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3. The property of truth is needed to account for truth-value gaps. Some apparent assertions fail to be either true or false. Such assertions suffer from truth-value gaps. Some examples of truth-value gaps involve failures of presuppositions of various kinds:
(5) The present king of Texas is bald.
(6) It is taboo to step on the emperor's shadow.
(7) It is noon on the surface of the sun.
Since there is no present king of Texas, it is neither true nor false to say of him that he is bald. If the supposed property of being taboo doesn't really exist, then it is neither true nor false to say that it characterizes some action. Statements about the hour of the day presuppose that the statement is being made within some time zone on the Earth's surface.
Some philosophers have argued that statements involving vague predicates or referring to vague entities are neither truth nor false:
(8) Robert Duvall is bald.
(9) Austin is a large city.
Deflationists have trouble accounting for truth-value gaps. It is natural for deflationists to explain the use of the word ‘false’ in a way parallel to their account of ‘true’:
(10) ‘Snow is white’ is false if and only if snow is not white.
(11) ‘Snow is green’ is false if and only if snow is not green.
If deflationists accept the laws of classical logic, including the Law of Excluded Middle, they will be forced to endorse every instance of the following argument:
Either p or not-p (the Law of Excluded Middle).
S is true if and only if p. (Tarski's schema, where ‘S’ is the name for ‘p’.)
S is false if and only if not-p. (the falsity counterpart to Tarski's schema.)
Either S is true or S is false.
In order to avoid this argument, deflationists have to restrict the application of Tarski's schema to those sentences that succeed in expressing a proposition. Then they could attribute truth-value gaps to (5) through (9) by denying that those sentences succeed in expressing propositions. However, deflationists would then owe some account of what a proposition is. It would seem to be part of the very essence or nature of a proposition that it be true or false, and such a metaphysical account of propositions would again require truth to be a natural property.
4. A property of truth is required for the contextualist solution to the Paradox of the Liar. Finally, we turn to the ancient Paradox of the Liar (first discussed by the Greek philosopher Epimenides). Consider (12):
(12) Statement (12) is not true.
If we assume that (12) is true, then we quickly find ourselves in the contradiction that (12) is both true and not true. However, it is equally difficult to affirm that (12) is not true (for whatever reason), since the non-truth of (12) seems to be exactly what (12) is affirming. If (12) is not true, then things are as (12) states them to be, so (12) is true after all. Tarski proved this result formally:
‘Statement (12) is not true’ is true if and only if statement (12) is not true. (T-schema)
‘Statement (12) is not true’ = statement (12). (Stipulated identity of ‘statement (12)’)
Statement (12) is true if and only if statement (12) is not true. (1, 2, the substitution of identicals)
Introducing truth-value gaps won't resolve this problem. Suppose we say that (12) fails to express a proposition. If (12) doesn't express a proposition, then it isn't true. But this is just what (12) says—so it seems to express a proposition, in fact a true one, after all.
There have been many attempts to solve this paradox, but there is no consensus about which is correct. Most approaches to the Liar are consistent with deflationism, but there is one popular approach that is not: the context-sensitive solution of Burge (1979), Barwise and Etchemendy (1987), Koons (1992), Simmons (1993), and Glanzberg (2001, 2004). On the context-sensitive approach, when we assert that some assertion is true, we are claiming that it corresponds to some part of the world. Which part of the world we can refer to shifts from one context to another. Thus, we must interpret statement (12) as tacitly asserting something of this form:
(12S) Statement (12S) does not correspond to any part of S.
(Where S is some contextually indicated part of the world.)
We can now recognize that statement (12S) is true because it corresponds to some part S' of the world outside of S. S' is a part of the world that includes (as S does not) the fact that no part of S corresponds to (12S). This solution requires the existence of truthful correspondence as a real relation between statements (or propositions) and parts of the world and is therefore inconsistent with deflationism.
5. The theory of truth is not a conservative extension of truth-free theories. Stewart Shapiro (1998) points out that reasoning by means of truth “non-conservatively extends” theories that do not involve truth. That is, we are able to reach novel conclusions by deploying obvious facts about truth. For example, the mathematician Kurt Gödel proved that no consistent mathematical system (like Peano arithmetic) can prove its own consistency. However, we can use truth to do so:
All of the axioms of Peano arithmetic are true.
Every set of truths is mutually consistent (in the sense that no contradiction can be proved formally from them).
Therefore, Peano arithmetic is consistent.
Deflationists can account for the truth of premise 1: asserting 1 is simply to assert once again the axioms of arithmetic themselves. However, premise 2 is a problem for deflationists. It is a mathematical assertion about a class of sets of sentences. The axioms of Peano arithmetic are complicated enough that it is by no means obvious that one couldn't derive a contradiction from them, assuming that one sets aside the facts expressed by premises 1 and 2. Hence, the property of truth provides us with real insight into mathematics in a way that deflationists cannot explain.
Deflationists could respond to this argument (as have Field 2001 and Azzouni 2008) by claiming that the device of using the word ‘true’ enables us to express more than we could express without it, even though there is no real property of truth. Consider especially premise 2 of the argument above. Deflationists should claim that we can assert premise 2, even though the content of premise 2 goes beyond our present mathematical knowledge, because we know in advance that however knowledgeable we become about mathematics in the future, we will never assert two contradictory claims about the numbers. We are in effect committing ourselves (and all future mathematicians) to a certain general policy: never to assert both of a pair of contradictory statements.
The problem for deflationists is explaining how it is reasonable for us to commit ourselves in advance to such an open-ended policy. Shouldn't we have to consider each contradictory pair on a case-by-case basis, if deflationism is true? This raises an even more fundamental problem for deflationists of explaining how we know the fundamental laws of logic, like the law of non-contradiction. The logical pioneer Gottlob Frege classified these fundamental laws as “laws of truth,” and with apparently good reason. It is because we grasp something about the property of truth (and falsity) that we can say with confidence that no proposition whatsoever could be both truth and false. In light of that confidence, we can sensibly affirm the law of non-contradiction as a general law without ever having to consider the specific cases to which we apply it. Deflationists lack any similar story about what grounds our knowledge of the absolute generality of classical logic. They can always posit that such knowledge is simply constitutive of rationality, but such a stipulation counts against the simplicity of their theory.
We can appeal here to a principle known as ‘Ockham's Razor’, after the English scholastic philosopher William of Ockham. Ockham's Razor directs us to prefer the simplest theory consistent with the known facts. One way in which a theory can be simpler is in positing fewer basic postulates of reason. This is the first corollary of Ockham's Razor:
Principle of Methodology (PMeth) 1 Ockham's Razor. Other things being equal, adopt the simplest theory.
PMeth 1.1 First Coro
llary of Ockham's Razor: Minimizing Rational Postulates. Other things being equal, prefer the theory that posits the fewest primitive, underivable postulates of reason.
2.4 Truthmaker Maximalism
We have seen some reason to believe that there are truthmakers, so let's suppose that there are truthmakers. (We worry about the reasons, and consider an alternative to classical truthmakers, in Section 2.4.1.) The simplest, most natural understanding of Truthmaker Theory is the view that every truth has a truthmaker. This is ‘Truthmaker Maximalism’ (sometimes just ‘Maximalism’):
2.1T.1 Truthmaker Maximalism. Every truth has a classical truthmaker.
Truthmaker Maximalism possesses some impressive theoretical virtues. In particular, it has great simplicity and explanatory power. Given Maximalism, no question arises about how it is that any truth is true, and Maximalism accounts for each truth in a uniform, straightforward manner. We will need to keep these virtues in mind as we consider alternatives to Maximalism.
2.4.1 Fundamentality and logically complex propositions
The first worry for Maximalism stems from the intuition that there must be a set of fundamental truths upon which all other truths depend. Take, for example, the sentence ‘The cat is on the mat, and THP is sitting’. According to Maximalism, we must believe there is a truthmaker for this sentence over and above the truthmakers for ‘The cat is on the mat’ and ‘THP is sitting’. This is counter-intuitive. Shouldn't the existence of the truthmakers for ‘The cat is on the mat’ and ‘THP is sitting’ be enough? The conjunctive truth just seems less fundamental than the two simpler, subject-predicate truths, a seeming that is emphasized when we realize that we don't seem to need another truthmaker for it. Why go in for the extra truthmaker, when the two we had are already sufficient?
However, there is a relatively simple change to Truthmaker Maximalism that will take care of this problem: require that, for every true proposition, there is either one thing that makes it true or there are some things that jointly make it true. In the case of a conjunction like ‘The cat is on the mat, and THP is sitting’, we can suppose that there are two facts that jointly make the conjunction true without having to suppose that there is a single, fundamental conjunctive fact.
What about negations? Suppose that Fido is not a cat or is not gray. What would the truthmakers for (13) or (14) have to be like?
(13) Fido is not a cat.
(14) Fido is not gray.
Given Maximalism, even of the modified variety, one would apparently need metaphysically fundamental negative truthmakers. Raphael Demos (1917) suggested that these could be made true by positive facts about Fido, facts that are incompatible with Fido's being a cat or being gray. Fido's being a dog or Fido's being white seem to fit the bill nicely. However, there are two problems with this suggestion. First, as Bertrand Russell pointed out (Russell 1918–1919, 213–215), it is not clear that we can make sense of the incompatibility relation without making use of purely negative truthmakers. Second, there are some negative predications that seem to be pure privations, in the sense that they don't require the thing to have any positive property at all. Consider, for example, (15) and (16):
(15) John is not thinking of anything right now.
(16) Mary does not remember Paris.
John does not have to be doing anything in order not to be thinking. Similarly, there doesn't have to be any relevant, positive state of affairs involving Mary's mind for it to be true that she simply doesn't remember Paris. Thus, negative propositions seem to require special, negative truthmakers. This is a serious enough problem to deserve its own section, which we will move to next.
2.4.2 The problem of negative existentials
Negative existentials, such as ‘There are no unicorns’ or ‘There are no golden mountains’, are an especially serious version of the negativity problem for Truthmaker Maximalism. (This problem involves as well universal statements, like ‘Obama is the only president of the United States’. This sentence is equivalent to, ‘There does not exist any president of the United States other than Obama’.) A truthmaker for such a sentence must be something that, by its very nature, excluded the possibility of adding a unicorn (or a golden mountain) to the world. In addition, it would have to be something that, as a matter of metaphysical necessity, had to exist whenever there are no unicorns (or golden mountains). Positing such truthmakers involves populating our theory with a large number of brute metaphysical necessities connecting separate things. The existence of the truthmaker for ‘unicorns do not exist’ somehow excludes the existence of any unicorn, and the absence of truthmakers for the existence of unicorns somehow entails the existence of the truthmaker of the negative existential claim. But Ockham's Razor demands that, other things being equal, we should minimize the class of brute necessities that we posit; this is the second corollary of Ockham's Razor:
PMeth 1.2 Second Corollary of Ockham's Razor. Other things being equal, adopt the theory with the fewest brute, inexplicable impossibilities and necessities.
Truthmaker Maximalism requires many such brute, inexplicable impossibilities and necessities in order to account for the truth of negative existentials. These necessary connections between positive and negative truthmakers are the sort of things that a simple and elegant metaphysical theory must minimize.
Further, Maximalism entails that at least one contingent thing exists. For each possible but non-actual contingent being, there must exist a truthmaker for the claim that that being does not exist. Here is an example. Let ‘Winnie’ name a particular unicorn that might have existed. The sentence ‘Winnie doesn't exist’ is true, since Winnie doesn't exist. According to Maximalism, there exists a truthmaker that makes this sentence true. Call this truthmaker, ‘Un-Winnie’. Un-Winnie is a contingent thing. If Winnie had existed, the sentence ‘Winnie does not exist’ would have been false. But that sentence can't be false if Un-Winnie exists, since Un-Winnie necessitates the truth of that sentence, since Un-Winnie is that sentence's truthmaker. Thus it is possible that Un-Winnie fails to exist, and this is because Winnie might have existed. So Un-Winnie is a contingent being. It follows, then, that for every contingent being that fails to exist, there exists a truthmaker for the claim that that very contingent being fails to exist. These truthmakers are themselves contingent. Thus, at least one contingent being must exist. If Winnie doesn't exist, then Un-Winnie does, and vice versa. Even God couldn't create a world devoid of contingent beings, on the assumption that Maximalism is true.
On Maximalism, one cannot create a new possibility simply by thinning down an old one. If one deletes an entity, one must simultaneously add a truthmaker that makes it true that the deleted entity doesn't exist. This seems implausible. It should be possible to thin out the population of entities without being forced to introduce new ones in the process. Putting these together, Truthmaker Maximalism entails that there is some number N such that the number of contingent beings that exists is necessarily N. Implausible, indeed.
Another worry is that Maximalism requires a truthmaker for every universal generalization. Consider (17):
(17) Every living organism is terrestrial.
Suppose that (17) is true and that every living organism lives on the Earth. What would the truthmaker for (17) have to be like? At the very least, the truthmaker for (17) would have to include a part that is a truthmaker for proposition (17x), for every existing thing x:
(17x) Either x is not a living organism, or x is terrestrial (or both).
However, even this massive truthmaker (call it ‘Max’) is not sufficient to be a truthmaker for (17). The existence of Max is consistent with the existence of a new entity, one that does not in fact exist at all (call it ET), and that is both alive and non-terrestrial. Max determines that all the things that actually exist are either terrestrial or not alive, but it does not exclude the existence of additional entities, entities that could but do not in fact exist. Since Max does not exclude such “new” entities, it is compatible with a world th
at is just like this one, except that some new entities exist that are living extra-terrestrials.
Thus, we have to add to Max what David Armstrong calls “the totality fact,” a truthmaker that guarantees that nothing exists except the things that actually exist. Suppose the set T contains every actually existing thing. Then the totality fact is the fact that nothing exists except what is in T. The totality fact “says,” in effect, “That's all, folks. Nothing but the members of T.” However, the totality fact is quite a strange entity. As we've seen with negative facts, to believe in the totality fact we would have to believe in brute necessities. The existence of totality would have to be metaphysically incompatible with the existence of anything outside of T.
In addition, (as Armstrong admits) it is hard to see how we could ever come to know the totality fact in any detail. Only an omniscient being (like God) could possibly know the set T, the set containing exactly the things that exist. However, if we do not know the totality fact, how could we know any universal generalization, even one as simple and everyday as (18)?
(18) All ravens are black.
The truthmaker for (18) will also include the totality fact, which is needed to exclude the existence of any “new” entities that are non-black ravens. If we can't know the totality fact, we can't know the truthmaker of (18), which seems a problematic result. We could put the problem in the form of a dilemma: either there is just One Big Totality Fact for all of reality, or there are many, merely local totality facts, like the fact that S contains every raven, or that S′ contains every mammal in this building. If there were only One Big Totality Fact, then every bit of knowledge that we have of negative existential facts would have to involve some familiarity with this Big Fact, which seems wildly implausible. On the other horn of the dilemma, if we suppose that there are many localized totality facts, then knowledge of negativities is unproblematic, but we must posit a huge number of brute necessary connections between the different totality facts. For example, if there is the fact that S contains every raven, and another fact that T contains every existing thing, then it is necessary that S be a subset of T. Moreover, if E is an atomic fact, for example, the fact that some particular raven, Edgar, is a raven, then E must (of necessity) be a part of S (the actual totality fact for ravens). Other things being equal, we should try to minimize the class of necessary connections (PMeth 1.2). Finally, as Merricks has pointed out, truths like (18) do not seem to be about the positive character of the whole universe. (18) is not about how many quasars there are, for example. Yet the totality fact would include every detail about every part of the universe.