8.2: Oxygenation Basic Concepts
Several body systems contribute to a person’s oxygenation status, including the respiratory, cardiovascular, and hematological systems. These systems are reviewed in the following sections.
Respiratory System
The main function of our respiratory system is to provide the body with a constant supply of oxygen and to remove carbon dioxide. To achieve these functions, muscles and structures of the thorax create the mechanical movement of air into and out of the lungs called ventilation . Gas exchange occurs at the alveolar level where blood is oxygenated and carbon dioxide is removed, which is called respiration . Several respiratory conditions can affect a patient’s ability to maintain adequate ventilation and respiration, and there are several medications used to enhance a patient’s oxygenation status. Use the following hyperlinks to review information regarding the anatomy and physiology of the respiratory system, common respiratory conditions, and classes of respiratory medications.
Read additional information about the “ Respiratory System ” in Open RN Nursing Pharmacology or use the following hyperlinks to go to specific subsections of the chapter:
-
- Review the anatomy and physiology of the respiratory system.
- Learn about common respiratory disorders .
- Read about common respiratory medications .
Cardiovascular System
In order for oxygenated blood to move from the alveoli in the lungs to the various organs and tissues of the body, the heart must adequately pump blood through the systemic arteries. The amount of blood that the heart pumps in one minute is referred to as cardiac output . The passage of blood through arteries to an organ or tissue is referred to as perfusion . Several cardiac conditions can adversely affect cardiac output and perfusion in the body. There are several medications used to enhance a patient’s cardiac output and maintain adequate perfusion to organs and tissues throughout the body. Use the following hyperlinks to review information regarding the anatomy and physiology of the cardiovascular system, common cardiac disorders, and various cardiovascular system medications.
Read additional information about the cardiovascular system in the “ Cardiovascular & Renal ” chapter in Open RN Nursing Pharmacology or use the following hyperlinks to go to specific subsections of this chapter:
- Review the anatomy and physiology of the cardiovascular system.
- Learn about common cardiac disorders.
- Read about common cardiovascular system medications.
Hematological System
Although the bloodstream carries small amounts of dissolved oxygen, the majority of oxygen molecules are transported throughout the body by attaching to hemoglobin within red blood cells. Each hemoglobin protein is capable of carrying four oxygen molecules. When all four hemoglobin structures contain an oxygen molecule, it is referred to as “saturated.” [1] See Figure 8.1 [2] for an image of hemoglobin protein within a red blood cell with four sites for carrying oxygen molecules.
When oxygenated blood reaches tissues within the body, oxygen is released from the hemoglobin, and carbon dioxide is picked up and transported to the lungs for release on exhalation. Carbon dioxide is transported throughout the body by three major mechanisms: dissolved carbon dioxide, attachment to water as HCO3-, and attachment to the hemoglobin in red blood cells. [3] See Figure 8.2 [4] for an illustration of carbon dioxide transport. [5]
Measuring Oxygen, Carbon Dioxide, and Acid Base Levels
Because the majority of oxygen transported in the blood is attached to hemoglobin, a patient’s oxygenation status is easily assessed using pulse oximetry, referred to as SpO2 . See Figure 8.3 [6] for an image of a pulse oximeter. This reading refers to the amount of hemoglobin that is saturated. The target range of SpO2 for an adult is 94-98%. [7] For patients with chronic oxygenation conditions such as COPD, the target range for SpO2 is often lower at 88% to 92%. Although SpO2 is an efficient, noninvasive method for assessing a patient’s oxygenation status, it is not always accurate. For example, if a patient is severely anemic, the patient has a decreased amount of hemoglobin in the blood available to carry the oxygen, which subsequently affects the SpO2 reading. Decreased perfusion of the extremities can also cause inaccurate SpO2 levels because less blood delivered to the tissues causes a false low SpO2. Additionally, other substances can attach to hemoglobin such as carbon monoxide, causing a falsely elevated SpO2.
A more specific measurement of oxygen and carbon dioxide in the blood is obtained using an arterial blood gas (ABG) . ABG results are often used for patients who have deteriorating or unstable respiratory status requiring emergency treatment. An ABG is a blood sample that is typically drawn from the radial artery by a respiratory therapist. ABG results indicate oxygen, carbon dioxide, pH, and bicarbonate levels. The partial pressure of oxygen in the arterial blood is referred to as PaO2 . PaO2 measures the pressure of oxygen dissolved in the arterial blood and how well oxygen is able to move from the lungs into the blood. The normal PaO2 level of a healthy adult is 80 to 100 mmHg. The PaO2 reading is more accurate than a SpO2 reading because it is not affected by hemoglobin levels. The partial pressure of carbon dioxide in the arterial blood is the PaCO2 level. The PaCO2 level measures the pressure of carbon dioxide dissolved in the blood and how well carbon dioxide is able to move out of the body. It is typically used to determine if sufficient ventilation is occurring at the alveolar level. The normal PaCO2 level of a healthy adult is 35-45 mmHg. The normal range of pH level for arterial blood is 7.35-7.45, and the normal range for the bicarbonate ( HCO3- ) level is 22-26. The SaO2 level is also calculated in ABG results, which is the calculated arterial oxygen saturation level. [8]
Hypoxia and Hypercapnia
Hypoxia is defined as a reduced level of tissue oxygenation. Hypoxia has many causes, ranging from respiratory and cardiac conditions to anemia. Hypoxemia is a specific type of hypoxia that is defined as decreased partial pressure of oxygen in the blood (PaO2) indicated in an arterial blood gas (ABG) result.
Early signs of hypoxia are anxiety, confusion, and restlessness. As hypoxia worsens, the patient’s level of consciousness and vital signs will worsen with an increased respiratory rate and heart rate and decreased pulse oximetry readings. Late signs of hypoxia include bluish discoloration of the skin and mucous membranes called cyanosis. See Figure 8.4 [9] for an image of cyanosis.
Hypercapnia , also referred to as hypercarbia, is an elevated level of carbon dioxide in the blood. This level is measured by the PaCO2 level in an ABG test and is indicated when the PaCO2 level is greater than 45. Hypercapnia is caused by hypoventilation or when the alveoli are ventilated but not perfused. In a state of hypercapnia, the accumulation of carbon dioxide in the blood causes the pH of the blood to drop, leading to a state of respiratory acidosis. You can read more about respiratory acidosis in the “ Acid-Base Balance ” section of the “Fluids and Electrolytes” chapter. Patients with hypercapnia have symptoms such as tachycardia, dyspnea, flushed skin, confusion, headaches, and dizziness. If the hypercapnia develops gradually over time, symptoms may be mild or may not be present at all. Hypercapnia is managed by addressing its underlying cause. A noninvasive positive pressure device such as a BiPAP may be used to help eliminate the excess carbon dioxide, but if this is not sufficient, intubation may be required. [10] You can read more about BiPAP devices and intubation in the “ Oxygen Therapy ” chapter in Open RN Nursing Skills .
It is important for a nurse to recognize early signs of respiratory distress and report changes in patient condition to prevent respiratory failure. See Table 8.2a for symptoms and signs of respiratory distress. [11]
| Signs and Symptoms | Description |
|---|---|
| Shortness of breath (Dyspnea) | Dyspnea is a subjective symptom of not getting enough air. Depending on severity, dyspnea causes increased levels of anxiety. |
| Restlessness | An early sign of hypoxia. |
| Tachycardia | An elevated heart rate (above 100 beats per minute in adults) can be an early sign of hypoxia. |
| Tachypnea | An increased respiration rate (above 20 breaths per minute in adults) is an indication of respiratory distress. |
| Oxygen saturation level (SpO2) | Oxygen saturation levels should be above 94% for an adult without an underlying respiratory condition. |
| Use of accessory muscles | Use of neck or intercostal muscles when breathing is an indication of respiratory distress. |
| Noisy breathing | Audible noises with breathing are an indication of respiratory conditions. Further assess lung sounds with a stethoscope for adventitious sounds such as wheezing, rales, or crackles. Secretions can plug the airway, thereby decreasing the amount of oxygen available for gas exchange in the lungs. |
| Flaring of nostrils | Nasal flaring is a sign of respiratory distress, especially in infants. |
|
Skin color
( Cyanosis ) |
Bluish changes in skin color and mucus membranes is a late sign of hypoxia. |
| Position of patient | Patients in respiratory distress often automatically sit up and lean over by resting arms on their legs, referred to as the tripod position. The tripod position enhances lung expansion. Conversely, patients who are hypoxic often feel worse dyspnea when lying flat in bed and avoid the supine position. |
| Ability of patient to speak in full sentences | Patients in respiratory distress may be unable to speak in full sentences or may need to catch their breath between sentences. |
| Confusion or change in level of consciousness (LOC) | Confusion can be an early sign of hypoxia and changing level of consciousness is a worsening sign of hypoxia. |
Treating Hypoxia and Hypercapnia
Hypoxia and/or hypercapnia are medical emergencies and should be treated promptly by calling for assistance as indicated by agency policy.
Failure to initiate oxygen therapy when needed can result in serious harm or death of the patient. Although oxygen is considered a medication that requires a prescription, oxygen therapy may be initiated without a physician’s order in emergency situations as part of the nurse’s response to the “ABCs,” a common abbreviation for airway, breathing, and circulation. Most agencies have a protocol in place that allows nurses to apply oxygen in emergency situations and obtain the necessary order at a later time. [12]
In addition to administering oxygen therapy, there are several other interventions a nurse can implement to assist an hypoxic patient. Additional interventions used to treat hypoxia in conjunction with oxygen therapy are outlined in Table 8.2b.
| Interventions | Additional Information |
|---|---|
| Raise the head of the bed. | Raising the head of the bed to high Fowler’s position promotes effective chest expansion and diaphragmatic descent, maximizes inhalation, and decreases the work of breathing. |
| Use tripod positioning. | Situate the patient in a tripod position. Patients who are short of breath may gain relief by sitting upright and leaning over a bedside table while in bed, which is called a three-point or tripod position. |
| Encourage enhanced breathing and coughing techniques. | Enhanced breathing and coughing techniques such as using pursed-lip breathing, coughing and deep breathing, huffing technique, incentive spirometry, and flutter valves may assist patients to clear their airway while maintaining their oxygen levels. See the “Enhanced Breathing and Coughing Techniques” section below for additional information regarding these techniques. |
| Manage oxygen therapy and equipment. |
If the patient is already on supplemental oxygen, ensure the equipment is turned on, set at the required flow rate, and is properly connected to an oxygen supply source. If a portable tank is being used, check the oxygen level in the tank. Ensure the connecting oxygen tubing is not kinked, which could obstruct the flow of oxygen. Feel for the flow of oxygen from the exit ports on the oxygen equipment. In hospitals where medical air and oxygen are used, ensure the patient is connected to the oxygen flow port.
Various types of oxygenation equipment are prescribed for patients requiring oxygen therapy. Oxygenation equipment is typically managed in collaboration with a respiratory therapist in hospital settings. Equipment includes devices such as nasal cannula, masks, Continuous Positive Airway Pressure (CPAP), Bilevel Positive Airway Pressure (BiPAP), and mechanical ventilators. For more information, see the “ Oxygenation Equipment ” section of the “Oxygen Therapy” chapter in Open RN Nursing Skills . |
| Assess the need for respiratory medications. | Pharmacological management is essential for patients with respiratory disease such as asthma, COPD, or severe allergic response. Bronchodilators effectively relax smooth muscles and open airways. Glucocorticoids relieve inflammation and also assist in opening air passages. Mucolytics decrease the thickness of pulmonary secretions so that they can be expectorated more easily. |
| Provide suctioning, if needed. | Some patients may have a weakened cough that inhibits their ability to clear secretions from the mouth and throat. Patients with muscle disorders or those who have experienced a stroke (i.e., cerebral vascular accident) are at risk for aspiration, which could lead to pneumonia and hypoxia. Provide oral suction if the patient is unable to clear secretions from the mouth and pharynx. See the “ Tracheostomy Care and Suctioning ” chapter in Open RN Nursing Skills for additional details on suctioning. |
| Provide pain relief, If needed. | Provide adequate pain relief if the patient is reporting pain. Pain increases anxiety and metabolic demands, which, in turn, increase the need for more oxygen supply. |
| Consider side effects of pain medication. | A common side effect of pain medication is respiratory depression. For more information about managing respiratory depression, see the “ Pain Management ” section of the “Comfort” chapter. |
| Consider other devices to enhance clearance of secretions. | Chest physiotherapy and specialized devices assist with secretion clearance, such as handheld flutter valves or vests that inflate and vibrate the chest wall. Consult with a respiratory therapist as needed based on the patient’s situation. |
| Plan frequent rest periods between activities. | Plan interventions for patients with dyspnea so they can rest frequently and decrease oxygen demand. |
| Consider other potential causes of dyspnea. | If a patient’s level of dyspnea is worsening, assess for other underlying causes in addition to the primary diagnosis. For example, are there other respiratory, cardiovascular, or hematological conditions occurring? Start by reviewing the patient’s most recent hemoglobin and hematocrit lab results, as well as any other diagnostic tests such as chest X-rays and ABG results. Completing a thorough assessment may reveal abnormalities in these systems to report to the health care provider. |
| Consider obstructive sleep apnea. | Patients with obstructive sleep apnea (OSA) are often not previously diagnosed prior to hospitalization. The nurse may notice the patient snores, has pauses in breathing while snoring, has decreased oxygen saturation levels while sleeping, or awakens feeling not rested. These signs may indicate the patient is unable to maintain an open airway while sleeping, resulting in periods of apnea and hypoxia. If these apneic periods are noticed but have not been previously documented, the nurse should report these findings to the health care provider for further testing and follow-up. A prescription for a CPAP or BiPAP device while sleeping may be needed to prevent adverse outcomes. |
| Monitor patient’s anxiety. | Assess patient’s anxiety. Anxiety often accompanies the feeling of dyspnea and can worsen it. Anxiety in patients with COPD is chronically undertreated. It is important for the nurse to address the feelings of anxiety in addition to the feelings of dyspnea. Anxiety can be relieved by teaching enhanced breathing and coughing techniques, encouraging relaxation techniques, or administering antianxiety medications. |
Enhanced Breathing and Coughing Techniques
In addition to oxygen therapy and the interventions listed in Table 8.2b, there are several techniques a nurse can teach a patient to use to enhance their breathing and coughing. These techniques include pursed-lip breathing, incentive spirometry, coughing and deep breathing, and the huffing technique. Additionally, vibratory positive expiratory pressure (PEP) therapy can be incorporated in collaboration with a respiratory therapist.
Pursed-lip Breathing
Pursed-lip breathing is a technique that decreases dyspnea by teaching people to control their oxygenation and ventilation. See Figure 8.5 [13] for an illustration of pursed-lip breathing. The technique teaches a person to inhale through the nose and exhale through the mouth at a slow, controlled flow. This type of exhalation gives the person a puckered or pursed-lip appearance. By prolonging the expiratory phase of respiration, a small amount of positive end-expiratory pressure (PEEP) is created in the airways that helps to keep them open so that more air can be exhaled. This subsequently reduces air trapping that commonly occurs in conditions such as chronic obstructive pulmonary disease (COPD). Pursed-lip breathing relieves the feeling of shortness of breath, decreases the work of breathing, and improves gas exchange. People also regain a sense of control over their breathing while simultaneously increasing their relaxation. [14]
Incentive Spirometry
An incentive spirometer is a medical device commonly prescribed after surgery to expand the lungs, reduce the buildup of fluid in the lungs, and prevent pneumonia. See Figure 8.6 [15] for an image of a patient using an incentive spirometer. While sitting upright, if possible, the patient should place the mouthpiece in their mouth and create a tight seal with their lips around it. They should breathe in slowly and as deeply as possible through the tubing with the goal of raising the piston to their prescribed level. The resistance indicator on the right side should be monitored to ensure they are not breathing in too quickly. The patient should attempt to hold their breath for as long as possible (at least 5 seconds) and then exhale and rest for a few seconds. Coughing is expected. Encourage the patient to expel the mucus and not swallow it. This technique should be repeated by the patient 10 times every hour while awake. [16] The nurse may delegate this intervention to unlicensed assistive personnel, but the frequency in which it is completed and the volume achieved should be documented and monitored by the nurse.
Using an incentive spirometer can feel monotonous to patients, resulting in the lack of performing this important activity to prevent pneumonia. It is helpful to encourage patients to create easy reminders to complete the activity. For example, many patients watch television. Create the reminder to use the incentive spirometer each time they view a commercial. This is a helpful trigger to use the incentive spirometer frequently.
Coughing and Deep Breathing
Coughing and deep breathing is a breathing technique similar to incentive spirometry but no device is required. The patient is encouraged to take deep, slow breaths and then exhale slowly. After each set of breaths, the patient should cough. This technique is repeated 3 to 5 times every hour.
Huffing Technique
The huffing technique is helpful to teach patients who have difficulty coughing. Teach the patient to inhale with a medium-sized breath and then make a sound like “ha” to push the air out quickly with the mouth slightly open.
Vibratory PEP Therapy
Vibratory Positive Expiratory Pressure (PEP) Therapy uses handheld devices such as flutter valves or Acapella devices for patients who need assistance in clearing mucus from their airways. These devices require a prescription and are used in collaboration with a respiratory therapist or advanced health care provider. To use vibratory PEP therapy, the patient should sit up, take a deep breath, and blow into the device. A flutter valve within the device creates vibrations that help break up the mucus so the patient can cough and spit it out. Additionally, a small amount of positive end-expiratory pressure (PEEP) is created in the airways that helps to keep them open so that more air can be exhaled. See the supplementary video below regarding how to use the flutter valve device.
- This work is a derivative of Anatomy & Physiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/anatomy-and-physiology/pages/1-introduction ↵
- “ 2322_Fig_23.33-a.jpg ” by OpenStax is licensed under CC BY 3.0 ↵
- This work is a derivative of Anatomy & Physiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/anatomy-and-physiology/pages/1-introduction ↵
- “ 2325_Carbon_Dioxide_Transport.jpg ” by OpenStax is licensed under CC BY 3.0 ↵
- This work is a derivative of Anatomy & Physiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/anatomy-and-physiology/pages/1-introduction ↵
- “ OxyWatch_C20_Pulse_Oximeter.png ” by Thinkpaul is licensed under CC BY-SA 3.0 ↵
- This work is a derivative of StatPearls by Patel, Miao, Yetiskul, and Majmundar and is licensed under CC BY 4.0 ↵
- This work is a derivative of StatPearls by Castro and Keenaghan and is licensed under CC BY 4.0 ↵
- “ Cynosis.JPG ” by James Heilman , MD is licensed under CC BY-SA 3.0 ↵
- This work is a derivative of StatPearls by Patel, Miao, Yetiskul, and Majmundar and is licensed under CC BY 4.0 ↵
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology and is licensed under CC BY 4.0 ↵
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology and is licensed under CC BY 4.0 ↵
- "v4-460px-Live-With-Chronic-Obstructive-Pulmonary-Disease-Step-8.jpg” by unknown is licensed under CC BY-SA 3.0 . Access for free at https://www.wikihow.com/Live-With-Chronic-Obstructive-Pulmonary-Disease ↵
- This work is a derivative of StatPearls by Nguyen and Duong and is licensed under CC BY 4.0 ↵
- “ Incentive Spirometer.png ” by BruceBlaus is licensed under CC BY-SA 4.0 ↵
- Cleveland Clinic. (2018, May 2). Incentive spirometer. https://my.clevelandclinic.org/health/articles/4302-incentive-spirometer ↵
- NHS University Hospitals Plymouth Physiotherapy. (2015, May 12). Acapella . [Video]. YouTube. All rights reserved. https://youtu.be/XOvonQVCE6Y ↵


4.9: Relativity of Simultaneity
we turn around; our changing colleagues say Earth's clock flies forward
By this time James Fastlane has gotten his second wind. "I am still stuck in this Twin Paradox thing. The time for the outward trip is less as measured in the rocket frame than as measured in the Earth frame. But if relativity is correct, every free-float frame is equivalent. As you sit on the rocket, you feel yourself to be at rest, stationary, motionless; you measure our Earth watch-station clocks to be zipping by you at high velocity. Who cares about labels? For you these Earth clocks are in motion! Therefore the time for the outward trip should be less as measured on the ('moving') Earth clock than as measured on your ('stationary') rocket clock."
Rocket observer: Fewer Earth-clock ticks on outward trip . . .
We nod assent and he continues. "Nothing prevents us from supposing the existence of a series of rocket lookout stations moving along in step with your rocket and strung out at separations of one light-year as measured in your rocket frame, all with clocks synchronized in your rocket frame and running at the same rate as your rocket clock. Now, as Earth passes each of these rocket lookout stations in turn, won't those stations read and record the times on the passing Earth clock to be less than their own times? Otherwise how can relativity be correct?" 1
"Yes, your prediction is reasonable," we reply.
. . . also fewer Earth-clock ticks on return trip
"And on the return trip will not the same be true: Returning-rocket lookout stations will measure and record time lapses on the passing Earth clock to be less than on their own clocks?" 2
"That conclusion is inevitable if relativity is consistent."
"Aha!"' exclaims Mr. Fastlane, "Now I've got you! If Earth clock is measured by rocket lookout stations to show smaller time lapses during the outward trip - and also during the return trip - then obviously total Earth time must be less than rocket round-trip time. But you claim just the opposite: that total rocket time is less than Earth time. This is a fundamental contradiction. Your relativity is wrong!" Folding his arms he glowers at us.
There is a long silence. Everyone looks at us except Professor Bright, who has his head down. It is hard to think with all this attention. Yet our mind runs over the trip again. Going out . . coming back ... turning around .... that's it!
"All of us have been thinking the wrong way?" we exclaim. "We have been talking as if there is only a single rocket frame. True, the same vehicle, with its traveler, goes out and returns. True, a single clock makes the round trip with the traveler. But this vehicle turns around - reverses its direction of travel - and that changes everything.
Astronaut jumps from outgoing frame to returning frame
"Maybe it's simpler to think of two rockets, each moving without change of velocity. We ride on the first rocket going out and on the second rocket coming back. Each of these two is really a rocket frame : each has its own long train of lookout stations with recording clocks synchronized to its reference clock (Figure \(\PageIndex{1}\)). The traveler can be thought of as 'jumping trains' at Canopus - from outward-bound rocket frame to inward-bound rocket frame- carrying the calendar clock." 3
"Now follow Mr. Fastlane's prescription to analyze the trip in the rocket frame, but with this change: make this analysis using two rocket frames - one outward bound, the other inward bound."
Outgoing rocket: As it arrives at Canopus, Earth clock reads 3.96 years
"It is 20 years by outward-rocket time when the traveler arrives at Canopus. That is the reading on all lookout station clocks in that outward-rocket frame. One of those lookout stations is passing Earth when this rocket time arrives. Its clock, synchronized to the clock of the outward traveler at Canopus, also reads 20 years. What time does that rocket lookout-station guard read on the passing Earth clock? For the rocket observer Earth clock reads less time by the same factor that rocket clocks read less time (20 years at arrival at Canopus) for Earth observers (who read 101 years on their own clocks). This factor is \(20 / 101\) . Hence for the outward-rocket observer the Earth clock must read \(20 / 101\) times 20 years, or 3.96 years." 4
"What!" explodes Fastlane. "According to your plan, the turnaround at Canopus occurs at 101 years of Earth time. Now you say this time equals less than 4 years on Earth clock."
"No sir, I do not say that," we reply, feeling confident at last. "I did say that at the same time as the outgoing rocket arrives at Canopus, Earth clock reads 3.96 years as measured in that outgoing rocket frame . An equally true statement is that at the same time as the outgoing rocket arrives at Canopus, Earth clock reads 101 years as measured in the Earthbound frame . Apparently observers in different reference frames in relative motion do not agree on what events occur at the same time when these events occur far apart along the line of relative motion."
Once again Professor Bright supplies the label. "Yes, that is called relativity of simultaneity . Events that occur at the same time-simultaneously - judged from one free-float frame but far apart along the line of relative motion do not occur simultaneously as judged from another free-float frame."
"As an example of relativity of simultaneity, consider either chain of lookout stations strung along the line of relative motion. If all clocks in the lookout stations of one frame strike exactly at noon in that frame, these strikes are not simultaneous as measured in another frame in relative motion with respect to the first. This is called relative synchronization of clocks ."
"Incidentally, most of the so-called 'paradoxes' of relativity, one of which we are considering now, turn on misconceptions about relativity of simultaneity."
Dr. Short breaks in. "What about the returning rocket? What time on the Earth clock will the returning rocket lookout station measure as the traveler starts back?"
Returning rocket: As it leaves Canopus, Earth clock reads 198.04 years
"That shouldn't be too difficult to figure out," we reply. "We know that the clock on the returning rocket reads 40 years when we arrive home on Earth. And the Earth clock reads 202 years on that return. Both of these readings occur at the same place (Earth), so we do not need to worry about relativity of simultaneity of that reading. And during the return trip Earth clock records less elapsed time than rocket clocks' 20 years by the same factor, \(20 / 101\) , or a total elapsed time of \(20 \times 20 / 101=3.96\) years according to return rocket observations. Therefore at the earlier turnaround, return rocket observers will see Earth clock reading \(202-3.96=198.04\) years."
"Wait a minute!" bellows Fastlane. "First you say that the rocket observer sees the Earth clock reading 3.96 years at turnaround in the outward-bound frame. Now you say that the rocket observer sees the Earth clock read \(198.04\) years at turnaround in the inward-bound frame. Which one is right?"
"Both are right," we reply. "The two observations are made from two different frames. Each of these frames has a duly synchronized system of lookout-station clocks, as does the Earth-linked frame (Figure \(\PageIndex{1}\)). The so-called Twin Paradox is resolved by noticing that between the Earth-clock reading of 3.96 years, taken from the outward rocket lookout station at turnaround and the Earth-clock reading of 198.04 years, taken by the returning-rocket lookout station at turnaround, there is a difference of 1 94.08 years." 5
Forward “jump” in Earth clock results from frame change
"This 'jump' appears on no single clock but is the result of the traveler changing frames at Canopus. Yet this jump, or difference, resolves the paradox: For the traveler, the Earth clock reads small time lapses on the outward leg-and also small time lapses on the return leg - but it jumps way ahead at turnaround. 6 This jump accounts for the large value of Earth-aging during the trip: 202 years. In contrast the traveler ages only 40 years during the trip (Table \(\PageIndex{1}\))."
"And notice that the traveler is unique in the experience of changing frames; only the traveler suffers the terrible jolt of reversing direction of motion. In contrast, the Earth observer stays relaxed and comfortable in the same frame during the astronaut's entire trip. Therefore there is no symmetry between rocket traveler and Earth dweller, so no genuine contradiction in their differing time lapses, and the story of the twins is not a paradox."
|
Earth-clock reading observed by |
||||
|---|---|---|---|---|
| Event | Time measured in Earth-linked frame | Time measured by traveler | outgoing-rocket lookout stations passing Earth | return-rocket lookout stations passing Earth |
| Depart Earth | 0 years | 0 years | 0 years | |
| Arrive Canopus | 101 years | 20 years | 20 years \(\times\) 20/101 = 3.96 years | |
| Depart Canopus | 101 years | 20 years | 3.96 years | 202 - 3.96 = 198.04 years |
| Arrive Earth | 202 years | 40 years | 202 years |
All observers agree on result, disagree on reason
"In fact, the observer in each of the three frames - Earth-linked, outward-rocket, and inward-rocket - has a perfectly consistent and non-paradoxical interpretation of the sequence of events. However, in accounting for disagreements between his or her readings and those of observers in other free-float frames, each observer infers some misbehavior of measuring devices in these other frames. Each observes less elapsed time on clocks in the other frame than on his or her own clocks (time stretching or time dilation). Each thinks that an object lying along the line of relative motion and at rest in another frame is contracted (Lorentz contraction). 7 Each thinks that lookout-station clocks in other frames are not synchronized with one another (relative synchronization of clocks). As a result, each cannot agree with other observers as to which events far apart along the line of relative motion occur at the same time (relativity of simultaneity)."
"Boy," growls Fastlane, "all these different reference frames sure do complicate the story!"
Spacetime interval is universal language
"Exactly!" we exclaim. "These complications arise because observations from any one frame are limited and parochial. All disagreements can be bypassed by talking only in the invariant language of spacetime interval, proper time, wristwatch time. 8 The proper time from takeoff from Earth to arrival at Canopus equals 20 years, period. The proper time from turnaround at Canopus to re-arrival at Earth equals 20 years, period. The sum equals 40 years as experienced by the astronaut, period. On the Earth clock, the proper time between departure and return is 202 years, period. End of story. Observers in all free-float frames reckon proper times - spacetime intervals between these events - using their differing space and time measurements. However, once the data are translated into the common language of proper time, every observer agrees. Proper times provide a universal language independent of reference frame."
1 Rocket observer: Fewer Earth-clock ticks on outward trip...
2 ...Rocket observer: Fewer Earth-clock ticks on outward trip
3 Astronaut jumps from outgoing frame to returning frame
4 Outgoing rocket: As it arrives at Canopus, Earth clock reads 3.96 years
5 Returning rocket: As it leaves Canopus, Earth clock reads 198.04 years
6 Forward “jump” in Earth clock results from frame change
7 Forward “jump” in Earth clock results from frame change
8 Spacetime interval is universal language


Chapter 3: The Cellular Level of Organization
Cellular and developmental biologists study how the continued division of a single cell leads to such complexity and differentiation. Consider the difference between a structural cell in the skin and a nerve cell. A structural skin cell may be shaped like a flat plate (squamous) and live only for a short time before it is shed and replaced. Packed tightly into rows and sheets, the squamous skin cells provide a protective barrier for the cells and tissues that lie beneath. A nerve cell, on the other hand, may be shaped something like a star, sending out long processes up to a meter in length and may live for the entire lifetime of the organism. With their long winding appendages, nerve cells can communicate with one another and with other types of body cells and send rapid signals that inform the organism about its environment and allow it to interact with that environment. These differences illustrate one very important theme that is consistent at all organizational levels of biology: the form of a structure is optimally suited to perform particular functions assigned to that structure.
-
- 3.1: Introduction
- You developed from a single fertilized egg cell into the complex organism containing trillions of cells that you see when you look in a mirror. During this developmental process, early, undifferentiated cells differentiate and become specialized in their structure and function. These different cell types form specialized tissues that work in concert to perform all of the functions necessary for the living organism.
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- 3.2: The Cell Membrane
- Despite differences in structure and function, all living cells in multicellular organisms have a surrounding cell membrane. As the outer layer of your skin separates your body from its environment, the cell membrane (also known as the plasma membrane) separates the inner contents of a cell from its exterior environment. This cell membrane provides a protective barrier around the cell and regulates which materials can pass in or out.
-
- 3.3: The Cytoplasm and Cellular Organelles
- Now that you have learned that the cell membrane surrounds all cells, you can dive inside of a prototypical human cell to learn about its internal components and their functions. All living cells in multicellular organisms contain an internal cytoplasmic compartment, and a nucleus within the cytoplasm. Cytosol, the jelly-like substance within the cell, provides the fluid medium necessary for biochemical reactions. Eukaryotic cells, including all animal cells, also contain various organelles.
-
- 3.4: The Nucleus and DNA Replication
- The nucleus is the largest and most prominent of a cell’s organelles (Figure 3.3.1). The nucleus is generally considered the control center of the cell because it stores all of the genetic instructions for manufacturing proteins. Interestingly, some cells in the body, such as muscle cells, contain more than one nucleus, which is known as multinucleated. Other cells, such as mammalian red blood cells, do not contain nuclei at all.
-
- 3.5: Protein Synthesis
- Most structural components of the cell are made up, at least in part, by proteins and virtually all the functions that a cell carries out are completed with the help of proteins. One of the most important classes of proteins is enzymes, which help speed up necessary biochemical reactions that take place inside the cell. Some of these critical biochemical reactions include building larger molecules from smaller components and breaking down larger molecules into smaller components.
-
- 3.6: Cell Growth and Division
- Cells in the body replace themselves over the lifetime of a person. For example, the cells lining the gastrointestinal tract must be frequently replaced when constantly “worn off” by the movement of food through the gut. But what triggers a cell to divide, and how does it prepare for and complete cell division? The cell cycleis the sequence of events in the life of the cell from the moment it is created at the end of a previous cycle of cell division until it then divides itself, generating two
-
- 3.7: Cellular Differentiation
- How does a complex organism such as a human develop from a single cell—a fertilized egg—into the vast array of cell types such as nerve cells, muscle cells, and epithelial cells that characterize the adult? Throughout development and adulthood, the process of cellular differentiation leads cells to assume their final morphology and physiology. Differentiation is the process by which unspecialized cells become specialized to carry out distinct functions.


3.12: India's Early Medieval Age and the Development of Islamic States in India, 600-1300
The history of ancient India concludes with the decline of the Gupta Empire. The next major period, which lasts for roughly seven centuries (c. 600 – 1300), is the early medieval age. During these centuries, kingdoms in both the north and south proliferated and regularly turned over. Therefore, at any one time, India was fragmented by numerous regional kingdoms. As the rulers of these warred and formed alliances, they employed the system of paramountcy and subordination begun during the Gupta era, with some rulers being overlords and others vassals. Also, successful rulers demonstrated their power by granting land to officers, Brahmins, and temples. The outcome was a political pattern labeled Indian feudalism .
These rulers also demonstrated their power—and enhanced it—by patronizing Hindu institutions and developing local traditions in the regions where their courts resided. They adopted titles showing their devotion to the great Hindu deities, declared their intent to uphold dharma, built fabulous Hindu temples in urban centers, and charged Brahmins with attending to them and serving at their courts. One outstanding example of a feudal kingdom is the Chola Kingdom of southern India.
Lastly, at the end of this age, a new force appeared on the Indian scene. Muslim Arab and Turkic rulers of West and Central Asia made incursions into the subcontinent. Along with Arab traders arriving on India’s west coast for trade, they brought a new religion and type of rule to the landscape of early medieval India and forged new connections between the subcontinent and the rest of Afro-Eurasia.
Indian Feudalism
Feudalism is a term historians first used to describe the political, social, and economic system of the European Middle Ages (see Chapter Twelve). That system was the world of lords, vassal knights, and serfs characteristic of Europe from the tenth to thirteenth centuries. In exchange for homage and military service, vassals received land from their lords. These lands became their manors, and serfs worked them. The lords and their vassals constituted a privileged nobility, while the serfs lived in a state of servitude.
Historians also use feudalism to describe India during the early medieval age. But the usefulness of this term is much debated, because conditions on the ground varied from place to place, not only in Europe but also in India. Therefore, historians now only use the term in a general sense while also describing specific variations. In general, feudalism designates a political and economic scene characterized by fragmented authority, a set of obligations between lords and vassals, and grants of land (including those who work it) by rulers in exchange for some kind of service.
Authority on the early medieval Indian subcontinent was indeed fragmented, not only by the many regional kingdoms that existed at any one time but also, more importantly, within kingdoms. Because kingdoms incessantly warred with one another, their boundaries were fluid. Rulers usually closely administered a core area near the capital with a civil administration, while granting feudatories on the periphery. Having defeated the ruling lineage of a powerful neighboring state—such as a king, prince, or chief, victorious kings might allow them to retain noble titles and their lands, on the condition that they demonstrate allegiance to him and even supply tribute and military service. The overlord could then wield the title “Great King of Kings,” while the lesser rulers bore titles signifying their status as subordinate rulers who do obeisance.
Additionally, aside from granting these feudatories, medieval rulers also issued land grants to important persons and institutions in their realms, such as Brahmins, high officials, or temples. As opposed to receiving a cash salary, these recipients were permitted to retain revenue from villages on this land, as well as to exercise some level of judicial authority. Brahmins were so important to kings because they aided him in upholding the king’s dharma. The king’s duty was to protect the people, uphold the varna social order, sacrifice to the traditional Vedic deities, and show devotion to Shiva or Vishnu. As the religious leaders and intellectuals in the community, and the most prestigious varna , Brahmins could craft genealogies proving a king’s illustrious origins in the heroic lineages of the epic stories of ancient times, perform the sacrifices, and maintain temples. So rulers often generously gifted land to them or to the magnificent temple complexes rulers built.
Medieval India, then, consisted of a multitude of kingdoms, each of which governed a part of their realms through feudal arrangements, by granting feudatories and issuing land grants to nobility or prestigious religious and political leaders, in exchange for allegiance and assisting the ruler in demonstrating his being worthy of his sacred role. In most instances, given that society was patriarchal, rulers were male, but in many cases queens inherited the throne. Rudramadevi, for instance, was chosen by her father to accede to the throne of a kingdom in central India, likely because he had no sons or living brothers. Inscriptions refer to her as a king; indeed, she is said to have donned male attire while leading soldiers into battle. She is also portrayed seated on a lion, with a dagger and shield in hand. Thus, she was conformed to the expected role of a warrior, male king.
The Chola Dynasty
The Chola Kingdom illustrates well the grandeur of powerful regional states during the early medieval period. South India first came to prominence with the rise of the Satavahana Kingdom in the Deccan Plateau (c. the second century BCE). But even at that time, in the fertile hinterlands and along the seaboard of the southern tip of the subcontinent, Tamil states were forming. Tamil refers to the regional language spoken and written by Indian peoples of the far south, as well as to their local customs and culture. Powerful Tamil lineages divided up Tamil land among chiefdoms, and, over time, some evolved into small but impressive kingdoms. These Tamil states also adopted elements of the Aryan culture of the north, including the use of Sanskrit, the varna social system, Vedic Brahmanism, and the Hindu cults of Shiva and Vishnu.
The Cholas , for instance, date back to Satavahana times, but they don’t become significant for India’s political history until the ninth century CE, when they show up as a feudatory of a neighboring Tamil kingdom. Beginning with King Aditya I (r. 871 – 907), the Cholas began a process of expansion that would eventually make it the most powerful kingdom of the south up until the thirteenth century (see Map \(\PageIndex{1}\)). Two of the most powerful Chola rulers were Rajaraja I (r. 985 – 1012) and his son Rajendra I (1012 – 1044). During their reigns, most of south India was conquered, including Sri Lanka, and a royal administration was built. Chola kings directly administered a core area of provinces and districts with royal officials, but also granted feudatories to allegiant chieftains and land grants to Brahmins. At the local level, these authorities worked with village assemblies and town associations, both of which were remarkable for the level of freedom they had to manage local affairs.
The Chola kings proved their greatness through not only the success of their imperial ambitions but also the temples they built. These temples, some of the most impressive structures in India, testify to the kings’ piety. Eulogies to Chola kings describe them not only as great warriors and conquerors but also as protectors of the dharma, destroyers of evil, and generous givers of gifts. Because they were built in honor of the great Hindu deities, temples put those latter qualities on display. In 1010 CE, Rajaraja I completed the Brihadeshwara Temple in Tanjavur, the Chola capital and ceremonial center (see Figure \(\PageIndex{1}\)). Atop the main sanctuary stands a sixty-two meter tall tower carved out of a block of granite weighing eighty-one tons. Numerous representations portray the supreme lord Shiva in his various manifestations, including one located in the inner sanctuary. On the profusely ornamented tower, Shiva appears repeatedly in his form as destroyer of the cities of demons. Clearly, Rajaraja I wished to raise the cult of Shiva to a pre-eminent position in his kingdom, and built this temple to project Chola power.
Like other great regional powers with their unique histories and architectural traditions during the early medieval period, the Chola kingdom peaked for about two centuries and then declined. In the thirteenth century, neighboring kingdoms nibbled away at its power and it came to an end.
The Rise of Islamic States in North India
One of the most important developments during India’s early medieval age is the arrival of Muslim Arab and Turkic traders and conquerors and the eventual establishment of Islamic states and communities in India. This story begins with the life of Muhammad (570 – 632 CE), the Prophet of Islam, whose revelations were recorded in the Quran (see Chapter 8). During his lifetime and after, an Arab community and state governed by the principles of Islam were founded in Arabia. Over the course of the seventh and eighth centuries, this Arab Islamic state grew into an empire that included much of West Asia and North Africa. This size meant that many different ethnic groups—Egyptians, Persians, and Turks, for instance—fell under its governing umbrella.
The ruler of this empire was the caliph, a man designated political successor to Muhammad, as the leader of the Islamic community. The caliph’s government is called the caliphate. The first caliphs were friends and relatives of Muhammad, but eventually long-lasting dynasties formed that made the position hereditary. The first was the Umayyad Dynasty (661 – 750 CE) and the second the Abbasid Dynasty (750 – 1258 CE). It was during these caliphates that Islam and Islamic rule made their way into India.
During the Umayyad Caliphate, systematic reconnaissance of the northwest coast of India was undertaken because conquests brought the empire just to the west of the Indian subcontinent. When pirates plundered an Arab vessel at the mouth of the Indus River, the caliph authorized punitive measures, and Umayyad rulers sent an invasion force. In 711, the Sindh (the lower Indus) was seized from a Hindu ruler and incorporated into the Islamic Empire. An Islamic community then began to set roots in this part of India.
The Umayyad Caliphate ruled from 661 – 750 CE. Note the inclusion of part of northwest India, near the Indus River. The next major event didn’t transpire until the tenth century, at a time when Turkic peoples had become important to the history of Islamic states in India. By this time, the Abbasid Dynasty had replaced the Umayyad as the caliphs of the Islamic Empire.  Today, the word Turkic might normally be associated with the country of Turkey, but in fact, Turkic peoples and their language family—Turkish—originated in Central Asia. That is where the expanding Islamic empire first encountered and began incorporating Turks into their governing.
The significance of these Turkic slave soldiers for India is that the Abbasids employed them as governors of the eastern end of their empire. This end included parts of Afghanistan, the neighbor to northwest India. In the tenth century, however, the caliphate was falling apart, and Turkic military governors took advantage of this dissolution by establishing an independent state in Afghanistan. The family that did so was the Ghaznavids. Ruling from a fortress in Ghazna, the Ghaznavids forged an empire that included much of Iran and northwest India (see Map \(\PageIndex{3}\)).
The Ghaznavid ruler who first made forays into India was Mahmud of Ghazna (971 – 1030). In some historical writing, Mahmud has been portrayed as a brutal plunderer who descended on India seventeen times with hordes of Turkic cavalry, shocking wealthy cities of the north with the sword of Islam by destroying their Hindu temples and returning to his capital with their stolen wealth.
But the reality may have been otherwise. Recent work suggests that Mahmud was neither interested in spreading Islam nor caused massive destruction. Rather, northwestern India had always been closely linked to Central Asia, as well as being the location of both repeated invasions and kingdoms that crossed over into the mountains. Furthermore, by the tenth century, Muslim communities had already become a part of the Indian scene, along the west coast and in this region. Therefore, Mahmud’s incursions were hardly something new. Nor were his motives. In medieval India, kings often waged war not only for revenue but also because such was their custom. Mahmud was likely no different. His empire was experiencing instability; he therefore sought to prove his mettle as a warrior ruler and to secure his legacy by using Indian wealth to build palaces and mosques in Ghazna.
Ghaznavid control over India didn’t extend much beyond the Punjab, lasting less than two centuries. In 1186, Muhammad of Ghur—chieftain of a minor hill state in Afghanistan that was subordinate to the Ghaznavids—overthrew his overlords in Ghazna and proceeded to forge his own empire. Desiring to extend it across northern India, he found that his greatest foes were the Rajputs. The Rajputs were clans located in northern and central India that claimed descent from renowned Kshatriya (warrior) lineages of ancient times. While Muhammad of Ghuri was planning his military expeditions, the Rajputs were governing several regional Hindu kingdoms from large fortresses they had built. In 1192, Muhammad’s forces defeated a confederation of Rajput rulers at the Battle of Tarain. His slave general and commander-in-chief, Qutb-ud-din Aybak, then achieved a string of victories across northern India, making it a part of the Ghurid Empire.
Muhammad of Ghur returned to his Afghan homeland, leaving northern India to Aybak, who then proceeded to set up his headquarters in Delhi, one of the most important cities in South Asia, and also the capital of today’s nation of India. When Muhammad died in 1206, Aybak took control of these Indian possessions and established a state of his own called the Delhi Sultanate (see map \(\PageIndex{4}\)). A sultanate is the government of a sultan, and a sultan is an Islamic ruler who governs a country largely independently of the caliphs, but without claiming their title. The Delhi sultans, then, were the sovereign rulers of the first major Muslim state in India, one that would last for three hundred years.
Looking ahead to our own time, the nation of India today is both culturally and religiously diverse. Approximately 80% of the population practices Hinduism while 15% practices Islam, making these the two largest religious traditions in India today. For this reason, relations between peoples adhering to these two different faiths have been an important issue in the history of South Asia. As we have seen, the history of Islam and Islamic communities in the subcontinent begins during the early medieval period. Therefore, historians pay close attention to how Delhi sultans governed an overwhelmingly Hindu population, as well as how Islamic communities fit into it.
Ruling as they were over an ancient and vast agrarian civilization, the Turkic sultans worked out an accommodation with India, adapting to the pattern of Indian feudalism. Outside the highest levels of government, Hindu society and its traditional leaders were largely left in place, so long as tax revenue was submitted. With a long history of conquest behind them, Islamic rulers had learned the benefits of adopting a pragmatic approach to non-Muslims, and these sultans were no exception. They had little interest in forcibly converting people to the faith, and rather adopted a principle from the Quran whereby non-Muslim peoples with a scriptural tradition of their own can live amidst the Islamic community and state so long as they pay a higher tax. At the highest levels, however, sultans placed Turkic military nobility and educated Persians in charge, often compensating them with land grants. In fact, because Persians became so important to Sultanate administration, Persian was adopted as the language of government.
 At the lower levels of society, Muslim traders and artisans became an important presence in Indian towns and cities, as did Indian converts who saw an advantage to converting to this faith. Thus, Hindu and Muslim communities increasingly interacted with each other during the early medieval age, adopting elements of each other’s way of life. For that reason, historians speak of a fusion of Islamic and Indian culture.


1.6: CHAPTER 5 FLETCHER’S SITUATION ETHICS
Fletcher’s Situation Ethics
Every man must decide for himself according to his own estimate of conditions and consequences… 1
People like to wallow or cower in the security of the law . 2
1. Situation Ethics Introduction
In the introduction to
The Situation Ethics: The New Morality
Joseph Fletcher
(1905–1991) develops what he calls an ethical non-system. His book caused a “fire storm” amongst the public because it legitimised the general post-war dissatisfaction with authority. At the time it was written it seemed to make some radical claims such as that it is not wrong to have extramarital sex, to be homosexual, or to have an abortion. All that said, Fletcher’s work is not widely discussed nor respected in philosophical circles. It is badly argued, idiosyncratic and rehashes old ideas.
Although there is the clothing of religion in the book — Fletcher uses religious terms such as “
agápe
” and cites famous theologians such as
Rudolf Bultmann
(1884–1976) — the central ideas do not rely on the truth of any particular religion. As he says his argument has “…nothing special to do with theological…faith”
3
Fletcher calls this ethical “non-system”
Situationism
and a Bible story will illustrate the general point of the book. In Mark 3:1–6 we are told that Jesus healed a man with a withered hand in the Jewish Temple; an act which we would consider to demonstrate Jesus’s
love
for all. However, the Pharisees tell him off because he has performed this healing on the Sabbath day and the Jewish law says that no one can work on the Sabbath.
Fletcher’s work is an attempt to show how acts can be morally acceptable even if they go against so-called moral laws (if you’ve read Chapter 3 on Aristotle you might already have an answer to this). Fletcher says that Jesus’ act is morally acceptable — despite going against the Jewish law — because he acted to bring about the most love.
2. Fletcher’s Overall Framework
Fletcher says there are two unattractive views in ethics: “ Legalism ” and “ Antinomianism ”, and one attractive view which sits in between them: “ Situationism ”.
Legalism
Someone who is following the system of Legalism is someone who “blindly” observes moral rules without being sensitive to the situation. Fletcher has in mind a simple minded deontologist who holds that actions are right and wrong irrespective of the consequences. For example, we ought to tell the truth in all situations, even if this means that, say, millions of people die.
Various Christian sects are legalistic; for instance, some might refuse medical help — such as blood transfusions — when someone in their community is ill because they think it is against God’s commands. Or consider an example of Islamic Legalism (obviously, just as in the Christian sect, these are not wholly representative of either religion). In 2002 the religious police of Saudi Arabia refused to let a group of girls escape from a burning building because they were wearing “inappropriate” clothing, which was against the will of Allah. One witness said he saw three policemen “beating young girls to prevent them from leaving the school because they were not wearing the abaya ”. 4 Fifteen girls died.
Antinomianism
The other extreme is Antinomianism (“anti” meaning against; “nominalism” meaning law). This is the view that says that an agent can do whatever he or she wants in a situation. Fletcher calls this an “ existential ” view because it is one that says that people are always free to choose what they want. Any supposed laws and rules limiting the actions of people are simply a way of trying to comfort them because they are scared of absolute freedom. If Antinomianism is right and if an agent believes that something is right, then it is. Antinomianism means the moral agent is erratic and random, is unpredictable, and any decisions taken are ad hoc . There are no laws nor guiding principles, just agents and their conscience and the institutions in which they find themselves.
A Middle Ethics: Situationism
We might think that Legalism and Antinomianism exhaust the possibilities. If we reject moral laws then are not we forced into lawless moral anarchy? Fletcher thinks not.
Fletcher says that there is a moral law, and hence he rejects Antinomianism. But there is only
one
moral law, so he rejects Legalism. Fletcher’s one moral law is that we ought to always act so as to bring about the most love for the most people (“
Agápē
Calculus”). Fletcher’s Situationism is then a
teleological
theory. It is directed at the consequences that will determine whether an action is right or wrong.
Of course,
any
teleological theory will ask us to look at the details of the situation; consider Chapter 1 where we talk about Bentham and Mill’s Utilitarianism. So, Fletcher’s view is not unique. What makes his view different is the centrality of “love”, or as he calls it
agápē
.
Fletcher thinks that there
can
be moral principles but that these differ from
laws
. Principles are generalizations which are context-sensitive and which derive from the one law regarding maximizing love. For example, we might have a moral principle that we ought not to murder. This is a
principle
because we might think in that
in general
murder is wrong because it does not bring about the most love. However, it is not a
law
because for Fletcher, murder is not wrong in
all
situations. This then is similar to the discussion of Rule-Utilitarianism in Chapter 1.
For example, a situation might arise where the child of a terrorist would have to be murdered in order to get information to stop a nuclear attack. Fletcher would say that here is a situation where we ought not to follow the principle do not murder but rather do the most loving thing, which in this case turns out to be murder. From the universal law we can only derive principles, not other universal laws. As Fletcher puts it: “we cannot milk a universal from a universal”.
5
This mean that for Fletcher it might, on occasions, be morally acceptable to break the Ten Commandments. In fact, he says something stronger, that in some situations it is our duty to break these commandments. He thinks that there are four working principles of Situationism.
3. The Four Working Principles of Situationism
Principle 1. Pragmatism
The situationalist follows a strategy which is
pragmatic
. What does that mean? Well it does
not
mean that Fletcher is a pragmatist. “Pragmatism” is a very specific and well worked-out philosophical position adopted by the likes of
John Dewey
(1859–1952),
Charles Peirce
(1839–1914) and
William James
(1842–1910). Fletcher does not want his theory associated with these views and rejects all the implications of this type of “Pragmatism”.
What makes his view pragmatic is very simple. It is just his attraction to moral views which do not try to work out what to do in the abstract (e.g. Kant’s Categorical Imperative (see Chapter 2)), but rather explores how moral views might play out in each
real life situations
.
Principle 2: Relativism
Even with his rejection of Antinomianism and his acceptance of one supreme principle of morality, Fletcher, surprisingly, still calls himself a relativist. This does not mean he is a relativist in the sense that we can simply choose what is right and wrong rather it is just an appeal for people to stop trying to “lay down the law” for all people in all contexts. If situations vary then consequences vary and what we ought to do will change accordingly. This is a very simple, unsophisticated idea, like his ideas on pragmatism, and Fletcher just means that what is right or wrong is related to the situation we are in.
Principle 3: Positivism
His use of “positivism” is not the philosophical idea with the same name but rather is where:
Any moral or value judgment in ethics, like a theologian’s faith propositions, is a decision — not a conclusion. It is a choice, not a result reached by force of logic… 6
So when challenged as to how he can justify that the only law is to maximize love, Fletcher will say that he cannot. It is not a result of logic or reasoning, rather it is a decision we take, it is like the “theologian’s faith”.
Principle 4: Personalism
Love is something that is experienced by people . So Personalism is the view that if we are to maximize love we need to consider the person in a situation — the “who” of a situation. Summing up this Fletcher says:
Love is of people, by people, and for people. Things are to be used; people are to be loved… Loving actions are the only conduct permissible. 7
These then are his “four working principles”: pragmatism, relativism, positivism and personalism.
4. How to Work out What to Do: Conscience as a Verb not a Noun
For Fletcher “conscience” plays a role in working out what to do. He says “conscience” is a verb and not a noun. This sounds complicated but it really is not (for complex and sophisticated discussions of conscience see Chapter 9).
First consider what he means when he says conscience “is not a noun”. Conscience is not the name of an internal faculty nor is it a sort of internal “moral compass”. This is how people typically think of conscience and it is often portrayed in cartoons with a devil and angel sitting on someone’s shoulder whispering into her ears.
Rather for Fletcher conscience is a verb. Imagine we have heard some bullies laughing because they have sent our friend some offensive texts and we are trying to decide whether or not to check his phone to delete the texts before he does. The old “noun” view of conscience would get us to think about this in the abstract, perhaps reason about it, or ask for guidance from the Holy Spirit, a guardian angel etc.
According to Fletcher this is wrong. Instead, we need to be in the situation, and experience the situation, we need to be doing (hence “verb”) the experiencing. Maybe, we might conclude that it is right to go into our friend’s phone, maybe we will not but whatever happens the outcome
could not have been known beforehand
. What our conscience would have us do is revealed when we live in the world and not through armchair reflection.
5. The Six Propositions of Situation Ethics
Fletcher gives six propositions (features) of his theory.
1: Only one ‘thing’ is intrinsically good; namely, love, nothing else at all
There is one thing which is intrinsically good, that is good irrespective of context, namely love. If love is what is good, then an action is right or wrong in as far as it brings about the most amount of love. Echoing Bentham’s Hedonic Calculus (see Chapter 1) Fletcher defends what he calls the:
agapeic calculus, the greatest amount of neighbor welfare for the largest number of neighbors possible. 8
Notice that here he talks about “ welfare ” rather than “love”. Fletcher does this because of how he understands love which, importantly, is not about having feelings and desires. We discuss this below.
2: The ruling norm of Christian decision is love, nothing else
As we have seen in the first proposition, the
only
way to decide what we ought to do (the ruling norm) is to bring about love. We need to be careful though because for Fletcher “love” has a technical meaning.
By love Fletcher means “
agápē
” — from ancient Greek.
Agápē
has a very particular meaning. Initially it is easier to see what it
is not
. It is not the feeling we might have towards friends or family member which is better described as brotherly love (
philēo
). Nor is it the erotic desire we might feel towards others (
érōs
).
Rather
agápē
is an
attitude
and not a feeling at all, one which does not expect anything in return and does not give any special considerations to anyone.
Agápē
regards the enemy in the same way as the friend, brother, spouse, lover. Given our modern context and how people typically talk of “love” it is probably unhelpful to even call it “love”.
Typically people write and think about love as experiencing an intense feeling. In cartoons when a character is in love their hearts jump out of their chest, or people “in love” are portrayed as not being able to concentrate on things because they “cannot stop thinking” about someone.
This is not what love means for Fletcher. In the Christian context
agápē
is the type of love which is manifest in how God relates to us. Consider Christ’s love in saying that he forgave those carrying out his execution or consider a more modern example. In February 1993, Mrs Johnson’s son, Laramiun Byrd, 20, was shot in the head by 16-year-old Oshea Israel after an argument at a party in Minneapolis, Minnesota. Mrs Johnson subsequently forgave her son’s killer and after he had served a 17 year sentence for the crime, asked him to move in next door to her. She was not condoning his actions, nor will she ever forget the horror of those actions, but she does love her son’s killer. That love is agápē.
3: Love and justice are the same, for justice is love distributed, nothing else
For Fletcher, practically all moral problems we encounter can be boiled down to an apparent tension between “justice” on the one hand and “love” on the other. Consider a recent story:
|
Trevell Coleman, better known as the rapper G Dep, was a rising star on the New York hip-hop scene and had been signed to P Diddy’s Bad Boy record label. He also had a wife, Crystal, and twin boys.
Yet Trevell, who was brought up a Catholic and always retained his faith, had a terrible secret, as an 18-year-old, he had mugged and shot a man. He never knew what happened to his victim, yet 17 years later, in 2010, he could no longer bear the guilt and went to the police — a step almost unimaginable for someone from the Hip Hop world. A police search of their cold case files revealed the case of John Henkel — shot and killed in 1993 at exactly the same street corner in Harlem where Trevell says he committed his crime. He is now serving a jail sentence of 15 years to life for Henkel’s murder. Yet he has no regrets; “I wanted to get right with God”, he says. Trevell’s choice was perhaps hardest to bear for his wife Crystal, who now has to bring up their teenage boys on her own. |
This could be expressed as a supposed tension between “love” of family and doing the right thing — “justice”. Fletcher thinks that most other moral problems can be thought of in this way. Imagine we are trying to decide what is the best way to distribute food given to a charity, or how a triage nurse might work in a war zone. In these cases we might put the problem like this. We want to distribute fairly, but how should we do this?
Fletcher says the answer is simple. To act justly or fairly is precisely to act in love. “Love is justice, justice is love”.
9
4: Love wills the neighbor’s good when we like him or not
This is self-explanatory. As we noted above, agápē is in the business of loving the unlovable. So related to our enemies:
Christian love does not ask us to lose or abandon our sense of good and evil, or even of superior and inferior; it simply insists that however we rate them, and whether we like them nor not, they are our neighbors and are to be loved. 10
5: Only the ends justify the means, nothing else
In direct rejection of the deontological approaches Fletcher says that any action we take, as considered as an action independent of its consequences is literally, “meaningless and pointless”. An action, such as telling the truth, only acquires its status as a means by virtue of an end beyond itself.
6: Love’s decisions are made situationally, not prescriptively
Ethical decisions are not cut and dried most of the time and they exist in a grey area. No decision can be taken before considering the situation. Fletcher gives the example of a women in Arizona who learned that she might “bear a defective baby because she had taken thalidomide”. What should she do? The loving decision was not one given by the law which stated that all abortions are wrong. However, she travelled to Sweden where she had an abortion. Even if the embryo had not been defective according to Fletcher her actions were “brave and responsible and right” because she was acting in light of the particulars of the situation so as to bring about the most love.
6. Problems with Fletcher’s Situationism
Fletcher’s Situationism is a hopelessly confused and confusing moral theory. Fletcher’s work has the annoying tendency to present trivially true claims as if they are profound philosophical insights.
At the most general level, Fletcher commits the
fallacy of appealing to authority
. This is simply the mistake of thinking that an argument is strengthened by saying that someone else — normally someone in “authority”, holds it.
Fletcher uses many quotations from famous theologians and mentions famous philosophers, such as Aristotle, as a substitute for argument. Unfortunately simply appealing to others is not an argument. To see how useless this approach is consider the following: “Walker’s crisps are healthy because Gary Lineker says so”.
The other concern throughout Fletcher’s work is that he is simply unclear and inaccurate, especially when dealing with the two central ideas: “love” and “situation”.
In some places he talks about love being an “attitude”. In other places he says it is what we ought to
bring about
as an
end point
. Which is it? Is it a loving “attitude” in virtue of which we act? Or is it about bringing about certain consequences?
To see why this might be problematic, consider a case where we act out of the attitude of
agápē
but the consequence is one of great death and destruction. Suppose we act in good “conscience” as Fletcher calls it but our act brings about horrendously dire consequences. According to Fletcher have we done right or wrong? It is not clear.
If he does say that what we did is “wrong” then fine,
agápē
should not be thought of as an attitude, but rather some feature of consequences. This reading is of course in line with his
agápē
calculus. Ok, so then imagine the devil acting out of hatred and malice but — due to his lack of knowledge — happens to bring about a vast amount of love in the world. Has the devil acted in the morally right way? If the “
agápē
calculus” is used then “yes”. So, according to Fletcher has the devil done the right thing? It is not clear.
Notice it is no good saying “well we cannot decide because it depends on the situation!” Because we have just given you the details of the situation. If you need more information, just make some up and then reframe the question. So what Fletcher means by “love” is not clear. Nor is what he means by “situation”.
If you were writing a book on Situationism you would expect a clear and extended discussion of these concepts. However, there is no discussion of it in his key text and this is an important omission. To see how thorny the issue actually is consider the following. A politician stands up and says “given the current situation we need to raise taxes”. Our first response is probably going to be “what situation?” The point, simply put, is that there is no obvious way of knowing what is meant by “situation”. What we will choose to consider in any situation will depend on what is motivating us, what our dispositions are, what agendas we have.
Consider a moral example. A terminally ill patient wants to die; given the situation what ought we to do? The point is what does, and does not, get considered in “the situation”, will be dependent on what we
already think
is important. Do we consider his religious views, the fact that he has three cats which depend on him? What about the type of illness, the type of death, who he leaves behind, the effect it might have on the judicial system, the effect on the medical profession etc.
So then, as a way of actually working out what we ought to do, Fletcher’s prescription that we should “ask what will bring about the most love in the situation” is singularly unhelpful. It seems perfectly plausible that one person might see the situation in one way and someone else see it in another, and hence we get two different claims about what we ought to do. You might think this is OK, on Fletcher’s account. But recall he rejects Antinomianism (Relativism).
It is in fact quite easy to generate lots and lots of worries about Fletcher’s account. This is because his theory is based on a very crude form of Utilitarianism. Have a look at Chapter 1 where we suggest some problems and simply replace “happiness” with
agápē
. Here is one example.
Utilitarianism is accused of being counter intuitive. If we could only save our dad or five strangers from drowning, the utilitarian would argue we should save the strangers because five lots of happiness is better than one. But is not it admirable and understandable to save a loved one over strangers?
The situationalist will have exactly the same problem. We might imagine that saving five strangers would bring about more “love” than saving your dad. In which case we ought to save the strangers over your dad. But is not it admirable and understandable to save a loved one over strangers?
You can simply repeat this substitution for most of the problems we cited regarding Utilitarianism, e.g. it being “too demanding” and hence generate a whole host of problems for Fletcher.
We leave you with the following quotation from Graham Dunstan writing in the Guardian, regarding Fletcher’s book:
It is possible, though not easy, to forgive Professor Fletcher for writing his book, for he is a generous and lovable man. It is harder to forgive the SCM Press for publishing it.
SUMMARY
|
Fletcher’s Situational Ethics gained a popular following as it allowed the religious believer to fit their views into the rapidly changing and nuanced moral and political landscape of the 1960s. Fletcher's position has a central commitment to God’s love —
agápē
. It is this central focus on
agápē
as the moral guide for behaviour that allows Fletcher to claim that an action might be right in one context, but wrong in a different context — depending on the level of
agápē
brought about. In fact, Fletcher thinks that sometimes what might be morally required of us is to break the Ten Commandments.
|
COMMON STUDENT MISTAKES
• Mixing up Fletcher’s use of “Positivism” with Ayer’s use of “positivism”.
• Thinking that Fletcher’s is a “pragmatist”.
• Think that situation ethics allows you do to anything you want.
• Think that love is about feelings.
• Think that by “conscience” Fletcher means a “moral compass”.
ISSUES TO CONSIDER
1. Why do you think Fletcher’s book was so popular at the time of publication?
2. If an alien visited earth and asked “What is love?” how would you answer them?
3. How does Situationism differ from “Utilitarianism” if at all?
4. If we act from love, does that mean we can do anything?
5. What does it mean to say that conscience is a verb rather than a noun?Do you think we have a conscience? If you do, should we think of it as a verb or a noun?
6. Why does Fletcher say that his theory is: “fact-based, empirical-based, data-conscious and inquiring”?
7. What do you think a Christian would make of Fletcher’s theory?
8. What do you think “situation” means?
9. What does Fletcher mean by “positivism”?
10. What is the “fallacy of appealing to authority”? Can you give your own example?
11. Pick one challenge to Utilitarianism, and reform the challenge as one towards Situationism.
KEY TERMINOLOGY
Agápē
Agápē calculus
Eros
Legalism
Pragmatic
Conscience
Consequentialism
References
Fletcher, Joseph F., Situation Ethics: The New Morality (Louisville and London: Westminster John Knox Press, 1966).
Kirk, Kenneth E., Conscience and Its Problems: An Introduction to Casuistry (Louisville: Westminster John Knox Press, 1999).
‘Saudi Police “Stopped” Fire Rescue’, BBC News (15 March 2002), freely available at http://news.bbc.co.uk/1/hi/world/mid...st/1874471.stm
|
1 K. E. Kirk, Conscience and its Problems , p. 331. 2 J. F. Fletcher, Situation Ethics . 3 Ibid. , p. 15. 4 ‘Saudi Police “Stopped” Fire Rescue’, BBC News, http://news.bbc.co.uk/1/hi/world/mid...st/1874471.stm 5 J. F. Fletcher, Situation Ethics , p. 27. 6 Ibid. , p. 47. 7 Ibid ., p. 51. 8 Ibid ., p. 95. 9 Ibid ., p. 89. 10 Ibid ., p. 107. |


15.6: Appendix F- Mathematical Properties of State Functions
A state function is a property of a thermodynamic system whose value at any given instant depends only on the state of the system at that instant (Sec. 2.4).
F.1 Differentials
The differential \(\df\) of a state function \(f\) is an infinitesimal change of \(f\). Since the value of a state function by definition depends only on the state of the system, integrating \(\df\) between an initial state \(1\) and a final state \(2\) yields the change in \(f\), and this change is independent of the path: \begin{equation} \int_{f_1}^{f_2}\!\df=f_2-f_1=\Del f \tag{F.1.1} \end{equation} A differential with this property is called an exact differential. The differential of a state function is always exact.
F.2 Total Differential
A state function \(f\) treated as a dependent variable is a function of a certain number of independent variables that are also state functions. The total differential of \(f\) is \(\df\) expressed in terms of the differentials of the independent variables and has the form \begin{equation} \df = \Pd{f}{x}{}\dx + \Pd{f}{y}{}\dif y + \Pd{f}{z}{}\dif z + \ldots \tag{F.2.1} \end{equation} There are as many terms in the expression on the right side as there are independent variables. Each partial derivative in the expression has all independent variables held constant except the variable shown in the denominator.
Figure F.1 interprets this expression for a function \(f\) of the two independent variables \(x\) and \(y\). The shaded plane represents a small element of the surface \(f = f(x,y)\).
Consider a system with three independent variables. If we choose these independent variables to be \(x\), \(y\), and \(z\), the total differential of the dependent state function \(f\) takes the form \begin{equation} \df = a\dx + b\dif y + c\dif z \tag{F.2.2} \end{equation} where we can identify the coefficients as \begin{equation} a=\Pd{f}{x}{y,z} \qquad b=\Pd{f}{y}{x,z} \qquad c=\Pd{f}{z}{x,y} \tag{F.2.3} \end{equation} These coefficients are themselves, in general, functions of the independent variables and may be differentiated to give mixed second partial derivatives; for example: \begin{equation} \Pd{a}{y}{x,z} = \frac{\partial^2 f}{\partial y \partial x} \qquad \Pd{b}{x}{y,z} = \frac{\partial^2 f}{\partial x \partial y} \tag{F.2.4} \end{equation} The second partial derivative \(\partial^2 f/\partial y\partial x\), for instance, is the partial derivative with respect to \(y\) of the partial derivative of \(f\) with respect to \(x\). It is a theorem of calculus that if a function \(f\) is single valued and has continuous derivatives, the order of differentiation in a mixed derivative is immaterial. Therefore the mixed derivatives \(\partial^2 f/\partial y\partial x\) and \(\partial^2 f/\partial x\partial y\), evaluated for the system in any given state, are equal: \begin{equation} \Pd{a}{y}{x,z} = \Pd{b}{x}{y,z} \tag{F.2.5} \end{equation} The general relation that applies to a function of any number of independent variables is \begin{equation} \Pd{X}{y}{} = \Pd{Y}{x}{} \tag{F.2.6} \end{equation} where \(x\) and \(y\) are any two of the independent variables, \(X\) is \(\partial f/\partial x\), \(Y\) is \(\partial \f/\partial y\), and each partial derivative has all independent variables held constant except the variable shown in the denominator. This general relation is the Euler reciprocity relation, or reciprocity relation for short. A necessary and sufficient condition for \(\df\) to be an exact differential is that the reciprocity relation is satisfied for each pair of independent variables.
F.3 Integration of a Total Differential
If the coefficients of the total differential of a dependent variable are known as functions of the independent variables, the expression for the total differential may be integrated to obtain an expression for the dependent variable as a function of the independent variables.
For example, suppose the total differential of the state function \(f(x,y,z)\) is given by Eq. F.2.2 and the coefficients are known functions \(a(x,y,z)\), \(b(x,y,z)\), and \(c(x,y,z)\). Because \(f\) is a state function, its change between \(f(0,0,0)\) and \(f(x',y',z')\) is independent of the integration path taken between these two states. A convenient path would be one with the following three segments:
-
The expression for \(f(x,y,z)\) is then the sum of the three integrals and a constant of integration.
Here is an example of this procedure applied to the total differential \begin{equation} \df = (2xy)\dx + (x^2+z)\dif y + (y-9z^2)\dif z \tag{F.3.1} \end{equation} An expression for the function \(f\) in this example is given by the sum \begin{equation} \begin{split} f & = \int_0^{x'}\!(2x\cdot 0)\dx + \int_0^{y'}\![(x')^2+0]\dif y + \int_0^{z'}\!(y'-9z^2)\dif z + C\cr & = 0 + x^2y + ( yz-9z^3/3 ) + C\cr & = x^2y + yz - 3z^3 + C \end{split} \tag{F.3.2} \end{equation} where primes are omitted on the second and third lines because the expressions are supposed to apply to any values of \(x\), \(y\), and \(z\). \(C\) is an integration constant. You can verify that the third line of Eq. F.3.2 gives the correct expression for \(f\) by taking partial derivatives with respect to \(x\), \(y\), and \(z\) and comparing with Eq. F.3.1.
A different kind of integration can be used to express a dependent extensive property in terms of independent extensive properties. An extensive property of a thermodynamic system is one that is additive, and an intensive property is one that is not additive and has the same value everywhere in a homogeneous region (Sec. 2.1.1). Suppose we have a state function \(f\) that is an extensive property with the total differential \begin{equation} \df = a\dx + b\dif y + c\dif z + \ldots \tag{F.3.3} \end{equation} where the independent variables \(x,y,z,\ldots\) are extensive and the coefficients \(a,b,c,\ldots\) are intensive. If the independent variables include those needed to describe an open system (for example, the amounts of the substances), then it is possible to integrate both sides of the equation from a lower limit of zero for each of the extensive functions while holding the intensive functions constant: \begin{equation} \int_0^{f'}\!\!\df = a\int_0^{x'}\!\!\dx + b\int_0^{y'}\!\!\dif y + c\int_0^{z'}\!\!\dif z + \ldots \tag{F.3.4} \end{equation} \begin{equation} f' = ax'+by'+cz'+\ldots \tag{F.3.5} \end{equation} Note that a term of the form \(c\dif u\) where \(u\) is intensive becomes zero when integrated with intensive functions held constant, because \(\dif u\) is this case is zero.
F.4 Legendre Transforms
A Legendre transform of a state function is a linear change of one or more of the independent variables made by subtracting products of conjugate variables.
To understand how this works, consider a state function \(f\) whose total differential is given by \begin{equation} \df=a\dx+b\dif y+c\dif z \tag{F.4.1} \end{equation} In the expression on the right side, \(x\), \(y\), and \(z\) are being treated as the independent variables. The pairs \(a\) and \(x\), \(b\) and \(y\), and \(c\) and \(z\) are conjugate pairs . That is, \(a\) and \(x\) are conjugates, \(b\) and \(y\) are conjugates, and \(c\) and \(z\) are conjugates.
For the first example of a Legendre transform, we define a new state function \(f_1\) by subtracting the product of the conjugate variables \(a\) and \(x\): \begin{equation} f_1 \defn f-ax \tag{F.4.2} \end{equation} The function \(f_1\) is a Legendre transform of \(f\). We take the differential of Eq. F.4.2 \begin{equation} \df_1 = \df - a\dx - x\dif a \tag{F.4.3} \end{equation} and substitute for \(\df\) from Eq. F.4.1: \begin{equation} \begin{split} \df_1 & = (a\dx+b\dif y+c\dif z) - a\dx - x\dif a \cr & = -x\dif a + b\dif y + c\dif z \end{split} \tag{F.4.4} \end{equation} Equation F.4.4 gives the total differential of \(f_1\) with \(a\), \(y\), and \(z\) as the independent variables. The functions \(x\) and \(a\) have switched places as independent variables. What we did in order to let \(a\) replace \(x\) as an independent variable was to subtract from \(f\) the product of the conjugate variables \(a\) and \(x\).
Because the right side of Eq. F.4.4 is an expression for the total differential of the state function \(f_1\), we can use the expression to identify the coefficients as partial derivatives of \(f_1\) with respect to the new set of independent variables: \begin{equation} -x = \Pd{f_1}{a}{y,z} \qquad b = \Pd{f_1}{y}{a,z} \qquad c = \Pd{f_1}{z}{a,y} \tag{F.4.5} \end{equation} We can also use Eq. F.4.4 to write new reciprocity relations, such as \begin{equation} -\Pd{x}{y}{a,z} = \Pd{b}{a}{y,z} \tag{F.4.6} \end{equation}
We can make other Legendre transforms of \(f\) by subtracting one or more products of conjugate variables. A second example of a Legendre transform is \begin{equation} f_2 \defn f - by - cz \tag{F.4.7} \end{equation} whose total differential is \begin{equation} \begin{split} \df_2 & = \df - b\dif y - y\dif b - c\dif z - z\dif c \cr & = a\dx - y\dif b - z\dif c \end{split} \tag{F.4.8} \end{equation} Here \(b\) has replaced \(y\) and \(c\) has replaced \(z\) as independent variables. Again, we can identify the coefficients as partial derivatives and write new reciprocity relations.
If we have an algebraic expression for a state function as a function of independent variables, then a Legendre transform preserves all the information contained in that expression. To illustrate this, we can use the state function \(f\) and its Legendre transform \(f_2\) described above. Suppose we have an expression for \(f(x,y,z)\)—this is \(f\) expressed as a function of the independent variables \(x\), \(y\), and \(z\). Then by taking partial derivatives of this expression, we can find according to Eq. F.2.3 expressions for the functions \(a(x,y,z)\), \(b(x,y,z)\), and \(c(x,y,z)\).
Now we perform the Legendre transform of Eq. F.4.7: \(f_2=f-by-cz\) with total differential \(\df_2=a\dx-y\dif b-z\dif c\) (Eq. F.4.8). The independent variables have been changed from \(x\), \(y\), and \(z\) to \(x\), \(b\), and \(c\).
We want to find an expression for \(f_2\) as a function of these new variables, using the information available from the original function \(f(x,y,z)\). To do this, we eliminate \(z\) from the known functions \(b(x,y,z)\) and \(c(x,y,z)\) and solve for \(y\) as a function of \(x\), \(b\), and \(c\). We also eliminate \(y\) from \(b(x,y,z)\) and \(c(x,y,z)\) and solve for \(z\) as a function of \(x\), \(b\), and \(c\). This gives us expressions for \(y(x,b,c)\) and \(z(x,b,c)\) which we substitute into the expression for \(f(x,y,z)\), turning it into the function \(f(x,b,c)\). Finally, we use the functions of the new variables to obtain an expression for \(f_2(x,b,c)=f(x,b,c)-by(x,b,c)-cz(x,b,c)\).
The original expression for \(f(x,y,z)\) and the new expression for \(f_2(x,b,c)\) contain the same information. We could take the expression for \(f_2(x,b,c)\) and, by following the same procedure with the Legendre transform \(f=f_2+by+cz\), retrieve the expression for \(f(x,y,z)\). Thus no information is lost during a Legendre transform.


7: Proof Techniques III - Combinatorics
Tragedy is when I cut my finger.
–Mel Brooks
-
- 7.1: Counting
- The title of this section, “Counting,” is not intended to evoke the usual process of counting sheep, or counting change. What we want is to be able to count some collection in principle so that we will be able to discover a formula for its size. There are two principles that will be indispensable in counting things: the “multiplication rule” which tells us when we should multiply, and the “addition rule” which tells us when we should add.
-
- 7.2: Parity and Counting Arguments
- This section is concerned with two very powerful elements of the proofmaking arsenal: “Parity” is a way of referring to the result of an even/odd calculation; Counting arguments most often take the form of counting some collection in two different ways – and then comparing those results. These techniques have little to do with one another, but when they are applicable they tend to produce really elegant little arguments.
-
- 7.3: The Pigeonhole Principle
- The word “pigeonhole” can refer to a hole in which a pigeon roosts (i.e. pretty much what it sounds like) or a series of roughly square recesses in a desk in which one could sort correspondence. Whether you prefer to think of roosting birds or letters being sorted, the first and easiest version of the pigeonhole principle is that if you have more “things” than you have “containers” there must be a container holding at least two things.
-
- 7.4: The Algebra of Combinations
- A binomial is a polynomial with two terms. It seems likely that you will have already seen the arrangement of these binomial coefficients into a triangular array – known as Pascal’s triangle. The thing that makes this triangle so nice and that leads to the strange name “binomial coefficients” for the number of k-combinations of an n-set is that you can use the triangle to (very quickly) compute powers of binomials.