25 Jun 2018

Melville (4) The Confidence-Man: His Masquerade, ch.4 ‘Renewal of Old Acquaintance,’ summary in brief

 

by Corry Shores

 

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[The following summarizes parts of the text, with my commentary in brackets. Boldface is my own. Proofreading is incomplete, so I apologize for my typos.]

 

 

Summary (in Brief) of

 

Herman Melville

 

The Confidence-Man: His Masquerade

 

4

Renewal of Old Acquaintance

 

 

Brief summary:

The Man with a Weed encounters the Country Merchant on the boat. The man with a weed makes it clear that he knows all the basic information about the merchant that would be found on his business card, which makes it apparent that the man with a weed is quite possibly the Crippled, Freed African Slave Beggar without his black make-up on. (The beggar secretly obtained the merchant’s business card in Ch.3.) We learn that the man with a weed calls himself “John Ringman” and the merchant’s name is Henry Roberts. The man with a weed tries to convince the merchant they met once before. When the merchant insists he cannot recall it, the man with the weed tries to convince the merchant that he had once had some brain injury that could have caused the memory loss. The merchant says there was only a brain fever that he had around that time of the supposed meeting, and the man with a weed pushes that theory for how they could have meet before without the merchant having any semblance of a recollection of it. The man with a weed then says that like the merchant, he is also a free-mason, and he tells a sad story that leads to the merchant handing over money as aid. Later the man with a weed informs the merchant of a business opportunity: there is a president of a coal company looking to sell many stocks well under their real value. Were the merchant to buy them, he would make a lot of money fast (supposing they are authentic).

 

 

 

 

 

Summary (in brief)

 

The Man with a Weed greets the Country Merchant by name (Mr. Roberts), even though the merchant does not recognize him, despite the man with a weed’s insistence he remembers the merchant vividly (26). Nonetheless, the merchant recalls the Crippled, Freed African Slave Beggar’s naming of a man with a weed as one of his character witnesses (see Ch.3), so the merchant is at least looking for the man with a weed. The man with a weed indicates that he knows the beggar and says he assured the other passengers of his character (27). The man with a weed persists in trying to get the merchant to remember him. At that point, he gives more information about the merchant, which would also be information one would find on a business card, and in fact makes this obvious by saying the information would actually be on one of his business cards:

“[...] Are you not, sir, Henry Roberts, forwarding merchant, of Wheeling, Pennsylvania? Pray, now, if you use the advertisement of business cards, and happen to have one with you, just look at it, and see whether you are not the man I take you for.”

(27)

The man with a weed then tries to get the merchant to doubt his own memory, so that the merchant would believe that in fact he met the man with the weed, but lost that memory. And he asks for the merchant’s trust (confidence) in the man with a weed’s memory over his own.

“To come to particulars, my dear sir, I met you, now | some six years back, at Brade Brothers & Co’s office, I think. I was traveling for a Philadelphia house. The senior Brade introduced us, you remember; some business-chat followed, then you forced me home with you to a family tea, and a family time we had. Have you forgotten about the urn, and what I said about Werter’s Charlotte, and the bread and butter, and that capital story you told of the large loaf. A hundred times since, I have laughed over it. At least you must recall my name—Ringman, John Ringman.”

“Large loaf? Invited you to tea? Ringman? Ringman? Ring? Ring?”

“Ah sir,” sadly smiling, “don’t ring the changes that way. I see you have a faithless memory, Mr. Roberts. But trust in the faithfulness of mine.”

(27-28)

The merchant continues to insist he has no memory of their previous encounter. To sow more doubts, the man with the weed says that the merchant may have had a brain injury that prevented him from recording the memory of their prior encounter. The merchant had no such injury at the time, but he notes that he did have a terrible brain fever. And despite the merchant’s objections, the man with the weed concludes that the fever is what is preventing him from remembering their previous encounter.

 

The man with a weed then takes the merchant to a less public area and begins to look very troubled. He also notes that the merchant is a free-mason, and indicates he is a fellow mason who is in need of help. The man with the weed then tells some calamitous tale that evokes the merchant’s sympathy, but the narrator does not tell us the details. This causes the merchant to give the man with a weed some money.

At every disclosure, the hearer’s commiseration increased. No sentimental pity. As the story went on, he drew from his wallet a bank note, but after a while, at some still more unhappy revelation, changed it for another, probably of a somewhat larger amount; which, when the story was concluded, with an air studiously disclamatory of alms-giving, he put into the stranger’s hands; who, on his side, with an air studiously disclamatory of alms-taking, put it into his pocket.

(32)

 

The man with a weed then parts ways with the merchant. But before getting to far off, the man with a weed then comes back to the merchant to inform him about a business opportunity. On account of some odd circumstances, the president of the Black Rapids Coal Company, who is on board, is ready to sell shares of the company at very low prices. This would enable the merchant to make money very fast, (supposing that the shares are legitimate) (33). When the merchant asks why the man with a weed did not buy the stocks, the man with a weed reminds him indirectly of his unfortunate circumstances that required the almsgiving from just before. He furthermore uses this incident to suggest that the merchant’s brain fever must have caused him to forget about the man with a weed’s dire financial situation. He also uses this incident to convince the merchant to have more charity and trust.

“To reproach a penniless man with remissness in not availing himself of an opportunity for pecuniary investment—but, no, no; it was forgetfulness; and this, charity will impute to some lingering effect of that unfortunate | brain-fever, which, as to occurrences dating yet further back, disturbed Mr. Roberts’s memory still more seriously.”

“As to that,” said the merchant, rallying, “I am not——”

“Pardon me, but you must admit, that just now, an unpleasant distrust, however vague, was yours. Ah, shallow as it is, yet, how subtle a thing is suspicion, which at times can invade the humanest of hearts and wisest of heads. But, enough. My object, sir, in calling your attention to this stock, is by way of acknowledgment of your goodness. I but seek to be grateful; if my information leads to nothing, you must remember the motive.”

He bowed, and finally retired, leaving Mr. Roberts not wholly without self-reproach, for having momentarily indulged injurious thoughts against one who, it was evident, was possessed of a self-respect which forbade his indulging them himself.

(34-35)

 

 

 

Chapter text [copied from Project Gutenburg]:

 

CHAPTER IV.

RENEWAL OF OLD ACQUAINTANCE.

“How do you do, Mr. Roberts?”

“Eh?”

“Don’t you know me?”

“No, certainly.”

The crowd about the captain’s office, having in good time melted away, the above encounter took place in one of the side balconies astern, between a man in mourning clean and respectable, but none of the glossiest, a long weed on his hat, and the country-merchant before-mentioned, whom, with the familiarity of an old acquaintance, the former had accosted.

“Is it possible, my dear sir,” resumed he with the weed, “that you do not recall my countenance? why yours I recall distinctly as if but half an hour, instead of half an age, had passed since I saw you. Don’t you recall me, now? Look harder.”

“In my conscience—truly—I protest,” honestly bewildered, “bless my soul, sir, I don’t know you—really, really. But stay, stay,” he hurriedly added, not without gratification, glancing up at the crape on the stranger’s hat, “stay—yes—seems to me, though I have [27] not the pleasure of personally knowing you, yet I am pretty sure I have at least heard of you, and recently too, quite recently. A poor negro aboard here referred to you, among others, for a character, I think.”

“Oh, the cripple. Poor fellow. I know him well. They found me. I have said all I could for him. I think I abated their distrust. Would I could have been of more substantial service. And apropos, sir,” he added, “now that it strikes me, allow me to ask, whether the circumstance of one man, however humble, referring for a character to another man, however afflicted, does not argue more or less of moral worth in the latter?”

The good merchant looked puzzled.

“Still you don’t recall my countenance?”

“Still does truth compel me to say that I cannot, despite my best efforts,” was the reluctantly-candid reply.

“Can I be so changed? Look at me. Or is it I who am mistaken?—Are you not, sir, Henry Roberts, forwarding merchant, of Wheeling, Pennsylvania? Pray, now, if you use the advertisement of business cards, and happen to have one with you, just look at it, and see whether you are not the man I take you for.”

“Why,” a bit chafed, perhaps, “I hope I know myself.”

“And yet self-knowledge is thought by some not so easy. Who knows, my dear sir, but for a time you may have taken yourself for somebody else? Stranger things have happened.”

The good merchant stared.

“To come to particulars, my dear sir, I met you, now [28] some six years back, at Brade Brothers & Co’s office, I think. I was traveling for a Philadelphia house. The senior Brade introduced us, you remember; some business-chat followed, then you forced me home with you to a family tea, and a family time we had. Have you forgotten about the urn, and what I said about Werter’s Charlotte, and the bread and butter, and that capital story you told of the large loaf. A hundred times since, I have laughed over it. At least you must recall my name—Ringman, John Ringman.”

“Large loaf? Invited you to tea? Ringman? Ringman? Ring? Ring?”

“Ah sir,” sadly smiling, “don’t ring the changes that way. I see you have a faithless memory, Mr. Roberts. But trust in the faithfulness of mine.”

“Well, to tell the truth, in some things my memory aint of the very best,” was the honest rejoinder. “But still,” he perplexedly added, “still I——”

“Oh sir, suffice it that it is as I say. Doubt not that we are all well acquainted.”

“But—but I don’t like this going dead against my own memory; I——”

“But didn’t you admit, my dear sir, that in some things this memory of yours is a little faithless? Now, those who have faithless memories, should they not have some little confidence in the less faithless memories of others?”

“But, of this friendly chat and tea, I have not the slightest——”

“I see, I see; quite erased from the tablet. Pray, [29] sir,” with a sudden illumination, “about six years back, did it happen to you to receive any injury on the head? Surprising effects have arisen from such a cause. Not alone unconsciousness as to events for a greater or less time immediately subsequent to the injury, but likewise—strange to add—oblivion, entire and incurable, as to events embracing a longer or shorter period immediately preceding it; that is, when the mind at the time was perfectly sensible of them, and fully competent also to register them in the memory, and did in fact so do; but all in vain, for all was afterwards bruised out by the injury.”

After the first start, the merchant listened with what appeared more than ordinary interest. The other proceeded:

“In my boyhood I was kicked by a horse, and lay insensible for a long time. Upon recovering, what a blank! No faintest trace in regard to how I had come near the horse, or what horse it was, or where it was, or that it was a horse at all that had brought me to that pass. For the knowledge of those particulars I am indebted solely to my friends, in whose statements, I need not say, I place implicit reliance, since particulars of some sort there must have been, and why should they deceive me? You see sir, the mind is ductile, very much so: but images, ductilely received into it, need a certain time to harden and bake in their impressions, otherwise such a casualty as I speak of will in an instant obliterate them, as though they had never been. We are but clay, sir, potter’s clay, as the good book says, [30] clay, feeble, and too-yielding clay. But I will not philosophize. Tell me, was it your misfortune to receive any concussion upon the brain about the period I speak of? If so, I will with pleasure supply the void in your memory by more minutely rehearsing the circumstances of our acquaintance.”

The growing interest betrayed by the merchant had not relaxed as the other proceeded. After some hesitation, indeed, something more than hesitation, he confessed that, though he had never received any injury of the sort named, yet, about the time in question, he had in fact been taken with a brain fever, losing his mind completely for a considerable interval. He was continuing, when the stranger with much animation exclaimed:

“There now, you see, I was not wholly mistaken. That brain fever accounts for it all.”

“Nay; but——”

“Pardon me, Mr. Roberts,” respectfully interrupting him, “but time is short, and I have something private and particular to say to you. Allow me.”

Mr. Roberts, good man, could but acquiesce, and the two having silently walked to a less public spot, the manner of the man with the weed suddenly assumed a seriousness almost painful. What might be called a writhing expression stole over him. He seemed struggling with some disastrous necessity inkept. He made one or two attempts to speak, but words seemed to choke him. His companion stood in humane surprise, wondering what was to come. At length, with an effort mastering [31] his feelings, in a tolerably composed tone he spoke:

“If I remember, you are a mason, Mr. Roberts?”

“Yes, yes.”

Averting himself a moment, as to recover from a return of agitation, the stranger grasped the other’s hand; “and would you not loan a brother a shilling if he needed it?”

The merchant started, apparently, almost as if to retreat.

“Ah, Mr. Roberts, I trust you are not one of those business men, who make a business of never having to do with unfortunates. For God’s sake don’t leave me. I have something on my heart—on my heart. Under deplorable circumstances thrown among strangers, utter strangers. I want a friend in whom I may confide. Yours, Mr. Roberts, is almost the first known face I’ve seen for many weeks.”

It was so sudden an outburst; the interview offered such a contrast to the scene around, that the merchant, though not used to be very indiscreet, yet, being not entirely inhumane, remained not entirely unmoved.

The other, still tremulous, resumed:

“I need not say, sir, how it cuts me to the soul, to follow up a social salutation with such words as have just been mine. I know that I jeopardize your good opinion. But I can’t help it: necessity knows no law, and heeds no risk. Sir, we are masons, one more step aside; I will tell you my story.”

In a low, half-suppressed tone, he began it. Judging [32] from his auditor’s expression, it seemed to be a tale of singular interest, involving calamities against which no integrity, no forethought, no energy, no genius, no piety, could guard.

At every disclosure, the hearer’s commiseration increased. No sentimental pity. As the story went on, he drew from his wallet a bank note, but after a while, at some still more unhappy revelation, changed it for another, probably of a somewhat larger amount; which, when the story was concluded, with an air studiously disclamatory of alms-giving, he put into the stranger’s hands; who, on his side, with an air studiously disclamatory of alms-taking, put it into his pocket.

Assistance being received, the stranger’s manner assumed a kind and degree of decorum which, under the circumstances, seemed almost coldness. After some words, not over ardent, and yet not exactly inappropriate, he took leave, making a bow which had one knows not what of a certain chastened independence about it; as if misery, however burdensome, could not break down self-respect, nor gratitude, however deep, humiliate a gentleman.

He was hardly yet out of sight, when he paused as if thinking; then with hastened steps returning to the merchant, “I am just reminded that the president, who is also transfer-agent, of the Black Rapids Coal Company, happens to be on board here, and, having been subpoenaed as witness in a stock case on the docket in Kentucky, has his transfer-book with him. A month since, in a panic contrived by artful alarmists, some credulous [33] stock-holders sold out; but, to frustrate the aim of the alarmists, the Company, previously advised of their scheme, so managed it as to get into its own hands those sacrificed shares, resolved that, since a spurious panic must be, the panic-makers should be no gainers by it. The Company, I hear, is now ready, but not anxious, to redispose of those shares; and having obtained them at their depressed value, will now sell them at par, though, prior to the panic, they were held at a handsome figure above. That the readiness of the Company to do this is not generally known, is shown by the fact that the stock still stands on the transfer-book in the Company’s name, offering to one in funds a rare chance for investment. For, the panic subsiding more and more every day, it will daily be seen how it originated; confidence will be more than restored; there will be a reaction; from the stock’s descent its rise will be higher than from no fall, the holders trusting themselves to fear no second fate.”

Having listened at first with curiosity, at last with interest, the merchant replied to the effect, that some time since, through friends concerned with it, he had heard of the company, and heard well of it, but was ignorant that there had latterly been fluctuations. He added that he was no speculator; that hitherto he had avoided having to do with stocks of any sort, but in the present case he really felt something like being tempted. “Pray,” in conclusion, “do you think that upon a pinch anything could be transacted on board here with the transfer-agent? Are you acquainted with him?” [34]

“Not personally. I but happened to hear that he was a passenger. For the rest, though it might be somewhat informal, the gentleman might not object to doing a little business on board. Along the Mississippi, you know, business is not so ceremonious as at the East.”

“True,” returned the merchant, and looked down a moment in thought, then, raising his head quickly, said, in a tone not so benign as his wonted one, “This would seem a rare chance, indeed; why, upon first hearing it, did you not snatch at it? I mean for yourself!”

“I?—would it had been possible!”

Not without some emotion was this said, and not without some embarrassment was the reply. “Ah, yes, I had forgotten.”

Upon this, the stranger regarded him with mild gravity, not a little disconcerting; the more so, as there was in it what seemed the aspect not alone of the superior, but, as it were, the rebuker; which sort of bearing, in a beneficiary towards his benefactor, looked strangely enough; none the less, that, somehow, it sat not altogether unbecomingly upon the beneficiary, being free from anything like the appearance of assumption, and mixed with a kind of painful conscientiousness, as though nothing but a proper sense of what he owed to himself swayed him. At length he spoke:

“To reproach a penniless man with remissness in not availing himself of an opportunity for pecuniary investment—but, no, no; it was forgetfulness; and this, charity will impute to some lingering effect of that unfortunate [35] brain-fever, which, as to occurrences dating yet further back, disturbed Mr. Roberts’s memory still more seriously.”

“As to that,” said the merchant, rallying, “I am not——”

“Pardon me, but you must admit, that just now, an unpleasant distrust, however vague, was yours. Ah, shallow as it is, yet, how subtle a thing is suspicion, which at times can invade the humanest of hearts and wisest of heads. But, enough. My object, sir, in calling your attention to this stock, is by way of acknowledgment of your goodness. I but seek to be grateful; if my information leads to nothing, you must remember the motive.”

He bowed, and finally retired, leaving Mr. Roberts not wholly without self-reproach, for having momentarily indulged injurious thoughts against one who, it was evident, was possessed of a self-respect which forbade his indulging them himself.

 

 

 

 

 

Melville, Herman. 1857. The Confidence-Man: His Masquerade. New York: Dix, Edwards.

PDF at:

https://archive.org/details/confidencemanhis00melvrich

Online text at:

https://www.gutenberg.org/ebooks/21816

https://www.gutenberg.org/files/21816/21816-h/21816-h.htm

 

URL of this post:

http://piratesandrevolutionaries.blogspot.com/2018/06/melville-4-confidence-man-his_25.html

 

 

.

22 Jun 2018

Priest (9.5) Introduction to Non-Classical Logic, ‘Tableaux for N4,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

 

 

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part I:

Propositional Logic

 

9.

Logics with Gaps, Gluts and Worlds

 

9.5

Tableaux for N4

 

 

 

 

Brief summary:

(9.5.1) The tableau rules for N4 are the same as for K4, except the rules for → will apply only at world 0.

 

 Double Negation

Development, True (¬¬D,+)

¬¬A,+i

A,+i

 

 Double Negation

Development, False (¬¬D,−)

¬¬A,−i

A,−i

 

Conjunction

Development, True (D,+)

A ∧ B,+i

A,+i

B,+i

 

Conjunction

Development, False (D,−)

A ∧ B,−i

↙   ↘

A,−i      B,−i

 

 Negated Conjunction

Development, True (¬D,+)

¬(A ∧ B),+i

¬A ¬B,+i

 

 Negated Conjunction

Development, False (¬D,−)

¬(A ∧ B),−i

¬A ¬B,−i

 

 Disjunction

Development, True (∨D,+)

A ∨ B,+i

↙   ↘

A,+i      B,+i

 

 Disjunction

Development, False (∨D,−)

  A B,-i

A,-i

B,-i

 

 Negated Disjunction

Development, True (¬D, +)

¬(A ∨ B),+i

¬A ¬B,+i

 

 Negated Disjunction

Development, False(¬D, -)

¬(A ∨ B),-i

¬A ¬B,-i

 

 Conditional

Development, True (→D,+)

A → B,+i

↙   ↘

A,-j      B,+j

.

j is every number that occurs on the branch (and this rule applies only to world 0)

 

 Conditional

Development, False (→D,−)

  A B,-i

A,+j

B,-j

.

j is a new number. (Here i will always be 0 and j will be 1)

 

 Negated Conditional

Development, True (¬D, +)

¬(A B),+i

A,+j

¬B,+j

.

j is a new number. (Here i will always be 0 and j will be 1)

 

 Negated Conditional

Development, False(¬D, -)

¬(A B),-i

↙     ↘

A,-j     ¬B,-j

.

j is every number that occurs on the branch (and this rule applies only to world 0. So i will always be 0)

(165, titles for the rules are my own additions. Note that the rules for double negation and disjunction are not in the text and are probably mistaken. Also, I am guessing about the conditionals, too.)

 

(9.5.2) Priest then gives an example of a formula that is valid in N4 but not in K4. (9.5.3) We construct counter-models from open branches in the following way. There is a world wi for each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A B, at non-normal worlds only, if p,+i or A B,+i occurs on the branch, set pρwi 1 or A Bρwi 1; if ¬p,+i or ¬(A B),+i occurs on the branch, set pρwi0 or A Bρwi o. There are no other facts about ρ. (9.5.4) “N4 is a sub-logic of K4, but not the other way around,” because all valid formulas of N4 are valid in K4, but not all valid formulas of K4  are valid in N4. (9.5.5) “The tableaux for N4 are sound and complete with respect to the semantics” (169).

 

 

 

 

 

Contents

 

9.5.1

[The Tableau Rules for N4]

 

9.5.2

[Tableau Example for N4]

 

9.5.3

[Counter-Models]

 

9.5.4

[N4 as a Proper Sub-Logic of K4,]

 

9.5.5

[The Soundness and Completeness of N4]

 

 

 

 

 

 

Summary

 

9.5.1

[The Tableau Rules for N4]

 

[The tableau rules for N4 are the same as for K4, except the rules for → will apply only at world 0.]

 

[Recall from section 9.2 that K4 is a possible worlds First Degree Entailment (and thus four value-situationed) system. In section 9.4 we noted that we would want our conditional to be able to express statements that suppose certain laws of logic be suspended and then say what might follow from that suspension. For this we appealed to non-normal worlds, which are worlds where the normal laws of logic might fail (see section 4). In section 9.4, Priest gave the semantic rules for such a system. They are the same rules for K4 except in non-normal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation (see section 9.4.8). This new non-normal worlds First Degree Entailment system is called N4. We now wonder how we construct tableaux in N4. Priest says we use the same rules as for K4, but the rules for → will apply only at world 0. As such, I will guess that the rules would be the following, but I am not certain this is correct.

 

 Double Negation

Development, True (¬¬D,+)

¬¬A,+i

A,+i

 

 Double Negation

Development, False (¬¬D,−)

¬¬A,−i

A,−i

 

Conjunction

Development, True (D,+)

A ∧ B,+i

A,+i

B,+i

 

Conjunction

Development, False (D,−)

A ∧ B,−i

↙   ↘

A,−i      B,−i

 

 Negated Conjunction

Development, True (¬D,+)

¬(A ∧ B),+i

¬A ¬B,+i

 

 Negated Conjunction

Development, False (¬D,−)

¬(A ∧ B),−i

¬A ¬B,−i

 

 Disjunction

Development, True (∨D,+)

A ∨ B,+i

↙   ↘

A,+i      B,+i

 

 Disjunction

Development, False (∨D,−)

  A B,-i

A,-i

B,-i

 

 Negated Disjunction

Development, True (¬D, +)

¬(A ∨ B),+i

¬A ¬B,+i

 

 Negated Disjunction

Development, False(¬D, -)

¬(A ∨ B),-i

¬A ¬B,-i

 

 Conditional

Development, True (→D,+)

A → B,+i

↙   ↘

A,-j      B,+j

.

j is every number that occurs on the branch (and this rule applies only to world 0)

 

 Conditional

Development, False (→D,−)

  A B,-i

A,+j

B,-j

.

j is a new number. (Here i will always be 0 and j will be 1)

 

 Negated Conditional

Development, True (¬D, +)

¬(A B),+i

A,+j

¬B,+j

.

j is a new number. (Here i will always be 0 and j will be 1)

 

 Negated Conditional

Development, False(¬D, -)

¬(A B),-i

↙     ↘

A,-j     ¬B,-j

.

j is every number that occurs on the branch (and this rule applies only to world 0. So i will always be 0)

(165, titles for the rules are my own additions. Note that the rules for double negation and disjunction are not in the text and are probably mistaken.)

Priest notes also that although the rules for → apply only at world 0, we will never need to assume that there is more than one normal world in a counter-model (but I am not sure how that works yet. It could be that 0 is always the normal world and any others will be the non-normal ones. See section 9.5.3 below.)]

Tableaux for N4 can be obtained by modifying those for K4. Specifically, the rules are exactly the same as those for K4, except that the rules for → apply at world 0 only. (It turns out that we never need to assume that there is more than one normal world in a counter-model.)

(168)

[contents]

 

 

 

 

9.5.2

[Tableau Example for N4]

 

[Priest then gives an example of a formula that is valid in N4 but not in K4.]

 

[Priest next gives an example of one that is valid in N4 but not in K4. Let us do both to compare them.

 

K4 ¬(p → p) → (q → q)

1.

.

2.

.

3.

.

4.

.

5.

.

¬(p → p) → (q → q),0–

¬(p → p),1+

(q → q),1

q,2+

q,2

×

P

.

1→–

.

1→–

.

3→–

.

3→–

(5×6)

valid

 

(not in the text, and probably mistaken)

 

Compare this with the formula in N4, given as quotation below.]

For example: ⊬ ¬(p p) → (qq):

 

N4 ¬(p → p) → (q → q)

1.

.

2.

.

3.

.

.

¬(p → p) → (q → q),0–

¬(p → p),1+

(q → q),1

 

P

.

1→–

.

1→–

(open)

invalid

(p.168, enumeration and step accounting are my own and are probably mistaken)

 

The tableau finishes there! (In K4 an application of the rule for untrue → to the last line would immediately close it.)

(168)

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9.5.3

[Counter-Models]

 

[We construct counter-models from open branches in the following way. There is a world wi for each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A B, at non-normal worlds only, if p,+i or A B,+i occurs on the branch, set pρwi 1 or A Bρwi 1; if ¬p,+i or ¬(A B),+i occurs on the branch, set pρwi0 or A Bρwi o. There are no other facts about ρ.]

 

[Recall from section 9.3.7 that in K4 we construct counter models from open branches in the following way: “There is a world wi for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρwi 1; if ¬p,+i occurs on the branch, set pρwi0. ρ relates no parameter to anything else” (p.166, section 9.3.7). Priest says we will construct a counter-model for N4 in the same way, except now only world 0 is normal, with the remainder being non-normal. Also, the stipulations above for the ρ relations hold for propositional parameters of all worlds and also for formulas of the form  A B at non-normal worlds. As we can see from our tableau above, there are no propositional parameters. So no ρ relation is given to them. However, we do have a conditional formula that is true, namely, ¬(p p) . That means we set p pρw10. (From this we can infer that the negation ¬(p p) is true at world 1. But since we have not stipulated a value for (qq), that means it is neither true nor false in world 1. That means that in ¬(p p) → (qq) the antecedent is true but the consequent is not true in world 1. And thus the whole formula is not true at world 0. For, recall the rule for the conditional:

A Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

A Bρw0 iff for some w′ ∈ W, Aρw1 and Bρw0

(p.164, section 9.2.4)

Since world 1 is one of the worlds, despite being non-normal, it determines the value of the formula in the normal world as false. I am not sure I have this right, so please see the quotation below.)]

We read off a counter-model from an open branch exactly as for K4 (9.3.7), except that the only normal world is w0 – all others are non-normal – and the recipe for determining ρ is applied to propositional parameters at all worlds, and to any formula of the form A B at non-normal worlds. Thus, in the tableau of the previous paragraph, W = {w0,w1}; N = {w0} and p pρw10, there being no other facts about ρ. Since ¬(p p) is true at w1, and q q is not true at w1, ¬(p p) → (qq) is not true at w0.

(168)

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9.5.4

[N4 as a Proper Sub-Logic of K4,]

 

[“N4 is a sub-logic of K4, but not the other way around,” because all valid formulas of N4 are valid in K4, but not all valid formulas of K4  are valid in N4.]

 

[Recall from section 8.4.13 that if all valid inferences of system A are valid in system B, then system A is a sub-logic of system B. And if in addition to that not all formulas of system B are valid in A, then system A is a proper sub-logic of system B. We saw above a formula in K4 that is not valid in N4. Thus “N4 is a sub-logic of K4, but not the other way around.”]

Since interpretations for K4 are special cases of interpretations for N4 (namely, when W N = φ), N4 is a sub-logic of K4, but not the other way around, as this example shows.

(168)

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9.5.5

[The Soundness and Completeness of N4]

 

[“The tableaux for N4 are sound and complete with respect to the semantics” (169).]

 

[Priest ends by noting that:]

The tableaux for N4 are sound and complete with respect to the semantics. This is proved in 9.8.8–9.8.9.

(169)

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From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

.