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What is Vacuous Truth? Peter Sutton & Wylie Breckenridge Some propositions are said to be vacuously true. Since there are no giraffes in Ithaca it is vacuously true that every giraffe in Ithaca is male. It is also vacuously true that every giraffe in Ithaca is female, that every giraffe in Ithaca is wearing a green beret and reading The Times, and so on. But we should not be concerned that these things are all true, because they are vacuously true. If these propositions are true they are not obviously so. Before our intuitions are tainted by a class in formal logic class we are reluctant to judge them true, and equally reluctant to judge them false. The intuition is that they can only be true or false if there are some giraffes in Ithaca. In technical terms, these propositions (or perhaps the sentences that express them, or perhaps the speakers who express them) presuppose that there are some giraffes in Ithaca. In this respect the proposition that every giraffe in Ithaca is make is very much like the proposition that the giraffe in Ithaca is male. Here again, before our intuitions are tainted by studying Russell we are reluctant to judge it true, and equally reluctant to judge it false: the proposition can only be true or false if there is exactly one giraffe in Ithaca – it, or any sentence that expresses it, or any speaker who expresses it, presupposes that there is exactly one giraffe in Ithaca. Russell famously argued that our intuitions about this latter case are wrong: it is false that the giraffe in Ithaca is male. According to Russell, it is true that the giraffe in Ithaca is male if and only if there is exactly one giraffe in Ithaca and it is male; there is not exactly one giraffe in Ithaca; so it is false that the giraffe in Ithaca is male. The matter is controversial. But whether or not Russell is right, what all parties to the controversy agree is that it is not true. Given the similarity of the cases, it is strange that anyone would think that it is true that every giraffe in Ithaca is male. The reason that we do, it seems, is that we are taught that it is true in logic class. We are taught to formalize ‘Every F is G’ as ‘x(Fx Gx)’, and it follows from the semantics of FOL that ‘x(Fx Gx)’ is true: if there are no F-s then ‘Fx’ is false of everything in the domain, so ‘(Fx Gx)’ is true of everything in the domain, so ‘x(Fx Gx)’ is true. Is this a good reason to think that every donkey in Ithaca is male? One reason to think not is that we easily turn things around: the formalization is incorrect, because it declares it to be true that every donkey in Ithaca is male, and this is not true (it is perhaps not false either). There are other formalizations that we could use instead: if, following Russell’s lead in the case of ‘The F is G’, we want ‘Every F is G’ to come out false if there are no F-s then we could formalize it as ‘(xFx x(Fx Gx))’; if, respecting intuition, we want it to come out neither true nor false then we could switch to using generalized quantifiers (for which there is independent motivation1), formalize it as [every x: Fx](Gx), and build into the meaning of ‘every’ that this is neither true nor false if there are no F-s. So the first point that we might to make is this: it is not obvious that it is even true that every giraffe in Ithaca is male. So too with every vacuous truth. 1 See Barwise and Cooper. No matter, we can just change the label. If it turns out to be false that every giraffe in Ithaca is male then rather than calling it vacuously true we can call it vacuously false, and if it turns out to be neither true nor false then we can call it vacuously neither true nor false. The point is: whatever it is, it is that vacuously. Let’s not commit and just say that the proposition is vacuously X, where ‘X’ means ‘true’, ‘false’ or ‘neither true nor false’ according to whether the proposition that every giraffe in Ithaca is male is true, false, or neither true nor false. What is it for a proposition to be vacuously X? At a minimum we want the following: the proposition that every F is G is vacuously X (at a world w) just in case nothing is F (at w). Thus since there are no donkeys in Ithaca (in the actual world) the proposition that every donkey in Ithaca is male is vacuously X (in the actual world). If something is F at w then it still might be X that every F is G at w, but it is not vacuously X. If we want it to be possible that some but not all truths are vacuously true, plausibly a sufficiency condition on the notion of vacuous truth being an interesting one, then we have to take a certain stand on the nature of propositions. We cannot, for example, follow one popular approach and take a proposition to be a set of worlds (the worlds at which it is true). On this view, the proposition that every donkey in Ithaca is male is identical to the proposition that every non-male donkey is outside Ithaca (they are the same set of worlds). So on this view the proposition that every non-male donkey is outside Ithaca is also vacuously X.2 Worse still, on this view every proposition that is X is vacuously X. Any proposition p is identical to the proposition that everything that is such that p is false is such that p is true.3 So if p is true at world w then it is vacuously true at w. So if propositions are sets of worlds then we get the result that every truth is vacuously true; moreover, necessarily, every truth is vacuously true. So if we want the notion of vacuous truth to be an interesting one then cannot take propositions to be sets of worlds. Contrapositively, if we take propositions to be sets of worlds then the notion of vacuous truth is not an interesting one. We can also argue that no proposition that is X is vacuously X.4 Any proposition p is identical to the proposition that everything that is self-identical is such that p is true. So if p is true at world w then it is vacuously true at w. So if propositions are sets of worlds then we get the result that no truth is vacuously true; moreover, necessarily, no truth is vacuously true. So being vacuously X is an incoherent concept? One alternative is to take them to be structured entities. Then we can allow the proposition that every giraffe in Ithaca is male to be distinct from the proposition that every non-male giraffe is outside Ithaca, and not be committed to the former being vacuously X entailing the latter also being vacuously X by claiming that they have a different structure. Abandoning Classical Logic Other, more run-of-the-mill (but less generalizable) examples of vacuous truths that are equivalent to nonvacuous truths include “All Dragons are Communists,” which is equivalent to “All non-Communists are not dragons,” and “All flying donkeys are black,” which is equivalent to both “All non-black flying things are non-donkeys” and “All non-black donkeys don’t fly.” These examples don’t generalize fully because of cases in which the consequent-clause applies to the whole domain of discourse, as in “All Dragons are self-identical,” the contrapositive of which, “All non-self-identical things are non-dragons,” is also vacuous. 3 Proof: 4 Thanks to Zachary Abrahams for helping to develop this point. 2 The logical equivalence between (2) and P depends on the truth of the law of excluded middle (the formal proof in footnote 2 exploits the rule of double-negation elimination, which is logically equivalent to the law of excluded middle). So by rejecting classical logic and giving our hearts to intuitionism we can avoid UniVac. But this is, for most of us, quite a bullet to bite, and at any rate it only solves half the problem. P and (2) are not intuitionistically equivalent, but P is intuitionistically equivalent to 1. x[(P & x=x) P] Thus, every true negation is logically equivalent to a vacuous truth, so, given the premises above, we can still show that all true negations expressible in first-order logic are vacuously true, which isn’t much better than UniVac. It’s overwhelmingly plausible that at least some true negations are nonvacuously true (if anything is nonvacuously true). I can imagine arguments for the vacuity of, e.g., “no dragons are Communists,” but this argument tells us that run-of-the-mill truths like “7 is not even,” “Eugene V. Debs is not the president,” and “the shortest man on earth is not seven feet tall” are all vacuously true, contrary to what we might have thought. This is quite a bullet to bite, especially if our mouths are already full of the intuitionist-logic-bullet. Abandoning Premise A If truth is a property of propositions (at worlds), then any way of being true is a way for a proposition to be true (at a world). Being vacuously true is a way of being true, so vacuous truth must be a property of propositions. We might still want to say that the sentence “All dragons are Communists” is vacuously true in virtue of expressing a vacuously true proposition, but it will not do to say that the sentence, but not the proposition it expresses, is vacuous. Thus, if anything is vacuously true, propositions must be.5 But we might want to reject the premise of this argument that being vacuous is a way of being true. Instead we could say that vacuity is a syntactic property of sentences rather than a semantic property of propositions. Of course it can’t be a purely syntactic property (the syntax of (1) does not entail the non-existence of dragons, and so cannot entail that (1) is vacuous) but perhaps it could be a result of interplay between syntax and semantics. One plausible suggestion is the following: P is a vacuously true expression iff P is universal in form and is derivable from a true negative existential.6 This at least seems to work in the standard cases. (1) is a universal that follows from the true negative existential “There are no dragons” and so is vacuously true. Of course “All non-Communists are non-dragons” is also vacuously true on this account, but that’s not too much of a bullet to bite.7 We actually find this suggestion quite appealing, and it would seem to solve our problem, but unfortunately appearances are deceiving. Even if we accept this definition of vacuous truth, we still have an argument that all truths are vacuous. It turns out that any true sentence P is clasically equivalent to: 5 6 7 One avenue that this last sentence seems to open up is that of denying that anything is vacuously true. Notice, first of all, that it isn’t clear whether this is a way of rejecting the Premise A or merely a different way of accepting it. If there are no vacuous truths, then isn’t it true that only propositions are vacuously true? Granted, it can’t be vacuously true, by hypothesis, but that’s no reason to think it’s false. Anyway, this response is akin to accepting UniVac, but denying its significance. Thus, we’ll return to it in the final section. So vacuous truth involves syntax in virtue of the derivability constraint and semantics in virtue of the fact that the negative existential in question must be true. Thanks to Theodore Korzukhin for this proposal. We might call the contrapositive of any vacuous truth a copious truth, following Baumer (1964), p. 183. 4. x(P x≠x) (4) is a negative existential that is equivalent to P8, and P is true by hypothesis, so (4) is also true. And, of course, P is derivable from it. Thus, even on the proposed modification of vacuous truth, Univac holds. A final way of rejecting premise A is this: we accept that sentences are not vacuously true in virtue of their semantic values (i.e. propositions) but rather in virtue of the semantic values of their parts (e.g. words and phrases). Perhaps (1) is vacuous in virtue of having the sub-sentential component 5. All dragons… (5) is a non-referring determiner phrase, so if any part of (1) makes it vacuously true, it must be (5). More generally, we could say that a sentence S is vacuous iff (a) S is of the form “All Fs are Gs,” and (b) “xFx” is true. There is nothing inconsistent about this account of vacuous truth, and it allows us to reject UniVac. But it is worrying to us that this response (and, it seems, any response that rejects premise A) makes vacuous truth an arbitrary property: a result of the words we choose to express ourselves rather than what it is we are actually saying9, which remains, presumably, propositional. Abandoning Premise B One obvious answer to King’s worry is to say that propositions must be composite things that have objects and properties as parts. So the number 2 is a part of the proposition that 2 + 2 = 4, but not part of the proposition that first-order logic is undecidable. Similarly, the city of Paris (or a proxy for it) and the property of being pretty are both parts of the proposition that Paris is pretty, whereas the city of Santa Monica is not. Of course, this initial response is too simplistic because it doesn’t distinguish the proposition that Mary killed John from the proposition that John killed Mary. So our composite propositions will have to be structured—presumably the properties have argument-places that have to be filled by the relevant objects in the right way. But this composite, structured proposition proposal still isn’t enough to get us out of premise B. Arguably, the propositions expressed by “all dragons are communists” and “all noncommunists are non-dragons” attribute precisely the same properties and relations to precisely the same objects.10 Similarly, our sentences P and (2) contain all the same non-logical vocabulary and so plausibly attribute the same properties and relations to the same objects. So in order to deny premise B, we will have to say that the logical structure of the sentence makes a difference to what proposition is expressed, even when it doesn’t alter the truth conditions. In fact, there are views of structured propositions that take all of the above (objects, properties, argument-placement, and logical structure) into account (e.g. King 2007), and can suffice for abandoning premise B, but many of the traditional views of structured propositions do Proof: First assume P. Now assume x(P x≠x). It follows that something is non-self-identical, thus, we reject our assumption and get (P x≠x). Now, assume x(P x≠x). Then assume P. Given P, any conditional with P as antecedent must be true, so we get P a≠a. Existential intro gets us x(P x≠x), which contradicts our first assumption. Thus, P, or P, QED. 9 We could say what (1) says by uttering “All non-Communists are non-dragons,” which doesn’t contain “All dragons…” or any other “empty” part. 10 And it is plausible that anybody who believes the former also believes the latter, and that they both have their truth-values in virtue of the same things. 8 not. Gottlob Frege, for example, believed that a proposition (or thought, as his ‘gedanke’ is usually translated) was a structured entity. In “On Sense and Reference” he claims that the thought is the sense of a sentence, and its parts are the senses of the parts of the sentence.11 So the sense of ‘Paris’ is a part of the thought that Paris is pretty, just as the word ‘Paris’ is a part of the sentence “Paris is Pretty.” But in his “Compound Thoughts” he claims that logically equivalent expressions like “B” and “not-not-B” express the same thought (i.e. proposition).12 And in “Concept and Object” he writes: [A] thought can be split up in many ways, so that now one thing, now another, appears as subject or predicate. The thought itself does not determine what is to be regarded as the subject…We must never forget that different sentences may express the same thought. (Frege 1960)13 So the founding father of structured propositions (and of propositions more generally, perhaps) seemed to think that such structure by no means guaranteed that P and (2) express distinct propositions. Indeed, given that they are first-order equivalent (like “A” and “not-not-A”) Frege would almost certainly have insisted that they do express the same proposition. But Frege is no infallible deity, and it remains wide open for us to abandon his view propositions in favor of one that allows us to abandon premise B. Perhaps our argument above can even be construed as an argument for such contemporary views as King’s. Thus, abandoning premise B seems to us to be the best way of responding to the argument short of accepting the conclusion. Accepting the Conclusion Aristotle apparently believed that vacuous truths aren’t true at all.14 The argument deals only with first-order expressible propositions, but not every truth is firstorder expressible. Maybe not, but it will not be terribly easy to produce plausible counterexamples. Notice that it is not sufficient to point out that certain sub-sentential expressions like “most” have no first-order translation. It is not necessary to translate every portion of a sentence in order to translate it into first-order logic, so a notoriously difficult sentence like: 6. Most Englishmen admire few of the Irishmen who admire only one another. Can easily be translated for our purposes as “Q,” which will be logically equivalent to x[(Q & x=x) Q]. Even sentences that are in English and about English (and so are in both in English 11 12 13 14 Frege 1960 pp. 62-64. Some quotes: “The thought loses value for us as soon as we recognize that the reference of one of its parts is missing” (p. 63) and “Subject and predicate (understood in the logical sense) are indeed elements of thought; they stand on the same level for knowledge. By combining subject and predicate, one reaches only a thought, never passes from sense to reference, never from thought to its truth value.” (p. 64) Frege (1963) pp. 10 & 15. He also says that “A or A” and “A and A” each express the same thought as A. The example sentences Frege has in mind as expressing the same thought are ‘there is at least one square root of 4,’ and ‘The concept square root of 4 is realized.’ These don’t bear directly on the case of P and (2), but given what Frege says in “Compound Thoughts” about simple logical equivalences, it is hard to believe that he would have taken P to express a different proposition from (2). Note that Aristotle would likely not accept our proof of the equivalence of P and (2), because he wouldn’t have accepted that universals like (1) and (2) are in fact universally quantified material conditionals. Thus, our argument unfortunately fails to shed any new light on this age-old disagreement between classical logicians and classic logicians. qua metalanguage and English qua object language) can typically be translated in this manner. Basically, any sentence that is either true or false has a simple first-order translation. Perhaps genuinely paradoxical examples like liar-sentences will resist this treatment, as might sentences with non-referring proper names and sentences attributing vague predicates to borderline cases, but it is little comfort to think that these are the only sentences with a chance of being nonvacuously true. References Barwise, J. and Cooper, R. 1981. “Generalized Quantifiers and Natural Language.” Linguistics and Philosophy, vol. 4: pp. 159-219. Baumer, William. 1964. “Confirmation without Paradoxes.” The British Journal for the Philosophy of Science, vol. 15, no. 59: pp. 177-195. Frege, Gottlob. 1963. “Compound Thoughts.” Mind, New Series, vol. 72, no. 285: pp. 1-17. Frege, Gottlob. 1960. Translations from the Philosophical Writings of Gottlob Frege. Translated and edited by P.T. Geach and Max Black. Oxford: Blackwell Publishers. King, Jeffrey. 2007. The Nature and Structure of Content. Oxford: Oxford University Press. If there are no giraffes in Ithaca, then is it true or false that every giraffe in Ithaca is male? It is sometimes said (a) that it is true, and (b) that it is vacuously true. If it is true then it is not obviously so. It is very tempting to say that it is neither true nor false, because it can only be either true or false if there are some giraffes in Ithaca. In other words, the proposition that every giraffe in Ithaca is male, or any sentence that expresses this proposition, or any speaker who expresses this proposition, presupposes that there are some giraffes in Ithaca. In this respect the question seems to be very much like the following one: if there is not exactly one giraffe in Ithaca, then is it true or false that the giraffe in Ithaca is male? Here also it is neither true nor false, because it can only be either true or false if there is exactly one giraffe in Ithaca. Bertrand Russell famously argued for a position according to which the correct answer to the latter question is false: if there is not exactly one giraffe in Ithaca, then it is false that the giraffe in Ithaca is male. There is still considerable debate about the matter – about whether it is false, or neither true nor false. But it seems clear to everyone in the debate that the answer is not true. Given the similarity of the cases, it seems strange that anyone would think that the answer to the former question is true. The reason seems to be that standard predicate logic, the formal language of choice among philosophers, forces this answer upon us. Because of the limited resources of PL we cannot formalize the English words ‘every’ and ‘some’, but only the English words ‘everything’ and ‘something’. So to formalize the sentence ‘Every donkey in Ithaca is male’ we first have to distort it into the approximate paraphrase ‘Everything is such that if it is a donkey in Ithaca then it is male’, or ‘Everything is such that it is either not a donkey in Ithaca or it is male’, or ‘It is not the case that something is such that it is a donkey in Ithaca and it is not male’, and so on. In the case of the second and third of these paraphrases it is pretty intuitive that they are true. In the first case, in order to But there is nothing stopping us formalizing it as ‘(x(Gx Ix) x((Gx Ix) Mx))’. Just as we might formalize ‘The giraffe in Ithaca is male’ as ‘(x(Gx Ix) xy((Gx Gy Ix Iy) x = y) x((Dx Ix) Mx))’ ; we are taught that something is such that if it is a giraffe in Ithaca then it is male just in case either it is not a giraffe in Ithaca or it is male; so we are taught that every giraffe in Ithaca is male just in case everything is such that either it is not a giraffe in Ithaca or it is male. If this is right, then it is true that every giraffe in Ithaca is male. Should we believe what we are taught in logic class? There are two key steps. I shall not adjudicate on the second: that is a matter of considerable debate. But I shall adjudicate on the first, because there seems to be some agreement on the matter. I shall grant that to the logicians. Consider the claim that most dogs have five legs. Following the same line of reasoning, this is true just in case most things are such that if they are dogs then they have five legs. That is, most things are such that either they are not dogs or they have legs. This is true. So it comes out true that most dogs have five legs. This suggests that the approach of logic is limited, and a different approach needs to be taken for ‘most’. But then why not take the same approach to ‘every’, if it can be made to work. And it can. Go generalized quantifiers: [most x: x is a dog](x has five legs), [every x: x is a giraffe in Ithaca](x is male). The following are logically equivalent, but only the first is vacuously true: Every giraffe in Ithaca is male x((Gx Ix) Mx) Every giraffe that is not male is not in Ithaca x((Gx Mx) Ix) Every thing in Ithaca that is not male is not a giraffe x((Mx Ix) Gx) Here is a proof that every truth is logically equivalent to a vacuous truth. ‘P’ is logically equivalent to ‘x((x = x P) (x = x P))’. So if we say that any truth which is logically equivalent to a vacuous truth is a vacuous truth, then every truth is a vacuous truth. Theory: It is vacuously true that every F is G just in case there are no Fs. Should we extend this to other determiners as well? Most Fs are G Few Fs are G Some Fs are G No Fs are G Should we allow there to be vacuous falsity as well? Some donkey in this room is red No donkey in this room is red What about the determiner ‘the’? The present king of France is bald – vacuously false? So it can never be vacuously true that the F is G, but it can be vacuously false What about empty names, instead of empty predicates: Phlogiston does not exist – vacuously true? Phlogiston is a liquid – vacuously false? What about vacuity in ‘G’? If there are no Gs, then is it vacuously true that no Fs are G? What about copious truth and falsity?