1. Introduction
What are truthmakers for negative truths? This is the central topic of this essay. It seems that propositions are made true by the following things: particulars, properties, relations, particulars having properties, or the relations holding among particulars.Foot note 2 (Let us call the last two things, particulars having properties and particulars being related to one another facts or states of affairs.) This is easy to see in the case of positive propositions, such as <Smith has blond hair>.Foot note 3 What makes this proposition true, or the truthmaker for this proposition, is Smith's (the particular's) having the property of having blond hair.Foot note 4 However, the search for truthmakers for true negative propositions has proved to be difficult. For instance, take the truths <Unicorns do not exist> and <Smith does not have dark hair>. What could be the truthmakers for these truths?
I shall present and discuss the answers to this problem offered by Bertrand Russell, Raphael Demos, and D. M. Armstrong. Then, I shall present my solution which involves the consideration of the world in which nothing exists. After that, I shall briefly talk about the implications of my solution on truthmaker theory and possibilities. Before going any further, however, I have first to present so-called entailment principle which will often be used in what follows.
2. Entailment Principle
When a proposition <p> entails another proposition <q>, we have the feeling that either of the following two things is going on. The content of <q> is identical to, or a part of, the content of <p>; the proposition <q> expresses the reality which is identical to, or a proper part of, the reality that the proposition <p> expresses. For example, <Smith is blond and Smith is 6 feet tall> entails that <Smith is blond> because the content of the latter proposition is merely a part of the content of the former. We think that the proposition <The fork is to the right of the spoon> and the proposition <The spoon is to the left of the fork> have different contents, but we think that one entails the other because these two propositions express the same reality.
It seems, then, that when <p> entails <q>, the truthmaker for <p> can be a truthmaker for <q>. Firstly, if the contents of <p> and <q> are exactly the same, what makes one true should also make the other proposition true. Secondly, if the content of <q> is only a part of the content of <p>, then whatever makes <p> true should also make <q> true. This is clear from the example used in the above. If some truthmaker makes <Smith is blond and Smith is 6 foot tall> true, then the same truthmaker should also make <Smith is blond> true. Thirdly, it seems trivial to say that <p> and <q> have the same truthmaker if they both express the same reality in different ways. For instance, <The fork is to the right of the spoon> and <The spoon is to the left of the fork> have the same truthmaker. Finally, if <q> expresses only a part of the reality that <p> expresses, it seems that the truthmaker for <p> can also be a truthmaker for <q>. Suppose there is a reality such that three objects, a knife, a fork and a spoon, are lined up from right to left in that order. Then, this reality is the truthmaker for the proposition <The knife is to the right of the fork and the fork is to the right of the spoon> as well as the proposition <The spoon is to the left of the knife>.
In the above we seem to have justified the entailment principle, which states: the truthmaker, T, which makes some proposition <p> true, also makes another proposition <q> true if <p> entails <q>. Following Armstrong's symbolization more or less, the principle can be represented as:
1. T → <p>
2. <p> entails <q>
3. T → <q>Foot note 5,Foot note 6
3. Negative Facts
Russell thought that negative truths are made true by the special type of facts, negative facts, but he did not investigate the structure of these special facts (1985, sec. III). So, we shall take up where Russell left off and try to come up with the logical form of negative facts here. In doing this, we will assume two things: that the only elements that constitute the world are particulars, properties, relations, and facts; that negative facts are distinct from positive facts which make positive truths true.
We might think that what makes negative facts distinct from positive facts is that they have negative particulars in their constituents, where negative particulars are the counterparts of regular particulars such that the presence of them in the facts make negative propositions true. So, the logical form of negative facts is negative particulars having positive properties or being related to one another by positive relations. To illustrate, suppose that <Smith does not have dark hair> is true. The truthmaker for this negative truth would be the fact that the negative Smith has dark hair.
Alternatively, we might try to introduce into our ontology negative properties and relations (e.g., being not-red and being not-left-of). By doing so, the logical form of negative facts would be particulars having negative properties or being related by negative relations. So, the truthmaker for the negative truth that Smith does not have dark hair would be the fact that Smith's hair has the property of being not-dark.
Objections to these two suggestions for the logical form of negative facts are as follows. First of all, postulating negative particulars or negative properties and relations would be ontologically uneconomical. Assuming that we ought to abide by Occam's razor, we should try to find another way to explain negative truths. Secondly, the employment of negative particulars or properties (and relations) seems like an ad-hoc solution; the only motivation that we have to postulate them is to explain negative truths.Foot note 7 Thirdly, both of the above suggestions for the logical form of negative facts cannot handle negative existential truths. For example, what would be the truthmaker for <Unicorns do not exist>? Employing the first approach to negative facts, we seem to need yet another type of particulars, non-existing particulars. This seems like a bad idea with regard to ontological economy, not to mention the strangeness of the notion of existing non-existing particulars. The second approach would be even worse off, under which view, the truthmaker for this negative truth would be each unicorn's having the property of being not-existing. Obviously, this is problematic. Unicorns do not exist; they cannot have any property, let alone the property of being not-existing.Foot note 8
After seeing the ontological cost incurred by introducing negative particulars or negative properties (and relations), we might try to economize by letting negative facts have a special relation, the not-relation, where the not-relation is a relation such that facts including this relation serve to be truthmakers for negative truths. Using the not-relation, we can think of negative facts as having the logical form, the not-relation relating positive particulars and positive properties (or relations). So, the truthmaker for the negative truth that Smith does not have dark hair is the fact of the not-relation relating Smith and the property of having dark hair.
This solution seems to be able to handle negative existential truths, for which the previous two suggested forms of negative facts are inadequate. We can say that the truthmaker for the negative truth that unicorns do not exist is the negative fact, the not-relation's holding between horsy things and the property of having a horn because this negative fact makes it true that horsy things do not have a horn, and this latter proposition seems to preclude the possibility of the existence of unicorns; thus, using the entailment principle, we can say that the truthmaker for <Unicorns do not exist> is the not-relation's holding between horsy things and the property of having a horn.
Nevertheless, the not-relation seems to be another artificial invention to explain negative truths, and for this reason, we should not use it to solve the problem of truthmakers for negative truths.
At this point, I am running out of suggestions for the logical form of negative facts. So, it is time for us to leave negative facts behind and move onto the next solution for the problem of negative truths, the incompatibility solution.
4. The Incompatibility Solutions
4.1. Demos's Incompatibilism
Raphael Demos's solution to the problem of truthmakers for negative truths, known as the incompatibility solution, is, very roughly, that negative propositions are just another way of expressing certain positive propositions, and since positive propositions have their own truthmakers, these truthmakers are the truthmakers for the negative propositions (Demos 1917, 188-196). Somewhat more precisely, a negative proposition <~p> means <Some proposition <q>, which is incompatible with <p>, is true>, where <~p> is a description of <q> in the sense that <~p> does not directly refer to any particular state of affairs. We may describe <q> by <~p> because <q> is related to a number of propositions by the incompatibility relation and one of the propositions that <q> is so related is <p>. Furthermore, since <~p> is another way of saying <q>, the truthmaker for <~p> is the truthmaker for <q>.
For instance, take the negative proposition, <Smith does not have dark hair>. This proposition means <Some proposition, which is incompatible with <Smith has dark hair>, is true>, and this proposition is a description of the proposition <Smith has blond hair>. When we say <Smith does not have dark hair>, what is essentially happening is that instead of directly asserting the proposition <Smith has blond hair>, we choose to describe it using the circumstance that <Smith has blond hair> is incompatible with <Smith has dark hair>. So, we might say <Some proposition, which is incompatible with <Smith has dark hair>, is true>, but as this proposition is rather long, we abbreviate it by saying <Smith does not have dark hair>. In this sense, the proposition <Smith does not have dark hair> is merely a description of <Smith has blond hair>, and the truthmaker that makes <Smith has blond hair> true, namely, Smith's having blond hair, also makes <Smith does not have dark hair> true; after all, the description is merely a different way of expressing the same state of affairs.
We may explain why the truthmaker for <Smith has blond hair> is also the truthmaker for <Smith does not have dark hair> by using the entailment. <Smith has blond hair> has Smith's having blond hair as its truthmaker. <Smith has blond hair> implies <Some proposition, which is incompatible with <Smith has dark hair>, is true>. Furthermore, <Some proposition, which is incompatible with <Smith has dark hair>, is true> means and hence implies <Smith does not have dark hair>. By the entailment principle, the fact of Smith's having blond hair makes it true that <Smith does not have dark hair>. We may write the argument in the following scheme.
1. T → <Smith has blond hair>.
2. <Smith has blond hair> implies <Some proposition, which is incompatible with <Smith has dark hair>, is true>.
3. <Some proposition, which is incompatible with <Smith has dark hair>, is true> means, and hence implies <Smith does not have dark hair>.
4. T → <Smith does not have dark hair>.
The strength of this solution is its intuitive appeal. We know that Smith does not have dark hair by looking at Smith's having blond hair. Then, it seems natural to say that it is Smith's having blond hair that is the truthmaker for the negative truth that Smith does not have dark hair. However, this solution is not objection-proof either.
4.2. A Possible Objection from the Negative Facts Camp
Here is a possible objection from the negative fact camp.Foot note 9 The proposition <p and q> is true when the two constituent propositions, <p> and <q>, are both true. Further, the condition for the truth of these two propositions is that they each have their respective truthmakers (i.e., T1 for <p> and T2 for <q>). Now, consider, <~p>. <~p> is true when <Some proposition q which is incompatible with p> is true, as the latter is the meaning of the former. <Some proposition q, which is incompatible with p, is true> can be symbolized as <q and p|q>.Foot note 10 Regarding <q and p|q>, we can say the following things: <q and p|q> is true when <q>and <p|q> are both true, the condition for <q> and <p|q> being both true is that <q> is true and <p> is false, and what makes <q> true and <p> false is the truthmaker for <q> and the falsemaker for <p>. Since the falsemaker for <p> is the truthmaker for <~p>, the truthmaker for <Some q, which is incompatible with p, is true> is actually the mereological sum of the truthmaker for <q> and the truthmaker for <~p>. It follows that the truthmaker for <q> by itself is not sufficient to make <Some q, which is incompatible with p, is true> true. Substituting <p> and <q> with <Smith has dark hair> and <Smith has blond hair> respectively, we have to say that the truthmaker for <Smith has blond hair> by itself is not a truthmaker for <Some proposition, which is incompatible with <Smith has dark hair>, is true>. Reminding ourselves that this last proposition is equivalent to <Smith does not have dark hair>, we have to say that Smith's having blond hair, which is the truthmaker for <Smith has blond hair>, is not a sufficient truthmaker for <Smith does not have dark hair>.
Those who endorse negative facts as the solution to the problem of negative truths would go on to say the following. As it is explained in the previous paragraph, the truthmaker for <Some q, which is incompatible with p, is true> is the mereological sum of the respective truthmakers for <q> and for <~p>. However, <Some q, which is incompatible with p, is true> means <~p>. It follows that the truthmaker for <~p> is the mereological sum of the respective truthmakers for <q> and for <~p>. This is clearly intolerable. For this reason, the incompatibility solution should be abandoned and the truthmaker for <~p> should be thought to be simply the negative fact for <~p>.
Our response to this objection on behalf of the incompatibilits is that the objection seems to make an unwarranted assumption. The objection unwarrantedly assumes that the condition for the truth of <q and p|q> is that <q> is true and <p> is false, and it concludes that what makes <q and p|q> true is the sum of the truthmaker for <q> and the falsemaker for <p> (i.e. negative fact that <~p>). However, the incompatibilists can just say that the condition for <q and p|q>'s being true is that <q> is true and <p|q> is true, and that the falsehood of <p> is a consequence of the truth of <q> and <p|q>. Thus, from the incompatibilists' point of view, what makes <q and p|q> true is the sum of the truthmaker for <q> and the truthmaker for <p|q>, rather than the sum of the truthmaker for <q> and the falsemaker for <p>.
From what has been said in response to the objection we have considered, however, we ought to fine-tune our view of the truthmaker for the negative truth <Smith does not have dark hair>. Initially, we contended that the truthmaker for this negative truth was Smith's having blond hair. That is, we thought that Smith's having blond hair made it true that <Some proposition, which is incompatible with <Smith has blond hair>, is true>. However, we have seen that the truthmaker for <q and p|q> (i.e., <Some proposition, which is incompatible with <Smith has dark hair>, is true>) under the present theory is the sum of the truthmaker for <q> (i.e., <Smith has blond hair>) and the truthmaker for <p|q> (i.e., <<Smith has dark hair> is incompatible with <Smith has blond hair>>). Therefore, we must say that the truthmaker for the above negative truth is the mereological sum of the states of affairs: Smith's having blond hair + the incompatibility holding between <Smith has blond hair> and <Smith has dark hair>.
To summarize, <Smith has blond hair> and <<Smith has blond hair> and <Smith has dark hair> are incompatible with each other> have the respective truthmakers: namely, Smith's having blond hair and the incompatibility relation's holding between <Smith has blond hair> and <Smith has dark hair>. Now, the two propositions, <Smith has blond hair> and <<Smith has blond hair> and <Smith has dark hair> are incompatible with each other>, together imply <Some proposition, which is incompatible with <Smith has dark hair>, is true>, which is the meaning of <Smith does not have dark hair>. Therefore, by the entailment principle, the respective truthmakers for these two propositions together make <Smith does not have dark hair> true. Using the formal scheme, we may represent the revised argument as:
1. T1: Smith's having blond hair → <q>: <Smith has blond hair>.
2. T2: The incompatibility holding between <Smith has blond hair> and <Smith has dark hair> → <p|q>: <<Smith has blond hair> is incompatible with <Smith has dark hair>>.
3. <q> and<p|q> together imply <Some proposition, which is incompatible with <Smith has dark hair>, is true>.
4. <Some proposition, which is incompatible with <Smith has dark hair>, is true> means and hence implies <Smith does not have dark hair>.
5. T1+T2 → <Smith does not have dark hair>.
Some may wonder what evidence we have for saying that the incompatibility relation exists. The evidence is that we cannot entertain the incompatible propositions, say, <Smith has blond hair> and <Smith has dark hair>, to be true at the same time. Because of the influence Immanuel Kant had upon me, I would like to say that the incompatibility among propositions has its ground in the mental structure of the mind. Incidentally, this view on the incompatibility has a nice consequence. It allows us to say that the incompatibility relation's holding among certain propositions supervenes upon the structure of the mind; consequently, the incompatibility relation as well as the facts about certain propositions being incompatible with each other are ontologically cost free.
4.3. Property Incompatibilism
So far, we have discussed Demos's incompatibility solution, which utilizes the idea that the incompatibility holds among propositions. At this point, I would like to introduce another type of incompatibility solution, which may be called property incompatibilism. This solution approaches the problem of negative truths in the same way as Demos's incompatibility solution does. The only difference is that instead of saying that some propositions are incompatible with each other, property incompatibilism states that some properties are incompatible with each other.
According to property incompatibilism, since the proposition <Smith has blond hair> (which is equivalent to <Smith has the property of having blond hair>) together with the proposition <The property of having blond hair is incompatible with the property of having dark hair> imply that <Some proposition, which is incompatible with <Smith has dark hair>, is true>, by the entailment principle, the truthmaker for the last proposition is the mereological sum of the respective truthmakers of the first two propositions. Furthermore, since <Smith does not have dark hair> means <Some proposition, which is incompatible with <Smith has dark hair>, is true>, the two propositions have the same truthmaker. Hence, the truthmaker for the negative truth <Smith does not have dark hair> is the mereological sum of the respective truthmakers for <Smith has blond hair> and <The property of having blond hair is incompatible with the property of having dark hair>, namely, Smith's having blond hair + the incompatibility relation's holding between the property of having blond hair and the property of having dark hair. Let us represent what has been said by using the formal scheme.
1. T1: Smith's having blond hair →<q>: <Smith has blond hair>.
2. T2: the incompatibility relation's holding between the property of having blond hair and the property of having dark hair → <p|q>: <There is an incompatible relation's holding between the property of having blonde hair and the property of having dark hair>.
3. <q> and <p|q> together imply <Some proposition, which is incompatible with <Smith has dark hair> is true>.
4. <Some proposition, which is incompatible with <Smith has dark hair>, is true> means and hence implies <Smith does not have dark hair>.
5. T1+T2 → <Smith does not have dark hair>.
4.4. Remarks on the Incompatibility Solutions
Prima facie, it seems that incompatibilism in general introduces an extra relation (i.e. the incompatibility relation) into our ontology, and if there is a theory for truthmakers for negative truths that achieves better ontological efficiency, by Occam's razor, the incompatibility solutions should be given up. However, this is, in fact, not the case.
As we mentioned earlier, in the case of Demos's incompatibilism, the incompatibility relation among propositions is not an ontological addition to the world if we think that the incompatibility among propositions have its basis in the structure of our mind. As for property incompatibilism, since it seems that the incompatibility relation is an internal relation holding among properties, this incompatibility relation (among properties) does not count as an ontological addition to the world.
What has been said here seems to suggest that incompatibilism is a better solution than the negative facts solution. For while the incompatibility solutions are ontologically cost free, the negative facts solution has to postulate some extra entity into our ontology; for it seems that the negative facts solution has to rely on some property or relation that makes negative facts negative facts (e.g., the not-relation).
4.4. A Difficulty with the Incompatibility Solutions
There is a proposition which it seems the incompatibilists cannot handle. It is true that some region of space is empty. Let us call such a region R. Then, the proposition <There is nothing in R> is true. Using Demos's incompatibilism, it is not clear how this proposition can be dealt with. We might try to say that this proposition is an abbreviation for <Some proposition, which is incompatible with <There is something in R>, is true>. Yet, the proposition that is incompatible with <There is something in R> seems to be the original proposition <There is nothing in R>. Thus, Demos's analysis of negative proposition cannot give the analysis of this proposition which yields a positive truth whose truthmaker is readily available.
Property incompatibilism does not fare well in finding the truthmaker for <There is nothing in R> either. The difficulty seems to arise from the circumstance that we cannot tell what property is involved in this proposition. If we cannot find the property involved in this proposition, then property incompatibilism, which requires the properties that are incompatible with each other in order to find truthmakers, is quite powerless.
5. Armstrong's Solution
5.1. General States of Affairs As Truthmakers for Negative Truths
The solution offered by D. M. Armstrong is to use general facts as truthmakers for negative truths (2004, 54-60). General facts can be used for truthmakers for negative truths because general truths imply negative truths. Suppose that a certain particular a has the properties, F and G, only. Then, the proposition <a is not H> is true, and this is true because in the list of the all the properties of a, H does not show up. Put differently, <F and G are the only properties of a> implies <H is not a property of a>. Thus, by the entailment principle, the truthmaker for the general truth <F and G are the only properties of a> is the truthmaker for the negative truth <a is not H>. For Armstrong, it turns out that the search for truthmakers for negative truths can be completed by finding truthmakers for general truths.
Now, what is the truthmaker for a general truth of the form, <X, Y, Z, etc. are the only F's>? It has been said that each member of X, Y, Z, etc.'s being F cannot be the truthmaker for this general truth. This seems to be correct if we consider the circumstance that the fact of Smith's having blond hair does not make it the case that Smith is the only one that has blond hair. As a result, Russell proposed that truthmakers for general truths are general facts; however, Russell did not know the logical form of general facts (1985, 104). Following Russell, Armstrong also thinks that there are such things as general states of affairs, and he proposed the logical form of general states of affairs to be the totality relation's holding between the mereological sum of the states of affairs consisting of X's being F, Y's being F, Z's being F, etc., and the property of being F (2004, 54, 72-75). So, in the case of the example mentioned above, the truthmaker for the general truth that F and G are the only properties of a is the state of affairs, the totality relation's holding between the mereological sum of Fa + Ga and the property of being a's property. (And this state of affair is the truthmaker for the negative truth that <a is not H>.)
Let me illustrate what has been said using our old friend, blond-haired Smith. The truthmaker for the negative truth that Smith does not have dark hair is the totality relation holding between the sum of the states of affairs involving Smith and the property of being a state of affairs involving Smith. This totality makes it true that a certain list of properties is all the properties of Smith, which implies the truth that the property of having dark hair is not in that list. That is, it implies that Smith does not have dark hair. Thus, by using the entailment principle, we can conclude that the totality relation's holding between the sum of the states of affairs involving Smith and the property of being a state of affairs involving Smith is the truthmaker for the negative truth that Smith does not have dark hair.
5.2. Remarks about Armstrong's Solution
It may be said that Armstrong's position runs into a trouble with the idea that there are no necessary connections among distinct states of affairs. To see why, consider the following case. As of 2005, Jean Paul Sartre and Le Duc Tho are the only people who have declined the Nobel prize. The truthmaker for the truth <Sartre and Tho are the only people who declined the Nobel prize> is the state of affairs, the totality relation's holding between the mereological sum of Sartre's declining the Nobel prize + Tho's declining the Nobel prize and the property of being the people who have declined the Nobel prize. As an abbreviation, we might symbolize this as Tot (S + Th, D). Now, John F. Nash, Jr. accepted the Nobel prize for economics so that there is a state of affairs, Nash's accepting the Nobel prize. This state of affairs seems to have some connection with Tot (S + Th, D); for the existence of Tot (S + Th, D) makes it somewhat necessary that Nash accepted the Nobel prize. I say it is only somewhat necessary because it is necessary given that there is another state of affairs that Nash was offered the Nobel prize; for if Nash had never existed or if he had not studied economics, then Tot (S + Th, D) would have existed without Nash's accepting the Nobel prize. Still, among the three states of affairs, Nash's being offered the Nobel prize, Nash's accepting the Nobel prize, and Tot (S + Th, D), a necessary connection seems to exist. This may be an unwelcome consequence of Armstrong's position, if one holds that there is no necessary connection among distinct states of affairs.
Perhaps, necessary connections among states of affairs are not all that bad. Some people think that there is a necessary connection between a cause and its effect.Foot note 11 At any rate, however, Tot (S + Th, D) is not a fact which is needed in our ontology. The proposition that Sartre and Tho are the only people who declined the Nobel prize has another truthmaker, the mereological sum of the following three states of affairs: (i) Sartre's declining the Nobel prize, (ii) Tho's declining the Nobel prize, and (iii) the totality relation's holding between the sum of the states of affairs which involve everyone minus Sartre and Tho and the property of being the states of affairs of everyone minus Sartre and Tho. For <Sartre declined his Nobel prize>, <Tho declined his Nobel prize>, and <Everyone else does not decline the Nobel prize> together amounts to <Sartre and Tho are the only people who have declined the Nobel prize>. Let us abbreviate this sum as S + Th + Tot (E, SofE). Now, Tot (S + Th, D), if it exists, seems to be a consequence of and supervene upon S + Th + Tot (E, SofE). Thus, for the purpose of coming up with the truthmaker for the truth that Sartre and Tho are the only people who have declined the Nobel prize, Tot (S + Th, D) is not needed. So, it seems wise not to postulate this state of affairs; for then we can avoid having the necessary connection between the distinct states of affairs.
Two things can be said against Armstrong's solution. First, the totality relation is quite artificial. Second, since my solution to the problem of negative truths, to which we now turn, can deal with negative truths as well as general truths without postulating the totality relation, my solution should be preferred over Armstrong's.
6. The World In Which Nothing Exists
6.1. My Hypotheses about Propositions
In order to talk about my solution for the problem for negative truths, I have to first present my hypothesis about propositions. My claim is that all propositions can be understood in terms of the existence of something. To begin with, all positive propositions are equivalent, in meaning, to propositions of the form <X exists, Y exists, Z exits, etc.>. For instance, the proposition <Smith has blond hair> is equivalent to <Smith exists, the property of having blond hair exists, and the state of affairs of Smith's having blond hair exists>.
Assuming that all positive propositions can be understood in terms of the existence of something, `not' denies the existence of at least one thing, and negative propositions have different meanings depending upon which existence `not' denies. For instance, `Not' in the proposition <Smith does not have dark hair> denies either the existence of Smith or the property of having dark hair or the state of affairs of Smith's having dark hair or all of the above. This seems clear if we consider the use of the negative proposition <An invisible man is not standing behind Smith>.Foot note 12 Some people would assert this proposition because they think that there is no such thing as an invisible man. For them, `not' is denying the existence of an invisible man and hence the state of affairs of an invisible man standing behind Smith. That is, the proposition is equivalent to <There is no invisible man, there is Smith, there is the relation of X standing behind Y, and there is no state of affairs of an invisible man's standing behind Smith>. (Notice here, the negative proposition implies the existence of Smith and of the standing-behind relation.) However, people who believe in invisible men might still assert <An invisible man is not standing behind Smith> if they think that an invisible man is not standing right behind Smith. In this case, `not' is denying the existence of the state of affairs, an invisible man's standing behind Smith. So, <An invisible man is not standing behind Smith> is equivalent to <There is an invisible man, there is Smith, there is the relation of X standing behind Y, but there is no state of affairs of an invisible man's standing behind Smith>. As we can see from these examples, `not' denies the existence of at least one thing. Moreover, the consideration of the above proposition makes it clear that negative propositions, though superficially having the same appearance, have more than one meaning, and that the meaning depends on the scope of `not.'
I must explain what I mean by «the scope of `not'». I say that the scope of `not' is the entire proposition, when `not' modifies the existence of all things that the proposition implies. If the scope of `not' for <Smith does not have dark hair> is the entire proposition, then the proposition is equivalent to <There is no Smith, no property of having dark hair, no state of affairs of Smith's having dark hair>. Accordingly, «the scope of `not' is partial» should be understood to mean that `not' does not modify the existence of all things that the proposition implies. Hence, <Smith exists, the property of having dark hair exists, but the fact of Smith's having dark hair does not exist> is an example of the propositions which are equivalent to <Smith does not have dark hair> in which `not' modifies only a part of the proposition.
6.2. The Wiwne Solution
Now that we have gone through my contention about propositions, we are ready for my thesis. No truthmaker is required for negative truths if the scope of `not' is the entire proposition, but if the scope of `not' is only a part of the proposition, their truthmakers are entities that these negative propositions imply exist. To see why, imagine the world in which nothing exists. I mean absolutely nothing, no particulars, no properties, no relations, and no facts. Call this world Wiwne, for short. About this world, we can truthfully say that there is nothing. Hence, the proposition about Wiwne that <Nothing exists> is true, and it is true despite the reality that there is nothing that makes this proposition trueFoot note 13. This consideration seems to show, at least, that for the truth of the proposition that <Nothing exists>, no truthmaker is needed. But about Wiwne, we can similarly assert such propositions as <No human beings exist>, <Unicorns do not exist>, <The earth does not exist>, etc., and those propositions seem to be true, again, despite the lack of things that make these propositions true. For this reason, we may conclude that a truthmaker is not required for negative existential propositions when we make an assertion about the world in which nothing exists.
We can further make the following observations about Wiwne: All positive propositions about Wiwne are false though no entity is making them false; the negative propositions about Wiwne are true if the scope of `not' is the entire proposition, and their truths do not require any truthmaker; when the scope of `not' is only a part of negative propositions, those negative propositions are false despite the reality that no entity is making them false.
All positive propositions about Wiwne are false. As I have said earlier, all positive propositions are equivalent in meaning to the propositions of the form <There is X, there is Y, there is Z, etc.>. The propositions of such a form about Wiwne are false, because nothing exists in Wiwne. Further, it is clear that no truthmaker is required to make those propositions false. Thus, all positive propositions about Wiwne are false and no entity makes them false.
All negative propositions about Wiwne with the scope of `not' the entire proposition are true. This is because negative propositions, if the scope of `not' is the entire proposition, are equivalent to propositions of the form <There is no X, there is no Y, there is no Z, etc.>. Since nothing exists in Wiwne, the propositions of this form are true, and they are true without any truthmaker. Thus, given that the scope of `not' is the entire proposition, negative propositions about Wiwne are true, although there are no truthmakers for these truths.
Finally, if the scope of `not' is only a part of the proposition, the negative propositions about Wiwne are false; for in that case, negative propositions imply the existence of at least one thing. When the scope of `not' is partial, `not' does not deny at least one existent. So, each of the propositions contains at least one `there is X', because of which the entire proposition implies the existence of at least one thing. Such a proposition is false without a falsemaker because the proposition implies the existence of something in Wiwne.
In order to further elucidate what has been said, an example is in order. Consider the negative proposition about Wiwne that <Smith does not have dark hair> in which `not' modifies the entire proposition. This proposition is equivalent to <There is no Smith, there is no property of having dark hair, and there is no state of affairs of Smith's having dark hair>. This proposition is true, and it is true without a truthmaker because there is nothing in Wiwne to serve as its truthmaker. Intuitively, we can treat any negative proposition about Wiwne with the scope of `not' being the entire proposition in a similar manner. Thus, we can conclude that all negative propositions about Wiwne are true without the need of a truthmaker as long as the scope of `not' is the entire proposition.
Suppose that the scope of `not' is only a part of <Smith does not have dark hair>. Then, the proposition must imply the existence of at least one thing. Hence, for example, the proposition is equivalent to <There is Smith, there is no property of having dark hair, and there is no state of affairs of Smith's having dark hair>. This proposition is false because there is no Smith in Wiwne. Further, since there is nothing in Wiwne, nothing is making it false.
At this point, imagine adding one thing to Wiwne. It seems, then, some of the propositions concerning the newly added entity are true, and the negations of those propositions are false. For instance, if we add Smith to Wiwne, the proposition <Smith exists> is true, and the negative proposition <Smith does not exist> is false. Furthermore, the propositions concerning Smith's internal properties are true and the negations of those propositions are false. For instance, the proposition that <Smith has blond hair> becomes true, and <Smith does not have blond hair> becomes false. (Notice, here, the scope of `not' does not really matter. <Smith does not have blond hair> is false whether the scope of `not' is the entire proposition or only a part of it.) As for the propositions concerning both the newly added entity and the entities that do not exist, the truth-value of those propositions may be true or false, depending on the scope of `not' in the proposition. For example, the proposition <Smith does not have dark hair> is false if the scope of `not' is the entire proposition. If the scope of `not' is the entire proposition <Smith does not have dark hair> is equivalent to <There is no Smith, there is no property of having dark hair, and there is no Smith's having dark hair>. This proposition is false because there is Smith in Wiwne at this point. However, if `not' denies only the existence of the property of having dark hair and the state of affairs of Smith's having dark hair, then the proposition is true; for in that case, <Smith does not have dark hair> is equivalent to <There is Smith, there is no property of having dark hair, and there is no Smith's having dark hair>.
As we have seen above, for some propositions about Wiwne, the change in the truth-values occurs due to the newly added entity, Smith. However, for other propositions, the truth-values are not affected. The truth-values of the propositions that do not concern the newly added entity, Smith, are unchanged. For instance, <Jones wears a hat> is still false because in Wiwne, there is still no Jones, no hat, no wearing relation, and no state of affairs of Jones's wearing a hat. Similarly, consider the negative proposition <Jones does not wear a hat>. The proposition is still true if `not' modifies the entire proposition, again, because there is no Jones, no hat, no relation of wearing, and no Jones's wearing a hat. If the scope of `not' is taken to be a part of the proposition, then the proposition is still false because the proposition imply the existence of at least one thing, may it be Jones, a hat, the wearing relation, or Jones' wearing a hat, while nothing except for Smith (and things concerning him) exists in Wiwne at this point. So, the introduction of Smith does not change the truth-value of the propositions that are not concerned with Smith and his internal properties.
Notice that those propositions whose truth-values are unchanged have the same truth-values by virtue of the same condition as before the introduction of Smith to Wiwne. Some of those propositions are true because the things that those propositions deny to exist do not exist, and the others are false because the things that those propositions imply exist do not exist. This means that the negative truths about Wiwne with only Smith in it are true without truthmakers, granted that their truth-values are unchanged with the introduction of Smith to Wiwne. Also notice that these true negative propositions have this characteristic: the scope of `not' is the entire proposition. Thus, it seems we can say that the negative truths about Wiwne with only Smith in it do not require truthmakers if the scope of `not' is the entire proposition.
As for the true negative propositions whose truth-values have changed from false to true since the addition of Smith, we can make the following two observations. First, their truthmakers are the entities that the propositions imply exist. (In particular, those are Smith or Smith's internal properties.) Second, the scope of `not' is only a part of the proposition. As we have seen before, <Smith does not have dark hair> has become true with the introduction of Smith to Wiwne, if the meaning of it is taken to be <There is Smith, there is no property of having dark hair, and there is no Smith's having dark hair>. The truthmaker for this proposition is just Smith, and the scope of `not' is only partial. Likewise, consider <Jones does not have blond hair> taken to mean <There is no Jones, there is the property of having blond hair, and there is no Jones's having blond hair>. This proposition is false at the initial condition of Wiwne. However, with the introduction of Smith to Wiwne, this proposition becomes true; for due to the circumstance that Smith is a blond haired individual, the property of having blond hair exists in Wiwne after the introduction of Smith to it, and hence <Jones does not have blond hair> (taken in the present sense) is made true by Smith's internal property, the property of having blond hair. Also, obviously, in this example, the scope of `not' in the proposition is only partial. By these examples, we can clearly see that for the negative propositions whose truth-values have changed from false to true by the introduction of Smith to Wiwne, their truthmakers are the entities that the propositions imply exist, and the scope of `not' is only partial. Further, it seems that we may conclude that if the scope of `not' is only a part of the proposition, then the truthmakers for the negative truths about Wiwne with only Smith in it are the entities that these truths imply exist.
Now, let us further suppose that we add another person, Jones, to Wiwne. It seems that we can go through a similar argument that we made above once again, and we can arrive at the same conclusion as before. That is, true negative propositions with the scope of `not' being the entire propositions are true without truthmakers, and truthmakers for true negative propositions in which `not' modifies only a part of the propositions are the things that those propositions imply exist. In fact, it seems that even if we add as many entities as we wish, we can always come to these two results. This is extremely important; for we can think of our actual world to be the world that we obtain by adding to Wiwne all the entities that exist in the actual world. The implication of this view of our actual world is that even in our world, negative truths are true without the need of truthmakers if the scope of `not' is the entire proposition, and if the scope of `not' is only partial, their truthmakers are the entities that the propositions imply exist.
To illustrate the last point, consider the following true negative propositions about the actual world: <Invisible men do not have invisible hair> and <Smith does not have dark hair>. The first proposition can be analyzed into <There is no invisible man, no property of having invisible hair, no state of affairs of invisible men's having invisible hair>. It is clear that no truthmaker is needed for the truth of this proposition and that the scope of `not' is the entire proposition. Next, <Smith does not have dark hair>, if true, is equivalent to <There is Smith, there is the property of having dark hair, and there is no state of affairs of Smith's having dark hair>. The truthmaker for this proposition is the mereological sum of Smith + the property of having dark hair. We can also see that Smith and the property of having dark hair are the entities that the proposition implies to exist, and that the scope of `not' in this proposition is only a part of the proposition.
6.3. General TruthsFoot note 14
The virtue of the above account for negative truths is that it can deal with general truths in a similar fashion without postulating a totality relation. To see how this can be done, suppose that we add Smith to Wiwne, as we did so in the previous section. Then, it seems true that Smith is the only human being that exists in Wiwne, and this is a general truth. Since we only added Smith in Wiwne, the only possible truthmaker for any truth in Wiwne is, either the thin particular, Smith, Smith's internal properties, or the state of affairs consisting of Smith and some of his internal properties. So, the only plausible candidate for the truthmaker for the general truth that Smith is the only human being is just this state of affairs, Smith's being human. Here, the totality relation does not seem to be needed.
Next, we add Jones in addition to Smith. Then, it is true that Smith and Jones are the only human beings in Wiwne, and this time, the truthmaker for this general truth is the sum of the states of affairs, Smith's being human + Jones's being human. Similarly, we add all the human beings that exist in the actual world to Wiwne, then the truthmaker for the general truth that they are all the human beings in Wiwne is the mereological sum of each of those human beings' being human. Furthermore, we may add some non-human entities to Wiwne, but the truthmaker for the general truth that all human beings in Wiwne are the only human beings in Wiwne seems to be the same as before. So, even if we add all the non-human entities in the actual world to Wiwne, the general truth that all the human beings in Wiwne are the only human beings in Wiwne is made true by the mereological sum of each human being's being human. As I have said before, the actual world is the world that we arrive at by adding to Wiwne all the entities that exists in the actual world. Thus, the truthmaker for the general truth about the actual world <a, b, c, etc. are the only human beings in the actual world (where a, b, c, etc. are the names for all the human beings in the world)> is just the mereological sum of each a, b, c, etc.'s being human. In general, contra Russell and Armstrong, the truthmaker for the general truth of the form, <X, Y, Z, and etc. are the only F's> is the mereological sum of each X, Y, Z, and etc's being F.
If this account of general truths is correct, we do not need the totality relation in our ontology. Consequently, in view of ontological economy, the Wiwne account of negative truths should be preferred over Armstrong's account of negative truths which uses the totality relation.
6.4. Truthmaker Maximalism and Truthmaker Necessitarianism
There are two important implications of my account to the truthmaking theory. First, the thesis that every truth has a truthmaker, called Truthmaker Maximalism, is false. This is because negative truths with the scope of not the entire proposition are true without truthmakers. Second, Truthmaker Neccesitarianism, which states that truthmakers necessitate truths, is false. We have concluded that the truthmaker for general truths of the form, <X, Y, Z, etc. are the only F's> is merely the sum of the states of affairs of each X, Y, Z, etc.'s being F. However, the sum of each X, Y, Z, etc.'s being F does not necessitate the truth <X, Y, Z, and etc. are the only Fs>; for there might have been more things that are Fs
6.5. Modalities in Wiwne
Let me briefly talk about modalities in Wiwne. Although nothing exists in Wiwne, intuitively, it seems that something might have existed. So, the modal proposition about Wiwne that something might have existed is true. However, since nothing exists in Wiwne, this truth does not seem to require a truthmaker. Similarly, we can truthfully say that Smith might have existed in Wiwne. Again, the truth of this proposition does not seem to require a truthmaker. Further, consider the proposition <Smith might have had dark hair>. This proposition can be understood as <There might have been Smith, there might have been the property of having dark hair, and there might have been Smith's having dark hair>, and this proposition about Wiwne seems true without a truthmaker. Therefore, the proposition about Wiwne, <Smith might have had dark hair>, is true, and it is true without a truthmaker. These considerations seem to show that modal truths about possibilities are true in Wiwne without truthmakers. Further, this result seems to suggest that possibilities in Wiwne are not dependent upon entities in Wiwne, and this in turn seems to point to the direction that even in the actual world possibilities do not depend upon entities. I am not willing to discuss the truthmakers for necessary propositions here, but I trust that the strategy similar to the one that I have used with respect to possiblities can be used in the case of necessities as well. So, I say that necessities in the actual world do not depend upon the entities in the actual world.
Someone might want to say that the proposition about Wiwne that Smith might have had dark hair is false; for since nothing exists in Wiwne, possibilities do not exist in Wiwne either. But this seems to presuppose that possibilities are thingy (i.e., they are entities of some sort). For my part, I am inclined to say that possibilities are not thingy. Moreover, if we assume that possibilities are thingy and that possibilities do not exist in Wiwne because nothing exists in Wiwne, then we have to conclude that it is necessary that nothing exists in the world, which we call Wiwne. But if it is necessary that nothing exists in that world and if we think that possibilities are thingy, then we have to think that the necessity that nothing exists in Wiwne exists in Wiwne as necessities and possibilities are closely related, and if one is thingy, then the other should also be thingy. However, since Wiwne is the world in which nothing exists, it is a contradiction to think that the necessity exists in Wiwne. Thus, by reductio ad absurdum, we should think that possibilities are not thingy.
7. Conclusion
In this essay, I have talked about four types of solutions to the problem of truthmakers for negative truths: the negative facts solution, the incompatibility solution, the general facts solution, and the Wiwne solution. I think that the Wiwne solution should be championed. The Wiwne solution has an upper hand over the negative facts solution and the general facts solution because it does not postulate any entities in order to come up with truthmakers for negative truths, and it is superior to the incompatibility solution because it can handle the negative truth that the incompatibility solution does not seem to be able to. However, accepting the Wiwne solution creates a problem for holding the correspondence theory of truths. By saying that negative truths with the scope of `not' the entire proposition do not require truthmakers, I am diverging from the correspondence theory of truths. So, it is appropriate for me to offer an alternative theory of truths, but at this point, I have not developed such a theory of truths yet.Foot note 15,Foot note 16
References
Yuki Miyoshi
Department of Philosophy
Simon Fraser University
Burnaby, B.C. Canada
V5A 1S6
yuki_miyoshi524 [at] hotmail [dot] com
[Foot Note 1]
This title was suggested by Dr. Bernard Linsky.
[Foot Note 2]
Particulars, properties, relations by themselves can be truthmakers. For example, a particular, Smith, can be the truthmaker for the proposition that Smith exists.
[Foot Note 3]
I will use the symbols `<' and `>' to mean whatever is enclosed by them is a proposition.
[Foot Note 4]
Although it is true that other things, such as the whole world, can also be truthmakers for <Smith has blond hair>, I say, `The truthmaker for <Smith has blond hair> is Smith's having blond hair'; for in this paper, I am mostly concerned with the minimal truthmakers.
[Foot Note 5]
See Armstrong 2004, 10.
[Foot Note 6]
I use `T→<p>' to mean T is a truthmaker for <p>.
[Foot Note 7]
Armstrong discusses these two objections. He further points out that the ontologically problematic nature of negative properties is highlighted by the fact that negative properties do not play any causal role (2004, 55).
[Foot Note 8]
Besides these problems, Dr. Linsky pointed out to me that negative particulars have this problem: if it is true that Smith is neither red nor green, negative Smith is both red and green; however, nothing is red and green at the same time.
[Foot Note 9]
I came up with this objection by thinking about what Armstrong might have meant when he said that Russell's objection to incompatibilism, which appears in the pages 69-70 of Russell's Logical Atomism, is `that a truth of incompatibility is itself a negative truth' (Armstrong 2004, 60). Contrary to what Armstrong said, my understanding of Russell's objection in these pages is that postulating the facts whose logical form is the incompatibility relation's holding between propositions is no more desirable than postulating negative facts; in fact, it is less desirable given that propositions are not existents and non-existing things cannot be the terms of real relations.
[Foot Note 10]
I am using `|' as the Sheffer stroke symbol.
[Foot Note 11]
Armstrong, in making the same point, cites C. B. Martin and George Molnar as examples (2004, 71).
[Foot Note 12]
This argument here is basically taken from Demos, although he used the argument to reach a different conclusion (Demos 1917, 190).
[Foot Note 13]
Armstrong reports that Bruin Christensen informed him of this insight. See Armstrong's book (2004, 91).
[Foot Note 14]
I owe a lot to Dr. Linsky for the argument in this section. Dr. Linsky noticed that if we added only Smith to Wiwne then <Smith is the only human being> would be true, while I was trying to convince him that general truths do not require the totality relation by using the entailment principle, and it was at that point that I grasped my argument clearly. Though my memory may not be 100% accurate, that is more or less how I came to that argument.
[Foot Note 15]
I thank D. M. Armstrong for pointing this out to me.
[Foot Note 16]
This essay was originally written for my honors essay project at University of Alberta in the Fall semester of 2005 under the supervision of Dr. Bernard Linsky. Naturally, I thank Dr. Linsky the most. I thank Stephen Latta and Dr. D. M. Armstrong for comments and suggestions. Dr. Armstrong was nice enough to read a draft of this essay even though I did not have any previous acquaintance with him. I should also mention that most of the things that I know about truthmakers are introduced to me by Dr. Armstrong's Truth and Truthmakers. I thank Dr. Kirstie Laird for her editorial help. Finally, I thank all the people who helped me write this essay one way or another.