title: Gerard John Schaefer

text: The following morning, Schaefer arrived at the bandstand at the prearranged time. On this occasion, he was not wearing his uniform and driving his own vehicle, although he convinced Trotter and Wells he was still on duty, having been switched to plain clothed, undercover duties and thus driving an unmarked vehicle.


title: Gerard John Schaefer

text: Shortly after the girls entered the vehicle, Shaefer deviated from their intended route on the pretext of showing the girls an "old Spanish fort" near Hutchinson Island. En route, he again briefly lectured the girls against accepting lifts from random strangers and the dangers of being "sold into white slavery" before stopping the vehicle close to a dilapidated shed deep inside a remote forest, where he handcuffed and gagged the girls. He then took one victim to a large cypress tree close to the Indian River, tying her legs to the trunk just below her knees before binding a noose around her neck, which he affixed to a branch in such a manner as to force her to stand upon the exposed roots to counter the pressure from the noose. Schaefer then took the other victim to another tree a short distance away where she too was bound in a similar manner in which she was forced to stand upon a narrow exposed tree root as a makeshift, sloping plinth to counter the pressure from the noose around her neck. Both were informed they were to be raped and murdered.


title: Gerard John Schaefer

text: At that moment, Schaefer received an urgent radio dispatch informing him to immediately report to the police station. He left both girls bound and standing upon their plinths, vowing that he would soon return and exclaiming to one of his captives, "I gotta go!" Both girls were warned not to "try to run away, 'cause I'm not going to be very far down the road", with Schaefer claiming he was to confer with the individual he intended to sell them to.


title: Gerard John Schaefer

text: When Schaefer returned to the forest approximately two hours later, he discovered that both girls had escaped; he immediately returned home to call his station, where he informed Sheriff Robert Crowder: "I've done something very foolish; you'll be mad at me." Schaefer then proceeded to explain that he had decided to teach two girls "a lesson" on the risks of hitchhiking but "overdid the job". He then proceeded to explain he had abandoned the two in the general swampland area of Hutchinson Island, not far from the Indian River.


title: Gerard John Schaefer

text: Crowder and Lieutenant Melvin Waldron immediately proceeded to Florida State Road A1A, where—close by the highway—they discovered a desperate, partially-gagged teenage girl with her hands pinioned behind her back swimming via a flutter kick in a subtropical river. As the officers slowed to a halt, they observed the distraught girl clamber from the riverbank with sections of her jeans and blouse shredded, attempting to gesticulate for their attention. Upon removing the gag from the girl's mouth, the officers heard her identify herself as "Nancy", sputtering her friend was somewhere in the forest. To Trotter's relief, she was informed that a truck driver had discovered Wells staggering through the woodland in the direction towards the highway approximately forty-five minutes earlier, and that her friend was already at the police station.


title: Gerard John Schaefer

text: Trotter was driven to the station, where she and Wells recounted their ordeal to Crowder. Both girls stated they had managed to escape from their bindings by gingerly writhing against their restraints and loosening their gags with their teeth as they maintained their balance upon the exposed tree roots; each stated the process of freeing themselves had taken considerable time, and that they had been acutely aware that, had they slipped, they would have hanged. Both girls provided a detailed description of their assailant and his vehicle, before formally identifying Schaefer as the individual responsible for their ordeal.


title: Gerard John Schaefer

text: Although Schaefer repeated his insistence that he had simply overreacted in his efforts to demonstrate the dangers of hitchhiking to the two young women, his story was not believed; he was dismissed from the force and placed under arrest, with Crowder instructing his officers to file charges of false imprisonment and aggravated assault against him.


title: Gerard John Schaefer

text: Approximately two weeks after his arrest, Schaefer posted a $15,000 bail, thus meaning he remained at liberty prior to his scheduled November 1972 trial. He returned to the house he and his second wife rented in Stuart; his wife and in-laws noted no change in his demeanor—believing his claim to have simply been "trying to teach [the hitchhikers] a lesson". As Schaefer awaited trial, he obtained menial employment at Kwik Chek.


title: Gerard John Schaefer

text: On September 27, 1972, Schaefer abducted two teenage friends named Susan Carol Place (17) and Georgia Marie Jessup (16). The two had encountered Schaefer while all three attended an adult education center in Fort Lauderdale. Schaefer introduced himself to the girls as "Jerry Shepherd", claiming to hail from Colorado and stating that he intended to return there following a trip to Mexico; he likely feigned interest in Jessup's fascination with the concept of reincarnation and ESP in addition to Place's love of music in order to ingratiate himself with both girls and gain their trust.


title: Gerard John Schaefer

text: On the afternoon of their disappearance, Place's mother, Lucille, arrived home to find her daughter "straightening her room" as Jessup sat upon a chair in the bedroom; both introduced Lucille to a man in his 20s whom they referred to as "Jerry". Place initially informed her mother that she, Jessup and "Jerry" intended to travel from Fort Lauderdale "to the beach and play guitar". Although Place's mother remained suspicious, "Jerry" assured her his intentions were noble; nonetheless, she noted the number of his 1969 Datsun. Place confirmed her mother's suspicions that she intended to leave home, although she tearfully assured her that she would be gone "just for a little while" and that she would remain in contact. The girls left the Place household with Schaefer at approximately 8:45 p.m.


title: Gerard John Schaefer

text: When Place had not returned after four days, Lucille first contacted Jessup's mother, Shirley, only to learn her daughter had "run away" on September 27, and that she likewise had not heard from either girl since. Both girls were subsequently reported missing to the Oakland Park police. Lucille provided investigators with the vehicle registration she had noted, in addition to a physical description of the man the girls had left her home with. The registration was traced to an entirely separate model of vehicle belonging to a St. Petersburg resident who did not resemble "Jerry Shepherd", and who had a firm alibi for the date of the girls' disappearance. The sole Jerry Shepherd registered as living in Fort Lauderdale was also eliminated from police inquiries and by early 1973, the teenagers' disappearance had largely become a cold case.


title: Gerard John Schaefer

text: Mary Alice Briscolina (14) and Elsie Lina Farmer (13) vanished while hitchhiking to a Commercial Boulevard restaurant from a Lauderdale-by-the-Sea motel on October 26, 1972, less than one month after Place and Jessup were last seen alive. Their bodies were separately recovered in undergrowth close to Sunrise Boulevard, not far from the city of Plantation, early the following year—both with legs spread apart. Briscolina had been extensively beaten about the head, with one blow to her skull proving fatal. Several of her fingernails had been torn from her body, indicating a ferocious physical struggle with her killer. Farmer had also been bludgeoned to death.


title: Gerard John Schaefer

text: Subsequent questioning of Briscolina's friends revealed she and Farmer frequently visited a Lauderdale-by-the-Sea apartment rented by the older sister of Briscolina's boyfriend, and that an individual these acquaintances recalled as a "Gary Shepherd" had been known to Briscolina. This individual—whom several identified as Schaefer—had also claimed to be an "ex-Wilton Manors police officer."


title: Gerard John Schaefer

text: Three months later, on January 11, 1973, hitchhikers Collete Marie Goodenough and Barbara Ann Wilcox (both 19) disappeared while hitchhiking from Sioux City, Iowa, to Florida. Both were last seen alive in Biloxi, Mississippi. Their disappearances occurred while Schaefer remained free prior to beginning his sentence for the abductions of Trotter and Wells. He is known to have made a long-distance phone call from Cedar Rapids, Iowa, to his Florida residence shortly before Goodenough and Wilcox's disappearance, and may have encountered the two while returning to Florida. Their scattered skeletal remains were discovered close to a large tree and an orange crate in January 1977. Both victims had been bound together with baling wire, and impressions upon the tree branches—coupled with the actual positioning of the orange crate—indicate one or both victims had been suspended from the tree as their murderer sat or stood upon the orange crate.


title: Gerard John Schaefer

text: In December 1972, Schaefer appeared in court in relation to the Trotter and Wells abductions. Due to a plea bargain his attorney strongly recommended he accept, Schaefer was able to plead guilty to just one charge of aggravated assault, for which he received a sentence of one year in jail with the possibility of parole after six months, to be followed by three years' probation.


title: Gerard John Schaefer

text: Upon passing sentence on December 22, Judge D. C. Smith lambasted Schaefer, informing him: "It is beyond the court's imagination to conceive how you were such a foolish and astronomic jackass as you were in this case." He allowed Schaefer's formal sentencing to be postponed until "after the holidays", and he began serving his sentence in Martin County jail on January 15, 1973. Upon leaving court on December 22, Schaefer informed several reporters: "I made a stupid mistake. There was no sex involved... no one was hurt."


title: Gerard John Schaefer

text: In March 1973, Lucille Place discovered a letter penned by "Jerry Shepherd" in her daughter's bedroom; she drove to the return address upon the letter—333 Martin Avenue in Stuart, Florida—only to learn from the building manager that "Jerry Shepherd" had registered at the property under his real name, Gerard Schaefer, and that he was recently sent to jail for the abduction and attempted hanging of two girls.


title: Gerard John Schaefer

text: As Lucille and her husband, Ira, drove around the county, she realized investigators had likely incorrectly noted Schaefer's license plate number as being registered in Pinellas County as opposed to Martin County, given the fact most license plates she observed in Martin County began with ""42"" and not the ""4"" of Pinellas County, which investigators had previously informed her was the location of the plate she had provided. Upon relaying this new information to police, Lucille discovered the plate she had noted actually belonged to a blue-green Datsun registered to one Gerard Schaefer, who resided at 333 Martin Avenue.


title: Gerard John Schaefer

text: When questioned, Schaefer denied ever having encountered Place or her parents, although Lucille positively identified a Wilton Manors personnel photograph of Schaefer as "Jerry Shepherd".


title: Gerard John Schaefer

text: On the afternoon of April 1, a father and son searching for discarded aluminum cans discovered the extensively decomposed remains of two individuals scattered within and around a hole dug among trees in Oak Hammock Park, Port Saint Lucie, Florida. The location of this grave was feet from the nearest dirt road; the grave itself was two feet, three inches (0.69 m) deep.


title: Gerard John Schaefer

text: Deep scratch marks were evident upon the base of the tree immediately alongside the gravesite, close to where sections of a torso had been bound to the base of the trunk. One victim wore the remnants of blue jeans emblazoned with a circular emblem of the Road Runner, whereas the other was completely nude, and a pile of clothing belonging to the decedents was discovered in nearby undergrowth. Sections of both bodies had evidently been disinterred and scattered by wildlife. The location of the discoveries was approximately six miles from where Trotter and Wells had been held captive prior to their escape the previous summer.


title: Gerard John Schaefer

text: Both girls had been bound and murdered, with their spinal cords severed at the lumbar and cervical section and several bones completely severed with a knife or machete. Their bodies had been decapitated after death and their jawbones had sustained numerous fractures. One set of remains, later identified as Place, had also sustained a gunshot wound to her lower jaw consistent with having been inflicted by a .22 caliber pistol. Furthermore, sections of wearing of tree bark upon a banyan tree approximately nine from the grave indicated one or both victims had been suspended from this tree long enough to leave welt impressions within the bark prior to their deaths, and the initials ""G.J."" had been carved into a tree trunk, the roots of which bore several deep knife, machete or ax incisions containing torn sections of clothing fibers.


title: Gerard John Schaefer

text: The bodies were taken to the Dade County Medical Examiner's Department, where Dr Richard Souviron formally identified the victims via dental records and healed bone fractures as Place and Jessup on April 5. Shortly thereafter, Schaefer was informed of the identifications; he immediately requested the representation of a public defender. The individual appointed as his legal representative was Elton Schwartz.


title: Gerard John Schaefer

text: The location of these discoveries and the decedents' identities, plus the similarities in the "modus operandi" of the method of abduction and murder of Place and Jessup to Schaefer's earlier captivity of Trotter and Wells, led police in Broward County and Martin County to obtain search warrants for Schaefer's house and vehicle, as well as the home of his mother after Place's mother formally identified Schaefer as being the man she had last seen with her daughter and Jessup.


title: Gerard John Schaefer

text: Inside a locked bedroom at the Fort Lauderdale residence of Schaefer's mother, police found 300 pages of lurid stories (occasionally accompanied by crude illustrations) Schaefer had both penned and typed over the course of several years. These stories detailed the kidnapping, humiliation, rape, and execution by hanging of a number of teenage girls and young women whom he routinely referred to as "whores", "sluts" and "harlots", including two named "Belinda" and "Carmen" and an unidentified woman whom he graphically describes hanging at an unknown location close to Powerline Road.


title: Gerard John Schaefer

text: Several of these narratives indicate Schaefer had forced his victims to drink beverages—typically beer or wine—as they stood upon makeshift plinths with a noose around their necks so that he could observe them urinate prior to their hanging. He had frequently returned to his crime scenes weeks or months after the actual murders in order to commit acts of necrophilia with buried and dismembered bodies, or to extract teeth from the skull. His writings also revealed his fascination with historical methods of torture and execution, and the pleasure he derived from observing acutely distressed females urinate and/or defecate prior to and at the time of their actual hanging.


title: Gerard John Schaefer

text: Also found at the Fort Lauderdale residence were eleven guns, bags filled with live and spent cartridges, thirteen hunting knives, sections of rope, and scores of softcore pornographic magazines which he had modified to depict nude, urinating women bound with ropes, hanging from trees or other makeshift gallows, or bearing bullet wounds. Other images recovered were thirty-seven black-and-white Polaroid photographs depicting women being hung and/or mutilated, typically within melaleuca thickets suspected to be in an undergrowth area in Davie, Florida, although the focus of these images was insufficient to permit identification of the subjects. Several other images depicted Schaefer dressed in female garments simulating his own hanging from a tree with fecal matter smeared across his buttocks.


title: Gerard John Schaefer

text: A letter dated July 20, 1971, from an individual in Brunswick East, Victoria, Australia, whom Schaefer had become acquainted with as the two traveled in Morocco in the summer of 1970, contained scores of Polaroid images this individual had taken of a "village in the Sahara" the two had encountered, following what his companion termed a "wog massacre" of both Europeans and Arabs. Several of these images depicted women who had been extensively disembowelled and otherwise mutilated with knives and axes.


title: Gerard John Schaefer

text: Inside a gold jewelry box, investigators discovered personal possessions such as jewelry, passports, and clothing belonging to several teenage girls and young women. Some of these items—such as a distinctive heart shaped charm inscribed with the initials ""M.T.N.""—investigators were unable to link to missing or murdered individuals; others were identified as belonging to young women who had been reported missing in recent years. One gold locket, inscribed with the name ""LEIGH"", was determined to belong to a missing woman named Leigh Hainline Bonadies, who had been a neighbor of Schaefer's when both were teenagers and who had been missing since September 1969; also recovered was a driver's license belonging to Barbara Ann Wilcox and a passport belonging to Collette Marie Goodenough—both of whom had been reported missing in January 1973. Furthermore, teeth and sections of bone later identified as belonging to at least eight victims were also recovered from the property.


title: Gerard John Schaefer

text: The April 6 search of Schaefer's Martin County residence yielded less physical evidence, although investigators did recover two human teeth stowed in a plastic capsule inside the master bedroom, several knives and firearms inside a utility shed, and an extensively bloodstained white pillowcase which had evidently been washed. Jessup's distinctive suede purse was discovered to have been in the possession of Schaefer's wife; she later informed police her husband had given her the item as a gift in "about November" of the previous year, but had attempted to persuade both her and his brother-in-law, Henry Dean, to discard the item upon learning of the April 1 discoveries in Oak Hammock Park with the explanation the police may use the item to "make up some kind of evidence" against him. Despite Schaefer's efforts, his brother-in-law had given the item to police.


title: Gerard John Schaefer

text: By May 12, investigators had gathered enough physical and circumstantial evidence to link Schaefer to nine murders and unsolved disappearances dating between 1969 and 1973. The same month, a periodical published a list of twenty-eight murdered or missing individuals believed to be linked to Schaefer. The majority of these individuals hailed from Florida, although two victims each hailed from Iowa and West Virginia. At a press conference held on May 14, Chief Investigator Lem Brumley Jr. informed the media that, "in terms of scope and bizarreness", the case was the biggest he had encountered in his career to date.


title: Gerard John Schaefer

text: On May 18, Schaefer was formally charged with first-degree murder for the killings of Place and Jessup; he was held without bond pending trial and transferred to Florida State Hospital in Chattahoochee to undergo thirty days of psychiatric examinations before being returned to St. Lucie County jail on June 20. The results of these examinations revealed Schaefer to be an individual suffering from paranoia, psychosis and acute sexual deviation who viewed himself as "an eliminator of women he deemed immoral", but nonetheless mentally competent to stand trial.


title: Gerard John Schaefer

text: At a circuit court hearing on June 21, District Attorney Robert Stone successfully argued before Judge Cyrus Pfeiffer Trowbridge that Schaefer was sane, and thus competent to stand trial. Schaefer vehemently protested his innocence, claiming the accusations against him were "a mistake" and informing one reporter he remained confident he would be exonerated.


title: Gerard John Schaefer

text: Schaefer was brought to trial on September 17, 1973. He was tried in St. Lucie County before Judge Trowbridge. The prosecution team consisted of Stone and Philip Shailer, assisted by Richard Purdy and Anthony Young. Schaefer was defended by Schwartz and Bruce Colton, assisted by James Brecker.


title: Gerard John Schaefer

text: As the murders of Place and Jessup had been committed at a time when the Supreme Court of Florida had declared the death penalty unconstitutional in the state, prosecutors sought a penalty of life imprisonment for Schaefer. The defendant pleaded not guilty to the charges against him, and frequently conveyed a distant and aloof demeanor throughout the official proceedings; often staring coldly at prosecution witnesses as they testified, or turning to smile at members of the press when a witness testified for his counsel.


title: Gerard John Schaefer

text: Although Schaefer did testify at the pretrial hearing (in which he denied any culpability in the murders of Place and Jessup and claimed to be unable to recall his whereabouts on the date of their abduction and murder), upon the advice of his defense counsel he did not testify at the trial itself, which formally began on September 19.


title: Gerard John Schaefer

text: In his opening statement to the jury on September 19, Stone outlined the prosecution's contention that the physical and circumstantial evidence to be presented—in addition to various forms of testimony—would clearly illustrate Schaefer's guilt in the murders of Place and Jessup, and that the state expected the jurors to return "a verdict of guilty of murder in the first degree as to both counts". Following the conclusion of Stone's opening statement, Schwartz requested the permission of the court to reserve his opening statement until the state had rested its case. This request was granted, with no objection from the prosecution.


title: Gerard John Schaefer

text: The first witnesses to testify on behalf of the state were the two individuals who had discovered the dismembered bodies of Place and Jessup in Oak Hammock Park; both men recounted their discoveries, observations, and contacting of authorities. Their testimony was followed by that of Lieutenant Patrick Duval of the St. Lucie County Sheriff's Office, who described the crime scene, the evidence retrieved, and the grave in which the victims had been buried. Duval also testified that the lack of erosion of the grave indicated the grave had been "dug into" at least twice; he also identified several crime scene photographs.


title: Gerard John Schaefer

text: Place's parents also took the stand on the first day of the trial to formally identify Schaefer as the man whom they had seen their daughter and Jessup in the company of prior to their disappearance. Despite efforts to discredit their identification by the defense counsel, both testified as to their certainty that Schaefer had been the individual who visited their house on the evening of their daughter's disappearance, and whom she and her friend had left their home in the company of. Jessup's parents and younger sister, Cheryl, also formally identified the distinctive suede purse found in his wife's possession as belonging to Jessup.


title: Gerard John Schaefer

text: Also to testify in the opening days of the trial was a clerk within a county courthouse, who revealed Schaefer's vehicle had been registered at his Martin Avenue address one month prior to the murders, thus proving the license plate Place's mother had noted indeed belonged to Schaefer. Next on the stand was Lieutenant David Yurchuk, who outlined the April 1973 search of Schaefer's mother's home and the evidence retrieved. Dr. Richard Souviron testified as to the autopsies he had conducted upon the victims, and their formal identification via dental records and distinctive healed bone fractures. Souviron withstood vigorous cross-examination from Schwartz as to the actual caliber of firearm which had inflicted the gunshot wound to Place's lower jaw, remaining adamant the bullet used to shoot Place was .22 caliber and not a .25 caliber cartridge. A Dade County medical examiner named Dr. Joseph Davis also detailed the restraining of the victims prior to their deaths, the dismemberment inflicted upon their bodies, and the numerous knife marks evident upon the victims' bones.


title: Gerard John Schaefer

text: Nancy Trotter and Paula Wells also testified as to their abduction and narrow escape from Schaefer during the first week of the trial. Trotter also recounted to the court a statement Schaefer had made to her after she had been bound to a cyprus tree: that he could dig a hole in which to bury her and Wells, and that nobody would ever find them. Their testimony was followed by a video presentation depicting the girls as they had been bound "in situ" at Hutchinson Island; this footage was admitted into evidence by Schwartz, who was eager to illustrate to the jury his contention there had been "no harm done" to the girls.


title: Gerard John Schaefer

text: Among the state's evidence presented before the jury were the actual tree limbs the victims had been suspended from and which bore sections of wearing upon the tree bark, indicating both girls had remained suspended at the murder location for a period of time sufficient to create these deep markings before their deaths. Also introduced into evidence were the actual tree roots upon which Place and Jessup had been forced to balance themselves to prevent themselves from being hung prior to their torture and murder. A microanalyst named Harold Nute testified the sections of clothing fibers discovered in the root were a precise match to one of the items of clothing at the crime scene, and that hemp fibers within the sections of wearing upon the tree limbs had likely sourced from rope.


title: Gerard John Schaefer

text: Following the entry into evidence of a manuscript penned by Schaefer in which he detailed how to "properly execute" women, Stone announced the state had presented their case. Schwartz immediately called for Schaefer's acquittal, claiming the state had not proven their case against his client and that all evidence presented was circumstantial. The court ruled against this motion.


title: Gerard John Schaefer

text: The strategy of the defense was to discredit previous witnesses' testimony regarding the identification of Schaefer, to contend one or both decedents had been alive after September 27, 1972, and to contend the gravesite had been dug on a date after Schaefer began serving his sentence in Martin County Jail on January 15, 1973. Several of these individuals—including Schaefer's sister and wife—testified as to his physical characteristics and attire differing from those described by Place's parents. Also to testify on Schaefer's behalf was a hunting companion named Edward Harris, who stated he always carried a purse similar to that identified by Jessup's parents and sister when the two went target shooting.


title: Gerard John Schaefer

text: One of the final witnesses to testify on behalf of the defense was a representative from the Fort Pierce City Water Plant, who testified as to the records the firm had retained of rainfall in an area close to Oak Hammock Park between September 1972 and April 1973. Upon cross-examination, this individual conceded the records presented related close to the vicinity of the crime scene, but not the vicinity of the crime scene itself.


title: Gerard John Schaefer

text: The final defense witnesses to testify delivered their testimony on September 26. The following day, both counsels delivered their closing arguments to the jury. Both counsels were granted two hours to deliver their argument. Prosecutor Philip Shailer delivered his argument first; he first referenced the fact the date of his closing argument was the first anniversary of Place and Jessup's disappearance before outlining the identification of Schaefer and his vehicle by Place's mother, the physical evidence linking the defendant to the victims, the evidence and paraphernalia recovered in the search warrants, and the similar fact evidence outlined, all collectively proving "not only beyond a reasonable doubt, but a shadow of a doubt" of Schaefer's guilt.


title: Gerard John Schaefer

text: Referencing the similarities between the abductions of Trotter and Wells to the murders of Place and Jessup, Shailer remarked the one "glaring dissimilarity" between the two incidents was that "thank God, Wells and Trotter escaped". He finished his closing argument by referencing the Constitution; outlining the fact Schaefer had received a fair trial, with legal representation, and the burden of proof upon the prosecution to prove his guilt. Referencing Place and Jessup, Shailer stated Schaefer had shown "no presumption of innocence" for the girls, but had acted as the "prosecutor, judge and executioner" in the commission of their murders.


title: Gerard John Schaefer

text: Schwartz attempted to discredit much of the state's evidence and witness testimony in his closing argument. He dismissed the accuracy of Lucille Place's identification of Schaefer and his vehicle before referencing speculation as to the actual date of the victims' deaths, which he suggested may have occurred after January 15, 1973 or on a date Schaefer is known to have been out of state. With regards to the personal artifacts of the decedents introduced into evidence, Schwartz inferred towards testimony from his client's family that the artifacts had actually belonged to Schaefer himself.


title: Gerard John Schaefer

text: In reference to grave in and around which the victims were discovered, Schwartz also alleged the grave must have been dug on a date after Schaefer's January 15 incarceration. To support this contention, Schwartz again referenced the "lack of erosion" at the burial site, despite over nineteen inches of rain having fallen in the vicinity between the date of the victims' disappearance and their discovery. Schwartz concluded his argument by contending the prosecution had not proven beyond a reasonable doubt Schaefer's guilt. Emphasizing his client's presumption of innocence, Schwartz stated in the closing stages of his argument: "Ladies and gentlemen, they have got the wrong man, and I hate to think of that, because somebody did this, but I don't believe it was Gerard John Schaefer Jr." Schwartz ended his closing argument by formally requesting the jury return a verdict of not guilty.


title: Gerard John Schaefer

text: Stone delivered his rebuttal argument on the afternoon of September 26. He began by dismissing one of Schwartz's tactics in his closing argument: the illustration of the circumstantial evidence as not being definitive proof of his client's guilt and of simply accusing him due to similar fact evidence regarding the Trotter and Wells incident. Stone then stated that the prosecution's case was constructed from both physical and circumstantial evidence—all collectively illustrating Schaefer's guilt. Referring to those who had testified, he began by defending the testimony of Lucille Place; emphasizing the crucial role in her accurate recording of Schaefer's license plate to ultimately linking him to the murders before inferring to the testimony of numerous individuals who had also testified, substantiating her identification of Schaefer and his vehicle. Stone also hearkened toward the positive identification of Jessup's purse by her mother, sister, and friend, who had each positively identified the item as belonging to Jessup, before inferring the defense witnesses who had also identified the item had lied to protect Schaefer, stating: "How many of them picked it up, looked at it? The first thing they [all] did was say, 'Yes, I recognize it. He brought it back from Morocco in 1970.'" He then referenced the testimony of Schaefer's own wife, who had stated to the court she had been given the item by her husband as a gift in November 1972, and had never seen the item prior.


title: Gerard John Schaefer

text: When referencing the fact ballistic testing had revealed none of the firearms recovered from Schaefer's utility shed had been a match for the .22 gunshot cartridge recovered at the crime scene, Stone referenced the testimony of Schaefer's brother-in-law, who had stated Schaefer also owned a .22 pistol he termed his "Saturday-night Special", which had not been found. Holding aloft and pointing toward one of Schaefer's handwritten manuscripts introduced into evidence titled ""How to Get Away with Murder"", Stone stated: "I submit to you ... you will never find that Saturday-night Special, because it is in a canal with [the victims'] skulls."


title: Gerard John Schaefer

text: In conclusion of his rebuttal argument, Stone referenced a comment his colleague had earlier made in the state's closing argument regarding Schaefer being the "prosecutor, judge and executioner" in the murders, adding "I submit to you one [more word]: God! Because he decided whether he was going to let them live or die. He became God!" Stone concluded his argument by stressing the evidence presented negated the defense contention of Schaefer's innocence before requesting a verdict of guilty.


title: Gerard John Schaefer

text: Following the closing arguments, the jury began their deliberations at 3:45 p.m. They deliberated for five hours and ten minutes before returning with two verdicts of first-degree murder, which Judge Trowbridge formally announced to the court at 11:05 p.m. Upon receipt of this verdict, Schaefer proclaimed his innocence, stating to reporters: "That's the roll of the dice. I had a good defense [but] I'm innocent."


title: Gerard John Schaefer

text: Closing arguments to determine the sentence Schaefer should receive began on October 3, and saw the defense argue he should be involuntarily institutionalized under the 1971 Baker Act, in contrast to the prosecution's argument of life imprisonment to be served in Florida State Prison. The following day, Schaefer was sentenced to two concurrent terms of life imprisonment. When asked if he had anything to say prior to sentencing, Schaefer proclaimed his innocence before requesting he be sent to a psychiatric hospital as opposed to prison.


title: Gerard John Schaefer

text: Place's mother expressed her satisfaction at Schaefer's sentence of life imprisonment as opposed to his receiving the death penalty, informing reporters: "At first I thought I'd like to see him dead, but I think people suffer more with confinement... death is the easy way out. Just as long as he's never on the streets again."


title: Gerard John Schaefer

text: Contemporary statutes indicated the possibility of parole for Schaefer after he had served between fourteen and nineteen years' imprisonment, although his presumptive parole release date was revised in May 1979 to indicate the likelihood of parole in 2016. Schaefer appealed his conviction, contending he had never been indicted by a grand jury and thus requesting a new trial. This appeal was rejected in June 1974.


title: Gerard John Schaefer

text: In the years following his conviction, Schaefer maintained his innocence in the Place and Jessup murders and denied his culpability in any others; insisting he was framed by what he termed "overzealous" prosecutors, corrupt law enforcement personnel and his own defense attorney, Elton Schwartz, who ultimately married Schaefer's second wife on December 21, 1973. Although maintaining his innocence and claiming to have never met either girl, Schaefer frequently—and falsely—described Place and Jessup as heroin users, police informants, and promiscuous. He was also known to have insulted their parents.


title: Gerard John Schaefer

text: While incarcerated, Schaefer developed a reputation among his fellow inmates as an aloof and imperious individual who many suspected of being an informant in efforts to earn privileges from authorities. In the early years of his confinement, he was placed in solitary confinement for prison mail violations on more than one occasion, occasionally for attempting to persuade members of the public to mail him female undergarments. He spent much of his time devoted to correspondence and what he termed a "new writing genre" called "killer fiction". Schaefer described this genre as one which does not glorify violence, but enables the reader to see the acts of murder and necrophilia in stark reality. Although he claimed these writings were fictional, several investigators believe many of these writings incorporate his own murders and assaults.


title: Gerard John Schaefer

text: In 1983, Schaefer was transferred to the Avon Park Correctional Institution; here, he assisted authorities in obtaining sufficient evidence to secure the conviction of pedophile Mervyn Eric Cross, who had been discreetly operating an international softcore child pornography operation while incarcerated. The information Schaefer provided also resulted in the arrest and conviction of two individuals in Seattle and Los Angeles (one a schoolteacher), the seizure of thousands of indecent images of children across four states and Cross being transferred to Florida State Prison and placed in solitary confinement, with his mail closely monitored. Schaefer was himself returned to Florida State Prison in August 1985.


title: Gerard John Schaefer

text: Schaefer also filed several frivolous lawsuits while incarcerated. One of these lawsuits was filed in 1993 against true crime writer Patrick Kendrick, who responded to a letter from Schaefer—masquerading via an outside contact as a college student—purporting to seek advice as to how to overcome his intimidation when meeting the "deadliest killer ever" who was believed to be "worse than Ted Bundy". In response to this letter, Kendrick had tersely responded there was little reason to be intimidated by Schaefer, whom he described as "a middle-aged, pale and doughy wimp, who preyed on victims that were physically and psychologically weaker than him." In response, Schaefer filed a $500,000 lawsuit which was ultimately settled out of court with prejudice. Kendrick agreed to provide Schaefer with a copy of the section of his manuscript which contained the revised list of his proposed victims, including those which Kendrick's research had revealed he could not have committed. He also sued true crime authors Sondra London, Colin Wilson, and Michael Newton and former FBI agent Robert Ressler for describing him as a serial killer in printed works.


title: Gerard John Schaefer

text: London, who had been Schaefer's girlfriend in high school shortly prior to his 1964 graduation, interviewed Schaefer at length following his conviction; she gradually served as his co-author in his writing genre, later publishing a compilation of his short stories and drawings entitled "Killer Fiction" in 1990. A second book, "Beyond Killer Fiction", followed two years later. The stories within "Killer Fiction" and "Beyond Killer Fiction" typically involve the savage, graphic torture, mutilation and murder of young women, usually penned from the perspective of the killer, who is often a rogue police officer.


title: Gerard John Schaefer

text: Schaefer remained adamant the stories he authored were pure fiction and that he had actually begun writing stories of this nature in the 1960s, having been inspired to do so after watching Herschell Gordon Lewis's 1963 splatter movie "Blood Feast". However, police and prosecutors viewed the content as thinly veiled descriptions of his actual crimes. In private letters to his attorneys, Schaefer admitted these speculations were true, claiming one story, "Murder Demons", specifically recounts the murders of Briscolina and Farmer.


title: Gerard John Schaefer

text: A revised edition of "Killer Fiction", published after Schaefer's death, included numerous stories and rambling articles from the first two books and a collection of letters to London, in which Schaefer claimed to be responsible for the murders of scores of women and girls. He also claimed that fellow serial killer inmate Ted Bundy, whom he labeled a "tyro" by comparison to himself, both admired and envied him; he also accused Bundy of "playing at copycat" regarding his own crimes.


title: Gerard John Schaefer

text: London noted that at the time Schaefer was corresponding with her, he was publicly proclaiming his innocence and threatening to sue anyone who labeled him a serial killer. In one letter, Schaefer claimed to have begun murdering women in 1965, when he was aged 19; in another, he claimed to be responsible for the December 1969 disappearance of two schoolgirls, nine-year-old Peggy Rahn and eight-year-old Wendy Stevenson, whom he claimed to have kidnapped from Pompano Beach and cannibalized. Publicly, Schaefer had denied any involvement.


title: Gerard John Schaefer

text: London ended her collaboration with Schaefer in 1991, shortly after publicly repudiating his claims that he was merely a "framed ex-cop" who wrote lurid fiction. Upon hearing that he had been publicly rebuked by London, Schaefer allegedly repeatedly threatened her life. One of the many frivolous lawsuits he filed was directed against her for publicly referring to him as a serial killer in print format. In support of London's defense against this lawsuit, she compiled an exhibit of photocopies of five hundred incriminating pages of Schaefer's handwritten correspondence. The judge immediately dismissed Schaefer's lawsuit. His aforementioned lawsuits against Newton and Wilson were likewise dismissed after London provided copies of the five-hundred page exhibit to them; his lawsuit against Kendrick was still ongoing at the time of Schaefer's 1995 murder.


title: Gerard John Schaefer

text: On December 3, 1995, Schaefer was stabbed to death on the floor of his cell. He had been stabbed over forty times about the face, head, neck and body, with his throat also being slashed, his right eye destroyed, and several ribs fractured. His body was discovered after a fellow inmate informed staff of his death.


title: Gerard John Schaefer

text: According to prison officials and prosecutors, a 32-year-old fellow inmate named Vincent Faustino Rivera had killed Schaefer following an argument over who received the final cup of hot water from a dispenser days prior to his murder. Rivera was convicted of Schaefer's murder in 1999; he received 53 years and ten months' imprisonment added to the sentence of life plus twenty years he was already serving for a double homicide committed in Tampa in 1990. Rivera never confessed to the crime nor gave any motive for the murder. It is suspected Schaefer was killed for being a prison informant, as in the year prior to his murder other inmates had repeatedly thrown human waste at him and twice set his cell on fire. Reportedly, Schaefer's classification officer confirmed that he was murdered in a direct response to his leaking confidential information to authorities pertaining to a well-respected and powerful inmate.


title: Gerard John Schaefer

text: Upon hearing of Schaefer's death, the mother of victim Georgia Jessup informed the media she considered Schaefer's own murder a case of long overdue justice, stating: "I've always believed he was going to get this. I just wish it had been sooner rather than later." The judge who had presided over his murder trial, Cyrus Trowbridge, remarked: "He's finally gotten the death sentence he ultimately deserved but couldn't be given."


title: Gerard John Schaefer

text: At the time of Schaefer's death, Broward County investigators were in the process of preparing to bring further murder charges pertaining to three unsolved murders—in part to ensure he would never be freed from prison.


title: Gerard John Schaefer

text: Sheriff Robert Crowder—who had worked alongside Schaefer for four weeks prior to his July 1972 abduction and assault of Trotter and Wells—would recollect of the case in 2009: "I think [the case] just made law enforcement more aware of the [actual] existence of serial murderers ... It made us aware of things to key on when we have a crime like this [where] there is the potential for it being one perpetrator and multiple victims." In reference to his victims, a fellow panelist at the meeting in which Crowder spoke also stated: "The [homicide] victims are not the only victims; the victims are also the families. The victims are also the law enforcement who work these cases. None of these people who worked this were ever the same."


title: Gerard John Schaefer

text: There is no consensus on when or where Schaefer began killing. As Schaefer himself was killed in prison in 1995, the true number of victims he had claimed will never be known. While incarcerated, he frequently encouraged speculation as to how many victims he had claimed, while simultaneously denying he had ever committed murder and claiming the numerous writings discovered at his mother's residence were purely fictional. Author Patrick Kendrick believes the most likely year Schaefer began killing is 1969, and that his true victim count stands at around eleven, although the commonly held belief is that he may have claimed up to twenty-eight victims.


title: Gerard John Schaefer

text: The initial list of twenty-eight potential victims was first published in a Florida periodical in the spring of 1973, although this list—which also includes seven males whose personal artifacts were recovered from his mother's home—contains the names of several individuals later discovered to have still been alive at the time of his arrest, or to have died at a later date. For example, one of the young women on this compiled list, 14-year-old Katrina Marie Bivens (reported missing in January 1970) was located alive in June of the same year, whereas one of the males upon the list, Michael Angeline, died in a plane crash in 1981.


title: Marquette, Michigan

text: Marquette ( ) is a city in Marquette County in the U.S. state of Michigan. The population was 20,629 at the 2020 United States Census, which makes it the largest city in the Upper Peninsula. Marquette serves as the seat of government of Marquette County. Located on the shores of Lake Superior, the city is a major port, known primarily for shipping iron ore. The city is partially surrounded by Marquette Charter Township, but the two are administered autonomously.


title: Marquette, Michigan

text: The land around Marquette was known to French missionaries of the early 17th century and the trappers of the early 19th century. Development of the area did not begin until 1844, when William Burt and Jacob Houghton (the brother of geologist Douglass Houghton) discovered iron deposits near Teal Lake west of Marquette. In 1845, Jackson Mining Company, the first organized mining company in the region, was formed.


title: Marquette, Michigan

text: The village of Marquette began on September 14, 1849, with the formation of a second iron concern, the Marquette Iron Company. Three men participated in organizing the firm: Robert J. Graveraet, who had prospected the region for ore; Edward Clark, agent for Waterman A. Fisher of Worcester, Massachusetts, who financed the company, and Amos Rogers Harlow. The village was at first called New Worcester, with Harlow as the first postmaster. On August 21, 1850, the name was changed to honor Jacques Marquette, the French Jesuit missionary who had explored the region. A second post office, named Carp River, was opened on October 13, 1851, by Peter White, who had gone there with Graveraet at age 18. Harlow closed his post office in August 1852. The Marquette Iron Company failed, while its successor, the Cleveland Iron Mining Company, flourished and had the village platted in 1854. The plat was recorded by Peter White. White's office was renamed as Marquette in April 1856, and the village was incorporated in 1859. It was incorporated as a city in 1871.


title: Marquette, Michigan

text: During the 1850s, Marquette was linked by rail to numerous mines and became the leading shipping center of the Upper Peninsula. The first ore pocket dock, designed by an early town leader, John Burt, was built by the Cleveland Iron Mining Company in 1859. By 1862, the city had a population of over 1,600 and a soaring economy.


title: Marquette, Michigan

text: In the late 19th century, during the height of iron mining, Marquette became nationally known as a summer haven. Visitors brought in by Great Lakes passenger steamships filled the city's hotels and resorts.


title: Marquette, Michigan

text: South of the city, K. I. Sawyer Air Force Base was an important Air Force installation during the Cold War, host to B-52H bombers and KC-135 tankers of the Strategic Air Command, as well as a fighter interceptor squadron. The base closed in September 1995, and is now the county's Sawyer International Airport.


title: Marquette, Michigan

text: Marquette continues to be a shipping port for hematite ores and, today, enriched iron ore pellets, from nearby mines and pelletizing plants. About 7.9 million gross tons of pelletized iron ore passed through Marquette's Presque Isle Harbor in 2005.


title: Marquette, Michigan

text: The Roman Catholic Bishop Frederic Baraga is buried at St. Peter Cathedral, which is the center for the Diocese of Marquette.


title: Marquette, Michigan

text: Lakeview Arena, an ice hockey rink in Marquette, won the Kraft Hockeyville USA contest on April 30, 2016. The arena received $150,000 in upgrades, and hosted the Buffalo Sabres and Carolina Hurricanes on October 4, 2016, in a preseason NHL contest. Buffalo won the game 2-0.


title: Marquette, Michigan

text: In addition to the Marquette #1 Post Office there is the "Northern Michigan University Bookstore Contract Station #384".


title: Marquette, Michigan

text: The first day of issue of a postal card showing Bishop Frederic Baraga took place in Marquette on June 29, 1984, and that of the Wonders of America Lake Superior stamp on May 27, 2006.


title: Marquette, Michigan

text: According to the United States Census Bureau, the city has a total area of , of which is land and is water.


title: Marquette, Michigan

text: The city includes several small islands (principally Middle Island, Gull Island, Lover's Island, Presque Isle Pt. Rocks, White Rocks, Ripley Rock, and Picnic Rocks) in Lake Superior. The Marquette Underwater Preserve lies immediately offshore.


title: Marquette, Michigan

text: Marquette Mountain, used for skiing, is located in the city, as is most of the land of Marquette Branch Prison of the Michigan Department of Corrections. The town of Trowbridge Park (under Marquette Township), is located to the west, Sands Township to the south, and Marquette Township to the northwest of the city.


title: Marquette, Michigan

text: The climate is a hemiboreal humid continental (Köppen: "Dfb") with four distinct seasons that are strongly moderated by Lake Superior and is located in Plant Hardiness zone 5b. Narrative below is based on chart below, reflecting 1991-2020 climate normals. Winters are long and cold with a January average of . Winter temperatures are slightly warmer than inland locations at a similar latitude due to the release of the heat stored by the lake, which moderates the climate. On average, there are 11.6 days annually where the minimum temperature reaches and 73 days with a maximum at or below freezing, including a majority of days during meteorological winter (December thru February).


title: Marquette, Michigan

text: Being located in the snowbelt region, Marquette receives a significant amount of snowfall during the winter months, mostly from lake-effect snow. Because Lake Superior rarely freezes over completely, this enables lake effect snow to persist throughout winter, making Marquette the third snowiest location in the contiguous United States as reported by the National Oceanic and Atmospheric Administration with an average annual snowfall of . The snow depth in winter usually exceeds . Marquette is the city with the deepest snow depths with a population of more than 20,000 in the US (and one of the largest in North America outside the western Cordillera or eastern Canada), as temperatures remain low throughout the winter and cold, dry air is intercepted by the Great Lakes.


title: Marquette, Michigan

text: The warmest months, July and August, each average , showing somewhat of a seasonal lag. The surrounding lake cools summertime temperatures and as a result, temperatures above are rare, with only 3.4 days per year. Spring and fall are transitional seasons that are generally mild though highly variable due to the alternation of air masses moving quickly. Spring is usually cooler than fall because the surrounding lake is slower to warm than the land, while in fall the lake releases heat, warming the area.


title: Marquette, Michigan

text: Marquette receives of precipitation per year, which is fairly evenly distributed throughout the year, though September and October are the wettest months with February and March being the driest. The average window for morning freezes is October 15 thru May 7. The highest temperature ever recorded in Marquette was on July 15, 1901, and the lowest was on February 8, 1861. Marquette receives an average of 2,294 hours of sunshine per year or 51 percent of possible sunshine, ranging from a low of 29 percent in December to a high of 68 percent in July.


title: Marquette, Michigan

text: The City of Marquette has received national attention for its measures to adapt to climate change, such as coastline restoration and moving portions of Lakeshore Boulevard which are flooded by Lake Superior 100 yards inland. Property owners are required to maintain “riparian buffers” of native plants along waterways. A county task force has created a guidebook in cooperation with the University of Michigan for landscaping which can reduce the habitat for disease-bearing ticks. A federally funded stormwater drain project will route runoff which flows into Lake Superior into restored wetlands. At the time of a 2014 NOAA climate study, climate change was expected to lead to rising temperatures, a longer growing season, and greater precipitation in Marquette.


title: Marquette, Michigan

text: As of the census of 2010, there were 21,355 people, 8,321 households, and 3,788 families residing in the city. The population density was . There were 8,756 housing units at an average density of . The racial makeup of the city was 91.1% White, 4.4% African American, 1.5% Native American, 0.9% Asian, 0.3% from other races, and 1.8% from two or more races. Hispanic or Latino of any race were 1.4% of the population.


title: Marquette, Michigan

text: There were 8,321 households, of which 18.6% had children under the age of 18 living with them, 33.3% were married couples living together, 9.0% had a female householder with no husband present, 3.3% had a male householder with no wife present, and 54.5% were non-families. 38.2% of all households were made up of individuals, and 11.8% had someone living alone who was 65 years of age or older. The average household size was 2.05 and the average family size was 2.71.


title: Marquette, Michigan

text: The median age in the city was 29.1 years. 12.2% of residents were under the age of 18; 30.6% were between the ages of 18 and 24; 22.3% were from 25 to 44; 21.9% were from 45 to 64; and 13% were 65 years of age or older. The gender makeup of the city was 51.8% male and 48.2% female.


title: Marquette, Michigan

text: At the 2000 census, there were 19,661 people, 8,071 households and 4,067 families residing in the city. The population density was 1,723.9 per square mile (665.3/km). There were 8,429 housing units at an average density of . The racial makeup of the city was 95% White, 0.8% African American, 1.7% Native American, 0.8% Asian, 0% Pacific Islander, 0.22% from other races, and 1.33% from two or more races. Hispanic or Latino of any race were 0.77% of the population. 15.5% were of German, 12.6% Finnish, 8.9% French, 8.5% English, 8.2% Irish, 6.8% Italian and 6.7% Swedish ancestry according to Census 2000.


title: Marquette, Michigan

text: There were 8,071 households, of which 23.0% had children under the age of 18 living with them, 37.2% were married couples living together, 10.2% had a female householder with no husband present, and 49.6% were non-families. 37.0% of all households were made up of individuals, and 11.5% had someone living alone who was 65 years of age or older. The average household size was 2.13 and the average family size was 2.81.


title: Marquette, Michigan

text: Age distribution was 16.8% under the age of 18, 25.9% from 18 to 24, 23.8% from 25 to 44, 19.7% from 45 to 64, and 13.8% who were 65 years of age or older. The median age was 31 years. For every 100 females, there were 94.4 males. For every 100 females age 18 and over, there were 92.4 males.


title: Marquette, Michigan

text: The median household income was $29,918, and the median family income was $48,120. Males had a median income of $34,107 versus $24,549 for females. The per capita income for the city was $17,787. About 7.2% of families and 17.0% of the population were below the poverty line, including 12.3% of those under age 18 and 5.1% of those age 65 or over.


title: Marquette, Michigan

text: Along with Northern Michigan University, the largest employers in Marquette are the Marquette Area Public Schools, UP Health System-Marquette (a regional medical center that is the only Level 2 Trauma center in the Upper Peninsula), Marquette Branch Prison, RTI Surgical, Charter Communications, and Blue Cross Blue Shield of Michigan.


title: Marquette, Michigan

text: Marquette is known for its breweries, including Ore Dock Brewing Company and Blackrocks Brewery. Five breweries were extant in the city ().


title: Marquette, Michigan

text: The city of Marquette has a number of parks and recreational facilities that are used by city and county residents. Presque Isle Park is Marquette's most popular park located on the north side of the city. It includes of mostly forested land and juts out into Lake Superior. The park was designed by Frederick Law Olmsted, noted for designing Central Park in New York City. Amenities include a wooden band shell for concerts, a park pavilion, a gazebo, a marina, a concession stand, picnic tables, barbecue pits, walking/skiing trails, playground facilities, and Moosewood Nature Center. The city has two popular beaches, South Beach Park and McCarty's Cove. McCarty's Cove, flanked by the red U.S. Coast Guard Station lighthouse on its south shore, serves as a reprieve from hot summer days, where city and county residents alike take advantage of the cool, but tolerable, water temperatures and the cooling effects of the lake-generated sea breeze. Both beaches have picnic areas, grills, children's playgrounds and lifeguard stands. Other parks include Tourist Park, Founder's Landing, LaBonte Park, Mattson Lower Harbor Park, Park Cemetery, Shiras Park, Williams Park, Harlow Park, Pocket Park, Spring Street Park and Father Marquette Park.


title: Marquette, Michigan

text: There are also numerous other recreational facilities located within the city. Lakeview Arena is best known for its use as an ice hockey facility, but it also hosts a number of public events. A skateboard park is located just outside the arena and open during the summer. Lakeview Arena was home to the Marquette Electricians and Marquette Senior High School's Redmen hockey team. In 1974, the arena replaced the historic Palestra, which had been located a few blocks away. Gerard Haley Memorial Baseball field home of the Marquette Blues and Reds is located in the north side along with numerous little league and softball fields. Marquette has the largest wooden dome in the world, the Superior Dome—unofficially but affectionately known as the YooperDome. During the football season, the Dome is used primarily for football on its newly renovated AstroTurf field. The turf was installed in July 2009. Northern Michigan University holds its home football games in the Dome, as does the Michigan High School Athletic Association with the upper peninsula's High School football playoffs. The dome also hosts numerous private and public events that draw in thousands from around the region. The Marquette Golf Club has brought international recognition to the area for its unique and dramatic Greywalls course, opened in 2005. The course features several panoramic views of Lake Superior and winds its way through rocky outcroppings, heaving fairways and a rolling valley, yet is located less than two miles (3 km) from the downtown area.


title: Marquette, Michigan

text: Marquette has an extensive network of biking and walking paths. The city has been gradually expanding the paths and has been promoting itself as a walkable and livable community. Cross Country ski trails are also located at Presque Isle Park and the Fit Strip.


title: Marquette, Michigan

text: The combination of hilly terrain (a vertical difference from top to bottom) and large area snow falls makes snowboarding and downhill skiing a reality on the edge of town.


title: Marquette, Michigan

text: Live theatrical productions are also provided through Northern Michigan University's Forest Roberts Theatre and Black Box Theatre, Marquette's Graveraet School Kaufman Auditorium and Lake Superior Theatre, a semi-professional summer stock theatre.


title: Marquette, Michigan

text: Marquette is served by American Eagle and Delta Connection out of Sawyer International Airport (MQT, KSAW) with daily flights to Chicago and Detroit. The airport is located south of downtown Marquette.


title: Marquette, Michigan

text: The city is served by a public transit system known as MarqTran, which runs buses through the city and to nearby places such as Sawyer International Airport and Ishpeming. The system operates out of a transit center in the adjacent Marquette Township in addition to a small transfer station in downtown. In addition, Indian Trails bus lines operates daily intercity bus service between Hancock and Milwaukee, Wisconsin. The line operates a stop at MarqTran's transit center.


title: Marquette, Michigan

text: Marquette has limited freight rail service by the Lake Superior and Ishpeming Railroad (LS&I). The Canadian National Railway also goes through nearby Negaunee. The LS&I serves the Upper Harbor Ore Dock, which loads iron ore pellets from nearby mining operations onto lake freighters for shipment throughout the Great Lakes.


title: Marquette, Michigan

text: Three of MDOT's state highways serve Marquette as did a former business route for US 41 and a former state highway.


title: Marquette, Michigan

text: The City of Marquette is served by the Marquette Area Public Schools. The district is the largest school district in the Upper Peninsula and Northern Wisconsin, with about 3,100 students and 420 faculty and Staff.


title: Muscle hypertrophy

text: Muscle hypertrophy or muscle building involves a hypertrophy or increase in size of skeletal muscle through a growth in size of its component cells. Two factors contribute to hypertrophy: sarcoplasmic hypertrophy, which focuses more on increased muscle glycogen storage; and myofibrillar hypertrophy, which focuses more on increased myofibril size. It is the most major part of the bodybuilding-related activities.


title: Muscle hypertrophy

text: A range of stimuli can increase the volume of muscle cells. These changes occur as an adaptive response that serves to increase the ability to generate force or resist fatigue in anaerobic conditions.


title: Muscle hypertrophy

text: Strength training (resistance training) causes neural and muscular adaptations which increase the capacity of an athlete to exert force through voluntary muscular contraction: After an initial period of neuro-muscular adaptation, the muscle tissue expands by creating sarcomeres (contractile elements) and increasing non-contractile elements like sarcoplasmic fluid.


title: Muscle hypertrophy

text: Muscular hypertrophy can be induced by progressive overload (a strategy of progressively increasing resistance or repetitions over successive bouts of exercise in order to maintain a high level of effort). However, the precise mechanisms are not clearly understood; currently accepted hypotheses involve some combination of mechanical tension, metabolic fatigue, and muscular damage.


title: Muscle hypertrophy

text: Muscular hypertrophy plays an important role in competitive bodybuilding and strength sports like powerlifting, football, and Olympic weightlifting.


title: Muscle hypertrophy

text: The best approach to specifically achieve muscle growth remains controversial (as opposed to focusing on gaining strength, power, or endurance); it was generally considered that consistent anaerobic strength training will produce hypertrophy over the long term, in addition to its effects on muscular strength and endurance. Muscular hypertrophy can be increased through strength training and other short-duration, high-intensity anaerobic exercises. Lower-intensity, longer-duration aerobic exercise generally does not result in very effective tissue hypertrophy; instead, endurance athletes enhance storage of fats and carbohydrates within the muscles, as well as neovascularization.


title: Muscle hypertrophy

text: During a workout, increased blood flow to metabolically active areas causes muscles to temporarily increase in size. This phenomenon is referred to as transient hypertrophy, or more commonly known as being "pumped up" or getting "a pump". About two hours after a workout and typically for seven to eleven days, muscles swell due to an inflammation response as tissue damage is repaired. Longer-term hypertrophy occurs due to more permanent changes in muscle structure.


title: Muscle hypertrophy

text: Biological factors (such as DNA and sex), nutrition, and training variables can affect muscle hypertrophy.


title: Muscle hypertrophy

text: Individual differences in genetics account for a substantial portion of the variance in existing muscle mass. A classical twin study design (similar to those of behavioral genetics) estimated that about 53% of the variance in lean body mass is heritable, along with about 45% of the variance in muscle fiber proportion.


title: Muscle hypertrophy

text: During puberty in males, hypertrophy occurs at an increased rate. Natural hypertrophy normally stops at full growth in the late teens. As testosterone is one of the body's major growth hormones, on average, males find hypertrophy much easier (on an absolute scale) to achieve than females, and, on average, have about 60% more muscle mass than women. Taking additional testosterone, as in anabolic steroids, will increase results. It is also considered a performance-enhancing drug, the use of which can cause competitors to be suspended or banned from competitions. Testosterone is also a medically regulated substance in most countries, making it illegal to possess without a medical prescription. Anabolic steroid use can cause testicular atrophy, cardiac arrest, and gynecomastia.


title: Muscle hypertrophy

text: In the long term, a positive energy balance, when more calories are consumed rather than burned, is helpful for anabolism and therefore muscle hypertrophy. An increased requirement for protein can help elevate protein synthesis, which is seen in athletes training for muscle hypertrophy. However, there is no scientific consensus on whether strength-training athletes have increased protein requirements.


title: Muscle hypertrophy

text: Training variables, in the context of strength training, such as frequency, intensity, and total volume also directly affect the increase of muscle hypertrophy. A gradual increase in all of these training variables will yield muscular hypertrophy.


title: Muscle hypertrophy

text: The message filters down to alter the pattern of gene expression. The additional contractile proteins appear to be incorporated into existing myofibrils (the chains of sarcomeres within a muscle cell). There appears to be some limit to how large a myofibril can become: at some point, they split. These events appear to occur within each muscle fiber. That is hypertrophy results primarily from the growth of each muscle cell rather than an increase in the number of cells. Skeletal muscle cells are however unique in the body in that they can contain multiple nuclei, and the number of nuclei can increase.


title: Muscle hypertrophy

text: Cortisol decreases amino acid uptake by muscle tissue, and inhibits protein synthesis. The short-term increase in protein synthesis that occurs subsequent to resistance training returns to normal after approximately 28 hours in adequately fed male youths. Another study determined that muscle protein synthesis was elevated even 72 hours following training.


title: Muscle hypertrophy

text: A small study performed on young and elderly found that ingestion of 340 grams of lean beef (90 g protein) did not increase muscle protein synthesis any more than ingestion of 113 grams of lean beef (30 g protein). In both groups, muscle protein synthesis increased by 50%. The study concluded that more than 30 g protein in a single meal did not further enhance the stimulation of muscle protein synthesis in young and elderly. However, this study didn't check protein synthesis in relation to training; therefore conclusions from this research are controversial. A 2018 review of the scientific literature concluded that for the purpose of building lean muscle tissue, a minimum of 1.6 g protein per kilogram of body weight is required, which can for example be divided over 4 meals or snacks and spread out over the day.


title: Muscle hypertrophy

text: It is not uncommon for bodybuilders to advise a protein intake as high as 2–4 g per kilogram of bodyweight per day. However, scientific literature has suggested this is higher than necessary, as protein intakes greater than 1.8 g per kilogram of body weight showed to have no greater effect on muscle hypertrophy. A study carried out by American College of Sports Medicine (2002) put the recommended daily protein intake for athletes at 1.2–1.8 g per kilogram of body weight. Conversely, Di Pasquale (2008), citing recent studies, recommends a minimum protein intake of 2.2 g/kg "for anyone involved in competitive or intense recreational sports who wants to maximize lean body mass but does not wish to gain weight. However athletes involved in strength events (..) may need even more to maximize body composition and athletic performance. In those attempting to minimize body fat and thus maximize body composition, for example in sports with weight classes and in bodybuilding, it's possible that protein may well make up over 50% of their daily caloric intake."


title: Muscle hypertrophy

text: Microtrauma is tiny damage to the muscle fibers. The precise relation between microtrauma and muscle growth is not entirely understood yet.


title: Muscle hypertrophy

text: One theory is that microtrauma plays a significant role in muscle growth. When microtrauma occurs (from weight training or other strenuous activities), the body responds by overcompensating, replacing the damaged tissue and adding more, so that the risk of repeat damage is reduced. Damage to these fibers has been theorized as the possible cause for the symptoms of delayed onset muscle soreness (DOMS), and is why progressive overload is essential to continued improvement, as the body adapts and becomes more resistant to stress.


title: Muscle hypertrophy

text: However, other work examining the time course of changes in muscle protein synthesis and their relationship to hypertrophy showed that damage was unrelated to hypertrophy. In fact, in one study the authors showed that it was not until the damage subsided that protein synthesis was directed to muscle growth.


title: Muscle hypertrophy

text: In the bodybuilding and fitness community and even in some academic books skeletal muscle hypertrophy is described as being in one of two types: Sarcoplasmic or myofibrillar. According to this hypothesis, during sarcoplasmic hypertrophy, the volume of sarcoplasmic fluid in the muscle cell increases with no accompanying increase in muscular strength, whereas during myofibrillar hypertrophy, actin and myosin contractile proteins increase in number and add to muscular strength as well as a small increase in the size of the muscle. Sarcoplasmic hypertrophy is greater in the muscles of bodybuilders because studies suggest sarcoplasmic hypertrophy shows a greater increase in muscle size while myofibrillar hypertrophy proves to increase overall muscular strength making it more dominant in Olympic weightlifters. These two forms of adaptations rarely occur completely independently of one another; one can experience a large increase in fluid with a slight increase in proteins, a large increase in proteins with a small increase in fluid, or a relatively balanced combination of the two.


title: Muscle hypertrophy

text: Examples of increased muscle hypertrophy are seen in various professional sports, mainly strength related sports such as boxing, olympic weightlifting, mixed martial arts, rugby, professional wrestling and various forms of gymnastics. Athletes in other more skill-based sports such as basketball, baseball, ice hockey, and football may also train for increased muscle hypertrophy to better suit their position of play. For example, a center (basketball) may want to be bigger and more muscular to better overpower his or her opponents in the low post. Athletes training for these sports train extensively not only in strength but also in cardiovascular and muscular endurance training.


title: Propositional calculus

text: Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.


title: Propositional calculus

text: Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.


title: Propositional calculus

text: Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" (but only when used to denote material conditional).


title: Propositional calculus

text: Both premises and the conclusion are propositions. The premises are taken for granted, and with the application of modus ponens (an inference rule), the conclusion follows.


title: Propositional calculus

text: As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those "atomic" statements with statement letters, which are interpreted as variables representing statements:


title: Propositional calculus

text: When is interpreted as "It's raining" and as "it's cloudy" the above symbolic expressions can be seen to correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference of this "form", which will be valid on the same basis this inference is.


title: Propositional calculus

text: Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of axioms and inference rules allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a "derivation" or "proof" and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.


title: Propositional calculus

text: When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as formula_2, formula_3 and formula_7) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.


title: Propositional calculus

text: In classical truth-functional propositional logic, formulas are interpreted as having precisely one of two possible truth values, the truth value of "true" or the truth value of "false". The principle of bivalence and the law of excluded middle are upheld. Truth-functional propositional logic defined as such and systems isomorphic to it are considered to be zeroth-order logic. However, alternative propositional logics are also possible. For more, see Other logical calculi below.


title: Propositional calculus

text: Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic, which was focused on terms. However, most of the original writings were lost and the propositional logic developed by the Stoics was no longer understood later in antiquity. Consequently, the system was essentially reinvented by Peter Abelard in the 12th century.


title: Propositional calculus

text: Propositional logic was eventually refined using symbolic logic. The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan—completely independent of Leibniz.


title: Propositional calculus

text: Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. Truth trees were invented by Evert Willem Beth. The invention of truth tables, however, is of uncertain attribution.


title: Propositional calculus

text: Within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently). Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce, and Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis. Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.".


title: Propositional calculus

text: In general terms, a calculus is a formal system that consists of a set of syntactic expressions ("well-formed formulas"), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.


title: Propositional calculus

text: When the formal system is intended to be a logical system, the expressions are meant to be interpreted as statements, and the rules, known to be "inference rules", are typically intended to be truth-preserving. In this setting, the rules, which may include axioms, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements.


title: Propositional calculus

text: The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see axiom schema). A formal grammar recursively defines the expressions and well-formed formulas of the language. In addition a semantics may be given which defines truth and valuations (or interpretations).


title: Propositional calculus

text: A "well-formed formula" is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar.


title: Propositional calculus

text: Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by , , and , propositional variables by , , and , and schematic letters are often Greek letters, most often , , and .


title: Propositional calculus

text: The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent, but differ in the details of:


title: Propositional calculus

text: Any given proposition may be represented with a letter called a 'propositional constant', analogous to representing a number by a letter in mathematics (e.g., ). All propositions require exactly one of two truth-values: true or false. For example, let be the proposition that it is raining outside. This will be true () if it is raining outside, and false otherwise ().


title: Propositional calculus

text: It is very helpful to look at the truth tables for these different operators, as well as the method of analytic tableaux.


title: Propositional calculus

text: Propositional logic is closed under truth-functional connectives. That is to say, for any proposition , is also a proposition. Likewise, for any propositions and , is a proposition, and similarly for disjunction, conditional, and biconditional. This implies that, for instance, is a proposition, and so it can be conjoined with another proposition. In order to represent this, we need to use parentheses to indicate which proposition is conjoined with which. For instance, is not a well-formed formula, because we do not know if we are conjoining with or if we are conjoining with . Thus we must write either to represent the former, or to represent the latter. By evaluating the truth conditions, we see that both expressions have the same truth conditions (will be true in the same cases), and moreover that any proposition formed by arbitrary conjunctions will have the same truth conditions, regardless of the location of the parentheses. This means that conjunction is associative, however, one should not assume that parentheses never serve a purpose. For instance, the sentence does not have the same truth conditions of , so they are different sentences distinguished only by the parentheses. One can verify this by the truth-table method referenced above.


title: Propositional calculus

text: Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values. A simple way to generate this is by truth-tables, in which one writes , , ..., , for any list of propositional constants—that is to say, any list of propositional constants with entries. Below this list, one writes rows, and below one fills in the first half of the rows with true (or T) and the second half with false (or F). Below one fills in one-quarter of the rows with T, then one-quarter with F, then one-quarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row. This will give a complete listing of cases or truth-value assignments possible for those propositional constants.


title: Propositional calculus

text: The propositional calculus then defines an "argument" to be a list of propositions. A valid argument is a list of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is modus ponens, one instance of which is the following list of propositions:


title: Propositional calculus

text: This is a list of three propositions, each line is a proposition, and the last follows from the rest. The first two lines are called premises, and the last line the conclusion. We say that any proposition follows from any set of propositions formula_9, if must be true whenever every member of the set formula_9 is true. In the argument above, for any and , whenever and are true, necessarily is true. Notice that, when is true, we cannot consider cases 3 and 4 (from the truth table). When is true, we cannot consider case 2. This leaves only case 1, in which is also true. Thus is implied by the premises.


title: Propositional calculus

text: Other argument forms are convenient, but not necessary. Given a complete set of axioms (see below for one such set), modus ponens is sufficient to prove all other argument forms in propositional logic, thus they may be considered to be a derivative. Note, this is not true of the extension of propositional logic to other logics like first-order logic. First-order logic requires at least one additional rule of inference in order to obtain completeness.


title: Propositional calculus

text: The significance of argument in formal logic is that one may obtain new truths from established truths. In the first example above, given the two premises, the truth of is not yet known or stated. After the argument is made, is deduced. In this way, we define a deduction system to be a set of all propositions that may be deduced from another set of propositions. For instance, given the set of propositions formula_12, we can define a deduction system, , which is the set of all propositions which follow from . Reiteration is always assumed, so formula_13. Also, from the first element of , last element, as well as modus ponens, is a consequence, and so formula_14. Because we have not included sufficiently complete axioms, though, nothing else may be deduced. Thus, even though most deduction systems studied in propositional logic are able to deduce formula_15, this one is too weak to prove such a proposition.


title: Propositional calculus

text: The "language" of formula_17, also known as its set of "formulas", "well-formed formulas", is inductively defined by the following rules:


title: Propositional calculus

text: In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called "natural deduction system". The particular system presented here has no initial points, which means that its interpretation for logical applications derives its theorems from an empty axiom set.


title: Propositional calculus

text: Our propositional calculus has eleven inference rules. These rules allow us to derive other true formulas given a set of formulas that are assumed to be true. The first ten simply state that we can infer certain well-formed formulas from other well-formed formulas. The last rule however uses hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulas to see if we can infer a certain other formula. Since the first ten rules don't do this they are usually described as "non-hypothetical" rules, and the last one as a "hypothetical" rule.


title: Propositional calculus

text: In describing the transformation rules, we may introduce a metalanguage symbol formula_65. It is basically a convenient shorthand for saying "infer that". The format is formula_66, in which is a (possibly empty) set of formulas called premises, and is a formula called conclusion. The transformation rule formula_66 means that if every proposition in is a theorem (or has the same truth value as the axioms), then is also a theorem. Note that considering the following rule Conjunction introduction, we will know whenever has more than one formula, we can always safely reduce it into one formula using conjunction. So for short, from that time on we may represent as one formula instead of a set. Another omission for convenience is when is an empty set, in which case may not appear.


title: Propositional calculus

text: One of the main uses of a propositional calculus, when interpreted for logical applications, is to determine relations of logical equivalence between propositional formulas. These relationships are determined by means of the available transformation rules, sequences of which are called "derivations" or "proofs".


title: Propositional calculus

text: In the discussion to follow, a proof is presented as a sequence of numbered lines, with each line consisting of a single formula followed by a "reason" or "justification" for introducing that formula. Each premise of the argument, that is, an assumption introduced as an hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premise" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from the previous ones by the correct application of a transformation rule. (For a contrasting approach, see proof-trees).


title: Propositional calculus

text: Interpret formula_105 as "Assuming , infer ". Read formula_106 as "Assuming nothing, infer that implies ", or "It is a tautology that implies ", or "It is always true that implies ".


title: Propositional calculus

text: We now prove the same theorem formula_107 in the axiomatic system by Jan Łukasiewicz described above, which is an example of a Hilbert-style deductive system for the classical propositional calculus.


title: Propositional calculus

text: The crucial properties of this set of rules are that they are "sound" and "complete". Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.


title: Propositional calculus

text: Note that the proofs for the soundness and completeness of the propositional logic are not themselves proofs in propositional logic ; these are theorems in ZFC used as a metatheory to prove properties of propositional logic.


title: Propositional calculus

text: We define a "truth assignment" as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible world) where certain statements are true and others are not. The semantics of formulas can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.


title: Propositional calculus

text: We define when such a truth assignment satisfies a certain well-formed formula with the following rules:


title: Propositional calculus

text: With this definition we can now formalize what it means for a formula to be implied by a certain set of formulas. Informally this is true if in all worlds that are possible given the set of formulas the formula also holds. This leads to the following formal definition: We say that a set of well-formed formulas "semantically entails" (or "implies") a certain well-formed formula if all truth assignments that satisfy all the formulas in also satisfy .


title: Propositional calculus

text: Finally we define "syntactical entailment" such that is syntactically entailed by if and only if we can derive it with the inference rules that were presented above in a finite number of steps. This allows us to formulate exactly what it means for the set of inference rules to be sound and complete:


title: Propositional calculus

text: Soundness: If the set of well-formed formulas "syntactically" entails the well-formed formula then "semantically" entails .


title: Propositional calculus

text: Completeness: If the set of well-formed formulas "semantically" entails the well-formed formula then "syntactically" entails .


title: Propositional calculus

text: Notational conventions: Let be a variable ranging over sets of sentences. Let and range over sentences. For " syntactically entails " we write " proves ". For " semantically entails " we write " implies ".


title: Propositional calculus

text: We note that " proves " has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If proves , then ...". So our proof proceeds by induction.


title: Propositional calculus

text: Notice that Basis Step II can be omitted for natural deduction systems because they have no axioms. When used, Step II involves showing that each of the axioms is a (semantic) logical truth.


title: Propositional calculus

text: The Basis steps demonstrate that the simplest provable sentences from are also implied by , for any . (The proof is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable—by considering each case where we might reach a logical conclusion using an inference rule—and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from "" we can derive " or ". In III.a We assume that if is provable it is implied. We also know that if is provable then " or " is provable. We have to show that then " or " too is implied. We do so by appeal to the semantic definition and the assumption we just made. is provable from , we assume. So it is also implied by . So any semantic valuation making all of true makes true. But any valuation making true makes " or " true, by the defined semantics for "or". So any valuation which makes all of true makes " or " true. So " or " is implied.) Generally, the Inductive step will consist of a lengthy but simple case-by-case analysis of all the rules of inference, showing that each "preserves" semantic implication.


title: Propositional calculus

text: By the definition of provability, there are no sentences provable other than by being a member of , an axiom, or following by a rule; so if all of those are semantically implied, the deduction calculus is sound.


title: Propositional calculus

text: We want to show: If implies , then proves . We proceed by contraposition: We show instead that if does not prove then does not imply . If we show that there is a model where does not hold despite being true, then obviously does not imply . The idea is to build such a model out of our very assumption that does not prove .


title: Propositional calculus

text: Thus every system that has modus ponens as an inference rule, and proves the following theorems (including substitutions thereof) is complete:


title: Propositional calculus

text: The first five are used for the satisfaction of the five conditions in stage III above, and the last three for proving the deduction theorem.


title: Propositional calculus

text: As an example, it can be shown that as any other tautology, the three axioms of the classical propositional calculus system described earlier can be proven in any system that satisfies the above, namely that has modus ponens as an inference rule, and proves the above eight theorems (including substitutions thereof). Out of the eight theorems, the last two are two of the three axioms; the third axiom, formula_116, can be proven as well, as we now show.


title: Propositional calculus

text: For the proof we may use the hypothetical syllogism theorem (in the form relevant for this axiomatic system), since it only relies on the two axioms that are already in the above set of eight theorems.


title: Propositional calculus

text: We now verify that the classical propositional calculus system described earlier can indeed prove the required eight theorems mentioned above. We use several lemmas proven here:


title: Propositional calculus

text: We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.


title: Propositional calculus

text: If a formula is a tautology, then there is a truth table for it which shows that each valuation yields the value true for the formula. Consider such a valuation. By mathematical induction on the length of the subformulas, show that the truth or falsity of the subformula follows from the truth or falsity (as appropriate for the valuation) of each propositional variable in the subformula. Then combine the lines of the truth table together two at a time by using "( is true implies ) implies (( is false implies ) implies )". Keep repeating this until all dependencies on propositional variables have been eliminated. The result is that we have proved the given tautology. Since every tautology is provable, the logic is complete.


title: Propositional calculus

text: An interpretation of a truth-functional propositional calculus formula_166 is an assignment to each propositional symbol of formula_166 of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of formula_166 of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables.