An Argument at Lakeshire
A story game of identifying logical argument forms.
College Math Games
Dear Editor,
The story that follows employs logical argument forms, yet I am admittedly inept at identifying these by name. It has come to my attention that you have been studying the subject as of late.
I seek your assistance in filling in the blanks (so to speak) of this manuscript, to recognize which argument is being employed in each deduction (or which fallacy is leading someone to a wrong conclusion).
As a professional, I know you will be unable to turn a page until you have filled out its blanks correctly. But when a page is complete (and correct), I am sure the page's top right corner will appear bent and ready for you to turn it.
You may find a summary of common argument forms and fallacies below. Please accept the challenge here , then drag the top right corner of this page.
Timothy S. Long
Liverpool, England
June 28, 1926
Addition
Subtraction
p
p ∧ q
q
p ∧ q
p
Constructive Dilemma (1)
p → q
r → s
p ∨ r
q ∨ s
Constructive Dilemma (2)
p → q
r → s
p ∧ r
q ∧ s
Modus Ponens
Modus Tollens
p → q
p → q
p
~q
q
~p
Disjunctive Syllogism
Transitivity
(Hypothetical Syllogism)
p ∨ q
p → q
~p
q → r
q
p → r
Fallacy of the Converse
Fallacy of the Inverse
p → q
p → q
q
~p
p
~q
An Argument at Lakeshire
By Timothy Scott Long
I can write, or I can quit. But I cannot quit. So I must write.
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If you read my tale, you might enjoy it! (You are reading now, and so you might.)
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1. The Notice
In the small, remote town of Lakeshire on the northeast coast, Dr. Logan was helping young James Erdwin prepare for winter exams when a sweaty Constable Harry unexpectedly barged into the room. The gentlemen jumped at this uncharacteristic entry, but before the Constable could catch his breath to explain himself, Dr. Logan spoke, "Come, Mr. Erdwin. We have a very wealthy woman to see about a murder."
The Constable was astonished. "How could you...?"
"You only visit when something is unsettled. That is, if you are here, then there is a case. Well... here you are."
"So there is a case," Erdwin finished, looking back at his pile of notes, " ."
"Yes, but-"
"With such excitement, uncharacteristic for a man of your decorum, it is clear that you have run here, panting and sweating profusely despite the winter cold. So the accused party must have much power or prestige, and since our small town of Lakeshire is far from the residence of any royalty or political influence, I assume this person is of incredible wealth. You were probably even hesitant to arrest her, unless there is some rather damning evidence and a severe crime. So by , we can choose to focus on the fact that there is a severe crime. Since a person of such wealth is not characteristically wont to steal, I would say the only conceivable crimes of such severity for her to commit are murder or treason."
"Let us take a moment to mull over the two possibilities, for an accusation of murder would imply someone has been killed, and that of treason would imply that the nation has been betrayed. Since we know the crime is murder or treason, tells us that we have a killed person, or a betrayed nation, so you are here on behalf of one of the two. And yet all this rumination is that and nothing more, for I am quite confident that the crime in question is murder."
He kindly looked to his protégé to continue, and Erdwin obliged. "'Lakeshire is a small town, far from any royalty or political influence,'" Erdwin reiterated, "A treason case here is highly unlikely. So it is murder or treason, but cannot be treason. Why then it must be murder, you see, by ."
"I cannot help but notice," fretted the Constable, "That there seems to be some speculation in your arguments."
"Ha! In the premises perhaps, but the arguments? No sir. I suggest that each statement listed in my argument is true, but one could quite feasibly believe that any of them is false; there may be other serious crimes to consider, and however unlikely, treason is not outright impossible, and so on. I am merely basing my assumptions, or premises, on probabilities; beyond that I can only tell you that our course of deduction is sound. To wit, no matter the truth value of any particular statement, I guarantee that any argument we are presenting of these common forms is valid- unless we explicitly declare it a fallacy, which is of course invalid by definition."
"I see. But how did you know it was a woman in the first place?"
"Easy enough. Your salty sweat notwithstanding, my keen nose detects the odour of over 12 different extracts."
Hoping for an explanation, the Constable turned his confused face to Erdwin, who obliged: "You smell like a woman's perfume."
"In fact, " Logan continued, "By a keen sense of smell alone, one could still make out two of the relevant details. Take as our first premise: if you smell of a woman's perfume, then the person in your custody is a woman. Now our second premise will be derived from the following pair of conditional statements: if you smell like 12 extracts, then you smell like quite an expensive bouquet, and if you smell like an expensive bouquet, then the suspect must be rich. But let me use to simplify the second premise to this: if you smell like 12 extracts, then the suspect much be rich. Now, since you do smell like a woman's perfume and you do smell of 12 different extracts, this argument of tells us the suspect must be a rich woman."
"Amazing! Your keen nose can really detect a perfume of that many extracts?"
"It is very highly trained, yes, but I also make fragrances a hobby."
"One of quite a few hobbies that seem to benefit from your expertise. For I know you are an expert in mathematics, an expert in logic, and now an expert in fragrances, so by simple you are an expert in these three, at the very least."
Logan continued, ignoring the Constable's praise, "If I concentrate, I can focus on one individual extract of the perfume and make it out. For example, I can sense coumarin; I can sense vanillin; I can sense heliotropin. So by I can assume this single perfume contains coumarin and vanillin and heliotropin (that is, assuming it came from a single perfume)."
"It did, indeed."
"I continue the process for all 12, and thus came the premise of the expensive bouquet."
"Astounding. And here my old nose cannot even tell me when supper is ready."
2. Into the Cold
"It shall be a cold walk to the station," Logan proclaimed. "I could take my hat, to keep my head warm. I could take my coat, to keep my body warm."
Constable gave a gaze as if Logan had lost his mind. "Then you can take both the hat and coat, and keep your entire self- that is, your head and your body- warm."
"Ho ho! Well done. I was merely having some fun, and verifying that our recent explanation of had stuck in your memory."
"You are a very odd man," the Constable stated.
Relieved with this newfound break from studying logical arguments, Erdwin let his mind wander as the other men prepared to leave. He thought about Lakeshire itself. "It has a beautiful lake in the southern area, but the coastal portion has some of the most beautiful beaches and cliffs on the whole seaboard, so why name it Lakeshire? Perhaps Cliffshire sounded too peculiar. Well, if you are going to name it after the lake or the coast, and you can't use the coast, I guess you have to choose the lake, -- oh, drats! I cannot escape this topic."
"Perhaps next time you could save us the trip," Logan said, gathering his hat and coat from the rack, "and bring the client here to us for questioning."
"What makes you assume I have not?" Constable asked, having some fun of his own, "Perhaps she is sitting in a cab outside, awaiting our company."
"No no. If you ran here, then you are traveling on foot. If you are traveling on foot, then you will of course let this prestigious woman wait for us in the warmth of the station. Thus by , if you ran here she is still at the station. Please," Logan urged, "Make this walk more enjoyable by enlightening us with the current status of this affluent femme fatale."
3. The Arrest
"Last Saturday around noon, a bicycle was brought to the station, apparently abandoned by the cliffs of Lakeshire Beach, not but a five minute walk from here. The couple who found it explained that they had been cycling in the after-noon of the previous day, planning a picnic near the beach when they came upon a woman yelling and crying hysterically. She would not explain the source of her anguish, but being good samaritans- and feeling unable to leave her by the cliffs in that state - they cycled with her all the way back to her home."
"'With her,' you say?" Logan asked, "Saying they 'cycled with her' implies that she was also on bicycle, so by I can assume you mean she was on bicycle as well?"
"Yes, this is true," the Constable answered, "She was on a second bicycle. After escorting her home, the couple got quite lost on their way back- a detour of over two hours in fact. It was only by luck that they eventually found themselves back at the cliffs- and indeed found the first bicycle, barely visible from within some bushes- but by then it was already dark, so they cycled home and decided to return to have their picnic in the forenoon."
"If they can make their way back, then they can make their way home," Erdwin said to himself, "If it is dark at the beach, they will postpone their date. They made their way back, but it was dark at the beach, and so by Constructive Dilemmas, they made their way home postponed their date. As planned, they returned at about 10 a.m., but they were surprised to find an abandoned bicycle not far off from where they found the lady. Recalling her demeanor they feared the worst-"
"But they did not dare seek the house again," Logan said.
"You guess right," the Constable declared.
"It is not a guess," Logan explained, "For, if they tried to find the house again, they would have undoubtedly been lost for hours once again, and if they were lost for hours, then would not have the time to bring the bicycle here by noon. So if they tried to find the house, they could not have brought the bicycle here at noon, as a matter of ."
"And you say they did bring it here at noon," Erdwin added, "Thus they did not even try to find the house. This now is a result of Try not to be confused by the negations."
"Indeed. This is the bicycle here," the Constable said as they approached the entrance of the station. Logan and Erdwin looked at the bicycle, a tall shiny beast, and Erdwin pointed to a tiny splotch of crimson sitting flatly on the horizontal crossbar, "That drop of blood aroused my senses as well," said the Constable, "So I asked the couple to describe the exact location where they found the girl, and I went to the cliffs to investigate. I searched the scene until I found more drops of blood, and it was at this location that I found two sets of footprints, in what appears to have been a struggle. One set- from a man's shoe- made impressions all the way to the edge, and then disappeared. I suspected foul play. Ah, but what of the bike; why would she dare return so soon? To cast herself as penance? To verify his demise? To lure her, I took out an ad that the bicycle had been found and gave my home address instead of the station."
Erdwin prattled on, "If you put the station's address, it would rouse her suspicions, and if it roused her suspicions, she would not come, so by Transitivity, if you put the station's address, . You could put the station's address or your own, but surely could not put the station's address; so due to Dysjunctive Syllogism, .
"I went into detail in the ad, saying I found it next to a confused man who claimed it was not his. I believed any hint that the man has survived their argument would surely compel her to come. As I was about to leave the house this forenoon, I heard a knock on my door. When the woman saw me answer in full uniform, she panicked and tried to dart away- an admission of guilt if I ever saw one. I caught up to her and placed her under arrest."
"Hm. If a person is guilty, they would likely run from the police," Logan agreed, "However, running from the police is not necessarily an indication of guilt. Concluding so would be a result of the ."
The Constable closed his eyes, as if to enjoy that first meeting all over again. "Even though I was taking her in on suspicion of murder, everything about this woman had an effect on me: her sweet perfume... her sun-kissed skin from ostensibly frequent trips to that sunny beach... her overall manner, as if she knew she was above me and everyone else in her presence." Looking worried now, he continued, "Then I learned her name is Helen Atler."
"The Atler family?" Erdwin asked, "Owners of the Atler Chemist's?"
"Indeed, the most successful business in Lakeshire. Now I come to a dilemma. I cannot let her go, or else I could be releasing a murderer. On the other hand, who knows what terrible fate I shall suffer if I have an innocent Atler in my custody."
"It is quite a conundrum, these thoughts of yours," Logan replied, "Whether you let her go or keep her in custody, you will release a murderer or be punished yourself. I see an argument of in your cogitation."
4. First Interview
Initially, the suspect refused to ring her own barrister, fearing scandal and hoping to be released before any news of her arrest spread. And so, perhaps more out of curiosity than trust, Logan acted as a proxy for her unofficial consultant in the matter. Erdwin watched in amusement as Logan squinted to make out the Constable's handwriting: "One count Assault. One count Murder."
"What's the charge?" Helen asked.
"Assault and murder" Logan answered, combining the crimes via in the interest of brevity.
"Murder?!" she gasped, countering his method with , focusing only on the count that she deemed worthy of anyone's scrutiny. She was taken aback, but subsequently her tone softened with a tinge of relief. "Well then I should be fine," she said. "It's a simple as this. If I committed the murder, then I will be charged, naturally. However, I swear to you that I did NOT commit the murder... and so I shall not be charged." She smiled and shrugged, ostensibly proud in the delivery of her argument.
"Miss Atler...", he implored,
"Mrs. Atler," she corrected, her eyes veering off in the distance.
"Apologies. Mrs. Atler..." he tried again, "I believe your argument is plagued with the . You see, though the severity of the punishment is most definitely affected by which charge or charges are upheld, a prison sentence only requires the prosecution to prove that you committed assault or murder. You say that it is impossible for them to prove that you committed the murder. Thus they will undoubtedly need to focus on the assault charge, as a matter of . Now, since you are being accused of assault or murder (or both), and you only protested the murder charge, I myself have come to the conclusion that the assault did indeed occur. Again, applies. And the fact is:
- If you committed assault, then you will be charged.
- You have committed assault.
- Therefore, you will be charged."
This is known as and is a direct argument which the prosecution will use against you."
"So now you know," Helen replied morosely, "that the assault occurred. Furthermore you must think me a murderer! And of course... a liar..."
Erdwin jumped in, sensing another miscalculation, "I think you may once again be jumping to conclusions. Yes, murder implies assault, but-"
"And the assault is true, as you two have deduced. So you must believe the murder charge is true as well."
"Ah! Well, no," Erdwin shouted proudly, "To admit so would be to admit to deception, of the . On the other hand, noting still that murder implies assault, if we could prove that no assault occurred, then would allow us to prove the murder charge is false as well."
"Murder implies assault?" she asked.
"One would assume so," Erdwin suggested, though her question cast doubt in his mind.
"Oh pish posh with your theoretical arguments," she replied, "I have no intention of lying to a jury. You have already deduced that I am guilty of assault. If I am caught lying- committing the crime of perjury!- then the jury will view me as a criminal. If the jury views me as a criminal, then I will be convicted as a murderer. It's , can you not see? If I am caught lying, then I will be convicted as a murderer. I have no choice. I will take my chances with the truth. It seems I will not be able to avoid prosecution in this matter. Please ring the Atler family barrister."
"I see it a little differently," Erdwin restated outside the questioning room, "If she tells the story as is, then she will surely be convicted of murder. If she lies- that is, changes her story from what she has reported, then we can assume she would be convicted of perjury. So no matter what- whether she lies or tells the truth- she will be convicted of either perjury or murder. A classic case of if ever there was one."
Logan and Erdwin returned to the lobby. "If perjury or murder was your accusation," Dr. Logan asked Erdwin as they walked around the building, "Which would you choose?"
"I'd prefer not have to choose either! But between the two, I certainly would not choose murder. So I suppose perjury then, by ."
"I as well. And yet she was adamant about speaking the truth. I don't believe she had any intention of hurting that man at all."
5. Second Interview
The investigators agreed to hold off on telling the husband. With the family barrister present Mrs. Atler openly divulged her experience.
"His name is Bart Quagmire. We were schoolmates, good friends in fact, but we lost touch when he joined the Royal Navy. Upon returning home, he invited me to picnic with him as we did in our youth, and so I met with him a few times. Well, perhaps more than a few."
"That explains the sun-kissed skin," Logan commented, looking at the blushing Constable. Mrs. Atler was unsure if this was flattery or just a very unusual acknowledgement at her tan. "Did your husband know of these meetings?"
"Oh dear no! He is a brute of a man, obsessively suspicious and jealous. Even innocent meetings with a gentlemen acquaintance would stir him into a mad rage. I dabble in creative writing, you see. So I told him I was going to the beach to work on a mystery novella every day, while he worked at the chemist's."
"For the record," said the barrister, keeping his key client in mind, "The coworkers at the chemist's have all verified that Mr. Atler works from exactly 8 a.m. to 5 p.m. every single Friday, like clockwork."
"On a recent picnic," Mrs. Atler continued, "I brought a bottle of sparkling wine that I had taken from our shop the previous night."
"Were you celebrating something?"
"Not that I knew of, at least not at first. Bart had asked me to bring it, as he knew I could pull a nice bottle from our shop's inventory without paying. Well, I brought a very expensive bottle and he still complained that it had already gone flat. Oh, if only that had been enough to deter his next sentence. Galvanized with the courage of a few sips that wine - Bart proposed marriage. I am of course already a married woman! I refused his proposal, but he became very aggressive. He tried again and again to kiss me, but I slapped him each time, until one of my nails drew blood from his cheek. He laughed loudly... then a little less... then sobbed. He tried again, but the words stopped coming out, as if he had just given up. He stumbled toward the cliff, confused and sad. I yelled and ran to him, but I swear to the heavens, he toppled over the edge before I was even close enough to touch him. I could see his mangled body on the rocks below, his head covered in blood from the impact. Then the waves pulled him under and I never saw him come back up."
"Did you expect him to come back up?"
"Well no. His head was cracked open. With his head in that state, I am sure he was already dead, and with him dead there would be no reason to expect him to re-emerge. So by , no; if his head was in that state, of course I did not expect him."
"But you kept looking, long enough to know for sure. So you didn't know for sure. If you knew, you would have no reason to keep looking to verify. But you did verify, so you did not know; "
"Are you saying you believe he is still alive, walking around with half of his head split open? Or simply that I am lying."
"I believe you."
"Well then you do think he's alive then- would suggest this. There are other reasons one would keep looking, you know."
"Perhaps your senses were slowed, due to the alcohol?"
"How dare you! No, I have never had a drop in my life. Bart was the one stumbling around drunk; I was sober as a judge."
6. Conclusion
"Tell me Erdwin," Logan said, "Was Mrs. Atler wearing a wedding ring at her interview?"
"I cannot tell you. I mean no offence, but if I did not notice, then I have nothing to say, and I did not notice, so by I simply cannot answer your question."
"And yet you can deduce that no such ring was present," Logan urged, "Since, as you often say, I never miss a detail."
"Let us see. If she was wearing a wedding ring, then I know you would have seen it. If you had seen it, then you would have known she was a married woman... and if you had known she was a married woman then you would have referred to her as Mrs., rather than offending her as Miss. And so, concisely, if she was wearing a wedding ring, you would have referred to her as Mrs."
"Quite so! Ah, that is the power of . It can consolidate not just 2 or 3 conditional statements, but as many as you will have! Now, let us take your conclusion as our hypothesis and work backwards:
- If she was wearing a wedding ring, I would have referred to her as Mrs.
- But- as you have observed- I did not."
"So she most definitely was not wearing a ring: !" Erdwin scratched his chin, considering the ring's whereabouts. "Perhaps it was lost, or stolen?"
"If so, she could have bought a thousand more. In fact, I am sure a woman of her wealth has various ostentatious substitutes that could be placed on the finger in the meantime."
"So you believe she had removed it intentionally?"
"Yes. I believe this lies in the Constable's fib that her male colleague was still alive. She thought she might see him again. Now listen, she goes to the cliffs often, in a very sunny location, and has gotten quite tan because of it. But I did not notice any tan line where the ring would have blocked the sun- again, I would have noticed. So I would insist that she never wore the ring at their picnics either. And I can see only one reason for this. He did not know she is married, and she did not want him to know."
"He never knew of the marriage? Well, I suppose someone enlisted in the Royal Navy might not have learned of his acquaintance's marriage while he was at sea. So not wearing a ring implies she did not want him to know she was married. And I suppose that implies that she actually did have a romantic interest in him."
"Indeed. That is- by - her not wearing a ring to the picnic implies she had a romantic interest in the gentleman. Perhaps this even bloomed into a full-fledged affair, with the gentleman believing it was proper courting. She despises her husband- a 'brute of a man'- but relishes her wealthy lifestyle with him."
"So when Bart proposed marriage, she realized this had gone too far, and she had to end things. But why resort to murder?"
"I do not believe she did. As much as I dislike attributing details to coincidence, the statement that the murder and the proposal both fell on that day is logical and nothing more. Let us consider the husband instead. Her husband is a 'suspicious and jealous' man, so by we can look at each of these claims individually. First, If he was suspicious, then he would have checked in on her at the beach, and if he did so, he would have certainly found out about the affair. So being a suspicious husband implies he would have found out about the affair, by . Second, if he was a jealous man, he would have sought revenge against Mr. Quagmire, or his own wife. Or perhaps both. But he could not kill his wife, as it would make her husband an obvious suspect. So Mr. Quagmire would have to be the target, by ."
" So being suspicious makes him aware. Being jealous makes him vindictive. And he is both suspicious and jealous, so by , he is both aware and vindictive. But he was not at the cliff that day. He was at the chemist's, as his coworkers have verified."
"He did not commit the murder at the beach in the after-noon. He committed it at his house in the forenoon. You see, a chemist would have both knowledge of and access to chemicals that could be used to poison someone. That is, if he has medical knowledge, he knows which elixirs are toxic in certain doses. If he works at a chemist's, he has access to various ingredients. Since he does have medical knowledge and does work at a chemist's, tells us that he knows some toxic "recipes", and has the ingredients. "
"Poison?"
"Yes. Mrs. Atler mentioned Bart was 'stumbling around drunk.' Have you ever known a 'few sips of sparkling wine' to achieve such an effect? On a man of the Royal Navy, no less! Perhaps this latter point is a matter of opinion, but I believe we can assume there was something more than sparkling wine in that bottle. Here is what I theorize: Her husband found out about the affair, as we have deduced. Early on Friday, before work, he searched her picnic basket and found the bottle of sparkling wine. He knows the wine must be for her or someone else. But she does not drink! So she must be meeting someone, by ."
"If it was for her, then she had been lying to him all her life about drinking," Erdwin added, "If it was for someone else, then she had been lying to him about her daily meetings. And as you say, it must be for her or someone else. So by she is lying about one thing or the other. Such a jealous man would likely loathe being lied to; it would only fuel his anger."
"So he knew she would be meeting with the man today, and that he would be the only one drinking from the bottle. Mr. Atler injected a concoction of toxins into the bottle before leaving for work, knowing that he would have an alibi for when the toxins took effect - and in doing so he must have removed and returned the cork, which caused the drink to start going flat. Perhaps he assumed his wife would leave the body, return home, and never speak of it. Or perhaps she would feel guilty, assuming she somehow caused the death, and take the blame. Either result would have satisfied her husband's jealousy. Of course, the only way to prove my theory is to find the bottle."
Logan explained the solution to the Constable, and posed that if the police could find the poisoned bottle it would prove his theory. A subsequent search of the Atler estate turned up the bottle, which Mrs. Atler had brought back home.
It would take many weeks before it was determined (through the assistance of a lab in America) that the bottle indeed contain the conjectured toxins. And so by the prime suspect became Mr. Atler, rather than his wife.
"We have determined what I believe is the true nature of the situation", Logan said. "In the end, perhaps both husband and wife hold some responsibility in this man's death. But that is up to the jury to decide."
The End
Dear Editor,
Thank you for completing my story. You have shown incredible prowess in the matters of logical arguments and their common forms. The full case is resolved and the story is complete, thanks to you.
Thank you!
The End
Dear Editor,
Thank you for completing my story. You have shown incredible prowess in the matters of logical arguments and their common forms.
Even in the more difficult edition, with more questions and more argument forms, you have excelled! Hurrah! The full case is resolved and the story is complete, thanks to you.