with the idea anew, but it has not really been developed to the point where one could say it is pleasant to work with. Maybe the day will come, but I have yet to be convinced. So my advice is to continue with the two-valued logic because it is easy to understand and easy to use in applications; then when someone has made the other logic workable a switch should be reasonably painless." In his "Advice on Modal Logic," Philosophical Problems in Logic, edited by Karel Lambert, D. Reidel (Dordrecht, Holland, 1970), p. 153. '' See my Ontology and the Vicious-Circle Principle, Chapter I . 2! This version of the paradox is classified as a "strengthened Liar" (see
SEMANTIC PARADOXES
The only sentence in section 7 of this paper which begins with 'The' and in which the Arabic numeral denoting seven occurs is not true. It can be verified that L is the only sentence in section 7 of this paper which begins with 'The' and in which the Arabic numeral denoting- seven occurs. So if L is true, then what is said to be the case by L is in fact the case, that is, it is the case that the only sentence . . . is not true. So if L is true, L is not true. Thus, L is not true. But in that case, what is said to be the case by L is not the case. Thus, L is true. Evidently, what is crucial to the above chain of reasoning is, essentially, the principle: [Tr] A sentence is true if, and only if, what is said to be the case by the sentence is in fact the case.23 Admittedly, [Tr] is somewhat vague. But when we are dealing with sentences of certain sorts, such as those of the form 'A is F', where 'A' is to be replaced by a referring expression and 'F is to be replaced by a predicate of a n appropriate kind, the import of the principle may be quite clear.24In particular, the following restricted version of the principle is not vague: [TI If a is a sentence consisting of the referring expression p immediately followed by the words 'is not true', and if p denotes the sentence y, then a is true if, and only if, y is not true.25 The Paradox o f the Liar, edited by Martin, p. xiv). T h e present analysis can also be carried out when the Liar is stated in terms of propositions, statements, assertions and the like. It is primarily the advantages of brevity and simplicity that prompt me to disc& the paradox within the framework of the theory that takes truth and falsity to be predicates of sentences. For more on this point, see "A Diagnosis," p. 69. Alfred Tarski, in his "The Concept of T r u t h in Formalized Languages," Logic, Semantics, Metamathematics, translated by J . H. Woodger, Oxford a t the Clarendon Press (Oxford, 1956), p. 155, expressed the principle with the words: "a true sentence is one which says that the state o f affairs is so and so, and the state of affazrs indeed is so and so." 24 Cf. Tarski's view: "From the uoint of view of formal correctness. clarity. ,. and freedom from ambiguity of expressions occurring in it, the above formulation obviously leaves much td be desired. everth he less, its intuitive meaning and general intention seem to be quite clear and intelligible." Ibid. "'he idea of restricting the scope of the principle to get a clearer and more precise statement of what is intended is, of course, to be found in Tarski's classic paper (ibid.). Not surprisingly, the above statement is close in spirit to Tarski's Convention T.
CHARLES CHIHARA
Now why are we so inclined to accept [Tr]? T o bring out the significance of this question for our investigations, I shall consider first the analogous question about another version of the Liar-a version formulated in such a way that this analogous question can be easily answered. T h e paradox begins with a definition: We are told that a sentence is true* if, and only if, what is said to be the case by the sentence is in fact the case. Hence, we can state a principle, [Tr*], which differs from [Tr] only by the fact that 'true*' occurs in [Tr*] where, and only where, 'true' occurs in [Tr]. Furthermore, when we ask for clarification of the definition, we are informed that [Tr*] tells us that 'Snow is white' is true* if, and only if, snow is white. We are also told that the following is implied by the principle: If a is a sentence consisting of the referring expression P immediately followed by the words 'is not true*', and if p denotes the sentence y, then a is true* if, and only if, y is not true*. Thus, we are able to state a principle, [T*], which differs from [TI only in the fact that 'true*' occurs in [T*] where, and only where, 'true' occurs in [TI. We then construct the sentence L*: Exactly one sentence in section 7 of this paper begins with the word 'Exactly' and also contains a n occurrence of the Arabic numeral denoting seven; and that sentence is not true. A paradox (which I shall call 'the Defined Liar') can now be generated by essentially the sequence of steps found in the earlier version. In giving a diagnosis of the Defined Liar, we shall want to ask about [Tr*] what we asked, in the beginning of this paragraph, about [Tr]: Why are we so inclined to accept the principle? This time, the answer is obvious: It is because [Tr*] states what was laid down in the definition of 'true*'. Thus, as in all the previously analyzed paradoxes, we have a crucial premise that is thought to be true by fiat. And since it can be shown that [T*], and hence [Tr*], conflict with certain facts of reference, we again have a situation in which a premise thought to be true by definition, convention, or agreement of some sort, turns out not to be true a t all. So in basic structure, the Defined Liar turns out to be essentially no different from the others. We can confirm this diagnosis via the first-order test: Let the universe of discourse be the set of sentences of this paper; and
SEMANTIC PARADOXES
interpret the unary predicate 'T'to hold of all those sentences that are true*. Let the binary predicate 'IT be interpreted to hold of an ordered pair (x, y ) just in the case in which x is a sentence consisting of a referring expression 8 immediately followed by the words 'is not true*', where 8 denotes y. Let the individual constant 'a' denote the sentence L*. Then translating [T*] into this framework yields: [ l l (x) Cy) (Hxy 0 (Tx- -TY) Also, by direct inspection, we get: [2] Haa As the set consisting of [I] and [2] is unsatisfiable, the test is positive. Returning to the undefined Liar, we see that there is no explicit definition of 'true' which can be said to generate the paradox. So the diagnostic problem is more difficult, and there is considerable room for speculation regarding why we are so inclined to accept [Tr]. One reason might be: "We are inclined to accept it because it is a generalization which has received a great deal of empirical confirmation, that is, we have examined a large number of sentences, and those found to be true were such that what was said to be so was in fact so; those found not to be true were such that what was said to be the case was in fact not the case." More sophisticated versions of this might suggest that [Tr] is an empirical hypothesis which has received confirmation via its role in explanation and predication, or perhaps by some Bayesian confirmation process. But I have doubts about this general position, for it doesn't account for the special attractiveness of [Tr], its apparent necessity, and its resistance to rejection (there seems to be a definite incoherence involved in rejecting it). The view that our reasons for accepting [Tr] are simply empirical in nature goes hand in hand with another basic position regarding the character of the Liar-a position that is widely held but rarely, if ever, explicitly defended. The vast majority of solutions to the Liar offered thus far presuppose what I shall call 'the consistency view of truth', viz., the view that an accurate statement of what 'true' means will be logically consistent with all known facts, and in particular with all known facts of reference. Indeed, some philosophers build such a consistency view into their statement of adequacy conditions for a solution. Thus,
CHARLES CHIHA RA
John Pollock has written: "A solution to the Liar paradox must consist of an explanation of the meaning of 'true' and a demonstration that, given that analysis of 'true', the Liar sentence is not paradoxical." 26 So far as I can see, few philosophers have even argued for the consistency view; it seems to be a sort of unquestioned axiom. Thus, the consistency view is presupposed in much of Donald Davidson's writings on theory of meaning; yet he never seems to feel any need to justify it.27There is, however, something in the literature which might be construed as an argument for the view. Charles Parsons has suggested that if an irresolvable antinomy arose via formulations of the Liar within the technical language of semantics, then "it might be possible to attribute the contradiction to the technical semantical concepts and to avoid the claim that ordinary speakers of English are saddled with a n incoherent conceptual a p p a r a t u ~ . "This ~ ~ quotation intimates that an inconsistency view of truth would saddle ordinary speakers with
