I have argued elsewhere that such beliefs are necessary for gratitude for two reasons (Manela 2019, p. 299). The first is that without them, the feelings and behaviors typical of gratitude likely would not arise. After all, an affective disposition of goodwill, a desire to thank someone, and a desire to benefit that particular person more than one did before, do not typically come about spontaneously, *ex nihilo* in us. And without these affective and behavioral elements of the grateful response, a beneficiary could not count as grateful.

A second reason to think such beliefs are necessary for gratitude is this: even if the affective disposition of goodwill and the behavioral disposition of grateful beneficence could somehow arise in a beneficiary who did not believe his benefactor had benefited him benevolently, without such beliefs there would be nothing defining those feelings and behaviors as feelings and behaviors *of gratitude*, as opposed to, say, feelings and behavior of contrition, or love, or solidarity, or empathy, or concern. After all, the feelings and behaviors typical of gratitude—a desire to see someone fare well and a motivation to benefit them are the same sorts of feelings and behaviors as those of empathy, loyalty, solidarity, filial piety, concern, love, and other attitudes. What makes them feelings and behaviors *of gratitude* in particular, when that is what they are, is their connection to a benefactor's having done or tried to do something good for the beneficiary. This connection appears in two related ways. Part of what makes feelings and behaviors feelings and behaviors *of gratitude to R* is that R's being a benefactor plays an essential role in the *justification* a grateful beneficiary has for why he is disposed to feel those feelings and act beneficently toward R. In other words, insofar as I am grateful to someone, when I am asked why I bear them goodwill and am disposed to treat them beneficently, my response should reference their having been my benefactor—their having benefited me or tried to benefit me benevolently. If my justification for goodwill and beneficence toward someone did not reference their benevolent benefaction but, rather, just their suffering, that attitude might be *compassion* or *sympathy*. If my justification for goodwill and beneficence toward someone referenced not their benevolent benefaction but the fact that they were my parent, that might make such feelings and behaviors those of *filial piety*. If it referenced not their benevolent benefaction but just the fact that they were a victim of my harms in the past, that might make such feelings and behaviors those of *contrition*. So a person with gratitude-typical behaviors and feelings must be inclined to cite a benefactor's benevolent benefaction as part of the justification for those behaviors and feelings in order for them to constitute gratitude.

A second way these behaviors and feelings must be connected to a benefactor's benevolence is this: the details of the benefactor's benevolent benefaction should modulate the amount and content of the behaviors and the intensity of the feelings. In other words, insofar as a person is grateful, if they find out that a benefactor showed more benevolence

to them than they had originally suspected, their affective disposition of goodwill and their tendency toward grateful beneficence should consequently become more robust. If Y's feelings and behaviors toward R didn't change at all in response to changes in how Y perceived the benevolent benefaction of R, it would seem strained to call those behaviors and feelings behaviors and feelings of gratitude to R.

What emerges from this is an argument for why belief in a benefactor's benevolent benefaction is a necessary part of gratitude (or at least necessary for gratitude). Without the belief that a benefactor benevolently benefited or tried to benefit the beneficiary, gratitudetypical feelings and behaviors would seem to lack the connection to benevolent benefaction that defines them as feelings and behaviors of gratitude. Without such a belief, the argument goes, a beneficiary's justification for gratitude-typical feelings and behaviors could not reference a benefactor's benevolent benefaction. Nor could the magnitude of those feelings and behaviors be influenced by the details of the benevolent benefaction. And so those gratitude-typical feelings and behaviors literally would not be behaviors and feelings of gratitude. So belief in a benefactor's benevolent beneficence is necessary for gratitude, because without it, gratitude-typical feelings and behaviors likely would not arise in a beneficiary, and if they did, those feelings and behaviors still would not constitute gratitude.

Now, if Y must have certain beliefs about R's status as a benefactor—certain beliefs about R's motives and intentions in φ-ing—in order for Y to be grateful to R for φ-ing, then it seems it should also be necessary for Y to believe the claims presupposed by those beliefs. And those presuppositions include the belief that R exists. If Y did not believe R existed, Y could not believe that R benefited Y or tried to benefit Y benevolently, and then, for the reasons I just mentioned in the last two paragraphs, Y would not have gratitude-typical feelings and behaviors concerning R, or Y's feelings and behaviors toward R would not be feelings and behaviors of gratitude. And without feelings and behaviors of gratitude toward R, Y could not be grateful to R.

This argument leaves us with *prima facie* reason to think that belief in a benefactor's existence is necessary for gratitude to that benefactor. It also helps define the challenge that must be met by anyone who wants to show that gratitude to doubted benefactors is possible: one must show that it is possible for goodwill and beneficence to arise in a beneficiary, and somehow be definable as feelings and behaviors of gratitude, in the absence of belief that that benefactor exists.

To get a sense for how this challenge might be met, it's worth reflecting on several other attitudes that one might have toward entities whose existence one doubts.

#### **5. Attitudes toward Entities Whose Existence One Doubts**

A little reflection shows that we can have a variety of attitudes toward entities whose existence we doubt. We can fear things, for instance, whose existence we doubt. A person might believe vampires do not exist but still fear them. Such a person might, despite his lack of belief, still hang garlic on his door every night and experience feelings of fear and anxiety on nights when he realizes he forgot to do so, just in case vampires do turn out to exist. That fear might be irrational, but it is nonetheless fear. People can experience fear toward an object if they think such an object might exist—that its existence is possible—and if they have a mental representation of that object as having certain characteristics (e.g., a mental representation of vampires as posing danger to oneself). That belief in the mere possibility can be enough to give rise to certain feelings and to motivate certain behaviors (like hanging garlic on one's doorknob). Those feelings and behaviors are feelings and behaviors of fear because they are justified or explained by referring to a fearsome object: vampires.

Gratitude, though, is unlike fear in at least one important way: being grateful to others is not an attitude about what they might do to us in the future; gratitude is, at least in part, an attitude about *our* treatment *of them* in the future. As I noted in the last section, being grateful to someone includes grateful beneficence: an enhanced motivation to do them a favor or help them out or spare them from harm in the future, as well as an affective disposition of goodwill: a hope that they fare well in the future. In this regard, gratitude is

less like fear and more like *concern for* someone (what we might call *prepositional concern*). But a little reflection shows that concern for someone, like fear, is an attitude one can have toward entities whose existence one doubts. Imagine a golfer who believes, with good reason, that she is alone on a well-secured private golf course. She hits a shot that flies unexpectedly off-course toward a thick grove of trees. Though she believes with a high degree of confidence that nobody is in the trees, she is not certain that no such person exists, and she worries that if there were someone in the trees, they'd be in danger of being struck. Imagine that this thought causes her to shout a warning—not out of habit, and not out of fear for her own legal liability, but purely for altruistic reasons. Her justification for her actions and her feelings of worry center on the harm her action might cause someone, just in case there happens to be someone in the grove of trees. Now imagine it turns out, despite her initial belief, that there was a person in the grove of trees. It seems fair in this case to say that the golfer *felt concern* and *acted out of concern* for someone—and *was concerned* for someone—whose existence she doubted. So we can have some attitudes toward others whom we might affect in the future even if we do not firmly believe the objects of those attitudes exist, so long as we think that such objects might exist and, if they do, that our actions might impact them.

Gratitude, though, is unlike concern in an important way: gratitude is an attitude toward other persons qua *agents*, who do things, not merely persons qua *patients*, who can suffer or be benefited. More specifically, being grateful to someone is an attitude we typically have when see someone as having done something benevolent for us. That is a thought that must make it into a person's justification for gratitude-typical feelings and behaviors in order for those feelings and behaviors to constitute gratitude (as opposed to empathy or filial piety or loyalty). So gratitude requires more content in the cognitive attitude of its bearer than fear or concern do. Gratitude must represent its object not just as potentially open to harm and benefit, but also simultaneously as a past agent who did something rather specific: benevolently benefitted the beneficiary.

In order to show that gratitude to doubted benefactors is possible, what's needed is a plausible case of a beneficiary who is disposed to have gratitude-typical feelings and behaviors toward someone, and to cite that person's past benevolence toward him in his justification of why he has those feelings and behaviors, but who nonetheless never believes such a person ever existed. In the following section, I describe such a case.

#### **6. A Vignette about Gratitude to a Doubted Benefactor**

Imagine Yardley is a world-class professional tracker. In fact, he's the best tracker who ever lived, having successfully tracked hundreds of the world's craftiest human fugitives through dozens of natural environments on seven continents. For most of his life, he has lived a solitary existence. A moderate cynic, he believes that the average person is inherently selfish and not willing to benefit others for their own sakes. Yardley does, however, believe that some people are altruistic, and so even though he himself does not care much for the average person, he typically responds with reasonable levels of genuine gratitude toward those who benevolently help him.

Imagine that one day, Yardley survives the sinking of a ship at sea and washes ashore on a small uncharted island. He searches the island somewhat thoroughly, and, seeing no evidence of anyone else on the island, he comes to believe with a high degree of confidence that he is the only person on the island. The only source of food he notices at first is fish in a small creek in the island's small forest. He tries for hours to catch some fish at the creek, but he fails. He then finds some tools that washed ashore with him, and with those tools, he climbs some tall palm trees and cuts down fruit that otherwise would be inaccessible. He amasses a large hoard of fruit, but although he can cut down enough fruit to survive, he realizes that he will be malnourished if that is all he has to eat. So the next day, he returns to the creek, intent on figuring out how to catch fish. To his surprise, when he arrives at the spot where he had tried to fish the day before, he finds a fish lying on the bank exactly where he had stood the day before.

His first thought is that someone caught and placed the fish there. But then Yardley remembers that he searched the island somewhat thoroughly and found no evidence of anyone on the island. He searches the area around the creek again, more thoroughly, and finds no evidence of any human presence (aside from his own). While he acknowledges it's *possible* that there is someone else on the island with him, on reflection, he maintains his belief that there is no one on the island but him.13 He is, after all, the world's best tracker, and so the odds of there being someone on this island who can evade him are vanishingly small. He thus comes to believe that this fish, fortuitously, must have flopped out of the creek at this exact spot. He eats the fish and returns to the same spot by the creek the next day to teach himself to fish. When he arrives at the spot where he'd found the fish the day before, he finds a crooked pointy branch, which appears to have snapped off a nearby tree. He picks up the branch and, on a whim, tries to use the pointy end as a spear to spearfish in the creek. To his surprise, the branch works perfectly as a spear—its strange angles perfectly correcting for the light diffraction of the creek's surface. Yardley catches as many fish as he can carry. As he fishes, he starts to wonder if this suspiciously effective stick may have been placed here for him to find by someone who wanted him to catch fish. Again, he searches the terrain around the creek, and again, he reaffirms the belief that he is alone on the island—the fish and the stick being mere fortuitous coincidences.

Still, though, he continues to believe in the *possibility* that someone else is on the island—specifically, someone who saw him trying to catch fish and caught one for him and left it for him in case he returned, and then left him a means to catch more fish. And that scenario remains salient in Yardley's mind.

Yardley returns to his camp on the beach, where his large hoard of fruit awaits him. As he eats and enjoys the fish, he thinks about the possibility of a benevolent neighbor. He realizes that if such a person exists and has been on the island for a long time, he probably has not been able to get at the fruit on the tall trees. After all, Yardley was only able to get at that fruit using the tools that washed ashore with him. As a result of having these thoughts, Yardley finds himself motivated to bring an armful of fruit to the creek where he found the fish—just in case there really is a benevolent benefactor who helped him. He does this not out of a selfish desire to keep his possible benefactor's benevolence flowing; rather, he does it out of the same sort of disposition that motivated him, throughout his life, to help benefactors who'd benevolently helped him—that is, out of an intrinsic desire to see his benevolent benefactors be happy or satisfied. Yardley feels a bit silly doing this, since he does not believe there is any such benefactor. But, he reasons, there is a chance there *could* be such a benefactor; and Yardley really has plenty of free time on his hands. Though it makes him feel very silly, he cannot resist the urge to shout, "If you're out there, thank you!" as he leaves the creek. And as he's falling asleep that night, even though he is pretty sure there is no such benefactor—even though he *believes* there is no such benefactor—he imagines a possible world in which there was such a benefactor, and he is satisfied at the thought of his benefactor's being happy when he discovers the fruit.

Now imagine that in the middle of the night, Yardley is awakened by rumbling sounds, and he realizes that the island is a volcano, which has just begun to erupt. Fortunately for him, he is spotted by the crew of a boat that had been searching for him, and he is rescued just in time. He watches from the safety of the boat as the entire island is covered in molten lava and the sea around the island begins to boil. He asks the crew if they saw anyone else on the island, and the reply they did not. Yardley still doubts there was anyone else on the island. In fact, Yardley still *believes*, on reflection, that he was alone on the island. But when he imagines the possible benefactor who benevolently caught him a fish, and reflects on the horrible end that that benefactor would have just suffered, it makes Yardley sad—and sadder than he would be if Yardley had learned of a typical selfish stranger who passed away.

Now imagine that it turns out, despite a lack of evidence, that there really was another person, Randolph, on the island with Yardley. Randolph, who had washed up on shore years earlier, had grown paranoid after years alone on the island, and so he'd taken measures to hide his existence from Yardley. He was able to do this successfully because he himself had been a skilled tracker, like Yardley. Randolph had been watching Yardley ever since Yardley started exploring the island. He had seen Yardley trying to catch fish and, out of benevolence, caught a fish (which he'd learned to do over the years) and left it for Yardley in case Yardley came back. He also left Yardley a fishing stick he had made. He left both these gifts out of genuine benevolence for Yardley. And he left both these gifts in such a way as to make them seem like strokes of good luck. When Yardley came back with fruit, Randolph was surprised (because he did not think he'd left Yardley with sufficient evidence to believe he, Randolph, existed), but he was deeply happy, because he'd never been able to cut down fruit from the tall palm trees. Imagine too that, just as Yardley had feared, Randolph ultimately died when the island's volcano erupted.

Here is a case of a beneficiary and a benefactor where the beneficiary doubts the benefactor's existence.14 And though the case is a hypothetical one, Yardley's reactions in this scenario are not incomprehensible. We can understand what it's like to be Yardley in this case, and many of us might find ourselves responding just as Yardley does if we were in his position. Yardley responds to the fish and fishing spear next to the creek in the ways a grateful beneficiary should respond: Yardley acts the way a typical grateful beneficiary would act (finding a way to reciprocate), and does so for the sort of reasons (with the sort of motivation) that characterizes a typical grateful beneficiary. He does not just leave the fruit randomly, or as a way to get rid of a burdensome surplus of fruit at his camp, or for any selfish reason at all, but, rather, because he has an intrinsic desire to see his benefactor fare well—for the benefactor's sake. And he feels the way a grateful beneficiary should feel: when he sees the island engulfed in lava, he feels sad at the thought of his benefactor suffering and dying. Now, if he were pressed later to justify his behaviors and feelings—to say why he left the fruit by the creek, and why he felt sad when he saw the volcano erupt—his justification would cite or reference his hypothetical benefactor's benevolence. What motivated him was not the thought that the hypothetical neighbor was someone he had hurt, or someone he feared he might hurt, or someone he shared an identity with; it was someone he saw as having benevolently helped him. This thus seems to be a case where Yardley, a beneficiary, feels and acts gratefully—*is grateful*—to someone, Randolph, whose existence he doubts.

In the previous section, I presented an argument for thinking that belief in a benefactor's existence was necessary for gratitude. According to that argument, belief in a benefactor's existence was necessary for gratitude because such a belief was presupposed by the belief that the benefactor acted benevolently, and *that* belief was necessary to motivate a beneficiary's grateful feelings and behaviors. Belief in a benefactor's existence was also necessary for the benefactor's benevolence to appear in the beneficiary's justification for those gratitude-typical feelings and behaviors, and that in turn was necessary for those feelings and behaviors to constitute gratitude (as opposed to other attitudes). But now we can see where that argument goes wrong. Having thoughts of being benevolently benefited may be necessary for gratitude-typical behaviors and feelings to arise and count as gratitude. But belief in a benefactor's existence is not necessary for such thoughts, because another mental state might sometimes be enough to bring those thoughts about. What is that mental state? It's the mental state Yardley had vis à vis Randolph: Belief that Randolph *might* exist, together with a representation of Randolph, if he did exist, as having acted benevolently toward Yardley. That combination of elements, enlivened and made more vivid by the benefits Yardley received, could give rise to behaviors and feelings typical of gratitude. And it could place a benefactor's benevolence in a central role in the beneficiary's justification for why he is disposed to feel and act in gratitude-typical ways.

If this is true, then it would seem that failing to believe in the existence of one's benefactor does not make gratitude to that benefactor incoherent or incomprehensible. Indeed, one could be grateful to a benefactor he does not believe exists, who he *believes does not exist*, as long as he believes that that benefactor *might* exist, and thinking of that possibility helps lead to grateful behavior and an affective disposition of goodwill.15

#### **7. Gratitude to a Benefactor One Hopes Does Not Exist**

The case of Yardley and Randolph shows that doubting a benefactor ever existed is compatible with being grateful to that benefactor. The case suggests another implication worth noting: that gratitude to a benefactor is compatible with *hoping* that benefactor does not exist.

To see this, imagine Yardley is on the rescue boat watching the island become engulfed in lava. He realizes that if Randolph exists, Randolph would be suffering a horrible death right now. And because he hates the thought of Randolph suffering, he hopes that this is not the case. And because he hopes that is not the case, he finds himself hoping Randolph does not exist. Still, though, it seems fair to say he remains grateful to Randolph.16 After all, he is disposed to feel sad at the thought of Randolph's suffering terribly in the eruption, and insofar as that disposition arose from his seeing Randolph as benevolent, that would seem to be a feeling of gratitude.

We could imagine other reasons that Yardley might hope Randolph turns out not to exist. One of those might be that Yardley, who prides himself on his survivalist abilities, likes the thought that he was able to survive for a time on the desert island all alone, dependent on no one but himself. Relatedly, we might imagine Yardley hoping Randolph does not exist because if he did, that would mean Yardley's tracking skills were further from perfection than he'd thought—and he prides himself on having excellent tracking skills. Imagine that for these reasons, Yardley hopes that Randolph turns out not to exist. These reasons might also lead Yardley to suppress his thoughts about Randolph, or suppress motivation to bring Randolph his surplus fruit; and if that happened, then it would seem questionable whether Yardley was grateful to Randolph. But Yardley's hope that Randolph not exist in this case need not make his gratitude to Randolph impossible. Notwithstanding these hopes, we could imagine that even while he believes he is alone on the island, the possibility of someone else on the island still tempts him enough that he's motivated to leave surplus fruit at the place where he found the fish, just in case benefactor Randolph does turn out to exist. It thus seems fair to say that a beneficiary can be grateful to a benefactor, even when he does not believe his benefactor exists, and even when he hopes his benefactor does not exist because of his pride in his skills or in his independence of the help of others. Now, insofar as this sort of pride constitutes a lack of proper humility, this implication has an interesting upshot: a beneficiary can be grateful to a benefactor while failing to be properly humble vis à vis that benefactor and his contributions to the beneficiary's wellbeing. And while this is consistent with some characterizations of gratitude,17 it may be in tension with accounts of gratitude according to which gratitude requires humility (where this is understood as one's acceptance of one's imperfections and interdependence on others).18

#### **8. Implications for Gratitude to Supernatural Entities**

My case in Section 6 demonstrated that beneficiaries can be grateful to benefactors whose existence they doubt. In other words, the mere fact that a beneficiary does not believe a particular entity ever existed does not mean that beneficiary is incapable of gratitude to that entity. I demonstrated this in the case of gratitude to a human benefactor. But the same generalization could also be extended to apply to nonhuman benefactors. In theory, it could be extended to apply to supernatural benefactors, like ancestral spirits, natural spirits, ghosts, angels, gods, and God.19 In the final two sections of this essay, I will explore the consequences of my generalization in one particular spiritual or religious worldview: a monotheistic view, according to which there is one supreme god (God) who benevolently benefits every human being in various ways but leaves human beings with room to reasonably doubt he exists.

The scenario I described in the previous section's vignette could be seen as a close analog of this monotheistic worldview—at least if we accept two assumptions about people's relationships to God. One of these assumptions is that God is benevolent to human beings in roughly the same way human beings can be benevolent to each otherthat is, God does good things for human beings, for our sake, out of an ultimate desire to see us flourish. Another assumption is that gratitude to God is roughly the same as gratitude to human beings (i.e., gratitude to God and gratitude to humans are not fundamentally different concepts). More specifically, this involves the assumption that gratitude is the fitting response to God's benevolence, and that this gratitude to God involves the same sort of affective dispositions (hoping to see the benefactor happy, not disappointed) and grateful beneficence (behaviors aimed at pleasing the benefactor, because benefactor was benevolent) that gratitude to human beings consists of. If we accept these assumptions, then my vignette about Yardley and Randolph becomes a close analog of doubters' gratitude to God.20

In the vignette, Randolph is analogous to God, and Yardley is analogous to a human being who doubts God's existence. And just as my vignette shows that doubt in a human benefactor's existence does not necessarily disqualify someone from being grateful to the doubted person, my vignette implies that doubt in God's existence does not necessarily disqualify someone from being grateful to God. In other words, a person could doubt that God exists but still in theory be grateful to God. This might happen if a doubter experiences a series of unlikely, unexpected benefits, and is able to vividly imagine the possibility that those benefits were benevolently caused by God. Motivated by the thought of this possibility, such a person might go on to act in ways that he thinks would please God, and he might be pleased when he sees things that would please God. Insofar as thoughts of God's benevolence justify those feelings and those behaviors on his part, such a person would be grateful to God—even while he doubted God's existence.

There are three categories of doubters to whom this generalization might apply: religious people in periods of doubt, agnostics, and atheists. Religious people in periods of doubt are people who spend much of their lives believing in God but also go through periods of uncertainty. Some such people, for some period of time, may cease to believe that God exists. My vignette implies, however, that such people are still, during their periods of doubt, capable of gratitude to God. Such people would be like a person in Yardley's position who most of the time believed there probably *was* a benefactor helping him out, but who occasionally went through stretches of doubt. Imagine such a person went through a period of doubt while he was carrying his surplus fruit to the site where he found the fish, but he nonetheless continued to bring his surplus fruit to that spot through his period of doubt. Imagine he was motivated by his thoughts of his hypothetical benefactor's benevolence and a desire to please his benefactor just in case that benefactor turns out to exist, just the way the constant doubter Yardley was in the original vignette. Such a person would still be acting gratefully, and thus would still be grateful, to Randolph during his period. Imagine again that this occasional doubter in Yardley's position goes through a moment of doubt as he sees the island's volcano erupt. Despite that, he might still feel sadness at the mere possibility of his benefactor existing and dying a terrible death. He would thus have feelings of gratitude even in his moments of doubt. By the same token, a religious person in a period of doubt who continues to do things that will please God, motivated by thoughts of God's benevolence, is likewise still acting gratefully. And such a person who still feels saddened by thoughts of God's disappointment, and uplifted by thoughts of God's pleasure, might still have feelings of gratitude in periods of doubt. Religious people in periods of doubt are thus capable of being grateful to God through such periods.

This possibility should come as a relief to those religious people who worry that they have been ungrateful to God during their periods of doubt. After all, in many monotheistic sects, being grateful to God is an essential part of leading a good life, and so subscribers to such sects who go through periods of doubt might worry that during those periods, they have failed to lead good lives. If my arguments are correct, though, then such periods need not worry every religious person who goes through periods of doubt, since periods of doubt need not be periods of ungratefulness. Indeed, it is possible for a religious person to

go through periods of doubt but still have been continuously grateful to God throughout his life.

Another class of people who doubt God's existence are agnostics, whom, for the purposes of this essay, I will understand as people who do not have a belief one way or the other on the question of whether God exists. Agnostics, as I classify them here, are people who suspend judgment on the question of God's existence. They take themselves to lack sufficient evidence to believe he exists and to lack sufficient evidence to believe he does not. On the question of whether God exists, they are like someone facing the question of whether a coin toss that is about to happen will come up heads or tails. They do not believe it will come up heads and they do not believe it will come up tails. Agnostics are analogous to someone in Yardley's position who, upon considering all the evidence, does not come to believe that a benefactor exists, and does not come to believe that a benefactor does *not* exist on the island. He is unsure, and so he suspends judgment. Despite this suspended judgment, though, such a person might find motivation to carry surplus fruit to the site where he found the fish—motivation that comes from his thoughts of his hypothetical benefactor's benevolence. And he might still lament the thought of such a benefactor perishing when he sees the island's volcano erupt. Such a person was still acting gratefully and having grateful feelings, and thus was still grateful, to Randolph, despite not believing Randolph existed. By the same token, an agnostic who, motivated by thoughts of God's benevolence, does things that will please God, or grieves when he thinks of God's disappointment, is likewise still acting gratefully and feeling gratitude, and thus *being* grateful, to God. Agnosticism need not entail ungratefulness to God.

This possibility could play an important role in agnostics' deliberation about whether they believe God exists. After all, our beliefs are not always immediately obvious to us upon introspection, and many agnostics might occasionally wonder whether they really lean toward theism or atheism. Some agnostics who feel gratitude to God might, consciously or unconsciously, reason along the following lines: *I can't tell just from introspection whether I believe in this God, who might exist. But I do feel grateful to this possibly existing God. So I must believe God exists, since one can only be grateful to entities one believes exist.* As I have shown, though, the final premise in this argument can be questioned, and so the argument, even though valid, is unsound. Gratitude to God need not entail belief that God exists. Anyone who thinks otherwise, and reasons along the lines of the argument I just articulated, commits what we might call the *grateful agnostic's fallacy*. 21

A final group of doubters worth considering is atheists. Atheists, as I will define them, are people who believe that God does not exist. Some atheists are *certain* God does not exist. Those atheists, I believe, may not be capable of gratitude to God. But atheists need not be certain God does not exist in order to be atheists, just like one need not be certain climate change is occurring in order to believe climate change is occurring. Some atheists, in other words, might put some credence in the possibility that God does exist and have a rather clear representation of what God, if he did exist, would be like.22 These *uncertain atheists* are analogous to Yardley in the vignette from Section 6. And as I showed, it was plausible that Yardley was still able to find motivation to carry surplus fruit to the site where he found the fish, motivated by his thoughts of his hypothetical benefactor's benevolence. And we could imagine Yardley weeping at the thought of such a benefactor perishing when he sees the island's volcano erupt. Yardley was still acting gratefully and having grateful feelings, and thus was still grateful, to Randolph, despite being pretty sure that Randolph did not exist. By the same token, an uncertain atheist who, motivated by thoughts of God's benevolence, does things that he thinks would please God, or grieves when he thinks of God's disappointment, could likewise still be acting gratefully and feeling gratitude—could thus *be* grateful, to God. Atheism need not entail ungratefulness. And gratitude to God need not entail belief in God.

This possibility is significant for at least two reasons. The first is parallel to a remark I made about agnosticism above: this possibility could play an important role in uncertain atheists' deliberation about whether they are grateful to God. Many such atheists might, consciously or unconsciously, reason along the following lines: *I can't tell just from introspection whether I feel any gratitude to God. But I do not believe God exists, so I cannot feel grateful to God, because one can only feel grateful to entities one believes exist.* As I have shown, though, the final premise in this argument is not safe to assume, and so the argument, even though valid, is unsound. Belief that God does not exist need not entail an absence of gratitude to God. We can think of this unsound argument as the *uncertain atheist's fallacy*.

The possibility that uncertain atheists can be grateful to God also has implications for a line reasoning about the existence and appropriateness of a mysterious kind of gratitude some philosophers call "cosmic gratitude."23 This line of reasoning appeals to the experiences of atheists who, despite believing God does not exist, nevertheless feel deep, pervasive gratitude to something big, profound and non-human. Such people often find themselves reflecting on certain good things that have befallen them, or all good things that have befallen them, that seem not to have come from human benevolence. And these reflections sometimes seem to give rise to experiences of gratitude: feelings of gratitude and motivation to "make a return" by benefiting other people. An uncertain atheist who has such experiences might be tempted to reason as follows: *As an atheist*, *I believe God does not exist. But I feel deep, pervasive gratitude to something bigger, broader and more profound than any human entity. Since I do not believe God exists, it cannot be gratitude to God that I feel, because we cannot be grateful to entities we believe do not exist. So the gratitude I feel must be a special kind of gratitude, distinct from gratitude to human beings and gratitude to God. So there must exist a special kind of gratitude, "cosmic gratitude," which is substantially different from gratitude to human beings and gratitude to God.* The argument, however, is not sound, because we can doubt the premise that says we cannot be grateful to entities we believe do not exist. Once that premise is removed, the argument's conclusion no longer follows from the remaining premises. Another possibility emerges that could explain why at least some atheists seem to feel deep gratitude toward something profound and superhuman: they may be experiencing feelings of gratitude triggered by (1) good things in their lives, together with (2) a mental representation they have of a benevolent non-human benefactor (whose existence they doubt but regard as possible).24 As my vignette about Yardley suggested, these two elements could potentially induce significant experiences of gratitude even in people who are highly confident no agential benefactor exists. Insofar as this explanation is simpler than positing the fittingness of a mysterious sort of gratitude (e.g., gratitude to a cosmic non-agent like the universe, the laws of physics, or existence itself), the experiences of such atheists leave us with less reason to take seriously the possibility of "cosmic gratitude."25 I any event, my arguments suggest that uncertain atheists should take care not to jump too quickly from their experiences of gratitude and their disbelief in God's existence to conclusions about the existence or fittingness of mysterious sorts of gratitude.26

I've shown a number of interesting implications that follow from the possibility that nonbelievers are capable of gratitude to God. I want to close this section by pointing out three more implications that the Yardley vignette has for gratitude to God that are worth mentioning. The first two of these correspond to the two implications at the end of the previous section.

The first of these implications is that some degree of gratitude to God is compatible with hope that God turns out not to exist. There might be believers, agnostics and atheists who hope God does not exist because they look around the world and lament how disappointed God would be if he did exist. These people are like Yardley who, upon watching the island become engulfed in lava, hopes that Randolph turns out not to exist, because he hates the thought of his benefactor suffering—and thus prefers a world in which that benefactor did not exist. Despite that, his disposition to be sad at the thought of Randolph's suffering shows him to be grateful to Randolph. By the same token, a person's disposition to lament God's disappointment reveals such a person still to be grateful to God (at least insofar as that disposition arises from a representation of God as having been benevolent). And so believers, agnostics and atheists may still be grateful to God even if they hope he does not exist.

There are other reasons someone might hope God turns out not to exist—besides sadness at thoughts of God's disappointment. Some people might hope God does not exist because they take pride in the thought that they are solely responsible for the good they have achieved, and are averse to the thought of their own dependence on something so much more powerful than themselves. Some people might hope God turns out not to exist because their reason has led them to that conclusion, and they dislike the thought that their reason could have led them to a false conclusion. These people are like Yardley hoping Randolph does not exist because he'd prefer a world where he survived on the island all on his own, with help from no one, or because he'd prefer a world where his tracking skills never led him astray. Despite that, Yardley's disposition to be sad at the thought of Randolph's suffering, and his motivation to bring Randolph surplus fruit, show him to be grateful to Randolph. By the same token, a person's disposition to lament God's disappointment and to be motivated to please God in certain ways, just in case he turns out to exist, would reveal such a person still to be grateful to God (at least insofar as that disposition arises from a representation of God as having been benevolent). Gratitude to God, then, is compatible not only with believing God does not exist, but *hoping* God does not exist as well.27

This last point suggests a second implication about doubters' gratitude to God worth considering: it suggests that some degree of gratitude to God is compatible with a lack of proper humility vis à vis God. Many people think that proper humility vis à vis God requires people to accept and welcome the fact that they are dependent on him for the good things they receive and the good things they achieve. My arguments imply, however, that a person may prefer God not exist because he prefers a world in which he did not depend on God, and yet still have grateful feelings and exhibit grateful behaviors toward God. So it may be possible to be grateful to God, at least to some extent, without being fully humble vis à vis God.

These considerations highlight a final implication my arguments entail: that gratitude to God is consistent with lack of faith that God exists, and lack of faith in God.28 This follows from the fact that Yardley's gratitude to Randolph is consistent with Yardley's lack of faith *that* Randolph exists, and lack of faith *in* Randolph. Now, in order to have faith that *p*, I take it that one must have (1) some positive cognitive attitude (like belief, acceptance or reliance) toward *p*, and (2) a positive conative orientation toward *p* (e.g., a desire that *p* be the case).29 But Yardley lacks both of these. He lacks (1), insofar as he *dis*believes Randolph exists. And he lacks (2) insofar as he hopes Randolph turns out not to exist. By the same token, an atheist might be grateful to God, but lack faith that God exists because he believes God does not exist and hopes God turns out not to exist.

Faith *that* God exists is different from faith *in* God. In order to have faith *in* some entity, R, one must place trust in R, have some confidence that R will come through when needed—enough so that one's behavior should be shaped or changed by this trust and confidence. In other words, faith in R should motivate ventures of trust. But this need not be the case with Yardley. Imagine that after bringing the fruit to the clearing, Yardley finds some delicious looking mushrooms at the bottom of a pit near the stream. He wants to get the mushrooms, but he realizes that if he goes into the pit, he may not be able to get out on his own. The thought crosses his mind that if Randolph did exist and heard Yardley's cries, Randolph might come to help him. But Yardley has so little confidence in Randolph's existence that he passes up the chance to get the mushrooms. In fact, the whole time he's on the island, he never counts on Randolph helping him out, never undertakes a risky venture that Randolph could save him from. This is still consistent, though, with his behaving and feeling gratefully toward Randolph in the ways I described in the vignette. So Yardley can be grateful to Randolph even while he lacks faith in Randolph. By the same token, an atheist could be grateful to God even while lacking faith in God. This can happen

if an atheist has grateful behaviors and feelings vis à vis God, but never trusts in God's existence enough to take risky ventures relying on God's help.

#### **9. Broader Implications and Open Questions**

In this essay, I have argued that it is possible to be grateful to a benefactor whose existence you doubt. Belief that a benefactor *might* exist can sometimes be enough to motivate gratitude-typical feelings and behaviors, and can do so in a way that makes such feelings and behaviors genuinely manifestations of gratitude. This can happen if a person believes the benefactor might exist, has a representation of that benefactor as benevolent, and comes to see certain past benefits as possibly having been bestowed by such a benefactor. Just as we can be grateful to human benefactors whose existence we doubt, it seems we can also be grateful to supernatural entities whose existence we doubt, insofar as we see those entities as possibly existing, have representations of them as benevolent, and think that certain past benefits might have come from them. And this means that doubters (religious people in periods of doubt, agnostics, and even some atheists) are capable of gratitude to God. Indeed, not only can people be capable of gratitude to a God they doubt; they can also be grateful to a God they *hope* turns out not to exist, and they can also be grateful to God when they lack faith in God or faith that God exists. I showed how these conclusions have important implications for doubters of various kinds. I also showed how my arguments help us understand the concept of gratitude more generally—by, e.g., revealing that belief is not a necessary cognitive element of the grateful response.

In this final section of my essay, I would like to highlight several questions my arguments raise as salient, and suggest some broad implications my arguments might have for doubters' relationship to God and to religion more generally.

Perhaps the most obvious questions my conclusion raises are empirical questions. I have shown that it is theoretically possible for doubters of various kinds to be, to some degree, grateful to God. How prevalent are grateful feelings and grateful behaviors among different stripes of nonbelievers? What psychological and social factors tend to predict whether a particular doubter will show certain degrees of gratitude toward God? These are questions that cannot be answered from an armchair. And insofar as people have assumed belief in a benefactor was necessary for gratitude to that benefactor, these questions likely struck empirical researchers as nonstarters. My conclusions in this essay, however, suggest that they are not.

My arguments in this essay also highlight the urgency of certain theoretical and normative questions about gratitude to doubted benefactors. Some of these are conceptual questions about other conditions, besides belief in a benefactor's existence, that make gratitude possible. Given that doubt in God's existence does not in itself disqualify a doubter from being grateful to God, we might wonder: are there other disqualifiers specific to doubters that would make gratitude to God impossible or attenuated? One possibility, for instance, is that agnostics and atheists typically lack a clear representation of God's interests, motives and values, and so they may lack a clear sense of how to reciprocate a benefit from God, or a clear understanding of the conditions under which God would be disappointed. And that could make being grateful to God impossible for such doubters. Another possibility is that agnostics and atheists may lack an understanding of how an omnipotent God sacrifices in benefiting them; and insofar as benefactor sacrifice is necessary for gratitude, they may thus be incapable of having a representation of God as worthy of gratitude. Another possibility is that doubters might have somewhat negative representations of God. Indeed, many doubters become doubters because of events that make them angry at God. To what extent is it possible for a doubter to be grateful to God for certain benefits while being angry with him for other things? Each of these questions is important for understanding the extent to which doubters are capable of gratitude to God, and the answer to each of them requires conceptual work on the nature of gratitude.

Perhaps even more pressing than these conceptual questions is a normative question that my conclusions in the essay make salient: Insofar as doubters are capable of gratitude to benefactors whose existence they doubt, when (if ever) *should* they be? And what are the conditions under which doubters *ought* to be grateful to God? Is it always irrational, inappropriate or unreasonable to be grateful to benefactors whose existence one doubts? Or can it sometimes be rational, acceptable and reasonable? Can it ever be *good* to be grateful to benefactors whose existence one doubts? And can it ever be *bad* to *fail* to be grateful to benefactors whose existence one doubts? These are obviously intriguing and important questions. And though they are too large to answer adequately here, I believe that my case of Yardley and Randolph may be helpful in answering them.

Leaving gratitude behind, my arguments in this essay suggest a number of other counterintuitive possibilities about doubters' relationships with God and other supernatural entities. If one does not need to believe in God, or even have faith in God, in order to be grateful to him, perhaps belief and faith are not necessary for other complex social attitudes either. And that suggests the possibility that some doubters (namely, those who believe God might exist) could, to one degree or another, be contrite, forgiving, praising, compassionate, loving, and perhaps even obedient or loyal to God. Now, insofar as these attitudes, together with gratitude, make up the core of a healthy relationship with God, some doubters might be able to go a surprisingly long way toward having a healthy relationship to God.30 And that in turn means, surprisingly, that theists and nontheists, those with faith and those without, or believers and nonbelievers vis à vis any particular religious tradition, share a piece of common ground.31

More generally, this raises the possibility that people who neither believe in nor have faith in the existence of supernatural entities could still nonetheless have robustly religious lives. Even without faith or belief in such entities, certain religious rituals, rites, and practices could still count as genuine expressions of certain important attitudes toward supernatural entities.32 The extent to which this is possible would require careful analyses of each of these attitudes and the extent to which they can be had toward doubted entities. My method in this essay may provide a helpful roadmap for such analyses.

**Funding:** This research was funded by the Templeton Foundation, grant number 61513.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** I would like to thank the faculty of the Siena College Philosophy Department, participants in Biola University's Gratitude to God project, and three anonymous reviewers for helpful insights on earlier versions of this article.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Notes**

and though (for such skeptics) these cases represent only a subset of prepositional gratitude, my analysis of only these cases will still be sufficient to convince such skeptics that one can be grateful without believing one's benefactor exists.

behavioral dispositions that make it up are real. And it's gratitude *to God* because it's the representation of God as having certain characteristics that brings those feelings and behaviors about, and that influences their content and magnitude. The prepositional phrase *to God* need not pick out anything that really exists; it only picks out and defines the representation. Now, if God does not exist, that might make gratitude to God unfitting, inappropriate, or irrational. And if God does not exist, it may be impossible to objectively confirm the extent to which one's representation of God, which sets the parameters of one's gratitude, is accurate. So different people's gratitude to God could manifest very differently—in different behaviors, for instance. But none of that stops gratitude to God from being real gratitude, and people who experience it as being really grateful, in the prepositional or targeted sense.Now, if the objector remains convinced that gratitude to non-existent entities is not real gratitude, there are still versions of my thesis he might be prepared to accept. One statement of my thesis was *failure to believe in a benefactor doesn't always necessarily imply that gratitude to that benefactor is impossible*. The objector should still endorse this statement of my thesis as applying when the benefactor is God, since if God does not exist, then it's not a person's failure to believe in God that makes gratitude to God impossible: it's God's non-existence. I have also stated my thesis more sweepingly, as *it is possible to be grateful to a benefactor whose existence one doubts*. This is a version the objector might not accept as applying when the benefactor is God, because insofar as God doesn't exist, gratitude to God is never possible. But the objector should be prepared to accept a qualified version of this statement of my thesis—one that reads: *insofar as gratitude to some benefactor is possible, it may be possible to be grateful to that benefactor without believing he exists*. That's a version of my thesis that that both the objector and others should be willing to accept as applying to cases where God is a benefactor.

God), when motivated by an atheist's conscience, can be sufficient to make an atheist welcome in heaven. Some Catholics might doubt that this could be sufficient. They might reason that even people who conscientiously do good things must have some relationship with God in order to be welcome in heaven, and atheists, because of their lack of belief, can have no such relationship. My arguments in this essay suggest that line of reasoning is flawed. Nonbelievers, including atheists, might indeed be able to have a relationship with God—a relationship constituted by important attitudes like gratitude and contrition. Such attitudes may turn out not to be necessary to get into heaven. It may be the case that consistent good actions governed by a conscience really are sufficient. That is a question for clerics and theologians to settle. But if such attitudes do turn out to be necessary, that need not exclude nonbelievers from heaven. For more on Francis's remarks about nonbelievers entering heaven, see https://www.ncronline.org/news/vatican/francis-chronicles/my-dad-heaven-little-boy-asks-pope (accessed on 20 December 2021).

32 Here my arguments point in the same direction as remarks made by Wolterstorff (2018).

#### **References**

Berger, Fred. 1975. Gratitude. *Ethics* 85: 298–309. [CrossRef]

Fitzgerald, Patrick. 1998. Gratitude and Justice. *Ethics* 109: 119–53. [CrossRef]

McAleer, Sean. 2012. Propositional Gratitude. *American Philosophical Quarterly* 49: 55–66.

Solomon, Robert C. 2006. *Spirituality for the Skeptic: The Thoughtful Love of Life*. New York: Oxford University Press.

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**1. Introduction**

Chronic kidney disease (CKD) is characterized by progressive loss of kidney function. The main risk factors of developing CKD are the combination of obesity, diabetes and hypertension, which is commonly referred to as metabolic syndrome. Other contributors are autoimmune diseases (e.g., glomerulonephritis), environmental exposures and genetic risk factors [1,2]. Morphologically, persistent low-grade renal inflammation and tubulointerstitial fibrosis are key hallmarks of CKD [3,4]. The complex interplay of fibroblasts, lymphocytes, tubular, and other cell types in the kidney lead to excessive extracellular matrix deposition and the further deterioration of renal function [5,6]. Although unspecific treatments strategies are available (e.g., medications lowering blood pressure), CKD progression is still poorly controlled.

In recent years, novel drug targets, such as transient receptor potential cation channel, subfamily C, and member 6 (TRPC6), emerged [7,8]. *TRPC6* mutations lead to glomerular injury and proteinuria, presumably involving the Ca2+ signaling pathway and resulting in progressive kidney failure [9–12]. Both TRPC6 gain-of-function and loss-of-function cause familial forms of focal segmental glomerulosclerosis (FSGS) [11,13]. Interestingly, in a murine model of kidney injury (unilateral ureteral obstruction (UUO)), *Trpc6*−/<sup>−</sup> deficiency and pharmacological blockade with BI-749327 ameliorated renal fibrosis in C57BL/6J mice [7,8]. Remarkably, these beneficial effects were not observed in the acute stage of kidney injury (AKI) [14]. Thus, TRPC6 inhibition may have effects on renal fibrogenesis during AKI-to-CKD transition. Given this state of affairs, TRPC6 inhibition seems to represent a promising new therapeutic approach to combat progressive renal failure since it potentially affects CKD at later stages after kidney injury. However, it is unknown whether TRPC6 inhibition is effective for inhibiting progressive tubulo-interstitial fibrosis in hypertension and metabolic syndrome.

Recently, by the chemical diversification of (+)-larixol originating from *Larix* decidua resin traditionally used for inhalation, its methylcarbamate congener, named SH045, was developed as a novel, highly potent, subtype-selective inhibitor of TRPC6 [15]. In the present study, we hypothesized that this novel selective TRPC6 inhibitor (SH045) [15] could ameliorate renal fibrogenesis in the New Zealand obese (NZO) mouse model, which is a polygenic model of metabolic syndrome [16]. We studied the therapeutic effects of the in vivo inhibition of TRPC6 by the novel blocker SH045 in the UUO mouse model of accelerated renal fibrogenesis utilizing these mice.


**2.3 Theoretical review**

The exchange rate channel is certainly effective in a given economy with a shallow money market, as well as a deep market for foreign. Overall, random changes in the rate of exchange were observed for Nigeria, specifically in all the years for the period 1980–1990 [2].

#### *2.3.1 Purchasing power parity (PPP)*

Mordi [18] opined that the purchasing power parity (PPP) presupposes "those exchange rates between two currencies at a given period are equal if the ratio of the level of price in a certain amount of products produced" and flowing in two countries and the rate of exchange among those two countries are equal. Adeoye and Saibu [2] observed that purchasing power parity hypothesis is the most useful reason proffered as regards the stability in the long term as well as persistence of the rate of exchange in bilateral relations and that testing for PPP in the long term; this is useful for a lot of purposes: following many monetary models [19], that this rests on the reliability of the PP theory in the long run, in addition to a lot of models that describe the macroeconomy by adopting the PPP to express the nexus to foreign and local development; mostly when examining market economy such as Nigeria. Adenekan et al. [20] did observe that the "PPP theory was criticized for not considering the impact of international capital movements, and suffers from the choice of an appropriate price index used in price calculations." There is thus the need for theoretical postulations that can be all encompassing in application.

#### *2.3.2 Mundell-Fleming theory*

The Mundell–Fleming model is also called the IS-LM-BOP model (or IS-LM-BP) model based on the independent research of Mundell [21] and Fleming [22]. Though the Mundell-Fleming model is an offshoot of the IS-LM model, the original IS-LM model expresses the situation in a closed economy (autarky), this model expresses the situation that exists in an economy that is not closed, but small and open. The impossible trilemma of this theoretical model is the premise for the position that in an economy that is open, a country cannot follow the path of: a fixed rate of exchange, free mobility of capital, and the application of monetary policy that is independent

simultaneously. The operation of a managed float was not specifically captured at the beginning of this model, yet managed float is an imperfect free float or fixed exchange rate regime. The model states that two and not three of the trilemma can be operated simultaneously at any point in time.

Aizenman [23] observed that the financial crises of 1990 have "induced emerging markets to converge to the trilemma's middle ground—managed exchange-rate flexibility, restricted financial integration, and an effective but less prominent monetary independence" and that the crisis of flight of capital has led to financial stability becoming an additional goal in the operation of the trilemma's targeted goals. Hence, the present scenario goes beyond the textbook description unveiled by the trilemma. This encapsulates the theoretical foundations of our present study.

## **2.4 Empirical review**

Ndung'u [24] assessed if the rate of exchange of Kenyan currency is influenced by the monetary authority's policy moves and whether the observed impact transient or consistent. The findings of this research reveal that the rate of exchange in nominal terms for the period 1970–1994 was impacted upon by income increase in real terms, the inflation rate, expansion in the supply of money, the circular movements in the volatility in the real rate of exchange, the cointegrating vectors, and the shocks. A similar research by Ubok-Udom [25] did study the nexus linking the total GDP growth rate in annual terms, GDP attributable to the non-oil sector, and the variations in the rate of exchange spanning 1991–1995. The results of this study further indicate that the rates of growth for total GDP and GDP from non-oil sources seem to reduce or rise with drops or spikes in the rate of exchange in nominal terms. Pattnaik et al. disclosed that the study shows that monetary policy has been useful achieving sustained factors in the exchange rate market revealing the pass-through via the exchange rate to the local inflation rate.

Similar studies such as Adebiyi [26] investigated the influence of the monetary authority's involvement in Nigeria's currency exchange market. The research did not focus on the link that connects shocks from monetary authority's policy moves and changes in the rate of exchange but looked at whether these involvements of the local monetary authority in the foreign exchange market are sterilized or not. The research by An and Sun [27] investigated the linkage between monetary policy, the monetary authority's intrusion into the foreign exchange market, and the rate of exchange for Japan in the form of a unifying model. The results from the study firstly lend support to the hypothesis referred to as "leaning-against-the –wind" as well as the hypothesis referred to as "signaling," though the presence of the "signaling" postulation happens to be rather minimal. Another finding is that the impact of intervention is not effective, possibly even a negative effect. Lastly, normal monetary policy seems to exert a major impact on both the rate of exchange and interventions in foreign exchange.

Cagliarini and Mckibbin [28] used the G-Cubed model for analyzing several sectors in an economy for a cross-country data; examining the possible impact of three shocks on US monetary policy, risk premia, and productivity to determine major relative price movement for the period 2002–2008. A very important summary of the study experimental exercise was done by this study, and it shows that monetary authority's policy moves seem to impact prices in relative terms for as much four years due to the fact that a non-permanent in real interest rates varies across sectors. Asad et al. [29] studied the influence of rate of exchange(in real and effective terms) on inflation on the economy of Pakistan by utilizing secondary data on GDP (in real and nominal terms), effective real rate of exchange, prices, and the supply of money that covered 1973–2007. The results from their study show that the exchange rate (in real terms) has a significant influence on the rate of inflation as

regards the economy of Pakistan; the nexus between the effective real rate of exchange and inflation was found to be positive and strong.

But Dickson [30] did study to examine the influence of volatility in the real rate of exchange on Nigeria's growth in output by adopting annual data from 1970 to 2009. The results of the study did reveal that the nexus between economic growth and rate of exchange volatility was positive in the short run; but in the long run, the relationship for both variables is negative. Adeoye and Saibu [2] analyzed that monetary policy shocks influence via movements in the policy instruments on volatility in the rate of exchange in Nigeria. Specifically the study focused on the relationship between volatility in the rate of exchange and shocks due to monetary policy in Nigeria. The short-run dynamics reveals that changes in monetary policy instruments correlate to the variations in the rate of exchange via process that is selfcorrecting without the involvement of the CBN. Furthermore, the findings from the test for causality that link the volatility in the rate of exchange and the monetary authority's policy tools indicate significant nexus between as regards historical values of the rate of exchange and monetary policy variables. It was observed that shifts in past values of policy tools result in changes in exchange rate volatility.

Nwachukwu et al. [31] modeled the long-run nexus that linked "the Bureau De Change rate of exchange and Nigeria's external reserves in a threshold vector error correction model (TVECM)" econometric methodology by utilizing daily data that spanned from 2014 to 2015. The study concluded that "the adjustment mechanism between the two variables flow from external reserves to BDC exchange rate." Ayomitunde et al. [32] examined the nexus that exists linking monetary authority's policy instruments and the rate of exchange rate Nigeria. The study adopted the Autoregressive Distributed Lag (ARDL) model to achieve the focused goal of the study. The results show that there was a negative nexus linking the Treasury bill rate, cash reserve requirement to the rate of exchange. On the other hand, the policy rate and money supply (broad) have a nexus with the exchange rate that is positive for Nigeria.

Miyajima [33] did carry out a study that was premised on an answer to the question "does the South African rand's relatively large volatility affect inflation?" The derived results indicate that when the volatility in the rate of exchange spikes, it leads to an increase in core inflation, though that impact is limited for the economy of the country under study.

The Nigeria experience shows that some research studies have been carried out on the nexus linking volatility in the rate of exchange and the actions of the Central Bank's stabilizing actions. The study is an attempt to determine the interventions in the markets for the buying and selling of foreign currency by the monetary authority; whether such intervention is sterilized or not sterilized has had the moderating impact on exchange rate volatility. The Nigerian experience from a few studies in the area of the interrelationship between exchange rate volatility and macroeconomic policies of the CBN have been carried out; yet the use of monthly data was not common in all these studies. This study did apply monthly data to reveal that nexus in a more timely manner.

## **3. Methodology**

#### **3.1 Study design**

The study utilizes an ex-post facto research design, the events have already taken place before the analysis, and hence there is no way the data can be manipulated. The "ex post facto design in its application is causal comparative and used when the

researcher aims to establish relationship between the independent and dependent variables with a view to establishing the causal link between them" [34, 35].


**5.2 Precipitation rate**

It is defined as water amount given per unit time in irrigation area (Connelan. 2002). It is generally expressed as mm/h. Main factors which affect precipitation rate are sprinkler flow, distance between sprinkler and distance between laterals. The average precipitation rate is calculated with the following equation.

$$\text{Pr} = 1000^{\circ} \text{q/} \text{S}^{\circ} \text{L} \tag{11}$$

Where.

Pr:The average precipitation rate. mm/h 1000: a constant which converts meters to mm. q:the total flow applied to the area by the sprinklers. m3/h S:the spacing between the sprinkler along lateral. m L: The spacing between rows of sprinkler. m

The flow rate of sprinkler heads automatically changes in case of their making irrigation in different angles. For example, when sprinkler angle decrease from 360° to 180° degree, flow rate increases doubled. Therefore in the system where heads having different angle values are used, the average precipitation rate is calculated by means of the following formula.

$$\text{Pr} \equiv 360000 \,\text{\textdegree q/} \not\text{\textquotesingle} \text{L} \tag{12}$$

Where.

Pr: Average precipitation rate of sprinkler. mm/h 360000:a constant related sprinkler's angel. q: flow rate of sprinkler. m3/h ɸ: Working angel of sprinkler. 0

Irrigation 315

Two different head loss are occurred in duration of water`s reaching from resource to plant in delivery of water in an irrigation system. One of them is friction head loss. Friction head loss occurs due to friction within pipes and it is related to length, dimension and coefficient of roughness of pipes. Friction losses are calculated by means of formula of Darcy-Weisbach

hf=λ(L/D)(v2/2g) (17)

Second local head losses occurring in system is called as geodetic and changes depending on

Flow in pipes is defined as functions of dimension of pipes and velocity of flow. While determining dimension of pipe in irrigation system investment costs can be minimized through choosing the possible smallest dimension of pipe. However keep in mind that reducing dimension of pipe shall increase the velocity as seen in following formula. Increase

Q=A\*V (19)

hl: kf\*V2/2g (18)

type of equipment used. Local head losses are calculated by means of formula below.

Operation unit is considered a number between Nmin and Nmax in planning step.

Nmax: maximum number of station

Ta: irrigation duration, h SA: irrigation interval, day.

**5.6 Hydraulic calculation**

hf:friction head loss m λ:friction coefficient L: length of line (m)

v: velocity of fluid (m/s)

hl: local head losses, m

V: velocity of fluid, m/s

g: acceleration due to gravity, m/s2

D:Inner diameter of pipe work (m)

g: acceleration due to gravity (m/s2)

kf: friction coefficient of irrigation equipment

**5.7 Matching water flow and pressure with pipe size**

of velocity means increase of loss of frictions occurring in pipes.

below.

Where.

Where.

Where.

Q:system flow (m3/h) A: line cross-section area: m2 V: velocity of flow. m/s

Tg: achievable irrigation duration per day, h/day

Precipitation rate requires considering soil infiltration rate in order to prevent runoff and deep seepage to be determined.

### **5.3 Irrigation duration**

Irrigation duration refers to required time for each irrigation. This duration is a function of net water application depth, precipitation rate of sprinkler and irrigation efficiency. Irrigation duration is calculated by the following formula

$$\text{Ta} \Leftarrow \text{dn} / (\text{Pr}^\text{\*} \text{Ea}) \tag{13}$$

Where.

Ta: Irrigation duration. h. dn: net water application depth. mm Pr: Precipitation rate. mm/h Ea: Application efficiency. % (80% can be taken for sprinkler)

### **5.4 Lateral flow**

Lateral flow changes according to the number of sprinkler which is planned to be placed on the lateral and their flow rate. Lateral flow which they need to carry increases similarly as number of heads and flow rate are increased. The lateral flow inlet is determined by equation 14.

$$\mathbb{Q}\models \mathsf{qs}^\*\mathsf{ns}\tag{14}$$

Where. Ql: lateral flow. l/s qs: sprinkler flow. l/s ns: the number of sprinkler on lateral

Sprinkler heads having wetting area in shape of circular, semi-circle and quarter circle on the same lateral are often used depending on geometrical shapes of irrigation area. In case of using heads having different flow rate on a lateral, heads which have the same flow rate are grouped and number of heads are multiplied and the total flow rate of lateral is defined.

#### **5.5 Operation unit**

Unit which is constituted from heads making irrigation in irrigation area is called as operating unit. Maximum operating unit is calculated by means of following formula.

$$\text{Nmax} = (\text{Tg/Ta})^{\text{a}} \text{SA} \tag{15}$$

Minimum operating unit is also calculated by means of following formula.

$$\text{Nmin} \equiv \text{Eq/Q} \tag{16}$$

Where. q:sprinkler flow m3/h Q:system flow m3/h

Precipitation rate requires considering soil infiltration rate in order to prevent runoff and

Irrigation duration refers to required time for each irrigation. This duration is a function of net water application depth, precipitation rate of sprinkler and irrigation efficiency.

Ta=dn/(Pr\*Ea) (13)

Lateral flow changes according to the number of sprinkler which is planned to be placed on the lateral and their flow rate. Lateral flow which they need to carry increases similarly as number of heads and flow rate are increased. The lateral flow inlet is determined by

Ql=qs\*ns (14)

Sprinkler heads having wetting area in shape of circular, semi-circle and quarter circle on the same lateral are often used depending on geometrical shapes of irrigation area. In case of using heads having different flow rate on a lateral, heads which have the same flow rate are grouped and number of heads are multiplied and the total flow rate of lateral is defined.

Unit which is constituted from heads making irrigation in irrigation area is called as operating unit. Maximum operating unit is calculated by means of following formula.

Nmax=(Tg/Ta)\*SA (15)

Nmin=Σq/Q (16)

Minimum operating unit is also calculated by means of following formula.

deep seepage to be determined.

Irrigation duration is calculated by the following formula

Ea: Application efficiency. % (80% can be taken for sprinkler)

**5.3 Irrigation duration**

Ta: Irrigation duration. h.

**5.4 Lateral flow**

equation 14.

Ql: lateral flow. l/s qs: sprinkler flow. l/s

**5.5 Operation unit**

q:sprinkler flow m3/h Q:system flow m3/h

Where.

Where.

Pr: Precipitation rate. mm/h

dn: net water application depth. mm

ns: the number of sprinkler on lateral

Where.

 Operation unit is considered a number between Nmin and Nmax in planning step.

#### **5.6 Hydraulic calculation**

Two different head loss are occurred in duration of water`s reaching from resource to plant in delivery of water in an irrigation system. One of them is friction head loss. Friction head loss occurs due to friction within pipes and it is related to length, dimension and coefficient of roughness of pipes. Friction losses are calculated by means of formula of Darcy-Weisbach below.

$$\text{hf} \! = \lambda (\text{L} / \text{D}) (\text{v} \text{2} / 2 \text{g}) \tag{17}$$

Where.

hf:friction head loss m λ:friction coefficient L: length of line (m) D:Inner diameter of pipe work (m) v: velocity of fluid (m/s) g: acceleration due to gravity (m/s2)

Second local head losses occurring in system is called as geodetic and changes depending on type of equipment used. Local head losses are calculated by means of formula below.

$$\text{lh!: } \text{kf\*V2/2g} \tag{18}$$

Where. hl: local head losses, m kf: friction coefficient of irrigation equipment V: velocity of fluid, m/s g: acceleration due to gravity, m/s2

#### **5.7 Matching water flow and pressure with pipe size**

Flow in pipes is defined as functions of dimension of pipes and velocity of flow. While determining dimension of pipe in irrigation system investment costs can be minimized through choosing the possible smallest dimension of pipe. However keep in mind that reducing dimension of pipe shall increase the velocity as seen in following formula. Increase of velocity means increase of loss of frictions occurring in pipes.

$$\mathbf{Q} \triangleq \mathbf{A}^\* \mathbf{V} \tag{19}$$

Where. Q:system flow (m3/h) A: line cross-section area: m2 V: velocity of flow. m/s

Irrigation 317

system units completely according to requirements in project step provides system performance being in high level. Drip and micro sprinkler irrigation methods should be considered as an alternative method in where sprinkler irrigation system is inadequate or

Allen. R.G. Pareira. L.S. Raes. D. Smith.M. 1998. Crop Evapotranspiration. Guidelines for Computing Crop Water Requirements. *FAO Irrigation and drainage paper 56*. Rome. Anonymous (2009). *Sprinkler Irrigation Systems*. Minister of Agriculture and Rural Areas

Anonymous. (2010). *Sprinkler Irrigation Systems and Design.* Arl Plastic Indistury. Istanbul.

ASCE. (1978). Describing irrigation efficiency and uniformity. *J. lrrig. and Drain. Engr*. 104(1).

Ayyildiz. M. (1990). *Irrigation Water Quality and Salinity Problems*. University of Ankara

Barrett. J. Vinchesi. B. Dobson. R. & Roche. P. (2003). *Golf Course Irrigation: Environmental*

Cardenas-Lailhacar. B. Dukes. M. D. & Miller. G. D. (2008). Sensor Based automation of

Christiansen. J.E. 1941. The uniformity of application of water by sprinkler system. *Agric.*

Cockerham. S. T. & Leinauer. B. (2011). *Turfgrass Water Conservation. (2nd Edition.).* University

Connellan. G. (2002). *Efficient Irrigation: A Reference Manual for Turf and Landscape.* Irrigation

Dines. N. & Brown. K. (2001). *Landscape Architect's Portable Handbook. McGraw-Hill.* New

Doorenboos J. & Pruit. W. O. (1977). Crop Water Requirements. *FAO Irrigation and Drainage*

Gungor. Y. Erozel. A. Z. & Yildirim. O. (2010). *Irrigation (4th Edition.).* University of Ankara

Harmancioglu N. Alparslan N. & Boeele E. (2001). *Irrigation. Health and the Environment: A*

Ingels. J. E. (2003). *Landscaping Principles and Practices(6th Edition.).* Thomson Delmar

Irrigation Association.( 2003.) *Certified golf irrigation auditor*. Falls Church. VA: Irrigation

Melby. P. (1995). *Simplified Irrigation Design: Professional Designer and Installer Version. (2nd*

Novotny. V. Ahern. J. & Brown. P. (2010). *Water Centric Sustainable Communities Planning. Retrofitting and Building The Next Environment*. John Wiley & Sons. USA

*Review of Literature from Turkey (Working paper)*. International Water Management

irrigation on bermudagrass. during wet weather conditions. *J. Irrig. and Drain.*

*Design and Management Practices*. John Wiley & Sons. NY. USA.

of California Agriculture & Natural Resource. CA.USA.

Best Management Training Program. Melbourne. Australia.

Agricultural Production Directorate of Operation of Adana and Education Center.

ineffective.

**7. References**

Adana. Turkey.

*Eng.* 22: 89-92.

York. NY. USA.

*Paper No 24*. Rome.

Learning. NY.USA.

Association.

Agriculture Faculty. Ankara. Turkey.

Agriculture Faculty. Ankara. Turkey.

*Edition.).* John Wiley & Sons. NY.USA.

Institute. Colombo Sri Lanka

*Engr.* 134 (2008). pp 120- 138.

Turkey.

35-41.

Possible minimum dimension of head should be chosen according to limitations of friction loss which are allowed. While choosing dimension of main pipe it is required not to exceed 15% of pressure of pumper. Pressure difference between starting and ending point in lateral shouldn`t be over 20%. Also total head losses in system shouldn`t exceed half of orifice pressure.

#### **5.8 Total dynamic head of the system and power requirement**

Pressure which is necessary for irrigation system is generally carried out through a pump installed at the beginning of system excluding the situations in which adequate elevation is not available between resource and irrigation area. Pump should be chosen at power which provides optimum pressure in the last head in the line which is called as critical line and has head losses at excessive rate in this system. Total dynamic height of pump in a system is calculated by means of following formula;

$$\mathbf{H}\mathbf{t} \mathbf{\overline{r}} \mathbf{s} \mathbf{s} + \mathbf{h}\mathbf{h} + \mathbf{h}\mathbf{e} + \mathbf{h}\mathbf{s}\mathbf{s}\mathbf{c}\tag{20}$$

Where.

Ht: total dynamic head of pump, m

hs: sprinkler operation pressure, m.

hh: total head losses, m.

he: elevation difference between highest point in irrigated area and pump, m

hsuc: suction line height if there is elevation difference between pump and water supply suction line height should be considered.

Also engine power of pump is required to be determined. Pumping Energy which is appropriate for system is determined by means of following formula.

$$\text{Np} = \text{Ht}^\* \text{Q} / 75^\* \,\mu 1 \mu 2 \tag{21}$$

Where. Np:Pump power, HP Ht: Total head loss, m. Q:system capacity, l/s µ1: pump efficieny, %

µ2: driver efficieny, %

Pumps which provide necessary pumping power are included in system by help of catalogues of relevant firms.

#### **6. Conclusion**

Irrigation is one of the main factors on plant growth and quality. Irrigation is applied in areas where evapotranspiration is not met by rain especially in semi-arid or arid climate. Pressurized irrigation methods are used generally in landscape irrigation.

The most common method between pressured irrigation methods is sprinkler irrigation method. Sprinkler irrigation method is preferred by reasons such as providing high corresponding water distribution, its use easily in any kind of soil and area where plants are grown, its low labor costs and fertilization proceeding`s being easily applied . Determining

Possible minimum dimension of head should be chosen according to limitations of friction loss which are allowed. While choosing dimension of main pipe it is required not to exceed 15% of pressure of pumper. Pressure difference between starting and ending point in lateral shouldn`t be over 20%. Also total head losses in system shouldn`t exceed half of orifice

Pressure which is necessary for irrigation system is generally carried out through a pump installed at the beginning of system excluding the situations in which adequate elevation is not available between resource and irrigation area. Pump should be chosen at power which provides optimum pressure in the last head in the line which is called as critical line and has head losses at excessive rate in this system. Total dynamic height of pump in a system is

Ht=hs+hh+he+hsuc (20)

hsuc: suction line height if there is elevation difference between pump and water supply

Also engine power of pump is required to be determined. Pumping Energy which is

Pumps which provide necessary pumping power are included in system by help of

Irrigation is one of the main factors on plant growth and quality. Irrigation is applied in areas where evapotranspiration is not met by rain especially in semi-arid or arid climate.

The most common method between pressured irrigation methods is sprinkler irrigation method. Sprinkler irrigation method is preferred by reasons such as providing high corresponding water distribution, its use easily in any kind of soil and area where plants are grown, its low labor costs and fertilization proceeding`s being easily applied . Determining

Pressurized irrigation methods are used generally in landscape irrigation.

Np=Ht\*Q/75\* µ1µ2 (21)

he: elevation difference between highest point in irrigated area and pump, m

appropriate for system is determined by means of following formula.

**5.8 Total dynamic head of the system and power requirement**

calculated by means of following formula;

Ht: total dynamic head of pump, m hs: sprinkler operation pressure, m.

suction line height should be considered.

hh: total head losses, m.

Np:Pump power, HP Ht: Total head loss, m. Q:system capacity, l/s µ1: pump efficieny, % µ2: driver efficieny, %

catalogues of relevant firms.

**6. Conclusion**

pressure.

Where.

Where.

system units completely according to requirements in project step provides system performance being in high level. Drip and micro sprinkler irrigation methods should be considered as an alternative method in where sprinkler irrigation system is inadequate or ineffective.

#### **7. References**

**1. Introduction**

works have been started.

demands of modern life.

**15**

*Turkey*

**Private Plantation Techniques**

The visual value of a town increases directly proportional to the density of her open and green spaces. Vertically and horizontally formed greenery is an indispensable part of urban design. However, with the advanced technology during the 20 th century, wide construction areas, highways, agricultural and industrial zones have developed in an unplanned manner, and natural resources were abused in an unsystematic way. Unfortunately, the number of natural elements in towns has decreased rapidly in recent years, and with the help of uncoordinated urbanization, the situation has turned for the worse for green areas. If we were to analyze this fact with figures, the example of Ankara, Turkey would prove to be more than enough. In physiological terms, according to oxygen exchange and leaf surface calculation, there is a theoretical need of 25–40 m² green area per person in an urban area. But in Ankara, this figure was 5.1 m² in 1950, 2.8 m² in 1965, and 1.8 m² in 1979. However, the urgency of the matter has been realized during recent years, and inner urban greenery

Improving the environmental conditions of the indoor and outdoor places where humans live, and also to arrange them to become suitable for living, has become a foremost priority. Today, extreme urbanization has become ever fast growing, and inner urban tree planting techniques are changing and improving accordingly too. Nowadays, it is necessary to make use of all new developments in technology and find ways to meet the ever increasing

The first time when large plants were uprooted and transferred to somewhere else was during the Munich Olympic games in Germany. Back then, a whole new Olympic village was created with immense greenery. At that time, this transplantation process was realized with much more labour force and time, also simpler techniques were used. However, the same could be done today with much time effort and time spent, through the use of modern techniques. A very important aspect of landscaping works is the time needed until it reaches an effective power, or in other words, the dimension of time. Trees and landscaping elements need on average 30–40 years to reach an effective power in terms of physics, visual, climactic etc. aspects. Therefore, it is very important to foresee the needs of the future, and do the landscaping planning accordingly. This is a difficult and compulsory responsibility to do. However when a planning is done, people of today believe that reaching a necessary green area needs to be done as rapidly as other advancements.

Ömer Lütfü Çorbac1 and Murat Ertekin2 *1Ankara University, Faculty of Agriculture 2Bartn University, Faculty of Forestry*


**2. Solution to antenna array problems**

Side lobe level, which is one of the key parameters to be minimized for effective performance of the antenna arrays, can actually be optimized or reduced in such a way that the system performance will not be adversely affected. Here, the use of enhanced firefly algorithm is considered.

#### **2.1 Side lobe level (SLL) optimization using enhanced firefly algorithm**

Though the use of enhanced firefly algorithm for effective reduction of side lobe level in antenna array will be presented here, it is however necessary to briefly describe firefly algorithm and state the reason for the choice of enhanced firefly algorithm instead of firefly algorithm itself.

#### *2.1.1 Firefly algorithm*

Humans have learnt so much from the natural world and they have applied such lessons in addressing some problems that face us. The behavior of some living things have been studied extensively and key aspect of their life mimicked by humans. One of such living things is the firefly, an insect that emits light, especially at twilight, to get the attention of other fireflies for matting. It uses that method to prey on other fireflies.

Experts mimic the way firefly flashes light to attract other fireflies to solve problems that are not linear, among others. Hence, there is firefly algorithm (FA) which rely on certain assumptions like: attraction between fireflies is independent of their sex: brightness of firefly is determined by objective function (OF) for such firefly algorithm, the most important things being the light intensity denoted by I and the attractiveness denoted by *β:* The extent of a firefly's brightness will determine how attractive it will be. The following equation gives the light intensity.

$$\mathbf{I}\_{\ } = l\_{\boldsymbol{\theta}} \mathbf{e}^{-\operatorname{yd}^2} \tag{11}$$

Where *lo* is the initial light intensity, y is coefficient of absorption of light while d is the distance between two given fireflies. Similarly, the attraction is given by:

$$
\beta = \beta\_o e^{-\gamma d^2} \tag{12}
$$

## *The Issue of Sidelobe Level in Antenna Array: The Challenge and the Possible Solution DOI: http://dx.doi.org/10.5772/intechopen.106344*

Where *βo* is the attractiveness when d = 0. One major drawback of firefly algorithm is that it takes longer to attain global optimization if the array is bulky. To address this drawback, the firefly algorithm is improved, giving rise to improved or enhanced firefly algorithm (EFA).

### *2.1.2 Enhanced firefly algorithm*

Enhanced firefly algorithm is adopted to address the problem of slow convergence, thereby overcoming the challenge of optimizing sidelobe level without serious consequence on the beam width.

A careful implementation of the flowchart for firefly algorithm can bring about real result in terms of reduction of sidelobe levels. Equations (11 and 12) are very important in the application of enhanced firefly algorithm (**Figure 2**).

#### **2.2 Solution to thinning problem in antenna array**

Addressing thinning issues also solves the problem of cost and power consumption, as these issues come about because of the bulky nature of some antenna arrays. One method that can be used to address the thinning problem associated with antenna array is the genetic algorithm. Compared to some frequently used techniques or methods, genetic algorithm offers great advantages owing to its robustness nature. Key advantages of genetic algorithm include the following:

One version of genetic algorithm that can be applied to solving antenna array thinning problems is the simple genetic algorithm (SGA) version. As the name implies, application of this version of genetic algorithm is actually simple. The steps, in the proper order, are as listed below:

*Antenna Arrays - Applications to Modern Wireless and Space-Born Systems*

#### **Figure 2.**

*Flowchart of enhanced firefly algorithm.*

*The Issue of Sidelobe Level in Antenna Array: The Challenge and the Possible Solution DOI: http://dx.doi.org/10.5772/intechopen.106344*

#### **2.3 Presentation of results and discusion**

This part of the research deals with the results of the work carried out, and the discussion is based on the result. Figures are presented and discussed accordingly.

As can be verified from **Figure 3**, side lobe level is smaller for the optimized thinned array when compared with the level of the sidelobe for a fully populated array. This shows that when optimizing sidelobe level for antenna array using the method of thinning, genetic algorithm can be used for such optimization. The plot shows that for a fully populated array at an angle of 90°, the sidelobe level was 12.967 dB while for the thinned array optimized with genetic algorithm for the same angle of 90°, the level of sidelobe reduced to 16.857 dB.

As **Figure 4** shows, in thinned array optimized with genetic algorithm, the halfpower beamwidth is higher compared to the level for fully populated array. While the half power beamwidth for thinned array optimized with genetic algorithm was 60° from **Figure 4**, for fully populated array the half power beamwidth was about 54.5°. This is an indication that when all the elements of an antenna array are ON, the gain is not the same as when some elements of the array are OFF. Thinning, therefore, is a key way of overcoming some challenges in antenna array, especially if thinning is optimized with genetic algorithm.

In **Figure 5**, the radiation pattern for antenna array of ten elements separated 0.5λ apart, having sidelobe level optimized using genetic algorithm is compared with the radiation pattern for antenna array of ten elements separated 0.5λ apart, having sidelobe level optimized using enhanced firefly algorithm. The plot shows that using enhanced firefly for sidelobe level optimization in antenna array is better than using the popular genetic algorithm. This is because optimizing using enhanced firefly has lower sidelobe level compared to optimizing using genetic algorithm.

#### **Figure 3.**

*A plot of SLL(dB) variation against tilt angle (deg.) for fully populated Array and optimized thinned Array using genetic algorithm.*

#### **Figure 4.**

*A plot of half-power beam width (deg.) variation against tilt angle (deg.) for fully populated Array and optimized thinned Array using genetic algorithm.*

**Figure 5.**

*An Array radiation pattern of ten elements spaced 0.5λ optimized with EFA.*

Presented in **Figure 6** is the radiation pattern for antenna array of ten elements separated 0.5λ apart, with normalized power plotted against radiation angle in degree. On one hand, the sidelobe level is optimized using genetic algorithm and on the other hand, the sidelobe level is optimized using enhanced firefly. The result of the optimizations using genetic algorithm and enhanced firefly algorithm are compared. Again, the plot reveals that using enhanced firefly for sidelobe level optimization in antenna array is better than using genetic algorithm. The reason for this inference is because

*The Issue of Sidelobe Level in Antenna Array: The Challenge and the Possible Solution DOI: http://dx.doi.org/10.5772/intechopen.106344*

**Figure 6.** *A plot of normalized power (dB) against angle (deg) for 10 elements optimized with EFA.*

optimization of sidelobe level using enhanced firefly has lower side lobe level compared to optimization using genetic algorithm.