Thursday, February 28, 2013

Leibniz on the Problem of Evil

First published Sun Jan 4, 1998; substantive revision Wed Feb 27, 2013

There is no question that the problem of evil vexed Leibniz as much as any of the problems that he engaged in the course of his philosophical career. This is manifest in the fact that the first and the last book-length works that he authored, the Philosopher's Confession (written at age 26 in 1672) and the Theodicy (written in 1709, seven years before his death) were both devoted to this problem, as well as in the fact that in the intervening years Leibniz wrote numerous short pieces on related issues––many of which may be found in Gr and will soon be available in English translations currently being undertaken by R. C. Sleigh, Jr.–– and one full-length work, the Dissertation on Predestination and Grace (DPG), which was only published in 2011. The fact that the Theodicy was the only book-length treatise that Leibniz published during his lifetime provides further evidence of the significance that he attributed to the topic. It is therefore appropriate that it has now become an interpretive commonplace that Leibniz's concern with the problem of evil was central to his overarching philosophical concerns throughout his philosophical career. [See Rutherford (1995) and Antognazza (2009).] Leibniz's approach to the problem of evil became known to many readers through Voltaire's lampoon in Candide: the link that Voltaire seems to forge between Leibniz and the extravagant optimism of Dr. Pangloss continues––for better or worse––to shape the popular understanding of Leibniz's approach to the problem of evil. In this entry we examine the two main species of the problem of evil that Leibniz addresses. The first, “the underachiever problem,” is raised by a critic who would argue that the existence of evil in our world indicates that God cannot be as knowledgeable, powerful, or good as traditional monotheists have claimed. The second, “the holiness problem,” is raised by the critic who would argue that God's intimate causal entanglements with the world make God the cause of evil. God is thereby implicated in evil to the detriment of his holiness.

1. Various Versions of the Problem of Evil in Leibniz

Before examining Leibniz's views on the problem of evil, it is necessary to do some stage-setting in order to locate just what sort of problem Leibniz thought evil presented. Consideration of any present-day introductory textbook of philosophy reveals that the problem of evil in contemporary philosophy is standardly regarded as an argument for atheism. The atheist contends that God and evil are incompatible, and given that evil clearly exists, God cannot exist. Some philosophers, conceding that the claimed incompatibility in the foregoing argument is too strong, contend, nevertheless, that even if the existence of God and the existence of evil should prove to be compatible, the existence (or duration, or amount, or pervasiveness) of evil provides us at the very least with compelling circumstantial evidence that God does not exist.

Framed in this way, the “atheistic problem of evil” invites certain sorts of responses. In particular, it invites the theist to explain how a being that is omniscient, omnibenevolent, and omnipotent can allow evil to exist. Present-day responses to the problem of evil therefore focus largely on presenting “theodicies,” that is, reasons why a perfect being does or might permit evil of the sort (or duration, or amount, or distribution) that we find in our world to exist.

When we consider, however, the works of medieval philosophers who address the problem of evil, the “atheistic problem” is not to be found. Since these figures believed that the arguments of natural theology demonstrated the existence of God, the problem that evil presented for them was different from that engaged by present-day philosophers. In present-day terminology, medieval philosophers did not engage the “evidential” problem of evil: rather, they engaged the “aporetic” problem of evil, in order to try to resolve the apparent logical incompatibility between God's attributes and the existence of evil. [On the distinction between these problems, see Adams and Adams (1990), pp. 1–3.] The problem, therefore, was taken to be that of explaining the compatibility of the existence of evil with divine moral purity or holiness. These philosophers believed that God is the author of everything that exists, and given that evil is one of the things that exists, it might seem that God is therefore the author of evil. And if an agent is an “author of evil,” he is therefore implicated in the evil and cannot be morally pure or holy. Thus, God cannot be morally pure nor holy. Let's call this version of the problem of evil the “holiness problem.” Before moving on, it should be noted that in light of the fact that Leibniz and his predecessors shared a commitment to God's existence, one might think that their approach to the problem of evil begs the question against the atheistic critic who charges that the existence of evil provides evidence that God does not exist. But this issue simply did not arise for Leibniz and his predecessors, given their antecedent belief in God's existence, and therefore it is inappropriate to charge these philosophers with begging the question.

Traditional theists held––and present-day theists still do hold––that God is the “author” or cause of everything in the cosmos in at least three different respects, so discussions of the holiness problem often branch off in three correspondingly different directions. First, God is regarded as the creative cause of everything in the cosmos. Everything that exists contingently is brought into existence by means of the creative activity of God. Second, it is held that God is the conserving cause of everything that exists. So God not only creates every created being, but every created being that continues to exist does so in virtue of God's continuously maintaining it in existence. Third, every action caused by a created being requires direct divine activity as concurrent cause. So every whack of a hammer, every strike of my fingertip on the keyboard, every tug of a magnet on a piece of iron, requires not only that the created being act, but also that the creator act concurrently with the created being in order to bring about the particular effect of the cause in question. [For a classic exposition of these various modes of divine causal involvement see St. Thomas Aquinas, Disputationes de Potentia Dei, Q.3, a.7, resp.]

Given that on this traditional account, God is intimately intertwined with the workings of the cosmos, the holiness problem seemed all the more intractable. In light of the intimate connections between God and the created world, the problem is not just that God created a world that happens to include evil, but that God seems to be causally (and thus morally) implicated in, for example, every particular act of murder, every earthquake, and every death caused by plague. Consequently, responses to the holiness problem sought to explain not only how God could remain holy despite having created a world such as ours, but also how he could remain holy despite conserving the world in existence and causally cooperating with all the events that occur in it.

In light of the fact that Leibniz lived in between these two eras, eras in which evil was taken to present different problems for the monotheistic philosopher, we are immediately led to wonder what sort of problem he sought to address. Leibniz expends a great deal of effort attempting to solve the holiness problem, but he also takes up something akin to the atheistic problem. It would be anachronistic, however, to claim that Leibniz was engaged with the atheistic problem, for in his time the existence of evil was taken to be an argument for an unorthodox form of theism rather than an argument for atheism. Thus, for example, a group of thinkers collectively known as the “Socinians” held, among other things, that the existence of evil was not incompatible with God's existence, but that it was incompatible with the existence of an omniscient God. The Socinians therefore held that God must not be omniscient, and that he must at the very least lack knowledge of future contingent events. [For Leibniz's view on the Socinians see Theodicy 364 (H343; G VI 318) et passim. More details on Socinianism can be found in Jolley, c.2, and Maclachlan.]

We might then characterize the problem raised by atheists in our own century and by the Socinians, to cite just one example from the seventeenth century, more broadly as the “underachiever problem.” According to the underachiever problem, if the sort of being that traditional monotheism identifies as God were to exist, the existence of this world would represent a vast underachievement on his part: therefore there is no such being. Atheists take this conclusion to prove that there is no God; the Socinians take it to show that God is not the sort of being that the traditional theist supposes him to be.

Although Leibniz is concerned about the underachiever problem, it is the Socinian, and not the atheistic, version of the problem that he engages. The winds of atheism had not reached the gale force proportions that they would in succeeding centuries. Consequently, this stronger conclusion was not yet taken as a serious, or at least the main, threat presented by the existence of evil.

It is important to distinguish between these versions of the problem of evil since we cannot understand Leibniz's treatment of evil in a given text until we know what problem it is that he means to be addressing in that text. Having set the stage in this way, we can now consider Leibniz's solutions to the problem of evil: we first consider the underachiever problem, and then turn to the holiness problem.

2. The Underachiever Problem

The core of Leibniz's solution to the underachiever problem is straightforward. Leibniz argues that God does not underachieve in creating this world because this world is the best of all possible worlds. Many thinkers have supposed that commitment to the claim that this world is the best of all possible worlds follows straightforwardly from monotheism. Because God is omnipotent and omniscient, nothing can prevent him from creating the best world, and his omnibenevolence obliges him to create the best world. So the created world is the best world.

Leibniz's reasoning to this conclusion does not, however, follow this straightforward path: among other things, this reasoning is not cogent as it stands. A number of seventeenth-century figures recognized that God would not be obliged to create the best world if there were no such thing as the best world. There would be no best world if the series of possible worlds formed a continuum of increasingly good worlds ad infinitum. And if there is no best world, God cannot be faulted for failing to create the best one since to do so would be as impossible as, say, naming the highest number. There is no such number of course, and likewise no such world. So while God may be obliged to create a world that has at least some measure of goodness, he cannot be obliged, on this view, to create the best. And therefore it might be the case that God simply chose arbitrarily to create one of infinitely many morally acceptable worlds. [This line of argument was common among certain Jesuit scholastics of the period. For discussions of this issue, see, for example, Ruiz de Montoya, Commentaria ac Disputationes in primam partem Summae Thologicae S. Thomae. De voluntate Dei et propiis actibus eius, Lyon 1630, disp. 9 and 10, and Diego Granado, Comentarii in primam partem Summae Theologicae S. Thomae, Pont-a-Mousson, 1624, pp.420–433.]

Leibniz was aware of this argument denying God's obligation to create the best, but he was firmly committed to rejecting it, in virtue of a central principle of his philosophical system, the Principle of Sufficient Reason. According to the Principle of Sufficient Reason, for any state of affairs, there must be a sufficient reason that explains why that state of affairs and not some other state of affairs obtains. When it comes to our world, then, there must be some reason that explains why it, and not some other world, obtains. But there can be no such reason if it is the case that the goodness of worlds increases ad infinitum. Leibniz therefore concluded that there can be no infinite continuum of worlds.

One might be tempted to resist Leibniz's argument by saying that even according to the view on which there is an “infinite continuum of good worlds,” there is something that can serve as the sufficient reason for existence of this world, namely, God's decree that this world be actual. But such a response, Leibniz observes, would merely push the problem back, because the Principle of Sufficient Reason applies to free choices just as it applies to any other event or state of affairs. Thus, we would have to provide a sufficient reason for God's choice of this world instead of some other world on the continuum of morally acceptable worlds. And it seems that such a sufficient reason cannot be given on the infinite continuum of good worlds view. Note that the sufficient reason cannot be derived from some feature or fact about the world that is actually chosen, for this would raise the obvious question: Why did this feature in particular serve as the sufficient reason for God's choice? The only possible answers, it appears, would be: (a) Because God arbitrarily selected that feature as the one he would favor in deciding which world to create; or (b) Because that feature made that world better than all its competitors. But notice that neither of these answers is acceptable. The first is inconsistent with the Principle of Sufficient Reason. The second is incompatible with the hypothesis at issue, that there is no “best world.”

One might think that declaring this world to be the best possible world does not constitute a valid response to the underachiever problem. Indeed, such a response might be taken to provide the basis for a new underachiever argument along the following lines:

  1. If God were all-powerful, all-knowing, and all-good, then this world would be the best possible world.
  2. But surely this world is not the best possible world.
  3. Thus, God is not all-powerful, all-knowing, and all-good.

Leibniz believes, however, that there was overwhelming evidence that the conclusion of this argument was false. He therefore must take one of the two premises in this argument to be false. Given that he himself is committed to the first premise, he must reject the second premise. And this is precisely what he does.

What reason, Leibniz asks, does the critic have for thinking that (2) is true? When Leibniz addresses this issue, he usually has the critic say something along the following lines:

Surely this world is not the best possible world since we can easily conceive of possible worlds that are better. Take some token instance of suffering: the tragic bombing of the Oklahoma City federal building. Surely a world without that event would be better than the actual world. And there is no reason why God couldn't have created the world without that event. Thus, this is not the best possible world. [See Theodicy 118–119 (H 188–191; G VI 168–172).]

Leibniz's response to this sort of criticism comes in two stages. First, Leibniz says that while we can think of certain token features of the world that in and of themselves might be better than they are, we do not know whether it is possible to create a better world lacking those features, because we can never be certain of the nature of the connections between the token events in question and other events in the world. If we could improve or eliminate the token event in question without otherwise changing the world, we might well have a better world. Unfortunately, we have no way of knowing whether such a change to the token event would leave the world otherwise unchanged, or might instead make things, on balance, worse. [See Theodicy 211–214 (H 260–2; G VI 244–7) and Gr, p.64f., for examples of this sort of response.]

Second, examples such as these are deceptive because they presume that God utilizes standards of world goodness that he may not in fact use. For example, it may presume that a world is good only if each part taken in isolation is good (a standard, we have seen, that Leibniz rejects), or it might presume that a world is good only if human beings enjoy happiness in it.

Leibniz argues in numerous texts that it is parochial to think that human happiness is the standard whereby the goodness of worlds is to be judged. A more reasonable standard, according to Leibniz, would be the happiness of all sentient beings. But once we admit this, it may turn out that the amount of unhappiness in the created realm is quite small, given that for all we know, the sentient beings on Earth might constitute a very small percentage of the sentient beings created by God. Here Leibniz includes not only preternatural beings such as angels, but also the possibility of extraterrestrial rational beings [Theodicy 19 (H 134–5; G VI 113–4)].

There is disagreement among Leibniz scholars about the basis for judging the goodness of worlds. Various scholars have defended one or more of the following:

  1. The best world is the one that maximizes the happiness (i.e., virtue) of rational beings.
  2. The best world is the one that maximizes the “quantity of essence.”
  3. The best world is the one that yields the greatest variety of phenomena governed by the simplest set of laws.

There is scholarly dispute about whether Leibniz believed that the maximization of the happiness or virtue of rational beings is one of the standards by which God judges the goodness of the world. [For supporters of this claim see Rutherford, c.3; Blumenfeld, Brown; for detractors see Russell, p. 199, Gale.] It is unlikely that Leibniz believed that (1) alone was the true standard of goodness of the world in light of following comment on an argument advanced by Bayle:

the author is still presupposing that false maxim … stating that the happiness of rational creatures is the sole aim of God. [Theodicy 120 (H 192; G VI 172)]

In part, the dispute over this standard hangs on whether or not (1) is compatible with the more metaphysical standards embodied in (2) and (3), since it is these more metaphysical standards that Leibniz seems to endorse most consistently. In some cases, Leibniz writes as if the standard of happiness is fully compatible with the more metaphysical criteria. For example, within a single work, the Discourse on Metaphysics, Leibniz entitled Section 5 “What the rules of the perfection of divine conduct consist in, and that the simplicity of the ways is in balance with the richness of effects,” and he entitled Section 36: “God is the monarch of the most perfect republic, composed of all minds, and the happiness of this city of God is his principal purpose.” So Leibniz seems to advance both standards (1) and (3) in the same work [For another example, see R p. 105 (K X pp.9–10)]. In other places however, he writes as if they compete with one another [See Theodicy 124 (H 197–8; G VI 178–9).]

Whatever position one comes to hold on this matter, Leibniz often points to the more metaphysical standards as the ones God utilizes in assessing the goodness of worlds. But there is further controversy over exactly which metaphysical standard, (2) or (3), Leibniz endorses. In general, Leibniz holds that God creates the world in order to share his goodness with created things in the most perfect manner possible [Gr 355–6]. In light of the fact that created beings, in virtue of their limitations, can mirror the divine goodness only in limited respects, God creates a variety of things, each of which has an essence that reflects a different facet of divine perfection in its own unique way. Since this is God's purpose in creating the world, it would be reasonable to think that maximizing the mirroring of divine goodness in creation is the goal that God seeks to achieve. And this in fact is one of the standards Leibniz seems to endorse. We might call this the “maximization of essence” standard. Leibniz seems convinced that the actual world meets this standard and that creatures are to be found that mirror the divine perfections in all the sorts of ways that creatures can do this. Thus, there are creatures with bodies and creatures without, creatures with freedom and intelligence and creatures without, creatures with sentience and creatures without, etc. [See, for example, MP pp. 75–6 and 138 (G VII 303–4 and 310).]

In some texts, however, Leibniz frames the standard of goodness in what some have taken to be a third distinct way. In these places he argues that the goodness of a world is measured by the ratio between the variety of phenomena that a world contains and the simplicity of the laws that govern that world. Here Leibniz emphasizes the fact that the perfection of a world that maximizes the variety of phenomena it contains is enhanced by the simplicity of its laws since this displays the intelligence of the creator who created it.

Some scholars have argued that one or the other of these two more metaphysical standards represents Leibniz's settled view on the true standard of goodness [Gale, for example]. Other scholars have argued that, in the end, the two standards are not exclusive of each other. [See Rutherford, cc.2–3 and Rescher, c.1 for two very different ways of harmonizing (2) and (3).]

Regardless of which of these interpretations is correct, if these are the standards by which God judges the world's goodness, it becomes much more difficult to defend the claim that this is not the best possible world. We can use standard (3) to illustrate. In order, for example, for God to eliminate the Oklahoma City bombing from the world, what would be required in order for him to do so? There are presumably a number of ways in which this might be done. The most obvious would involve miraculous intervention somewhere in the chain of events leading up to the explosion. God might miraculously prevent the explosives from detonating, or he might eliminate the truck and its contents from the world. But this sort of miraculous intervention would require that the laws governing the the world become more complex. Consequently, Leibniz, and others who share this view of what the goodness of a world consists in, such as Malebranche, think that miraculous intervention is generally repugnant and would require vastly outweighing goods to result from a miraculous intervention in order for such an intervention to be permissible. [See Theodicy 129 (H 192–3; G VI 182).]

In any event, Leibniz holds that we are simply unable to know how changing certain events would change the world's capacity to meet the standards of goodness described in (2) and (3). Thus, according to Leibniz, we are not justified in claiming that this world is not as good, all things considered, as some other possible world. According to Leibniz, then, the underachiever problem cannot get off the ground unless the critic is able to defend the claim that this world is not the best possible world. It should be noted that Leibniz's approach to the underachiever problem thus seems be immune to the line of criticism pressed by Voltaire in Candide, namely, that it is obvious that this world is not the best possible world because there are so many manifest evils in it. Leibniz does not believe that each individual event is the best possible event, and he does not think that it is possible for finite minds to demonstrate that every individual event must be a part of the best possible world: rather, he believes that the world as a whole is the best possible world. (That said, it should be noted that there is considerable scholarly controversy as to whether Voltaire's target in Candide is indeed Leibniz: it has been claimed, for example, that the “optimism” lampooned in Candide is closer to that of Pope (see Rutherford (1995); on the general reception of Leibniz in France, see Barber (1955)].) In any event, on Leibniz's view, our inability to know how changing certain events in the world would affect other events and our inability to know how such changes would affect the overall goodness of the world make it impossible to defend the claim that the manifest evils in the world constitute evidence that this is not the best possible world.

3. The Holiness Problem

Far less scholarly attention has been devoted to Leibniz's treatment of the holiness problem, if only because this conception of the problem has only recently been recognized by Leibniz scholars. As noted above, the main problem here is that God's character seems to be stained by evil because God causally contributes to the existence of everything in the world, and evil is one of those things. [For two recent treatments see Sleigh (1996) and Murray (2005)]

The standard solution adopted by medieval thinkers was to deny an assumption of the preceding argument, namely, that evil is “something.” Evil was claimed not to have any positive reality, but to be a mere “privation” or “lack” of being. On such a view, evil has no more reality than the hole in the center of a donut. Making a donut does not require putting together two components, the cake and the hole: the cake is all that there is to the donut, and the hole is just the “privation of cake.” It therefore would be silly to say that making the donut requires causing both the cake and the hole to exist. Causing the cake to exist causes the hole as a “by-product” of causing a particular kind of cake to exist. Thus, we need not assume any additional cause for the hole beyond that assumed for the causing of the cake.

The upshot of our pastry analogy is this: given that evil, like the hole, is merely a privation, it requires no cause (or as the medievals, and Leibniz, liked to say, it needs no “cause per se”). God does not “causally contribute to the existence of evil” because evil per se is not a thing and therefore requires no cause in order to exist. And since God does not cause the existence of evil, God cannot be causally implicated in evil. Thus, the holiness problem evaporates.

Early in his philosophical career, Leibniz, like other seventeenth-century philosophers, scoffed at this solution to the holiness problem. In a short piece entitled “The Author of Sin,” Leibniz explains why he thinks the privation response to the holiness problem fails. Leibniz argues that God is the author of all that is real and positive in the world, and that God is therefore also the “author” of all of privations in the world. “It is a manifest illusion to hold that God is not the author of sin because there is no such thing as an author of a privation, even though he can be called the author of everything which is real and positive in the sinful act” [A.6.3.150].

Leibniz explains why he takes this response to be a “manifest illusion,” through the consideration of an example. Suppose that a painter creates two paintings that are identical in every respect, except that the one is a scaled down version of the other. It would be absurd, Leibniz remarks,

… to say that the painter is the author of all that is real in the two paintings, without however being the author of what is lacking or the disproportion between the larger and the smaller painting… . In effect, what is lacking is nothing more than a simple result of an infallible consequence of that which is positive, without any need for a distinct author [of that which is lacking]. [A.6.3.151]

So even if it is true that evil is a privation, this does not have as a consequence that God is not the author of sin. Given that what is positively willed by God is a sufficient condition for the existence of the evil state of affairs, in virtue of willing what is positive in some state of affairs, God is also the author of what is privative in that state of affairs. [A similar early critique is found at A.6.3.544].

Leibniz therefore sought to develop a different strategy in order to clear God of the charge of being the author of sin. In the Philosopher's Confession, his most significant treatise on evil aside from the Theodicy, Leibniz claims that although God wills everything in the world, his will with respect to goods is decretory , whereas his will with respect to evils is merely permissive. And Leibniz argues that God's permissive willing of evils is morally permissible if and only if such permission of evil is necessary in order for one to meet one's moral obligations..

It should be noted that Leibniz does not think that the permission of evil is morally justified on the grounds that such permission brings about a greater good that may not otherwise be achieved. Such an explanation, according to Leibniz, would make it the case that God would violate in the the Biblical injunction “not to do evil that good may come” [Causa Dei 36 (S 121; G VI 444)]. Leibniz therefore claims that the evil that God permits is a necessary consequence of God's fulfilling his duty (namely, to create the best world). Leibniz characterizes (morally permissible) permission as follows:

P permits E iff:

  1. P fails to will that E
  2. P fails to will that not-E
  3. P brings it about that the state of affairs S obtains by willing that S obtains
  4. If S obtains then E obtains
  5. P knows that (4)
  6. P believes that it is P's duty to will S and that the good of performing one's duty outweighs the evil entailed by E's obtaining

[This account is distilled from A.6.3.129–131]

This, Leibniz believes, resolves any holiness problem that might arise in so far as God is considered as the creator of the universe. However, after writing the Philosopher's Confession, Leibniz became increasingly concerned that a tension might arise in his account when it was applied to the holiness problem in the context of concurrence. Recall that traditional theists held that God was not only creator and conserver of all created things, but that God also was the concurrent cause of all actions of created things.

There were heated debates in the sixteenth and seventeenth centuries concerning the nature of divine concurrence. The dispute centered on the respect in which God concurred with the free acts of creatures. This was an especially pressing problem for the obvious reason that positing too close a connection between God and created beings in cases where moral evils are committed runs the risk of implicating God in the evil, thus raising the holiness problem all over again. This debate often focused on a certain type of proposition and on what made this type of proposition true. The propositions in question are called “conditional future contingents”, propositions of the form:

If agent S were in circumstances C and time t, S would freely chose to f.

Propositions of this form were particularly important in discussions of philosophical theology in the sixteenth and seventeenth centuries because it was believed that it was necessary that God know propositions of this type in order to exercise providential control over the free actions of created beings. In order to exercise providential control over free actions in the created world, God must know precisely how each such being will choose to act in each circumstance in which it will find itself. If God, for example, did not know what Eve would choose to do when confronted by the serpent, or what I would choose to do when confronted with a tuna sandwich, God could not know in advance the order of events in the universe he deigns to create.

But how does God know whether or not a token proposition of this type is true? In general, disputants in this period held that there are only two possible answers to this question. God knows that a token proposition of this type is true either because he wills that that proposition be true, or he knows that proposition to be true because something independent of his will makes it true, and God, in virtue of his omniscience, therefore knows it to be true. Following recent scholarship, we will call the first view the “postvolitional view” (since the truth of the proposition is determined only after God wills it) and the latter view the “prevolitional” view (since the truth of the proposition is independent of what God wills). In his early writings on the topic, Leibniz seemed inclined to postvolitionalism. So take the token proposition:

If Peter were accused of consorting with Christ during Christ's trial, Peter would deny Christ.

The early Leibniz holds that this type of proposition is true because God decrees that it would be true: that is, God decrees that Peter would deny Christ under these circumstances [see C 26–7 and Gr 312–3]. Furthermore, those who held this view generally held that it was in virtue of divine concurrence that God makes the proposition true in the actual world. So, in virtue of causally influencing Peter at the moment of his decision, God brings it about that Peter denies Christ in these circumstances.

This view obviously faces a number of difficulties. For our purposes, the most pressing one is that it seems to undercut Leibniz's solution to the holiness problem based on permission. For if the above proposition is true because God wills that it be true, then it would seem that God wills that Peter sin, and if he wills that Peter sin, he cannot merely permit it, in light of condition (1) of the definition of permission given above. Consequently, it appears that Leibniz must abandon his initial answer to the question of “what makes conditional future contingents true” and adopt the alternative answer.

The alternative answer also raises problems. What does it mean to say that the truth of the proposition is determined independently of God's will? Defenders of this view usually hold that the human will cannot be determined. When an agent chooses freely, nothing can “determine” or “cause” the choice, for otherwise the ehoice would not be free. Thus, for those who defended this view, the answer to the question of “what makes conditional future contingents true” ought to be “nothing.” For if something made future contingents true, then that thing would determine the choice, and the choice would not be free.

Given his commitment to the Principle of Sufficient Reason, however, Leibniz could not endorse such a view. Does Leibniz, then, have an answer to this question that will rescue him from the holiness problem? There is scholarly disagreement about this issue. Some have held that Leibniz is obliged to hold the postvolitional view despite the difficulties that it raises for him. [See Davidson (1996), Sleigh (1994).] Others have held that Leibniz tried to forge a third alternative in order to avoid this seemingly intractable dilemma. [See Murray (2005); for an alternative to Murray (2005), see Cover and Hawthorne (2000)]. I will close by considering the latter suggestion.

According to Leibniz, free choice in humans is brought about through the activity of the human intellect and the human will working in concert with each other. The intellect deliberates about alternatives and selects the one that it perceives to be the best, all things considered. The intellect then represents this alternative to the will as the best course of action. The will, which Leibniz takes to be an “appetite for the good,”, then chooses the alternative represented to it as containing the most goodness [Theodicy, 311 (H 314; G VI 300–1].

On this view, it appears that there are two ways in which I might exercise “control” over my acts of will. First, I might be able to control what appears to me to be the best course of action, all things considered. That is, I might control the process of deliberation. Second, I might be able to control which alternative is presented to the will as containing the greatest good. Leibniz seems to accept both of these possibilities. In certain passages, he argues that by engaging in some sort of moral therapy, I can control which things appear to me to be good, and thus control the outcome of my deliberations. In other passages, he seems to say that while the will does “infallibly” choose that which the intellect deems to be the best, the will nevertheless retains the power to resist the intellect because the intellect does not “cause” the will to choose as it does. [Concerning the first strategy, see, for example, Reflections on Hobbes, 5 (H 396–7; G VI 391–1). For more on this aspect of Leibniz's view of freedom see Seidler (1985). Concerning the second strategy see, for example Theodicy 282 (H 298–300; G VI 284–5).]

Both suggestions face difficulties. Consider the first. How might I go about engaging in “moral therapy”? First, I would have to choose to do something to begin to bring about a change in how I see things. But of course I can make a choice to do this only if I first deliberate about it and see that making this change is the best thing for me to do. But did I have control over this process of “coming to see that a change is the best thing for me to do”? It seems that I may have control here only if I have control over the actions that led me to see things this way in the first place. But do I have control over those actions? If the answer is yes, it is only because I had control over my prior deliberations, and it looks as if this will lead us back in the chain of explanation to certain very early formative stages of my moral and intellectual life, stages over which it is hard to believe I had any control. It therefore seems that this line of reasoning will be difficult to sustain.

Let us consider the second alternative then, according to which I have control because the will is never “causally determined” to choose that which the intellect deems to be best in those circumstances. Leibniz holds that the will is not causally determined in the act of choice but merely “morally necessitated.” There is scholarly disagreement about the interpretation of this phrase. Some think it just means “causally necessitated.” But if this is right, it appears that God, who establishes the laws of nature, determines how creatures act, and this leads us back to the suggestion that Leibniz was a postvolitionalist in these matters. As we noted above, this is a troubling position for Leibniz to adopt since it seems to undermine his response to the holiness problem. [For various positions on the nature of “moral necessity,” see Adams, pp. 21–2, Sleigh (2000), Murray (1995), pp. 95–102, and (1996), esp. Section IV].

Others have held that moral necessity is a philosophical novelty, invented to explain the unique relationship between intellect and will. On this view, the will infallibly follows the determination of the intellect, without thereby being causally determined. Leibniz sometimes hints at this reading, as in the following example derived from Pierre Nicole:

It is considered impossible that a wise and serious magistrate, who has not taken leave of his senses, should publicly commit some outrageous action, as it would be, for instance, to run about the streets naked in order to make people laugh [Theodicy 282 (H299; G VI 284)]

Here, the wise magistrate is not causally determined to refrain from streaking to make people laugh. Instead, he just considers streaking to be so unseemly that “he can't bring himself to do it.” Something about his psychological constitution prevents him from seeing this as something that he might actually do, even though there is surely some sense in which he nevertheless could do it.

If we allow Leibniz to locate control over actions in a will that is only morally necessitated by the intellect, is there a way for him to avoid the postvolitional/prevolitional dilemma discussed earlier? The answer is not obvious. One would have to say that the will's infallibly choosing in accordance with the deliverances of the intellect is a fact whose truth is independent of God's will, while also saying that the deliverances of the intellect provide a sufficient reason for the will's choice. If this can be done, Leibniz may have a way of avoiding the difficulty posed by conditional future contingents.

However we might think these questions should be resolved, Leibniz himself appears to have thought that the prevolitional route was the one to take. He does not think that God makes it the case how human beings would act if they were created; rather, Leibniz believes God "discovers" in the ideas of the possibles how human beings would act if they were created [on this topic see Sleigh (1994).] [Leibniz speaks of these truths about how human beings will act as “limitations” that prevent God from making them, and the world that contains them, more perfect. In the end, it is these limitations, Leibniz argues, that prevent there from being a better world than the actual one. [On the notion of “limitations” see AG 60–2, 11, Theodicy 20 (H 86–7; G VI 114–5), Causa Dei 69–71 (S 128–30; 457–8).] If this interpretation is correct, then we might think that the permission strategy will work as a solution to the holiness problem both when it comes to defending God as creator and as concurrent cause of all effects in the cosmos.

Interestingly, however, Leibniz comes to favor, in later life, the scholastic “privation” view that he rejected in his earliest writings on the problem of evil. [See, for example, Theodicy 20, 30, 153 (respectively, H 86–7, 91–2, 219–20; G VI 114–5, 119–20, 201.] Leibniz's conception of privation in general, and the relation between his earlier and later views on the topic, has recently received a sustained and searching examination in Newlands (forthcoming), to which readers interested in the topic are directed.

The issues that arise in thinking about Leibniz's views on the problem of evil have only in the past couple of decades begun to receive the sustained scholarly attention that they deserve in virtue of their manifest significance for Leibniz. In the last few years in particular––probably not coincidentally, the three-hundredth anniversary of the publication of the Theodicy was celebrated in 2010––considerable interpretive attention has been devoted to the details of Leibniz's treatment of the problem of evil and related topics. [Rateau (2008) is the first book-length treatment of Leibniz's work on the problem of evil; the essays in Rateau (2011) and Newlands and Jorgensen (forthcoming) are devoted to particular topics related to Leibniz's treatment of the problem of evil.] Given the fact that Leibniz's treatment of the problem of evil draws on medieval sources and also was taken as a target by later writers such as Voltaire and Kant, renewed interest in Leibniz's treatment of the problem of evil, combined with the resurgence of interest in contextual history of philosophy, have inspired recent work on the general historical significance of Leibniz's work on the problem of evil that seeks to illuminate later approaches to the problem of evil as well as the nature of the problem of evil itself. [See, inter alia, Larrimore (2004), Neiman (2002), and the essays in Rateau (2009).] In light of the fact that new translations of Leibniz's central texts devoted to the problem of evil have either only relatively recently been published (CP) or are in process––a new edition and English translation of the Theodicy, by Sean Greenberg and R. C. Sleigh, Jr., is well underway and under contract with Oxford University Press–and given that other new texts, like DPW, that bear on this nest of issues may well be discovered, there is reason to expect that this topic will continue to be an active area of Leibniz scholarship, and therefore that any conclusions about Leibniz's views on the problem of evil must, for now, remain tentative and subject to revision.

Bibliography

Primary Sources

[A]Sämtliche Schriften und Briefe . Darmstadt and Berlin: Berlin Academy, 1923-. Cited by series, volume, and page.
[AG]Philosophical Essays. Roger Ariew and Daniel Garber (eds. and trans.), Indianapolis: Hackett, 1989.
[C]Louis Couturat (ed.), Opuscules et Fragments Inédits de Leibniz. Hildesheim: Georg Olms, 1966.
[CP]R. C. Sleigh, Jr. (editor and translator), with contributions from Brandon Look and James Stam, Confessio Philosophi: Papers Concerning the Problem of Evil, 1671–1678. New Haven: Yale University Press, 2005.
[DPW]Michael Murray (editor and translator), Dissertation on Predestination and Grace. New Haven: Yale University Press, 2011.
[G]Die Philosophischen Schriften von Gottfried Wilhelm Leibniz. Edited by C.I. Gerhardt. Berlin: Weidman, 1875–1890. Cited by volume and page.
[Gr]Textes Inédits. Edited by Gaston Grua. Paris: Presses Universitaires de France, 1948.
[H]Theodicy. Edited by Austin Farrer and translated by E.M. Huggard. New Haven: Yale UP, 1952.
[MP]Mary Morris and G.H.R. Parkinson (eds. and trans.), Leibniz-Philosophical Writings, London: J.M. Dent and Sons, 1973.
[R]G.W. Leibniz, Political Writings. Patrick Riley (ed. and trans.), Cambridge: Cambridge University Press, 1988.
[S]Paul Schrecker and Anne Martin Schrecker (eds. and trans.), Leibniz: Monadology and Other Philosophical Essays, Indianapolis: Bobbs-Merrill, 1965.

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  • Larrimore, Mark, 2004, “Autonomy and the Invention of Theodicy,” New Essays on the History of Autonomy: A Collection Honoring J. B. Schneewind, Natalie Brender and Lawrence Krasnoff (eds.), Cambridge: Cambridge University Press: 61–91.
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Theoretical Terms in Science

First published Mon Feb 25, 2013

The notion of a theoretical term may simply be understood as applying to expressions that refer to nonobservational entities. Paradigmatic examples of such entities are electrons, neutrinos, gravitational forces, genes etc. There is yet another explanation of theoreticity: a theoretical term is one whose meaning becomes determined through the axioms of a scientific theory. The meaning of the term ‘force’, for example, is seen to be determined by Newton's laws of motion and further laws about special forces, such as the law of gravitation. Theoreticity is a property that is commonly applied to both expressions in the language of science and the corresponding referents and concepts. Objects, relations and functions as well as concepts thereof may thus qualify as theoretical in a derived sense.

Several semantics have been devised that aim to explain how a scientific theory contributes to the interpretation of its theoretical terms and as such determines what they mean and how they are understood. All of these semantics assume the respective theory to be given in an axiomatic fashion. Yet, theoretical terms are also recognizable in scientific theories which have as yet resisted a satisfying axiomatization. This is due to the fact that these theories contain general propositions that have the logical form of universal axioms.

Theoretical terms pertain to a number of topics in the philosophy of science. A fully fledged semantics of such terms commonly involves a statement about scientific realism and its alternatives. Such a semantics, moreover, may involve an account of how observation is related to theory in science. All formal accounts of theoretical terms deny the analytic-synthetic distinction to be applicable to the axioms of a scientific theory. The recognition of theoretical terms in the language of science by Carnap thus amounts to a rejection of an essential tenet of early logical empiricism and positivism, viz., the demonstration that all empirically significant sentences are translatable into an observation language. The present article explains the principal distinction between observational and theoretical terms, discusses important criticisms and refinements of this distinction and investigates two problems concerning the semantics of theoretical terms. Finally, the major formal accounts of this semantics are expounded.

1. Two Criteria of Theoreticity

1.1 Reference to Nonobservable Entities and Properties

As just explained, a theoretical term may simply be understood as an expression that refers to nonobservable entities or properties. Theoreticity, on this understanding, is the negation of observabilty. This explanation of theoreticity thus rests on an antecedent understanding of observability. What makes an entity or property observable? As Carnap (1966, Ch. 23) has pointed out, a philosopher understands the notion of observability in a narrower sense than a physicist. For a philosopher, a property is observable if it can be ‘directly perceived by the senses’. Hence, such properties as ‘blue’, ‘hard’ and ‘colder than’ are paradigmatic examples of observable properties in the philosopher's understanding of observability. The physicist, by contrast, would also count quantitative magnitudes that can be measured in a ‘relatively simple, direct way’ as observable. Hence, the physicist views such quantities as temperature, pressure and intensity of electric current as observable.

The notion of direct perception is spelled out by Carnap (1966, Ch. 23) by two conditions. Direct perception means, first, perception unaided by technical instruments and, second, that the perception is unaided by inferences. These conditions are obviously not satisfied for the measurement of quantities like temperature and pressure. For the philosopher, only spatial positions of liquids and pointers are observed when these quantities are measured. To an even higher degree, we are unable to observe electrons, molecules, gravitational forces and genes on this narrow understanding of observability. Hence, expressions referring to such entities qualify as theoretical.

In sum, a property or object is observable (in the philosopher's sense) if it can be perceived directly, where directness of observation precludes the use of technical artifacts and inferences. Notably, Carnap (1936/37, 455; 1966, 226) did not think his explanation of the distinction to be sufficiently precise to result in a sharp line between observational and theoretical terms. He rather views the theory-observation distinction as being introduced into a ‘continuum of degrees of observability’ by choice. Prominent criticisms of the theory-observation distinction will be discussed in Section 2.1.

1.2 Semantic Dependence upon a Scientific Theory

The above explanation of theoreticity may be felt unsatisfactory as it determines the property of being theoretical only via negation of the property of being observable (Putnam 1962). This explanation does not indicate any specific connection between the semantics of theoretical terms and corresponding scientific theories. There is, however, also a direct characterization of theoreticity that complements the criterion of non-observability: an expression is theoretical if and only if its meaning is determined through the axioms of a scientific theory. This explanation rests on what has come to be referred to as the contextual theory of meaning, which says that the meaning of a scientific term depends, in some way or other, on how this term is incorporated into a scientific theory.

Why adopt the contextual theory of meaning for scientific terms? Suppose the notion of meaning is understood along the lines of the Fregean notion of sense. The sense of a term be understood as that what determines its reference (cf. Church 1956, 6n). It is, furthermore, a reasonable requirement that a semantic theory must account for our understanding of the sense and, hence, our methods of determining the extension of scientific terms (cf. Dummett 1991, 340). For a large number of scientific terms these methods rest upon axioms of one or more scientific theories. There is no way of determining the force function in classical mechanics without using some axiom of this theory. Familiar methods make use of Newton's second law of motion, Hooke's law, the law of gravitation etc. Likewise, virtually all methods of measuring temperature rest upon laws of thermodynamics. Take measurement by a gas thermometer which is based on the ideal gas law. The laws of scientific theories are thus essential to our methods of determining the extension of scientific terms. The contextual theory of meaning, therefore, makes intelligible how students in a scientific discipline and scientists grasp the meaning, or sense, of scientific terms. On this account, understanding the sense of a term is knowing how to determine its referent, or extension, at least in part.

The contextual theory of meaning can be traced back at least to the work of Duhem. His demonstration that a scientific hypothesis in physics cannot be tested in isolation from its theoretical context is joined with and motivated by semantic considerations, according to which it is physical theories that give meaning to the specific concepts of physics (Duhem 1906, 183). Poincare´ (1902, 90) literally claims that certain scientific propositions acquire meaning only by virtue of the adoption of certain conventions. Perhaps the most prominent and explicit formulation of the contextual theory of meaning is to be found in Feyerabend's landmark “Explanation, Reduction, and Empiricism” (1962, 88):

For just as the meaning of a term is not an intrinsic property but is dependent upon the way in which the term has been incorporated into a theory, in the very same manner the content of a whole theory (and thereby again the meaning of the descriptive terms which it contains) depends upon the way in which it is incorporated into both the set of its empirical consequences and the set of all the alternatives which are being discussed at a given time: once the contextual theory of meaning has been adopted, there is no reason to confine its application to a single theory, especially as the boundaries of such a language or of such a theory are almost never well defined.

The accounts of a contextual theory of meaning in the works of Duhem, Poincare´ and Feyerabend are informal insofar as they do not crystallize into a corresponding formal semantics for scientific terms. Such a crystallization is brought about by some of the formal accounts of theoretical terms to be expounded in Section 4.

The view that meaning is bestowed upon a theoretical term through the axioms of a scientific theory implies that only axiomatized or axiomatizable scientific theories contain theoretical terms. In fact, all formal accounts of the semantics of theoretical terms are devised to apply to axiomatic scientific theories. This is due, in part, to the fact that physics has dominated the philosophy of science for a long time. One must wonder, therefore, whether there are any theoretical terms in, for example, evolutionary biology which has as yet resisted complete axiomatization. Arguably, there are. Even though evolutionary biology has not yet been axiomatized, we can recognize general propositions therein that are essential to determining certain concepts of this theory. Consider the following two propositions. (i) Two DNA sequences are homologous if and only if they have a common ancestor sequence. (ii) There is an inverse correlation between the number of mutations necessary to transform one DNA-sequence S1 into another S2 and the likelihood that S1 and S2 are homologous. Notably, these two propositions are used to determine, among other methods, relations of homology in evolutionary biology. The majority of general propositions in scientific theories other than those of physics, however, have instances that fail to be true. (Some philosphers of science have argued that this so even for a large number of axioms in physics.) Formal semantics of theoretical terms in scientific theories with default axioms are presently being developed.

2. Criticisms and Refinements of the Theory-Observation Distinction

2.1 Criticisms

The very idea of a clear-cut theory-observation distinction has received much criticism. First, with the help of sophisticated instruments, such as telescopes and electron microscopes, we are able to observe more and more entities, which had to be considered unobservable at a previous stage of scientific and technical evolution. Electrons and a-particles which can be observed in a cloud chamber are a case in point (Achinstein 1965). Second, assume observability is understood as excluding the use of instruments. On this understanding, examples drawing on the use of cloud chambers and electron microscopes, which are adduced to criticize the theory-observation distinction, can be dealt with. However, we would then have to conclude that things being perceived with glasses are not observed either, which is counterintuitive (Maxwell 1962). Third, there are concepts applying to or being thought to apply to both macroscopic and submicroscopic particles. A case in point are spatial and temporal relations and the color concepts that play an important role in Newton's corpuscle theory of light. Hence, there are clear-cut instances of observation concepts that apply to unobservable entities, which does not seem acceptable (cf. Putnam 1962).

These objections to the theory-observation distinction can be answered in a relatively straightforward manner from a Carnapian perspective. As explained in Section 1.1, Carnap (1936/37, 1966) was quite explicit that the philosopher's sense of observation excludes the use of instruments. As for an observer wearing glasses, a proponent of the theory-observation distinction finds enough material in Carnap (1936/37, 455) to defend her position. Carnap is aware of the fact that color concepts are not observable ones for a color-blind person. He is thus prepared to relativize the distinction in question. In fact, Carnap's most explicit explanation of observability defines this notion in such a way that it is relativized to an organism (1936/37, 454n).

Recall, moreover, that Carnap's theory-observation distinction was not intended to do justice to our overall understanding of these notions. Hence, certain quotidian and scientific uses of ‘observation’, such as observation using glasses, may well be disregarded when this distinction is drawn as long as the distinction promises to be fruitful in the logical analysis of scientific theories. A closer look reveals that Carnap (1966, 226) agrees with critics of the logical empiricists' agenda, such as Maxwell (1962) and Achinstein (1965), on there being no clear-cut theory-observation distinction (see also Carnap's early (1936/37, 455) for a similar statement):

There is no question here of who [the physicist thinking that temperature is observable or the philosopher who disagrees, H. A.] is using the term ‘observable’ in the right or proper way. There is a continuum which starts with direct sensory observations and proceeds to enormously complex, indirect methods of observation. Obviously no sharp line can be drawn across this continuum; it is a matter of degree.

A bit more serious is Putnam's (1962) objection drawing on the application of apparently clear-cut instances of observation concepts to submicroscopic particles. Here, Carnap would have to distinguish between color concepts applying to observable entities and related color concepts applying to unobservable ones. So, the formal language in which the logical analysis proceeds would have to contain a predicate ‘red1’ applying to macroscopic objects and another one ‘red2’ applying to submicroscopic ones. Again, such a move would be in line with the artificial, or ideal language philosophy that Carnap proclaimed (see Lutz (2012) for a sympathetic discussion of artificial language philosophy.)

There is another group of criticisms coming from the careful study of the history of science: Hanson (1958), Feyerabend (1962) and Kuhn (1962) aimed to show that observation concepts are theory-laden in a manner that makes their meaning theory-dependent. In Feyerabend's (1978, 32) this contention takes on the formulation that all terms are theoretical. Hanson (1958, 18) thinks that Tycho and Kepler were (literally) ‘seeing’ different things when perceiving the sun rising because their astronomical background theories were different. Kuhn (1962) was more tentative when expounding his variant of the theory-ladenness of observation. In a discussion of the Sneed formalism of the structuralist school, he favored a theory-observation distinction that is relativized, first to a theory and second to an application of this theory (1976).

Virtually all formal accounts of theoretical terms in fact assume that those phenomena that a theory T is meant to account for can be described in terms whose semantics does not depend on T . The counter thesis that even the semantics of putative observation terms depends on a quotidian or scientific theory, therefore, attacks a core doctrine coming with the logical empiricists' and subsequent work on theoretical terms. A thorough discussion and assessment of theory-ladenness of observation in the works of the great historians of science is beyond the scope of this entry. Bird (2004), Bogen (2009) and Oberheim and Hoyningen-Huene (2009) are entries in the present encyclopedia that address, amongst other things, this issue.

2.2 Refinements

There is a simple, intuitive and influential proposal how to relativize the theory-observation distinction in a sensible way: a term t is theoretical with respect to a theory T, or for short, a T-term if and only if it is introduced by the theory T at a certain stage in the history of science. O-terms, by contrast, are those that were antecedently available and understood before T was set forth (Lewis 1970; cf. Hempel 1973). This proposal draws the theory-observation distinction in an apparently sharp way by means of relativizing that distinction to a particular theory. Needless to say, the proposal is in line with the contextual theory of meaning.

The distinction between T-terms and antecedently available ones has two particular merits. First, it circumvents the view that any sharp line between theoretical and observational terms is conventional and arbitrary. Second, it connects the theory-observation distinction with what seemed to have motivated that distinction in the first place, viz., the investigation how we come to understand the meaning of terms that appear to be meaningful in virtue of certain scientific theories.

A similar proposal of a relativized theory-observation distinction was made by Sneed in his seminal The Logical Structure of Mathematical Physics (1971, Ch. II). Here is a somewhat simplified and more syntactic formulation of Sneed's criterion of T-theoreticity:

Definition 1 (T-theoreticity)
A term t is theoretical with respect to the theory T, or for short, T-theoretical if and only if any method of determining the extension of t, or some part of that extension, rests on some axiom of T.

It remains to explain what it is for a method m of determining the extension of t to rest upon an axiom f. This relation obtains if and only if the use of m depends on f being a true sentence. In other words, m rests upon f if and only if the hypothetical assumption of f being false or indeterminate would invalidate the use of m in the sense that we would be lacking the commonly presumed justification for using m. The qualification ‘or some part of that extension’ has been introduced in the present definition because we cannot expect a single measurement method to determine the extension of a scientific quantity completely. T-non-theoreticity is the negation of T-theoreticity:

Definition 2 (T-non-theoreticity)
A term t is T-non-theoretical if and only if it is not T-theoretical.

The concepts of classical particle mechanics (henceforth abbreviated by CPM) exemplify well the notions of T-theoreticity and T-non-theoreticity. As has been indicated above, all methods of determining the force acting upon a particle make use of some axiom of classical particle mechanics, such as Newton's laws of motion or some law about special forces. Hence, force is CPM-theoretical. Measurement of spatial distances, by contrast, is possible without using axioms of CPM. Hence, the concept of spatial distance is CPM-non-theoretical. The concept of mass is less straightforward to classify as we can measure this concept using classical collision mechanics (CCM). Still, it was seen to be CPM-theoretical by the structuralists since CCM appeared reducible to CPM (Balzer et al. 1987, Ch. 2).

Suppose for a term t once introduced by a scientific theory T1 novel methods of determination become established through another theory T2, where these methods do not depend on any axiom of T1. Then, t would neither qualify as T1-theoretical nor as T2-theoretical. It is preferable, in this situation, to relativize Definition 1 to theory-nets N, i.e., compounds of several theories. Whether there are such cases has not yet been settled.

The original exposition of the theoreticity criterion by Sneed (1971) is a bit more involved as it makes use of set-theoretic predicates and intended applications, rather technical notions of what became later on labeled the structuralist approach to scientific theories. There has been a lively discussion, mainly but not exclusively within the structuralist school, how to express the relativized notion of theoreticity most properly (Balzer 1986; 1996). As noted above, Kuhn (1976) proposed a twofold relativization of theoreticity, viz., first to a scientific theory and second to applications of such theories.

Notably, Sneed's criterion of T-theoreticity suggests a strategy that allows us to regain a global, non-relativized theory-observation distinction: simply take a term t to be theoretical if and only if it holds, for all methods m of determining its extension, that m rests upon some axiom of some theory T. A term t is non-theoretical, or observational, if and only if there are means of determining its extension, at least in part, that do not rest upon any axiom of any theory. This criterion is still relative to our present stage of explicit axiomatic theorizing but comes nonetheless closer to the original intention of Carnap's theory-observation distinction, according to which observation is understood in the narrow sense of unaided perception.

3. Two Problems of Theoretical Terms

The problem of theoretical terms is a recurrent theme in the philosophy of science literature (Achinstein 1965; Sneed 1971, Ch. II; Tuomela 1973, Ch. V; Friedman 2011). Different shades of meaning have been associated with this problem. In its most comprehensive formulation, the problem of theoretical terms is to give a proper account of the meaning and reference of theoretical terms. There are at least two kinds of expression that pose a distinct problem of theoretical terms, respectively. First, unary predicates referring to theoretical entities, such as ‘electron’, ‘neutrino’ and ‘nucleotide’. Second, non-unary theoretical predicates, such as ‘homology’ in evolutionary biology and theoretical function expressions, such as ‘force’, ‘temperature’ and ‘intensity of the electromagnetic field’ in physics. Sneed's problem of theoretical terms, as expounded in (1971, Ch. II), concerns only the latter kind of expression. We shall now start surveying problems concerning the semantics of expressions for theoretical entities and then move on to expressions for theoretical relations and functions.

3.1 Theoretical Entities

A proper semantics for theoretical terms involves an account of reference and one of meaning and understanding. Reference fixing needs to be related to meaning as we want to answer the following question: how do we come to refer successfully to theoretical entities? This question calls for different answers depending on what particular conception of a theoretical entity is adopted. The issue of realism and its alternatives, therefore, comes into play at this point.

For the realist, theoretical entities exist independently from our theories about the world. Also, natural kinds that classify these entities exist independently from our theories (cf. Psillos 1999; Lewis 1984). The instrumentalist picture is commonly reported to account for theoretical entities in terms of mere fictions. The formalist variant of instrumentalism denies theoretical terms to have referents at all. Between these two extreme cases there is a number of intermediate positions.[1]

Carnap (1958; 1966, Ch. 26) attempted to attain a metaphysically neutral position so as to avoid a commitment to or denial of scientific realism. In his account of the theoretical language of science, theoretical entities were conceived as mathematical ones that are related to observable events in certain determinate ways. An electron, for example, figures as a certain distribution of charge and mass in a four-dimensional manifold of real numbers, where charge and mass are mere real-valued functions. These functions and the four-dimensional manifold itself are to be related to observable events by means of univsersal axioms. Notably, Carnap would not have accepted a charcterization of his view as antirealist or non-realist since he thought the metaphysical doctrine of realism to be void of content.

In sum, there are three major and competing characterizations of a theoretical entity in science that are in line with the common theoreticity criterion according to which such an entity is inaccessible by means of unaided perception. First, theoretical entities are characterized as mind and language independent. Second, theoretical entities are mind and language dependent in some way or other. Third, they are conceived as mathematical entities that are related to the observable world in certain determinate ways. We may thus distinguish between (i) a realist view, (ii) a collection of non-realist views and (iii) a Pythagorean view of theoretical entities.

Now, there are three major accounts of reference and meaning that have been used, implicitly or explicitly, for the semantics of theoretical terms: (i) the descriptivist picture, (ii) causal and causal-historical theories and (iii) hybrid ones that combine descriptivist ideas with causal elements (Reimer 2010). Accounts of reference and meaning other than these play no significant role in the philosophy of science. Hence, we need to survey at least nine combinations consisting, first, of an abstract characterization of the nature of a theoretical entity (realist, non-realist and Pythagorean), and, second, a particular account of reference (descriptivist, causal and hybrid). Some of these combinations are plainly inconsistent and, hence, can be dealt with very briefly. Let us start with the realist view of theoretical entities.

The realist view

The descriptivist picture is highly intuitive with regard to our understanding of expressions referring to theoretical entities on the realist view. According to this picture, an electron is a spatiotemporal entity with such and such a mass and such and such a charge. We detect and recognize electrons when identifying entities having these properties. The descriptivist explanation of meaning and reference makes use of theoretical functions, mass and electric charge in the present example. The semantics of theoretical entities, therefore, is connected with the semantics of theoretical relations and functions, which will be dealt with in the next subsection. It seems to hold, in general, that theoretical entities in the sciences are to be characterized in terms of theoretical functions and (non-unary) relations.

The descriptivist account, however, faces two particular problems with regard to the historic evolution of scientific theories. First, if descriptions of theoretical entities are constitutive of the meaning of corresponding unary predicates, one must wonder what the common core of understanding is that adherents of successive theories share and whether there is such a core at all. Were Rutherford and Bohr talking about the same type of entities when using the expression ‘electron’? Issues of incommensurability arise with the descriptivist picture (Psillos 1999, 280). A second problem arises when elements of the description of an entity being given by a predecessor theory T are judged wrong from the viewpoint of the successor theory T'. Then, on a strict reading of the descriptivist account, the corresponding theoretical term failed to refer in T. For if there is nothing that satisfies a description, the corresponding expression has no referent. This is a simple consequence of the theory of description by Russell in his famous “On Denoting” (1905). Hence, an account of weighting descriptions is needed in order to circumvent such failures of reference.

As is well known, Kripke (1980) set forth a causal-historical account of reference as an alternative to the descriptivist picture. This account starts with an initial baptism that introduces a name and goes on with causal chains transmitting the reference of the name from speaker to speaker. In this picture, Aristotle is the man once baptized so; he might not have been the student of Plato or done any other thing commonly attributed to him. Kripke thought this picture to apply both to proper names and general terms. It is hardly indicated, however, how this picture works for expressions referring to theoretical entities (cf. Papineau 1996). Kripke's story is particularly counterintuitive in view of the ahistorical manner of teaching in the natural sciences, wherein the original, historical introduction of a theoretical term plays a minor role in comparison to up-to-date textbook and journal explanations. Such explanations are clearly of the descriptivist type. The Kripkean causal story can be read as an account of reference fixing without being read as a story of grasping the meaning of theoretical terms. Reference, however, needs to be related to meaning so as to ensure that scientists know what they are talking about and are able to identify the entities under investigation. Notably, even for expressions of everyday language, the charge of not explaining meaning has been leveled against Krikpe's causal-historical account (Reimer 2010). The same charge applies to Putnam's (1975) causal account of reference and meaning, which Putnam himself abandoned in his (1980).

A purely causal or causal-historical account of reference does not seem a viable option for theoretical terms. More promising are hybrid accounts that combine descriptivist intuitions with causal elements. Such an account has been given by Psillos (1999, 296):

  1. A term t refers to an entity x if and only if x satisfies the core causal description associated with t.
  2. Two terms t' and t denote the same entity if and only if (a) their putative referents play the same causal role with respect to a network of phenomena; and (b) the core causal description of t' takes up the kind-constitutive properties of the core causal description associated with t.

This account has two particular merits. First, it is much closer to the way scientists understand and use theoretical terms than purely causal accounts. Because of this, it is not only an account of reference but also one of meaning for theoretical terms. In purely causal accounts, by contrast, there is a tendency to abandon the notion of meaning altogether. Second, it promises to ensure a more stable notion of reference than in purely descriptivist accounts of reference and meaning. Notably, the kind of causation that Psillos's hybrid account refers to is different from the causal-historical chains that Kripke thought responsible for the transmission of reference among speakers. No further explanation, however, is given of what a kind-constitutive property is and how we are to recognize such a property. Psillos (1999, 288n) merely infers the existence of such properties from the assumption of there being natural kinds.

Non-realist views

Non-realist and antirealist semantics for theoretical terms are motivated by the presumption that the problem of theoretical terms has no satisfying realist solution. What does a non-realist semantics of theoretical terms look like? The view that theoretical entities are mere fictions often figures only in realist portrays of antirealism and is hardly seriously maintained by any philosopher of science in the 20th century. Quine's comparison of physical objects with the gods of Homer in his (1951) seems to be an exception. If one were to devise a formal or informal semantics for the view that theoretical entities are mere fictions, a purely descriptive account seems most promising. Such an account could in particular make heavy use of the Fregean notion of sense. For this notion was introduced, amongst other objectives, with the intent to explain our understanding of expressions like ‘Odysseus’ and ‘Pegasus’. One would have to admit, however, that sentences with names that lack a referent may well have a truth-value and as such deviate from Frege. Causal elements do not seem of much use in the fiction view of theoretical entities.

Formalist variants of instrumentalism are a more serious alternative to realist semantics than the fiction view of theoretical entities. Formalist views in the philosophy of mathematics are ones which aim to account for mathematical concepts and objects in terms of syntactic entities and operations thereupon within a calculus. Such views have been carried over to theoretical concepts and objects in the natural sciences, with the qualification that the observational part of the calculus is interpreted in such a way that its symbols refer to physical or phenomal objects. Cognitive access to theoretical entities is thus explained in terms of our cognitive access to the symbols and rules of the calculus in the context of an antecedent understanding of the observation terms. Formalist ideas were sympathetically entertained by Hermann Weyl (1949). He was driven towards such ideas by adherence to Hilbert's distinction between real and ideal elements and the corresponding distinction between real and ideal propositions (Hilbert 1926). Propositions of the observation language were construed as real ones in the sense of this Hilbertian distinction by Weyl, whereas theoretical propositions as ideal ones. The content of an ideal proposition is to be understood in terms of the (syntactic) consistency of the whole system consisting of ideal and real propositions being asserted. This is the defining property of an ideal proposition.

The Pythagorean view

We still need to discuss the view that theoretical entities are mathematical entities being related to observable events in certain determinate ways. This theory is clearly of the descriptivist type, as we shall see more clearly when dealing with the formal account by Carnap in Section 4. No causal elements are needed in Carnap's Pythagorean empiricism.

It is fair to characterize the Pythagorean view in general by saying that it shifts the problem of theoretical terms to the theory of meaning and reference for mathematical expressions. The question of how we are able to refer successfully to electrons is answered by the Pythagorean by pointing out that we are able to refer successfully to mathematical entities. Moreover, the Pythagorean explains, it is part of the notion of an electron that corresponding mathematical entities are connected to observable phenomena by means of axioms and inference rules. The empirical surplus of theoretical entities in comparison to “pure” mathematical entities is thus captured by axioms and inference rules that establish connections to empirical phenomena. Since mathematical entities do not, by themselves, have connections to observable phenomena, the question of truth and falsehood may not be put in a truth-conditional manner for those axioms that connect mathematical entities with phenomenal events (cf. Section 4.2). Carnap (1958), therefore, came to speak of postulates when referring to the axioms of a scientific theory.

How do we come to refer successfully to mathematical entities? This, of course, is a problem in the philosophy of mathematics. (For a classical paper that addresses this problem see Benacerraf (1973)). Carnap has not much to say about meaning and reference of mathematical expressions in his seminal “The Methodological Character of Theoretical Concepts” (1956) but discusses these issues in his “Empiricism, Semantics, and Ontology” (1950). There he aims at establishing a metaphysically neutral position that avoids a commitment to Platonist, nominalist or formalist conceptions of mathematical objects. A proponent of the Pythagorean view other than Carnap is Hermann Weyl (1949). As for the cognition of mathematical entities, Weyl largely followed Hilbert's formalism in his later work. Hence, there is a non-empty intersection between the Pythagorean view and the formalist view of theoretical entities. Unlike Carnap, Weyl did not characterize the interpretation of theoretical terms by means of model-theoretic notions.

3.2 Theoretical Functions and Relations

For theoretical functions and relations, a particular problem arises from the idea that a theoretical term is, by definition, semantically dependent upon a scientific theory. Let us recall the above explanation of T-theoreticity: a term t is T-theoretical if and only if any method of determining the extension of t, or some part of that extension, rests on some axiom of T. Let f be such an axiom and m be a corresponding method of determination. The present explanation of T-theoreticity, then, means that m is valid only on condition of f being true. The latter dependency holds because f is used either explicitly in calculations to determine t or in the calibration of measurement devices. Such devices, then, perform the calculation implicitly. A case in point is measurement of temperature by a gas thermometer. Such a device rests upon the law that changes of temperature result into proportional changes in the volume of gases.

Suppose now t is theoretical with respect to a theory T. Then it holds that in order to measure t, we need to assume the truth of some axiom f of T. Suppose, further, that t has occurrences in f, as is standard in examples of T-theoreticity. From this it follows that, in standard truth-conditional semantics, the truth-value of f is dependent on the semantic value of t. This leads to the following epistemological difficulty: on the one hand, we need to know the extension of t in order to find out whether f is true. On the other hand, it is simply impossible to determine the extension of t without using f or some other axiom of T. This mutual dependency between the semantic values of f and t makes it difficult, if not even impossible, to have evidence for f being true in any of its applications.

We could, of course, use an alternative measurement method of t, say one resting upon an axiom ? of T, to gain evidence for the axiom f being true in some selected instances. This move, however, only shifts the problem to applications of another axiom of T. For these applications the same type of difficulty arises, viz., mutual dependency of the semantic values of ? and t. We are thus caught either in a vicious circle or in an infinite regress when attempting to gain evidence for the propriety of a single measurement of a theoretical term. Sneed (1971, Ch. II) was the first to describe that particular difficulty in the present manner and termed it the problem of theoretical terms. Measurement of the force function in classical mechanics exemplifies this problem well. There is no method of measuring force that does not rest upon some law of classical mechanics. Likewise, it is impossible to measure temperature without using some law that depends upon either phenomenological or statistical thermodynamics.

Though its formulation is primarily epistemological, Sneed's problem of theoretical terms has a semantic reading. Let the meaning of a term be identified with the methods of determining its extension, as in Section 1.2. Then we can say that our understanding of T-theoretical relations and functions originates from the axioms of the scientific theory T. In standard truth-conditional semantics, by contrast, one assumes that the truth-value of an axiom f is determined by the semantic values of those descriptive constants that have occurrences in f. Among these constants, there are theoretical terms of T. Hence, it appears that standard truth-conditional semantics does not accord with the order of our grasping the meaning of theoretical terms. In the next section, we shall encounter indirect means of interpreting theoretical terms. These prove to be ways out of the present problem of theoretical terms.

4. Formal Accounts

A few notational conventions and preliminary considerations are necessary to explain the formal accounts of theoretical terms and their semantics. Essential to all of these accounts is the division of the set of descriptive symbols into a set V-o of observational and another set V-t of theoretical terms. (The descriptive symbols of a formal language are simply the non-logical ones.) A scientific theory thus be formulated in a language L(V-o,V-t). The division of the descriptive vocabulary gives rise to a related distinction between T- and C-axioms among the axioms of a scientific theory. The T-axioms contain only V-t symbols as descriptive ones, while the C-axioms contain both V-o and V-t symbols. The latter axioms establish a connection between the theoretical and the observational terms. TC designates the conjunction of T- and C-axioms and A(TC) the set of these axioms. Let n1,…,nk be the elements of V-o and t1,…,tn the elements of V-t. Then, TC is a proposition of the following type:

(TC)       TC(n1,…,nk, t1,…, tn)

As for the domain of interpretation of L(V-o,V-t), Ramsey (1929) assumes there to be only one for all descriptive symbols. Carnap (1956, 1958), by contrast, distinguishes between a domain of interpretation for observational terms and another for theoretical terms. Notably, the latter domain contains exclusively mathematical entities. Ketland (2004) has emphasized the importance of distinguishing between an observational and a theoretical domain of interpretation, where the latter is allowed to contain theoretical entities, such as electrons and protons.

TC is a first-order sentence in a large number of accounts, as in Ramsey's seminal “Theories” (1929). Carnap (1956; 1958), however, works with higher-order logic to allow for the formulation of mathematical propositions and concepts.

4.1 The Ramsey Sentence

The Ramsey sentence of a theory TC in the language L(V-o,V-t) is obtained by the following two transformations of the conjunction of T- and C-axioms. First, replace all theoretical symbols in this conjunction by higher-order variables of appropriate type. Then, bind these variables by higher-order existential quantifiers. As result one obtains a higher-order sentence of the following form:

(TCR)       ?X1…?Xn TC(n1, …, nk, X1, …, Xn)

where X1, …, Xn are higher-order variables. This sentence says that there is an extensional interpretation of the theoretical terms that verifies, together with an antecedently given interpretation of the observation language L(V-o), the axioms TC. The Ramsey sentence expresses an apparently weaker proposition than TC, at least in standard truth-conditional semantics. If one thinks that the Ramsey sentence expresses the proposition of a scientific theory more properly than TC, one holds the Ramsey view of scientific theories.

Why should one prefer the Ramsey view to the standard one? Ramsey (1929, 120) himself seemed to have something like a contextual theory of meaning in mind when proposing the replacement of theoretical constants with appropriate higher-order variables:

Any additions to the theory, whether in the form of new axioms or particular assertions like a(0, 3) are to be made within the scope of the original a, ß, ?. They are not, therefore, strictly propositions by themselves just as the different sentences in a story beginning ‘Once upon a time’ have not complete meanings and so are not propositions by themselves.

a, ß, and ? figure in this explanation as theoretical terms to be replaced by higher-order variables. Ramsey goes on to suggest that the meaning of a theoretical sentence f is the difference between

  1. (TC ? A ? f)R

and

  1. (TC ? A)R

where A stands for the set of observation sentences being asserted and (...)R for the operation of Ramsification, i.e., existentially generalizing on all theoretical terms. This proposal of expressing theoretical assertions clearly makes such assertions dependent upon the context of the theory TC. Ramsey (1929, 124) thinks that a theoretical assertion f is not meaningful if no observational evidence can be found for either f or its negation. In this case there is no stock A of observation sentences such that (1) and (2) differ in truth-value.

Another important argument in favour of the Ramsey view was given later by Sneed (1979, Ch. III). It is easy to show that the problem of theoretical terms (Section 3.2) does not arise in the first place on the Ramsey view. For by TCR it is only claimed that there are extensions of the theoretical terms satisfying each axiom of the set A(TC) under a given interpretation of the observational language. No claim, however, is made by TCR as to whether or not the sentences of A(TC) are true. Nonetheless, it can be shown that TCR and TC have the same observational consequences:

Proposition 1   For all L(V-o) sentences f, TCR ? f if and only if TC ? f, where ? designates the relation of logical consequence.

Hence, the Ramsey sentence cannot be true in case the original theory TC is not consistent with the observable facts. For a discussion of empirical adequacy and Ramsification see Ketland (2004).

One difficulty, however, remains with the Ramsey view. It concerns the representation of deductive reasoning, for many logicians the primary objective of logic. Now, Ramsey (1929, 121) thinks that the ‘incompleteness’ of theoretical assertions does not affect our reasoning. No formal account, however, is given that relates our deductive practice, in which abundant use of theoretical constants is made, to the existentially quantified variables in the Ramsey sentence. One thing we lack is a translation of theoretical sentences (other than the axioms) that is in keeping with the view that the meaning of a theoretical sentence f is the difference between (TC ? A ? f)R and (TC ? A)R. As Ramsey observes, it would not be correct to take (TC ? A ? f)R as a translation of a theoretical sentence f since both (TC ? A ? f)R and (TC ? A ? ¬ f)R may well be true. Such a translation would not obey the laws of classical logic. These laws, however, are supposed to govern deductive reasoning in science. A proper semantics of theoretical terms must take the peculiarities of these terms into account without revising the rules and axioms of deduction in classical logic.

There thus remains the challenge of relating the apparent use of theoretical constants in deductive scientific reasoning to the Ramsey formulation of scientific theories. Carnap was well aware of this challenge and addressed it using a sentence that became labeled later on the Carnap sentence of a scientific theory (Carnap 1958; 1966, Ch. 23):

(AT)       TCR ? TC

This sentence is part of a proposal to draw the analytic-synthetic distinction at the global level of a scientific theory (as this distinction proved not to be applicable to single axioms): the analytic part of the theory is given by its Carnap sentence AT, whereas the synthetic part is identified with the theory's Ramsey sentence in light of Proposition 1. Carnap (1958) wants AT to be understood as follows: if the Ramsey sentence is true, then the theoretical terms be interpreted such that TC comes out true as well. So, on condition of TCR being true, we can recover the original formulation of the theory in which the theoretical terms occur as constants. For, obviously, TC is derivable from TCR and AT using modus ponens.

From the viewpoint of standard truth-conditional semantics, however, this instruction to interpret the Carnap sentence appears arbitrary, if not even misguided. For in standard semantics, the Ramsey sentence may well be true without TC being so (cf. Ketland 2004). Hence, the Carnap sentence would not count as analytic, as Carnap intended. Carnap's interpretation of AT receives a sound foundation in his (1961) proposal to define theoretical terms using Hilbert's epsilon operator, as we shall see in Section 4.3.

4.2 Indirect Interpretation

The notion of an indirect interpretation was introduced by Carnap in his Foundations of Logic and Mathematics (1939, Ch. 23–24) with the intention of accounting for the semantics of theoretical terms in physics. It goes without saying that this notion is understood against the background of the notion of a direct interpretation. Carnap had the following distinction in mind. The interpretation of a descriptive symbol is direct if and only if (i) it is given by an assignment of an extension or an intension, and (ii) this assignment is made by expressions of the metalanguage. The interpretation of a descriptive symbol is indirect, by contrast, if and only if it is specified by one or several sentences of the object language, which then figure as axioms in the respective calculus. Here are two simple examples of a direct interpretation:

‘R’ designates the property of being rational.

‘A’ designates the property of being an animal.

The predicate ‘H’, by contrast, is interpreted in an indirect manner by a definition in the object language:

?x(Hx ? Rx ? Ax)

Interpretation of a symbol by a definition counts as one type of indirect interpretation. Another type is the interpretation of theoretical terms by the axioms of a scientific theory. Carnap (1939, 65) remains content with a merely syntactic explanation of indirect interpretation:

The calculus is first constructed floating in the air, so to speak; the construction begins at the top and then adds lower and lower levels. Finally, by the semantical rules, the lowest level is anchored at the solid ground of the observable facts. The laws, whether general or special, are not directly interpreted, but only the singular sentences.

The laws A(TC) are thus simply adopted as axioms in the calculus without assuming any prior interpretation or reference to the world for theoretical terms. (A sentence f being an axiom of a calculus C means that f can be used in any formal derivation in C without being a member of the premisses.) This account amounts to a formalist understanding of the theoretical language in science. It has two particular merits. First, it circumvents Sneed's problem of theoretical terms since the axioms are not required to be true in the interpretation of the respective language that represents the facts of the theory-independent world. The need for assuming such an interpretation is simply denied. Second, the account is in line with the contextual theory meaning for theoretical terms as our understanding of such terms is explained in terms of the axioms of the respective scientific theories (cf. Section 1.2).

There are less formalist accounts of indirect interpretation in terms of explicit model-theoretic notions by Przelecki (1969, Ch. 6) and Andreas (2010).[2] The latter account proves to formally work out ideas about theoretical terms in Carnap (1958). It emerged from an investigation into the similarities and dissimilarities between Carnapian postulates and definitions. Recall that Carnap viewed the axioms of a scientific theory as postulates since they contribute to the interpretation of theoretical terms. When explaining the Carnap sentence TCR ? TC, Carnap says that, if the Ramsey sentence is true, the theoretical terms are to be understood in accordance with some interpretation that satisfies TC. This is the sense in which we can say that Carnapian postulates contribute to the interpretation of theoretical terms in a manner akin to the interpretation of a defined term by the corresponding definition. Postulates and definitions alike impose a constraint on the admissible, or intended, interpretation of the complete language L(V), where V contains basic and indirectly interpreted terms.

Yet, the interpretation of theoretical terms by axioms of a scientific theory differs in several ways from that of a defined term by a definition. First, the introduction of theoretical terms may be joined with the introduction of another, theoretical domain of interpretation, in addition to the basic domain of interpretation in which observation terms are interpreted. Second, it must not be assumed that the interpretation of theoretical terms results in a unique determination of the extension of these terms. This is an implication of Carnap's doctrine of partial interpretation (1958), as will become obvious at the end of this section. Third, axioms of a scientific theory are not conservative extensions of the observation language since they enable us to make predictions. Definitions, by contrast, must be conservative (cf. Gupta 2009). Taking these differences into account when observing the semantic similarities between definitions and Carnapian postulates suggests the following explanation: a set A(TC) of axioms that interprets a set V-t of theoretical terms on the basis of a language L(V-o) imposes a constraint on the admissible, or intended, interpretations of the language L(V-o,V-t). An L(V-o,V-t) structure is admissible if and only if it (i) satisfies the axioms A(TC) and (ii) extends the intended interpretation of L(V-o) to include an interpretation of the theoretical terms.

In more formal terms (Andreas 2010, 373; Przelecki 1969, Ch. 6):

Definition 3 (Set S of admissible structures)
Let A-o designate the intended interpretation of the observation language. Further, MOD(A(TC)) designates the set of L(V-o,V-t) structures that satisfy the axioms A(TC). EXT(A-o,V-t,D-t) is the set of L(V-o,V-t) structures that extend A-o to interpret the theoretical terms, where these terms are allowed to have argument positions being interpreted in a domain D-t of theoretical entities.

  1. If MOD(A(TC)) n EXT(A-o,V-t,D-t) ? Ø, then S := MOD(A(TC)) n EXT(A-o,V-t,D-t);
  2. If MOD(A(TC)) n EXT(A-o,V-t,D-t) = Ø, then S := EXT(A-o,V-t,D-t).

Given there is a range of admissible, i.e., intended structures, the following truth-rules for theoretical sentences are intuitive:

Definition 4 (Truth-rules for theoretical sentences)
?: L(V-o,V-t) ? {T, F, I}.

  1. ?(f) := T if and only if for all structures A ? S, A ? f;
  2. ?(f) := F if and only if for all structures A ? S, A ? f;
  3. ?(f) := I (indeterminate) if and only if there are structures A1, A2 ? S such that A1 ? f but not A2 ? f.

The idea lying behind these rules comes from supervaluation logic (van Fraassen 1969; Priest 2001, Ch. 7). A sentence is true if and only if it is true in every admissible structure. It is false, by contrast, if and only if it is false in every admissible structure. And a sentence does not have a determinate truth-value if and only if it is true in, at least, one admissible structure and false in, at least, another structure that is also admissible.

A few properties of the present semantics are noteworthy. First, it accounts for Carnap's idea that the axioms A(TC) have a twofold function, viz., setting forth empirical claims and determining the meaning of theoretical terms (Carnap 1958). For, on the one hand, the truth-values of the axioms A(TC) depend on empirical, observable facts. These axioms, on the other hand, determine the admissible interpretations of the theoretical terms. These two seemingly contradictory properties are combined by allowing the axioms A(TC) to interpret theoretical terms only on condition of there being a structure that both extends the given interpretation of the observation language and that satisfies these axioms. If there is no such structure, the theoretical terms remain uninterpreted. This semantics, therefore, can be seen to formally work out the old contextual theory of meaning for theoretical terms.

Second, Sneed's problem of theoretical terms (Section 3.2) does not arise in the present semantics since the formulation of this problem is bound to standard truth-conditional semantics. Third, it is closely related to the Ramsey view of scientific theories as the following biconditional holds:

Proposition 2   TCR if and only if for all f ? A(TC), ?(f) = T.

Unlike the Ramsey account, however, the present one does not dispense with theoretical terms. It can be shown rather that allowing for a range of admissible interpretations as opposed to a single interpretation does not affect the validity of standard deductive reasoning (Andreas 2010). Hence, a distinctive merit of the indirect interpretation semantics of theoretical terms is that theoretical constants need not be recovered from the Ramsey sentence in the first place.

The label partial interpretation is more common in the literature to describe Carnap's view that theoretical terms are interpreted by the axioms or postulates of a scientific theory (Suppe 1974, 86–95). The partial character of interpretation is retained in the present account since there is a range of admissible interpretations of the complete language L(V-o,V-t). This allows for the interpretation of theoretical terms to be strengthened by further postulates, just as Carnap demanded in his (1958) and (1961). To strengthen the interpretation of theoretical terms is to further constrain the range of admissible interpretations of L(V-o,V-t).

4.3 Direct Interpretation

Both the Ramsey view and the indirect interpretation semantics deviate from standard truth-conditional semantics at the level of theoretical terms and theoretical sentences. Such a deviation, however, was not felt to be necessary by all philosophers that have worked on theoretical concepts. Tuomela (1973, Ch. V) defends a position that he calls semantic realism and that retains standard truth-conditional semantics. Hence, direct interpretation is assumed for theoretical terms by Tuomela. Yet, semantic realism for theoretical terms acknowledges there to be an epistemological distinction between observational and theoretical terms. Tuomela's (1973, Ch. I) criterion of the theory-observation distinction largely coincides with Sneed's above expounded criterion. Since direct interpretation of theoretical terms amounts just to standard realist truth-conditions, there is no need for a further discussion here.

4.4 Defining Theoretical Terms

In Weyl (1949), Carnap (1958), Feyerabend (1962) and a number of further papers we can identify different formulations of the idea that the axioms of a scientific theory determine the meaning of theoretical terms without these axioms qualifying as proper definitions of theoretical terms. This idea has become almost constitutive of the very notion of a theoretical term in the philosophy of science. Lewis (1970), however, wrote a paper with the title “How to Define Theoretical Terms” (1970). A closer look at the literature further reveals that the very idea of explicitly defining theoretical terms goes back to Carnap's (1961) use of Hilbert's epsilon operator in scientific theories. This operator is an indefinite description operator that was introduced by Hilbert to designate some object x that satisfies an open formula f. So

ex f(x)

designates some x satisfying f(x), where x is the only free variable of f (cf. Avigad and Zach 2002). Now, Carnap (1961, 161n) explicitly defines theoretical terms in two steps:

(AT(0))       t = eX TC(X, n1,…,nk)

where X is a sequence of higher-order variables and t a corresponding instantiation. So, t designates some sequence of relations and functions that satisfies TC, in the context of an antecedently given interpretation of V-o. Once such a sequence has been defined via the epsilon-operator, the second step of the definition is straightforward:

(AT(i))       ti = ex (?u1 …?un(t = <u1,…,un> ? x = ui))

Carnap could show these definitions to imply the Carnap sentence AT. Hence, they allow for direct recovery of the theoretical terms for the purpose of deductive reasoning on condition of the Ramsey sentence being true.

Lewis (1970) introduced a number of modifications concerning both the language of the Carnap sentence and its interpretation in order to attain proper definitions of theoretical terms. First, theoretical terms are considered to refer to individuals as opposed to relations and functions. This move is made coherent by allowing the basic language L(V-o) to contain relations like ‘x has property y’. The basic, i.e., non-theoretical language is thus no observation language in this account. Yet, it serves as the basis for introducing theoretical terms. The set V-o of ‘O-terms’ is best described as our antecedently understood vocabulary.

Second, denotationless terms are dealt with along the lines of free logic by Dana Scott (1967). That means denotationless terms, such as an improper description, denote nothing in the domain of discourse. Atomic sentences containing denotationless terms are either true or false. Most notably, the free logic that Lewis refers to has it that an identity that contains a denotationless term on both sides is always true. If just one side of the identity formula has an occurrence of a denotationless term, this identity statement is false.

Third, Lewis (1970) insists on a unique interpretation of theoretical terms, thus rejecting Carnap's doctrine of partial interpretation. Carnap (1961) is most explicit about the indeterminacy that this doctrine implies. This indeterminacy of theoretical terms drives Carnap to using Hilbert's e-operator there, as just explained. For Lewis, by contrast, a theoretical term is denotationless if its interpretation is not uniquely determined by the Ramsey sentence. For a scientific theory to be true, it must have a unique interpretation.

Using these modifications, Lewis transforms the Carnap sentence into three Carnap-Lewis postulates, so to speak:

(CL1)
?y1 … ?yn ?x1 … ?xn (TC(n1,…,nk, x1,…,xn) ? y1 = x1 ? … ? yn = xn) ? TC(n1,…,nk, t1,…,tk)
(CL2)
¬?x1 … ?xn TC(n1,…,nk, x1,…,xn) ? ¬?x(x = t1) ? … ? ¬?x(x = tn)
(CL3)
?x1 … ?xn TC(n1,…,nk, x1,…,xn) ? ¬?y1… ?yn ?x1… ?xn (TC(n1,…,nk, x1,…,xn) ? y1 = x1 ? … ? yn = xn) ? ¬?x(x = t1) ? … ? ¬?x(x=tn)

These postulates look more difficult than they actually are. CL1 says that, if TC has a unique realization, then it is realized by the entities named by t1,…,tk. Realization of a theory TC, in this formulation, means interpretation of the descriptive terms under which TC comes out true, where the interpretation of the V-o terms is antecedently given. So, CL1 is to be read as saying that the theoretical terms are to be understood as designating those entities that uniquely realize TC, in the context of an antecedently given interpretation of the V-o terms. CL2 says that, if the Ramsey sentence is false, the theoretical terms do not designate anything. To see this, recall that ¬?x(x=ti) means, in free logic, that ti is denotationless. In case the theory TC has multiple realizations, the theoretical terms are denotationless too. This is expressed by CL3.

CL1–CL3 are equivalent, in free logic, to a set of sentences that properly define the theoretical terms ti (1 = i = n) :

(D-i)       ti = ?yi ?y1 … ?yi-1 ?yi+1… ?yn ?x1 … ?xn (TC(n1,…,nk, x1,…,xn) ? y1 = x1 ? … ? yi = xi ? … ? yn = xn)

ti designates, according to this definition schema, the i-th component in that sequence of entities that uniquely realizes TC. If there is no such sequence, ti (1 = i = n) is denotationless. Even so, the definitions of theoretical terms remain true if the complete language L(V-o,V-t) is interpreted in accordance with the postulates CL1–CL3, thanks to the use of free logic. Hence, all L(V-o,V-t) interpretations that extend the antecedently given interpretation of L(V-o) can be required to satisfy all definitions D-i.

A few further properties of Lewis's definitions of theoretical terms are noteworthy. First, they specify the interpretation of theoretical terms uniquely. This property is obvious for the case of unique realization of TC but holds as well for the other cases since assignment of no denotation counts as interpretation of a descriptive symbol in free logic. Second, it can be shown that these definitions do not allow for the derivation of any L(V-o) sentences except logical truths, just as the original Carnap sentence did. Lewis, therefore, in fact succeeds in defining theoretical terms. He does so without attempting to divide the axioms A(TC) into definitions and synthetic claims about the spatiotemporal world.

The replacement of theoretical relation and function symbols with individual terms was judged counterintuitive by Papineau (1996). A reformulation, however, of Lewis's definitions using second- or higher-order variables is not difficult to accomplish, as Schurz (2005) has shown. In this reformulation the problem arises that theoretical terms are usually not uniquely interpreted since our observational evidence is most of the time insufficient to determine the extension of theoretical relation and function symbols completely. Theoretical functions, such as temperature, pressure, electromagnetic force etc., are determined only for objects that have been subjected to appropriate measurements, however indirect. In view of this problem, Schurz (2005) suggests letting the higher-order quantifiers range only over those extensions that correspond to natural kind properties. This restriction renders the requirement of unique interpretation of theoretical terms plausible once again. Such a reading was also suggested by Psillos (1999, Ch. 3) with reference to Lewis's (1984) discussion of Putnam's (1980) model-theoretic argument. In that paper, Lewis himself suggests the restriction of the interpretation of descriptive symbols to extensions corresponding to natural kind properties.

One final note on indirect interpretation is in order. Both Carnap (1961) and Lewis (1970) interpret theoretical terms indirectly simply because any definition is an instance of an indirect interpretation. For this reason, Sneed's problem of theoretical terms (Section 3.2) does not arise. Yet, the pattern of Carnap's and Lewis's proposals conforms to the pattern of a definition in the narrow sense and not to the peculiar pattern of indirect interpretation that Carnap (1939) envisioned for the interpretation of theoretical terms. This is why the indirect interpretation semantics has been separated from the present discussion of defining theoretical terms.

5. Conclusion

The very existence of scientific terms whose semantics is dependent upon a scientific theory was already contended by Duhem and Poincare´. Such terms came to be referred to as theoretical terms in 20th century philosophy of science. Properties and entities that are observable in the sense of direct, unaided perception did not seem to depend on scientific theories as forces, electrons and nucleotides did. Hence, philosophers of science and logicians started to investigate the distinct semantics of theoretical terms. Various formal accounts resulted from these investigations, among which the Ramsey sentence by Ramsey (1929), Carnap's notion of indirect interpretation (1939; 1958) and Lewis's (1970) proposal of defining theoretical terms are the most prominent ones. Though not all philosophers of science understand the notion of a theoretical term in such way that semantic dependence upon a scientific theory is essential, this view prevails in the literature.

The theory-observation distinction has been attacked heavily and is presumably discredited by a large number of philosophers of science. Still, this distinction continues to permeate a number of important strands in the philosophy of science, such as scientific realism and its alternatives and the logical analysis of scientific theories. A case in point is the recent interest in the Ramsey account of scientific theories which emerged in the wake of Worral's structural realism (cf. Ladyman 2009). We have seen, moreover, that the formal accounts of theoretical terms work well with a theory-observation distinction that is relativized to a particular theory. Critics of that distinction, by contrast, have commonly attacked a global and static division into theoretical and observational terms (Maxwell 1962; Achinstein 1965). Note finally that Carnap assigned no ontological significance to the theory-observation distinction in the sense that entities of the one type would be existent in a more genuine way than ones of the other.

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