'lusion: Themes in the Philosophy of 'ry for a Corporeal Psychology. Oxford: ve and Being Objectified." In A Mind of

I and Objectivity, ed. Louise M. Antony

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Ice: The Very Idea. Cambridge: MIT.

and Natural Shame." In Explaining y: University of California Press. llity." Philosophy and Phenomenological lollers Let Him Go: Regulating Racist 'ound: Critical Race Theory, Assaultive [ari J. Matsuda, Charles R. Lawrence III, s Crenshaw. Boulder: Westview Press. in Neurosis. New York: International and Social Conventions." Scandinavian md Male Privilege: A Personal Account 19h Work in Women's Studies." In Race, Margaret L. Anderson and Patricia Hill lishing Company. Zur symbolischen Reproduktion sozialer )Us. 7)' of Desire: Theory and Practice in ;eton University Press.

ime as Moral Concepts." Proceedings of

:lbridge, Mass.: Harvard University Press.

In Explaining Emotions, ed. Amelie O.

ia Press.

Emotions: Pride and Shame as Causal

lociology ofEmotions, ed. Theodore D.

:w York Press.

,nstruction ofKnowledge, Authority, and dge. md Guilt Cultures." In Shame and Guilt: ed. Gerhart Piers and Milton B. Singer. er. ~d Guilt: Emotions of Self-Assessment. ysity. Berkeley: University of California. Philip Hugly and Charles Sayward Null Sentences 1. Introduction Consider the following classificationl of truth-valued sentences: First, null sentences: sentences which say nothihg whatsoever about what is or is not the case-sentences which in no way assert anything about how things are. Second, contentful sentences: sentences which say something about what is or is not the case-sentences which Jssert something about how things are. The notion of a null sentence h1s figured prominently in twentieth- century analytic philosophy. It plays ~ central role in logical positivism and I in the writing of Wittgenstein. However, the notion is subject to a vexing difficulty, which can be put as followsl: How can a sentence that says nothing be true? How can a sentence that say~ nothing be false? If it is true it must I say something which is true; but if i, says something which is true it says something. If it is false it must say something which is false; but if it says something which is false it says something. So how can there be any null sentences? 2. Null Sentences and Logical Positivism It is natural to think of logical POSitivi~m as centrally including the following I claim: if S is analytical, then if S is ~mowable, then S is knowable only a priori. This takes it that the notion of *n analytic statement itself is central to positivism. In place of this we suggest that it is the notion of a null sentence which is central. I Some important theses of logical positivism were that mathematical I sentences are null sentences, and that logically true sentences are null I sentences, etc. Another thesis of logical positivism was that null sentences are analytic. By this two things were meant. First, that what made a sentence null was that it was analytic. Second, that what justified the assertion of a i 43 @Iyyun, The Jerusalem Philosophical lunrtPT'!v48 (January 1999): 23-36 24 Philip Hugly and Charles Sayward null sentence was an analysis of its terms or of the concepts they expressed. I We suggest that the notion of analyticity be viewed as one once put to the task either of explaining how it is that null sentences are null or of explairting one way we might come to be justified in asserting a null sentence'IThe notion of analyticity has both an epistemic and a purely logical aspect.: Our own sense is that the notion of analyticity is a failed notion for the reasõ that it fails to provide an explanation either of what makes a null sentence nrill or of how we can come to be justified in asserting a null sentence. I I Yet another thesis of logical positivism was this: That what justified the assertion of a null sentence was never due to experience. That is, it was:held that experience is unable to justify any null sentence. I The thesis that experience is justificatorily irrelevant to null sent~nces seems to us unassailable. But it is not so obvious that there are [such sentences or that, e.g., the sentences of arithmetic are among them. The loss of the notion of analyticity would in no way diminish the power of the ~Iaim that experience is justificatorily irrelevant to null sentences or of the claim that some sentences are null. I 3. Null Sentences and Wittgenstein Wittgenstein held that there are null sentences-prominently including logical truths and the truths of arithmetic. ! In the Tractatus (4.461) Wittgenstein says that tautologies and contradictions are without sense (sinnlos), while at 4.4611 he denie~ that they are nonsensical (unsinnig). This is also Wittgenstein's view Of the I theorems of mathematics in 6.2-6.22: 6.2 Mathematics is logical method. The propositions of mathematics are equations, and therefore pseudopropositions. I 6.21 A proposition of mathematics does not express a thought. I 6.211 In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions, which do not belong to mathematics, to others which equally do not bel~ng to mathematics. ~ I (In philosophy the question "Why do we really use that word, that proposition?" constantly leads to valuable results.) I 6.22 The logic of the world, which the propositions of logic show in taut~logies, mathematics shows in equations. ' In his later lectures, as I theme that the provable Consider this passagt The proposition shows Who nothing. The tautology has no tn contradiction is on no conI Tautology and contradicti( (Like the point from whic/ (I know, e.g. nothing abou It is not obvious that trUi But our problem is not* that a tautology is true. Wittgenstein says a t unconditionally true is t regard truth and falsit languages. But surely it sentence is true or false sentence says nothing, i We are struggling w about logic and math. mathematics that they al are asking: If they say Il can they be true? Do the It might be thought I logic or mathematics t( Wittgenstein suggests tl example, we can view t (P&q):Jp as a rule of inference, n: p&q P written out in the form 0- to be a truth of logic is t On Wittgenstein's vi 5+7=12 -d rms or of the concepts they expressed. 'ticity be viewed as one once put to the null sentences are null or of explaining tied in asserting a null sentence. The temic and a purely logical aspect. Our ;ity is a failed notion for the reason that r of what makes a null sentence null or I asserting a null sentence. ivism was this: That what justified the . due to experience. That is, it was held Iy null sentence. ificatorily irrelevant to null sentences not so obvious that there are such )f arithmetic are among them. The loss 10 way diminish the power of the claim vant to null sentences -or of the claim :tein JIl sentences-prominently including letic. :genstein says that tautologies and nnlas), while at 4.4611 he denies that tis is also Wittgenstein's view of the ~: :s are equations, and therefore pseudos not express a thought. .1 proposition which we need, but we use

order to infer from propositions, which do

others which equally do not belong to

Why do we really use that word, that

valuable results.)

~ propositions of logic show in tautologies,

Null Sentences 25 I In his later lectures, as reported by G.E. Moore (1959, 266), he repeats the theme that the provable sentences of logic and mathematics say nothing. Consider this passage from the Tr~ctatus (4.461): .. h h' IThe propositIOn sows w at II says, the tautology and the contradiction that they say nothing. I

The tautology has no truth-conditions, for it is unconditionally true; and the

contradiction is on no condition true. .

Tautology and contradiction are without sense.

(Like the point from which two arrows go out in opposite directions.)

(I know, e.g. nothing about the weather, then 1 know that it rains or does not rain.)

It is not obvious that truths of logic and mathematics cannot serve to inform. But our problem is not with this. Our problem is with Wittgenstein's claim that a tautology is true. I Wittgenstein says a tautology is unconditionally true. Well, whatever is unconditionally true is true. What is true or false? It is currently common to regard truth and falsity as traits of sentences relativized to particular languages. But surely it is what a sentence says which is true or false, and a sentence is true or false only insofar!as what it says is true or false. So if a sentence says nothing, it is not true õ false. Or so it seems. We are struggling with these thre~ thoughts Wittgenstein has in talking about logic and mathematics. He: says of the theorems of logic and mathematics that they are true, that they are rules, that they say nothing. We are asking: If they say nothing, how tan they be true? If they are rules, how I can they be true? Do these thoughts cohere? It might be thought that one could understand the truth of a theorem of logic or mathematics to consist in ~ certain rule of inference being valid. WiUgenstein suggests that we view the equations as rules of inference. For example, we can view the tautology I (P&q)~p as a rule of inference, namely: p&q p written out in the form of a sentence. To understand what it is for this sentence to be a truth of logic is to grasp that in virtue of which the inference is valid. On Wittgenstein's view the equation 5+7=12 26 Philip Hugly and Charles Sayward is a sentential formulation of this rule of inference:

There are 5+7 As

There are 12 As

i To understand what it is for the equation to be a truth of arithtpetic is to grasp that in virtue of which the inference is valid. I We have doubts about this explanation. Consider a truth tjlble analysis of a tautology, e.g., I Snow is white If snow is white then snow is white true true I false true I Presumably, 'true' means the same in all three occurrences! Well, then, 'If snow is white then snow is white' is true in the same se~ that 'Snow is I white' is true. Both sentences are true in virtue of the way the world is. What fact makes it the case if snow is white then snow is white? Why, of course, l the fact that snow is white. That is clear from looking at the table. This makes it clear that one cannot go from That p makes it the case that S is true to S says that p 'If snow is white then snow is white', if it says anything at all, does not say that snow is white. But that snow is white is what makes it:the case that 'If snow is white then snow is white' is true. i More generally, from the fact that a sentence is true in virtue of the way the world is, it does not follow that it says something about the world or that it says anything at all. ! There is a problem with the idea that there are null sentences, and that tautologies are among them. Tautologies are paradigm e~amples of true sentences. But if a sentence is true it must say something rhich is true, in which case it must say something, in which case it is not nuU. Or so it seems. I 4. Null Sentences and Disquotational Truth I I The theme that the provable sentences of mathematics and logic say nothing was a theme ofWittgenstein early on. And it stayed with hi~ throughout. But how important for him was it that they are true? What turns on their being true? I Something Wittgens; Remarks on the Foundo./ does a proposition's 'bei Setting aside problems so-called disquotational In Word and Object C say that the statement 'I say that Brutus killed C< The utility of the truth pr cancellatory force of the tf' 'Snow is white' is tr Quotation makes all the di snow. By calling the senten ofdis quotation. (QUine 19' On this conception, atta( same effect as would b, This view is thus comr "'Snow is white' is true In Appendix I of Re 4ge) Wittgenstein eml grammatically declarati, . it says anything about h( A simple argument STh Questions and commanc does not entail being asSI Now if 'is true' is dis 'The King moves is simply to say The King moves 0 So if the second sentencl Similarly, to say 'Snow is white or is simply to say Snow is white or i which leaves it open wI rule or an assertion. Sayward lis rule of inference: equation to be a truth of arithmetic is to grasp rence is valid. !xplanation. Consider a truth table analysis of If snow is white then snow is white true true same in all three occurrences. Well, then, 'If 'hite' is true in the same sense that 'Snow is are true in virtue of the way the world is. What is white then snow is white? Why, of course, lat is clear from looking at the table. ~ cannot go from : that S is true s white', if it says anything at all, does not say now is white is what makes it the case that 'If hite' is true. fact that a sentence is true in virtue of the way "I that it says something about the world or that he idea that there are null sentences, and that . Tautologies are paradigm examples of true is true it must say something which is true, in hing, in which case it is not null. Or so it seems. Disquotational Truth sentences of mathematics and logic say nothing ~arly on. And it stayed with him throughout. But that they are true? What turns on their being true? Null Sentences 27 Something Wittgenstein says suggests that nothing turns on this. In Remarks on the Foundations ofMath~matics (1956, 50e) he says: "For what does a proposition's 'being true' mean? 'p' is true =p. (fhat is the answer.)" Setting aside problems having to do with use and mention, this is the so-called disquotational theory of truth. I In Word and Object Quine (1960, ~4) expresses this view as follows: "To say that the statement 'Brutus killed ~aesar' is true ... is in effect simply to say that Brutus killed Caesar." And in Philosophy ofLogic he writes: I The utility of the truth predicate is the cancellation of linguistic reference ... this cancellatory force of the truth predicate is explicit in Tarski's paradigm 'Snow is white' is true iff snow is White Quotation makes all the difference betwden talking about words and talking about snow. By calling the sentence true we call snow white. The truth predicate is a device of disquotation. (Quine 1970, 12) I On this conception, attaching 'is true'l to the quotation of a sentence has the same effect as would be obtained by simply erasing the quotation marks. This view is thus committed to the Ithesis that e.g., 'Snow is white' and "'Snow is white' is true" say the very same thing. In Appendix I of Remarks on the Foundations of Mathematics (1956, I 4ge) Wittgenstein emphasizes that from the fact that a sentence is I grammatically declarative it does not1follow that it is assertoric, that is, that . it says anything about how things are; A simple argument shows this: Questions and commands are not assertoric. Questions and commands can take dtklarative form. Thus, declarative form does not entail being assertoric. Now if 'is true' is disquotational then to say, for example, 'The King moves one space at ~ time' is true I is simply to say . The King moves one space at aitime So if the second sentence is a rule, as it appears to be, so is the first sentence. Similarly, to say I I 'Snow is white or it is not the Case that snow is white' is true is simply to say I

Snow is white or it is not the case that snow is white

which leaves it open whether the last displayed sentence has the force of a rule or an assertion. I 28 Philip Hugly and Charles Sayward So, if the disquotational theory of truth is true, defenders of null sentences, such as Wittgenstein, have a way out of the difficulty posed by such questions as these: If the theorems of logic ~nd mathematics say nothing, how can they be true? If they are rules, how;can they be true? An answer gleaned from the disquotational theory of truth; is that to say that S is true does not entail that S says something. To say S is true is simply to say what S says. If S says nothing then I S is true also says nothing. It is sometimes suggested that Snow is white and 'Snow is white' is true i do not say the same thing because they are about different things. The first l is about snow, and the second is about the sentence 'Snow is white'. But there are ever so many pairs of sentences that say the same but are about different things. Do not : ; The number of whales which are not mammals~ == 0 and I All whales are mammals falI into this category? Or take Bill's pencil is sharp and Bill has exactly one pencil and it is sharp i These say the same thing; but the first is about Bill's pencil, and the second is about Bill. There are a lot of counterexamples to t~e principle that saying the same thing entails being about the same thing. i If it is the case that it is one and the same thing to say that snow is white and to say that 'Snow is white' is true, then 'Snow is white' and "'Snow is white' is true" must have the same truth-conditions in English. It is necessary that if sentences have the same truth-conditions in ~nglish that they form a biconditional which is necessarily true in English. For if a biconditional can fail to be true in English, then it is possible for the truth-conditions in English of just one of its sentences to be satisfied, in which case they are different truth-conditions. I Since snow is white, and coal is black, and people speak English, these things are compossible. Further, it is clearly possible for those cOnditions to ; continue to language a English ju~ coal and n( Let w b, the sentenc Snov is true in EI in w. This: the sentenc English sIX But is th 'Sno' also true in The trutl it has in th English is t truth-condi true in (not in w, the se 'Sn01 is not true i The con, white' is ; disquotatiol 5.0; Objectivity not constit; epistemic Sl the whole) constitutes; But it is appears to I J For more d ~ard ry of truth is true, defenders of null lYe a way out of the difficulty posed by tleorems of logic and mathematics say they are rules, how can they be true? An onal theory of truth is that to say that S is Ilething. To say S is true is simply to say : they are about different things. The first about the sentence 'Snow is white'. But entences that say the same but are about :h are not mammals = 0 and it is sharp first is about Bill's pencil, and the second >unterexamples to the principle that saying .t the same thing. Id the same thing to say that snow is white s true, then 'Snow is white' and "'Snow is ~ truth-conditions in English. It is necessary ruth-conditions in English that they form a y true in English. For if a biconditional can ; possible for the truth-conditions in English ! satisfied, in which case they are different I is black, and people speak English, these it is clearly possible for those conditions to Null Sentences 29 i continue to hold even if, as is also poss,ible, English were to become the only language anyone speaks or understañs and to become different from actual English just in having 'coal' denote s!now and not coal and 'snow' denote coal and not snow. Let w be a possible world representing these possibilities. Note first that the sentence I Snow is white is true in English at w even though 'snow' denotes coal in wand coal is black in w. This is because snow is white inlw, and what gets assessed at w is not the sentence 'Snow is white' as used, in w, but rather that sentence as we English speakers actually use it. But is the sentence 'Snow is white' is true also true in English at WE The truth-conditions which a sentence has in a world are just those which it has in the languages of that world ! of which it is a sentence. So, since English is the only language spoken i~ w, 'Snow is white' has in w only the truth-conditions it has in the English ~poken in w. Thus, 'Snow is white' is true in (not at) w only if coal is white in w. Since coal is black and not white in w, the sentence 'Snow is white' is ~ot true in w. Thus 'Snow is white' is true is not true in English at w. I The conclusion is that the two sentences 'Snow is white' and "'Snow is white' is true" have different truth-conditions in English; hence, the disquotational theory of truth is false'r 5. Objectivity and Truth Objectivity is an epistemic fact if the~e is agreement on what does and does not constitute justification for a Jrtain range of propositions. In this epistemic sense, arithmetic, for example, is indisputably objective, since (on the whole) mathematicians agree õ whether a certain set of inferences constitutes a proof of some arithmetidal proposition. ! But it is also possible to speak of the objectivity of a proposition in what appears to be a quite different sense.! Here the idea is that a proposition is . I 1 For more detaIls see Hugly and Sayward (1993). I 30 Philip Hug/y and Charles Sayward objective just in case it says that things are some way and owes its truth or falsity to how things actually are. Some hold that there are truths in addition to obje~tive truths. If this is so,, then it is quite possible for a sentence to say nothing about how things are and to be true (or false). The idea now is that within 'the set of truth-valued statements there is a proper subset which coñists of non-objective statements. Null sentences would be included in this subset. How plausible is this idea that there are non-objective truth-valued statements in addition to objective truth-valued statements? We shall consider two sources of this idea. I One Source. One source of this idea is what might be called the provability theory of mathematical truth. I Against the thought that there is non-objective truth is the thought that truth is the same sort of thing in each area in w~ich we speak of truth (mathematics, logic, physics, politics, morality ...),' that knowledge is the same sort of thing in each area in which wei speak of knowledge (mathematics, logic, physics, politics, morality ...), and so forth. Certainly, many philosophers find it natural to think of the sent~nces of mathematics as truth-valued. Classically understood, for any definite (Le., non-ambiguous, non-vague, etc.) sentence p and name p of p, the sentence p is true iff p itself is a truth. The sentences of mathematics are paradigms of definite I sentences. Suppose then that some mathematical sentence p is neither true , nor false. Then the above biconditional links sentences respectively false and not false. In that case, the biconditional is not true. So, if, for the domain of mathematical sentences, some are neither true n'or false, mathematical sentences are not subject to the classical notion of tnlth (=objective truth). In this way application of the classical notion of tnlth to the sentences of mathematics presupposes that these sentences are one and all truth-valued. If there are no such things as numbers, the theorems of mathematics are not true in virtue of how things are with numbers. sci in what does their truth I consist? The provability theory says that their truth consists in their provability. There is nothing else for their truth to Consist in if they are not true in virtue of the way things are with numbers. I Outside of mathematics a proof establishes truth. But truth does not consist in proof. To prove that wild elephants still exist you have to search out one that the poachers or hUnters or park managers have not yet slaughtered the assertio way the wo Part of 1 mathemati( mathematic makes it tru Outside m: Within mat Here is a; axioms calli method can not somethiJ consists in t Here thi sentences OJ out proof. 1 numbers to the very po: world ofnu There art (1) The cannot be i( "Godel's tt mathematic. echoed in 01 The resul understand. the side and It comes theory inclu language oj derivable ff( 2 See, for ex~ 1972,229. 3 The Gtidel ~(Vx)A(x) is vward things are some way and owes its truth or ; in addition to objective truths. If this is so, lce to say nothing about how things are and lOW is that within the set of truth-valued iUbset which consists of non-objective be included in this subset. How plausible is tive truth-valued statements in addition to , We shall consider two sources of this idea. jea is what might be called the provability , is non-objective truth is the thought that in each area in which we speak of truth Iitics, morality ...), that knowledge is the rea in which we speak of knowledge itics, morality ...), and so forth. Certainly, to think of the sentences of mathematics as JOd, for any definite (Le., non-ambiguous, lame p of p, the sentence )f mathematics are paradigms of definite ae mathematical sentence p is neither true tionallinks sentences respectively false and itional is not true. So, if, for the domain of are neither true nor false, mathematical assical notion of truth (=objective truth). In ;sical notion of truth to the sentences of :lie sentences are one and all truth-valued. numbers, the theorems of mathematics are 'e with numbers. So in what does their truth { says that their truth consists in their : for their truth to consist in if they are not re with numbers. 'Oaf establishes truth. But truth does not rild elephants still exist you have to search hunters or park managers have not yet Null Sentences 31 I slaughtered. That would establish the truth of the assertion. But the truth of the assertion does not consist in its h~ving a proof; it is true in virtue of the way the world is with wild elephantsi Part of the content of the provability theory is that provability within mathematics is fundamentally different from provability outside mathematics. Outside of mathematics what establishes a sentence is not what makes it true. But within mathematics being true consists in having a proof. Outside mathematics proof establi~hes something beyond itself: truth. Within mathematics proof establishes nothing beyond itself. I Here is another way of putting the thought. A derivation from mathematical axioms cannot give an incorrect result: if correctly carried out. If a verification method can't give an incorrect result if correctly carried out, then that result is, not something the truth of which is due to the way things are. Instead, its truth consists in being the result of a correctly carried out verification procedure. Here there is no "gap" between :verification and truth. That is, for the sentences of pure mathematics truth is being the result of a correctly carried out proof. It thus becomes obvious ,that there is no need for a world of numbers to make for truth in mathematics. Against this it may be urged that the very possibility of defining truth Jor mathematical sentences requires a world of numbers. I There are two major objections to ,the provability theory. (1) The first objection is that Godel showed that mathematical truth cannot be identified with provability~ For example, Richard Jeffrey writes: "Godel's theorem dealt a deathblow to the theory which identified mathematical truth with provability" (Jeffrey 1967, 196). This theme is echoed in one logic text after another~2 The result of Godel of which Jeffrey speaks is actually pretty simple to understand. The complexities lie on the side of the proof. Let us put that to the side and just think about what he proved. It comes to this: That for any effec~ive and consistent axiomatization of a theory including at least elementary arithmetic there are sentences in the language of the theory such that reither they nor their negations are derivable from the axioms.3 ' 2 See, for example: Stoll 1961, 167; Pollock 1969, 229; Massey 1970, 129; Mates 1972,229. I 3 The Glidel result referred to is that if krithmetic is omega-consistent (if, that is, ,(\fx)A(x) is unprovable if each A(n) is! provable) then it is incomplete (there are I 32 Philip Hugly and Charles Sayward I This is a syntactical result. The notions of truth and falsity do not enter into it at all, either by way of the content of the the~rem itself or by way of its proof. In particular, that truth and falsity in mathe;matics go beyond proof and disproof is no part of what G6del proves. I So we do not think this objection poses a serious problem for the provability theory. I (2) The second objection goes thus: A proof in mathematics is a derivation from axioms. So, according to the provability theory, the truth of an axiom consists in its being derivable from itself. Is it not just obvious how implausible that is? Consider one of the Peano axioms: I Vx (O;o<sx) I How do we know that is true? The answer that it is derivable from itself is i not likely to satisfy anyone. And it should not satisfy anyone since every , statement is derivable from itself. I I And why is not one consistent set of axioms as good as any other on the account offered by the provability theory? Suppose that instead of the Pea no 1 axioms we had as our only axiom for arithmetic Vx(x=O) I Relative to this axiom a wholly different set of sentences is true. In Remarks on the Foundations of Mathematic~ Wittgenstein writes: "I should like to say mathematics is a motley of techniques of proof'(1956, 84; I his emphasis). We are sure Wittgenstein would haye denied that the motley of techniques of proof all reduce to derivations frõ axioms. But he gives no other account. Lacking such an account, it is impossible to say what the provability theory comes to. It is insufficiently clear to be accepted. So we think this second objection does pose 'a serious obstacle to the provability theory. I I A Second Source. A second source of the idea that there is non-objective truth and falsity is that fictional truth or falsity! is non-objective truth or falsity. Consider the sentence Pegasus was a winged horse. ----I sentences such that neither they nor their negations are provable). Rosser extended this: if arithmetic is consistent (if, that is, not every sentence is provable) then it is incomplete. i , Isn't it a tn There Ct sentence as which theSI about the al In these go, and so utterance, f only rightn, is what the More ge fall short c sentence 'F fiction requ ancient Gre call one of To this uttered in tl of 'Pegasm which it is. it, since it i is true. Fur belongs to: stories. So j to say of it But it is. in the telIin; entaiJed by the sentena and winged Ithaca' and if they are I different tyl That also is in fiction. F this holds i sentence to yward Ie notions of truth and falsity do not enter content of the theorem itself or by way of md falsity in mathematics go beyond proof Odel proves. jection poses a serious problem for the Jes thus: A proof in mathematics is a rding to the provability theory, the truth of ivable from itself. Is it not just obvious how : of the Peano axioms: he answer that it is derivable from itself is j it should not satisfy anyone since every t set of axioms as good as any other on the " theory? Suppose that instead of the Peano :l for arithmetic lifferent set of sentences is true. 'ns 0/ Mathematics Wittgenstein writes: "I a motley of techniques of proof'(1956, 84; ~enstein would have denied that the motley to derivations from axioms. But he gives no account, it is impossible to say what the insufficiently clear to be accepted. !ction does pose a serious obstacle to the Irce of the idea that there is non-objective 1 truth or falsity is non-objective truth or se. their negations are provable). Rosser extended that is, not every sentence is provable) then it is Null Sentences 33 Isn't it a true sentence? I There certainly are contexts in which we would regard a denial of that sentence as an error, and its affirmation as correct. And we pretty well know which these contexts are-the ones: in which we are retelling or talking about the ancient Greek myths. In these contexts the correctness of what is said is tied to how the myths go, and so the sentence 'Pegasus wJ a winged horse' makes for a correct utterance, for it gets the ancient myths right. But then what it possesses is only rightness relative to those myths:-not truth. That rightness, not truth, is what the generalization 'There were once winged horses' inherits. More generally, lots of sentences :which we perfectly well recognize to fall short of truth yet are, as we sometimes put it, "true in fiction." The sentence 'Pegasus was a horse with wings' is such a sentence. Its truth in fiction requires not that there once w~re horses with wings but only that the ancient Greek Pegasus stories are oñs that speak of horses with wings, and call one of them Pegasus. I To this it might be replied that what is true in fiction is a sentence as uttered in the course of story-telling, ~nd that we do not require for the truth of 'Pegasus was a winged horse' that'this sentence or any other sentence of I which it is a translation actually occu~ in any telling of any Pegasus story. So it, since it is not a sentence used in fiction, cannot be true in fiction. Still, it is true. Further, it is not a sentence that says that such and such a sentence belongs to some story, for it is not a sentence which says anything about any stories. So it also is not true o/fiction.! Still, it is true. So surely the right thing to say of it is just that it is true! ! But it is an error to suppose that a sentence is true in fiction only if it occurs in the telling of some fiction. A senteñ might also be true in fiction if it were entailed by sentences actually so used. If in the telling of a Greek myth we use the sentence 'And then that wonderful horse Pegasus leaped from the hillside and winged his way to Ithaca', then the sentences 'Pegasus winged his way to Ithaca' and'A horse winged its way to Ithaca' also are true in that fiction even if they are not used in the story telling. Or, to move to an example of a quite different type, consider a play in which the characters say and do certain things. That also is something in virtue of which a sentence can be true and thus be true in fiction. For example, it is true in HOmiet that Hamlet loved his mother. And this holds independently of whether br not 'Mom, I love you' or any other I sentence to the same effect is a line in the mouth of Hamlet in the play. ! 34 Philip Hugly and Charles Sayward The sentence 'Hamlet loved his mother' could be used to Isay something about some actual fellow named 'Hamlet'. So used, it might ~elI be true. But in certain contexts-the ones in which the play figures ~s a subject of discourse-the above sentence stands as one fit for cohect utterance independently of whether there ever was any such person ã the Hamlet of I that play, or of how things stand with anyone actually named .'Hamlet'. Here the play's the thing. Is the sentence then about the play? ~elI, in a way it isn't. For the sentence contains no term designating the play. But also, in a way it is, for we would say that someone who used that sentence in a context concerned with that play, had made an indirect reference to it. But however we decide about 'about', this is clear: in the sorts of cases; here at issue it would be by inquiring into the play that we would determine whether or not I the claim that Hamlet loved his mother was true. It is relative to the play that our sentence has such truth as it possesses in kinds of cont~xts with which we are here concerned. I Suppose that in a story the author writes Jones hated someone Suppose also nothing in the story entails any instance of this generalization, i.e., nothing in the story entails I Jones hated A I for any singular term A. How can 'Jones hated someone' be true under these imagined circumstances? It is true in the story because the author wrote it down as part of the story, but it is not true since nothing sat,isfies the schema 'Jones hated A'. Despite these remarks, there may remain a sense that the cases at hand do in a way really involve truth. And there is something to that. Consider for a moment not the sentence 'Hamlet loved his mother' but 'In Hamlet Hamlet loves his mother'. This seems to be straightforwardly true. :(If you don't find this interpretation of Hamlet plausible, switch examples, e.g., to 'In the ancient myths Pegasus was a winged horse'.) And so it is. The relevant semantic operation is that of applying a connective 'In Hamlet' to a sentence, thereby yielding another sentence which is true or not depending on how the play goes, independently of how the play is said to go in such stories as there may be in which it figures. By application of the connective we construct a sentence explicitly about the play. I We could take the occurrences of 'Hamlet loved his mother' in contexts concerning the playas elliptical for 'In Hamlet Hamlet loves his mother'. I Were we to do so, the I 'Someone loved his mo someone loved his mo And so it could be that horse' (in context) is the horse', and the truth fe horses' is the truth of 'II Let us sum up Our reI of non-objective truth .. denial of 'Pegasus Was correct. But this correctn true in virtue of how the any way with regard to world. The correctness certain myths go. It is tr truth. We need to distin horse' and 'Pegasus wa; the urge to call the lattel two sentences. 6. Final Remarks Wittgenstein was very nonsense in the way til gimble in the wabe" is nj (sinnlos); he denies that to say they are without things are. But he als( thoughts do not cohere. things are a certain way is false is to say that it ~ that way. University ofNebraska vward is mother' could be used to say something Hamlet'. So used, it might well be true. But 1 which the play figures as a subject of stands as one fit for correct utterance ver was any such person as the Hamlet of ,ith anyone actually named 'Hamlet'. Here Ice then about the play? Well, in a way it o term designating the play. But also, in a Imeone who used that sentence in a context de an indirect reference to it. But however clear: in the sorts of cases here at issue it ly that we would determine whether or not other was true. It is relative to the play that possesses in kinds of contexts with which hor writes entails any instance of this generalization, 'Jones hated someone' be true under these Ie in the story because the author wrote it not true since nothing satisfies the schema lay remain a sense that the cases at hand do j there is something to that. Consider for a t loved his mother' but 'In Hamlet Hamlet Je straightforwardly true. (If you don't find ausible, switch examples, e.g., to 'In the inged horse'.) And so it is. The relevant ring a connective 'In Hamlet' to a sentence, : which is true or not depending on how the the play is said to go in such stories as there Jplication of the connective we construct a '. s of 'Hamlet loved his mother' in contexts for 'In Hamlet Hamlet loves his mother'. Null Sentences 35 I Were we to do so, the generalization of 'Hamlet loved his mother', namely 'Someone loved his mother', would be elliptical (in context) for 'In Hamlet I someone loved his mother' -another sentence explicitly about the play. And so it could be that the truth feIt in the sentence 'Pegasus was a winged horse' (in context) is the truth of 'In th~ ancient myths Pegasus was a winged horse', and the truth felt in the generalization 'There once were winged horses' is the truth of 'Ill the ancient myths there were winged horses'. Let us sum up our reply to the thought that truth in fiction is an example of non-objective truth. There are codtexts in which we would regard the denial of 'Pegasus was a winged horse' as an error and its affirmation as correct. But this correctness is not trut~. 'Pegasus was a winged horse' is not true in virtue of how the world is in regard to Pegasus. The world is not in I any way with regard to Pegasus siñe 'Pegasus' refers to nothing in the world. The correctness of 'Pegasus {vas a winged horse' is tied to how certain myths go. It is true in fiction or true in myth. Truth in fiction is not truth. We need to distinguish 'In the ~ncient myths Pegasus was a winged horse' and 'Pegasus was a winged horse'. The former sentence is true, and the urge to call the latter sentence true might well arise from confusing the two sentences. 6. Final Remarks Wittgenstein was very clear that tãtologies and contradictions are not nonsense in the way that "Twas briilig an the slithy toves did gyre and gimble in the wabe" is nonsense. His thought was that they are without sense (sinnlos); he denies that they are nonsFnsical (unsinnig). What does it mean to say they are without sense? It means that they say nothing about how things are. But he also says they ire true and false, respectively. The thoughts do not cohere. For to say a sentence is true is to say that it says things are a certain way and that things are that way; and to say a sentence is false is to say that it says things are a certain way and that things are not that way. i University ofNebraska -Lincoln