祝融说。

Two Sentences Used the Same Set of Confirmations

Tang Ke circled R01, R02, and R03 from the historical table. She said these items both indicated checklist received and were confirmed within the target window; among past confirmed requests, most had this kind of message, so now that R17 has received the message, judgment should follow this majority.

Gu Ning asked her to write out the population the majority refers to. Among the five past confirmations, three had the specified message—that is three-fifths; among the four past requests with the message, three confirmed—that is three-quarters. Both questions circle the same three items, yet the denominators are five items and four items respectively.

Tang Ke fabricated no objects and misremembered no triple joint occurrence. The difficulty is that she looked back for messages from a group whose outcomes were already known, then handed this proportion to R17, for which only the message—and not yet the outcome—is known. Reading a sentence backward does not guarantee that the computational scope turns around completely with it.

This chapter is still a continuous thought experiment of the fictional Chengwan, using the eight requests already verified in the previous chapter, adding no new client replies or main-line confirmations. What is to be settled here is not how much the message is finally worth, but first identifying which direction's proportion can answer the question now being posed.

Fixed Letters, Unfixed Question Direction

A retains receiving complete confirmation of the first version within the original window. B retains receiving, after the corresponding Thursday 10:00 and up to 11:20, an explicit checklist-received message on the agreed channel. Only full acceptance of scope, payment conditions, and reception period belongs to A; B is only a limited message status.

Among the past eight requests, those satisfying both A and B are R01, R02, and R03. Those satisfying A but not B are R05 and R06; the one satisfying B but not A is R04; those satisfying neither are R07 and R08. None of these correspondences changed because Tang Ke posed another question.

This means one may look back from the outcome group at the condition, or look from the condition group at the outcome. Both are permitted, provided each states clearly within which population it computes. The error lies not in looking back at outcomes, but in treating the retrospective answer as though the forward question had been answered.

The eight requests' material and conditional table reached Gu Ning only at Thursday 12:00. She now has two checkable descriptions, but she did not thereby know at 10:00 what kind of message R17 would receive. Reverse analysis also cannot jump past the information's time of acquisition to supply material to an earlier version.

Looking Back at Messages from the Five Confirmations

To ask how many of the confirmers satisfy B, one first enters the five A objects. R01, R02, and R03 have B; R05 and R06 do not; the proportion is three over five. R04 has B but does not belong to this five-item outcome group, so it does not enter this denominator.

According to the conditional definition of the same probability model, the direction is written:

$$ P(B\mid A)=\frac{P(A\cap B)}{P(A)} $$

This step requires the probability of A to be greater than zero. It connects with Chapter 7's definition; the difference is that A is placed to the right of the vertical bar, indicating that the scope is first restricted to where A occurs. For the formula's basis, see the Cornell University CS2800 lecture notes on conditional probability.

If the eight requests are still given equal weight to build a descriptive empirical distribution, joint occurrence occupies three-eighths and A occupies five-eighths; dividing the two yields three-fifths. No new probability model was created here—only the question posed within the same table was changed.

Three-fifths can be used to describe the early messages of these historical confirmed objects. It is not erroneous data and need not be banned. If what is being studied is precisely the communication trajectory of confirmed objects, this direction fits the question; if the object of prediction has only B, then three-fifths cannot directly serve as the target probability.

Examining Outcomes from the Four Messages

R17 currently has a B-type message. To ask how many of those with this kind of message satisfy A, the historical correspondence scope is the four items R01 through R04; of these the first three were confirmed and R04 did not satisfy the original-window target, so the proportion is three over four.

When writing P(A given B), the right of the vertical bar is B, and the denominator returns to B's scope. The three joint occurrences remain, but the object R04—having the message yet unconfirmed—must be retained alongside. It must not disappear from the question just because the first three items' story runs more smoothly.

The two directions can be preserved in one small comparison table, without collecting two separate independent samples.

The three joint occurrences in the table are not six objects. Placing the two proportions side by side also does not provide two mutually independent supports. They share the same four-cell correspondence; this must still be remembered when sources are handled later.

When the Reverse Proportion Happens to Coincide

Tang Ke asked: if the two directions are sometimes equal, why separate them so strictly? Within the same model, when the joint occurrence has positive probability and both denominators are positive, the two ratios are equal when the denominator shares are equal. The equality comes from a quantitative relation; it does not make the two questions one question.

For example, as an independent comparison: among ten items, A has four and B has four, with two jointly shared. Both directions are one-half. If later B gains two items while A remains four and the joint remains two, the directions become one-half and one-third respectively.

This comparison was not written into Chengwan's history. It shows that equality may depend only on current shares. One cannot, after one coincidence, declare that backward reading is always permitted; nor can one, because the numbers are unequal, assert that some table was computed wrongly. One should first look at the respective denominators.

Another kind of equality is when the two groups share no objects: both directions are zero when the denominators are positive. This likewise provides no rule that the words are interchangeable. A mathematical result coinciding and the conditions meaning the same thing are two different layers of checking.

A Necessary Relation Is Not Yet a Sufficient Relation

Consider a hypothetical comparison even more liable to tempt backward reading: in some fully preserved finite set, all confirmed objects previously obtained message B. Looking back from group A, then, the proportion of B is one. If quite a few unconfirmed objects also obtained B, looking from group B at A would not likewise be one.

People easily compress "all confirmations had this message" into "this message is a marker of confirmation." The former sentence does not state how many objects with the message there are in total, nor exclude the message appearing among the unconfirmed. Even if the former is completely true within that set, the latter may still exceed the material.

In terms of object relations: A's range being contained within B's range does not mean B's range is contained within A's. To assert the two are identical within the same set, one must check the objects on both sides, not merely verify that the outcome group omits no messages.

Conversely, if in a finite set all B objects are confirmed, one still cannot say all confirmers must have B. Confirmed objects without B, like R05 and R06, would break the universal claim in the other direction. This states only the logical relation; it does not rewrite the fact that R04 in this book's original table has B and is unconfirmed.

This is not asking readers to reject affirmative judgments forever. If the material genuinely supports a containment relation within some scope, it may be adopted as such; but it first belongs to the already-verified scope. Extending everything within a limited history into any future object requires a separate account of the grounds for transfer.

Whose Story the Majority of Confirmers Leaves Out

Tang Ke circled the three items because they simultaneously satisfy the specified window's message condition and the original window's confirmation condition. If only these three are narrated, listeners may fill in a fluent process themselves: first received, then continued, finally confirmed. Yet the historical table has not verified item by item the order of receipt; this process cannot be treated as an established fact. R04 satisfies the same message condition but lacks the same window outcome, which also prevents this story from replacing the conditional count.

R04 is not an outlier fetched in to make a rebuttal; it was already among the four with B. Reverse judgment must return it, because the current question's denominator includes it. Whether it is rare does not change whether it should be counted; the limitation it brings and its count can be stated together.

Likewise, R05 and R06 remind us that the absence of this kind of early-window message does not preclude later confirmation. They do not belong to the current B group, yet they belong among the objects one must look at when explaining whether B is a necessary marker of confirmation. Different objections require different scopes; one does not always spread all eight items out evenly.

Chapter 3 explained how material enters; later chapters will examine selection mechanisms. This chapter first retains a more basic step: even with the entrance and the data entirely unchanged, merely changing the question posed may require letting objects not originally circled re-enter the judgment.

How Population Shares Affect the Backward Inference

Knowing only how common B is among confirmed objects, one still does not know how many of the B objects will confirm. Because unconfirmed objects may also show B, and the two outcome classes differ in their counts within the total. Only by returning to the same table can one know how many objects each class contributes to the B group.

To unfold this limitation, two entirely hypothetical hundred-item tables are posited below. They are neither new Chengwan material nor any industry statistic. Both tables set B at one-half within the confirmed group and one-tenth within the unconfirmed group, changing only the confirmed objects' share of the total.

The first hundred-item table has ten confirmed and ninety unconfirmed. The confirmed group has five items with B; the unconfirmed group has nine with B; all with B total fourteen. The confirmation proportion among those with B is five over fourteen, not one-half.

The second hundred-item table has ninety confirmed and ten unconfirmed. The confirmed group has forty-five items with B; the unconfirmed group has one; all with B total forty-six. The confirmation proportion among those with B is forty-five over forty-six, still not one-half.

In both tables, looking back from the confirmed group at B's proportion did not change, nor did B's proportion within the unconfirmed group. The backward-inference results differ greatly, because the two groups' population shares differ. This chapter displays the relation with explicit counts; the next chapter analyzes the message's relative discriminating role, and Chapter 10 systematically develops updating.

These two hypothetical tables cannot be used to fabricate a range for R17. Their role is to show that without the population share, one direction's proportion cannot fix the other direction; it is not to say Chengwan's actual probability lies between these two results.

What Should Be Asked Behind a Percentage

If someone hands over only the sentence "half of the confirmers have this kind of message," Gu Ning should first know what question he is actually answering. If the task is to describe the confirmers, the sentence can stand; if the task is to predict those with the message, one must additionally ask how messages appear among the unconfirmed, and how much of the total each class occupies.

Adding only "the total sample is one hundred" is still insufficient. Both the first and second tables have one hundred items, with results tending oppositely toward different directions. Total sample size cannot replace grouping structure, and large numbers cannot help one direction invert directly.

If the material provides only one of the two classes, Gu Ning should preserve the gap and not fabricate that the other class entirely lacks this message. Unknown is not zero. Nor can the unreported class be deleted from the population of the actual question simply because the other party did not report it.

Follow-up questions do not always require obtaining the full true material. One can complete the correspondence within a limited source, or explicitly posit an assumption and give a conditional computation. The key is to separate the obtained quantities from the assumed ones, so users know what the result varies with.

No Message Also Has Two Directions

The same error also occurs in negative phrasings. To ask how many unconfirmed objects lack B: in the original table, of the three unconfirmed, R07 and R08 lack B—that is two-thirds. To ask how many of those without B are unconfirmed: among the four without B, the same two are unconfirmed—that is one-half.

The two directions still share those two items, with denominators of three and four respectively. Not seeing the message readily invites a pessimistic interpretation, but it does not make the proportion usable in reverse. R05 and R06, lacking early B yet confirming, are precisely the objects that cannot be deleted from the latter question's denominator.

Not receiving the specified message type is not yet equal to having no contact at all, still less to an explicit rejection. B's negative scope retains its original time and content restrictions and must not be enlarged, when relayed in a table, into "no reply at all"; A's negative likewise retains only the non-holding of the window event and does not become a necessary failure of eventual cooperation.

Gu Ning therefore checks four complete interrogative forms, not merely two affirmative ones. Target and condition each taken positively or negatively—how they combine depends on the current question; whether the tone is optimistic or pessimistic, one must return to the actual denominator.

Inferring the Condition Backward Is Not Reversing Time

Looking back from group A at B used already-adjudicable target-window outcomes to select historical objects. This is retrospective description, permitted to consult past outcomes already obtained. It neither indicates that A occurred before B in time nor proves B preceded A; the actual order still requires item-by-item re-checking. The direction of conditioning itself proves neither that confirmation caused the message nor that the message caused confirmation.

Looking from group B at A is closer to the current prediction's direction of question, but the historical proportion itself was still compiled after the outcomes were known. Whether it can be used for R17 requires verifying that B was obtainable at the prediction time point and that the objects correspond closely enough; one cannot call a prospective test complete merely because the sentence was written in temporal order.

Causal direction, temporal order, and conditional-probability direction do not substitute for one another. An earlier condition can be examined backward given a later outcome; a later outcome can be studied given an earlier condition; the notation indicates only the scope and does not adjudicate the actual mechanism.

This point protects this volume's task. We first make the judgment's question clear, and later examine explanatory conditions; we neither treat retrospective stories of similarity directly as causal discoveries nor deny all descriptive association for lack of causal proof.

Restoring a Backward-Read Sentence into Questions

Gu Ning can require that every relay supply two complete sentences: what the material originally said—among whom, how much of what—and what is now to be judged—among whom, how much of what. If the two sentences' denominators are the same, check the numerators; if the denominators differ, return to the corresponding table rather than adopt directly on the strength of a shared keyword.

For example: "of the five confirmed, three have B"—the "whom" in the original sentence is the five confirmers. "Will R17, which now has B, confirm"—the "whom" corresponds to the historical four message-holders. The majority in the original sentence carries no R04; when turning to the current question, this object must be added back.

If only the original sentence exists, without the four-cell correspondence, Gu Ning need not immediately reject the entire material. She can first place it in the suitable direction, use it as limited description, and then state which direction's counts the current prediction still lacks. The material's use can be narrowed; one need not choose between believing all and discarding all.

The verification result should also be fed back to the relayer. When Tang Ke says "majority" in the future, she can simultaneously state which group the majority belongs to, rather than requiring listeners to guess. Clearer language adds no external facts, yet makes the same material harder to exceed its scope in transmission.

Complementary Numbers Must Stay Within the Same Denominator

Tang Ke might make another seemingly convenient conversion: three-fifths of confirmers have B, so the remaining two-fifths are those with B who do not confirm. The problem is that "the remainder" stays within the five confirmers, referring to R05 and R06 lacking B—not at all to objects like R04 that have B yet are unconfirmed.

Within the same given scope, the two proportions can sum to one only when the target is changed to its negation. Among confirmers, those with B are three-fifths and those without B are two-fifths; among those with B, confirmed is three-quarters and unconfirmed is one-quarter. The two pairs each close on themselves and cannot be spliced across.

This makes the check concrete. Upon seeing "how much is left," first ask which batch of objects "the remainder" is; do not assume the meaning is correct just because the arithmetic of one minus some number is correct. The numbers may complete the one while the semantics still exceed the denominator; only by holding to the given condition can one know who the filled-out portion is.

Complementary numbers also do not automatically yield an unknown proportion. If the original material states only the messages among confirmers and has not obtained the message group's total, subtraction cannot substitute for the other group's missing material. Preserving the unknown this way is more useful than filling in a computable but wrong number in the other direction.

Can Two Summaries Be Spliced into a Four-Cell Table

Suppose Tang Ke brings a summary saying that of five confirmations, three have the checklist-received message, and Ye Cheng brings another saying that of four message-holders, three confirmed. Even if the numbers happen to match the original table, one cannot conclude from identical numerators alone that they already correspond.

One must check whether the two summaries come from the same eight requests, whether the messages share the same cutoff, whether the confirmations accept the same first-version condition, and whether the three objects' identification numbers are the same. If one counts any acknowledgment at any time as B and the other counts only the early window, both sentences may hold within their own scopes, yet they cannot be poured directly into this chapter's same four-cell table.

If one summary comes from the eight requests and the other from some larger register, the three joint occurrences might be mere numerical coincidence. When totals lack provenance, one can assume neither overlap nor independence. Return to the objects first, then join the counts—avoid splicing two complete summaries into one complete body of data that does not exist.

After the correspondence passes, the row and column totals can be checked: the three joint occurrences belong respectively to five A's and four B's; the remaining two are A without B; one is B without A; of the eight, two remaining satisfy neither. If totals go negative or exceed the total, first check the definitions, duplicates, or records, rather than find explanations for a wrong table's percentages.

This restoration is, for this chapter's original table, only a re-check that adds no new facts. It offers the recipient another checking method: the direction of conditioning relies not only on remembering the vertical bar but on whether the various objects can be mutually consistent within the same population. Consistency is a necessary check, and it does not by itself prove the raw material true.

Taking "Very Common" Apart into Checkable Statements

A sentence like "confirmers often reply this way," with no proportion, no time point, and no object scope, may be only an impression left by a person. Gu Ning can first ask whether it comes from a complete compilation or from a few remembered cases; she cannot, for convenience of computation, fill "often" in by herself as three-fifths.

If the other party can point to only two concrete examples, one can preserve the examples' common process, but not claim to have obtained the confirmers' denominator. Only if the full five can be re-checked does one compute from the correspondence obtained. Memory is not necessarily false; it simply has not yet supplied the coverage statement this count requires.

Likewise, "rarely does one have this message yet not confirm"—the scope given by the original sentence may be the message-holders, or it may mean that within the entire register the two jointly occur rarely. Rare joint occurrence in the latter scope does not exclude a large share within an already small message group. One must ask "rare relative to whom" before deciding what the sentence supports.

Such clarification does not abolish a person's experience. It lets experience move stepwise from tellable examples into countable correspondences, retaining the steps not yet completed. A relayer who temporarily cannot give a denominator can keep verifying, or can remain at limited description; both are more faithful to the material than forcibly converting a vague tone into a probability.

Readers need not turn every daily conversation into an interrogation. If a sentence serves only casual talk, its limited expression can be understood; once it begins to support concrete predictions and will be handed to others for use, scope, direction, and quantity become contents that must be clarified. The judgment's use raises the requirements on explanation; familiarity does not substitute for basis.

Even once a proportion is obtained, the original counts should be retained when displaying it. Calling five-fourteenths "about thirty-something percent" suits some brief communication, but should not discard the correspondence of five items and fourteen items; when later reverse verification is needed, an approximate tone cannot restore all the cells. Nor can both directions be loosely written as "majority" as though the difference no longer mattered. Display precision may be chosen by use; the original record retains information sufficient to return to computation, preventing a rounded, similar-looking appearance from replacing the scope check.

What the Reverse-Condition Comparison Leaves Behind

At 12:10 on Thursday, Gu Ning saves the two-direction comparison beside the 12:00 historical table. The original eight objects and the first-version window remain; B still denotes only obtaining the checklist-received message within the specified window; R17's reception time awaits verification, with no new full-acceptance material added.

The comparison states explicitly that the numerator three is the same while the denominators are five and four; three-fifths describes messages within the confirmed group, three-quarters describes confirmations within the message group. The hypothetical hundred-item comparisons are saved separately, not mixed into the original register counts, nor serving as upper or lower bounds of the current estimate.

RC understands cognition as a representation that can continue to be generated under limited horizons. The limitation here refers not only to small sample size but also to which scope a sentence compresses away. Compressing the experience of the outcome group into the fate of all message-holders makes the representation omit the other part of objects the current question requires.

The direction of revision is not to abolish compression but to let compression return to scope and correspondence. Tang Ke may still narrate the three items' common process, but Gu Ning simultaneously preserves the position of R04 and of the confirmed objects without B; if the consensus is willing to accept this check, it can adjust its expression rather than permanently fix a familiar sentence as the answer.

What this chapter settles is the scope error in backward inference. R17 still has no final numerical prediction, and three-quarters remains only a candidate historical reference. The next chapter returns to the two groups of five confirmations and three non-holdings, comparing B's rates of appearance across the two classes: whether a message has discriminating power cannot be decided merely by its being common within one class.