Chapter 3: Discrete, Indeterminate, and Portional Claims

Chapter 2 showed how to reduce a simple claim to a subject and a predicate and then assign labels. This chapter adds another question: How many—or which—subjects does the claim include?

Compare:

“Daniel prayed.”
“At least one prisoner prayed.”
“Some prisoners prayed, and some did not.”

The first claim concerns one specific person. The second says that at least one person in a group prayed, without identifying who. The third divides the group: at least one prayed and at least one did not.

To quantify a claim is to indicate how many or which subjects it covers. A formula that does this is a quantified formula. Logic calls the reach of a claim across its subjects its subject scope. This chapter introduces three forms of subject scope that explicitly establish one or more subjects as present in the situation. Chapter 4 will examine two universal forms, which constrain a category without establishing that any member is present.

Main points in this chapter:

• A discrete claim concerns one fixed subject.
• An indeterminate claim uses + and means at least one, and possibly all.
• A portional claim uses % and means some but not all.
• In a discrete formula, the first label identifies one subject. In a quantified formula, both labels represent qualities or categories.
• Handles such as x′ and y′ help us show whether several qualities belong to the same subject or may belong to different subjects.
• A handled assignment does not introduce a subject. A designator or quantity operator supplies the required presence commitment.
• A claim may concern Reality or another specified universe, such as a story or hypothesis. Its quantity ranges over subjects present in the situation under consideration.

A Reminder About Situations

Chapter 1 defined a situation as a coherent, relevant set of facts that could obtain within a specified universe.

The universe may be Reality or another universe supplied by a story, parable, game, or hypothesis. The situation is limited to the subjects, qualities, events, times, and relationships introduced by—or required for understanding—the claims under consideration.

The factual situation is the coherent set of relevant facts that obtains within the specified universe. An asserted situation is what a claim or set of claims presents as being the case. When the claims leave several coherent variants open, the asserted situation includes several asserted possible situations concerning the same bounded factors.

For example, an inspection argument may concern only the crawlspace, the foundation, and the presence or absence of water. One asserted possible situation may include a dry crawlspace. Another may include water entering beneath the foundation. Unrelated facts elsewhere in Reality (e.g., the result of the recent fishing trip) are not part of either situation.

A subject may also be present in the factual situation of a story or in a stipulated hypothetical situation without existing in Reality. The wounded traveler is present in the universe of the Good Samaritan parable. A cracked beam may be present in a hypothetical inspection scenario. Neither claim, by itself, says that the traveler or beam exists in Reality.

The symbol @ records that a subject is present in the situation under evaluation. In this chapter, we will usually use the plainer phrase present in the situation, or just present.

From Individuals to Categories

A claim may concern one particular subject: “Ruth followed Naomi.” It may also concern members of a category:

“At least one widow needed assistance.”
“Some registered voters received ballots, and some did not.”

A category is a quality used to group subjects. The category widow includes subjects who have the quality of being widows. The category registered voter in the room includes subjects who have that status at the relevant time and place.

This is one sense in which quantified reasoning is abstract. We set aside the names of the individual members and reason about whoever fits the description.

Suppose:

We can ask whether at least one subject has both w and a, or whether some w have a while others do not. We do not need to list each widow by name or worry about other details.

The Five Basic Forms: A Roadmap

Philalethic Logic uses five common forms for basic subject scope:

Discrete: one fixed subject

Indeterminate: at least one unspecified subject, shown by +

Portional: some but not all, shown by %

Extensional universal: every member present in the situation has the predicate

Relational universal: having the first quality necessarily requires the second

Their compact patterns are:

Do not worry if the final two patterns are not yet familiar. They are included here only as a roadmap; Chapter 4 will explain the two universal forms in detail.

This chapter develops the first three forms. Chapter 4 develops the two universal forms and explains how their direction and existence commitments differ.

The letters in these clause patterns are placeholders for clause components. A real symbol key may use any upper or lowercase letters, although lowercase is preferred for ordinary subjects and predicates. As always, case is part of the chosen label: h and H are distinct labels.

The underscore in _t marks the complementary predicate: lacking t, or being non-t in the relevant respect. Chapter 5 will explain component negation more fully. For now, read _t as “lacks t.”

Keeping Clause Roles Visually Distinct

Chapter 2 used the simple pattern sp. In that pattern, s identifies one particular subject and p is the predicate asserted of that subject.

For example:

d = Daniel

	p = prayed
dp

d functions like a name. The technical term for such a label is a designator: it designates, or picks out, one subject.

Different generic letters help mark the different role in quantified formulas. In +ht and %ht, both h and t represent qualities or categories. For example:

In this case, +ht means that at least one subject has both qualities: being a student and having passed the examination. h does not name one particular student.

The construction tells us what job the label is doing:

• In discrete sp, s is a designator for one subject.
• In quantified +ht and %ht, h and t are predicates or category labels.

To make the difference easier to see, this chapter will often use s for a discrete subject-name and h and t for category predicates:

sp: one designated s has p
+ht: at least one subject has h and
t
%ht: at least one h has t, and at
least one h lacks t

Within one symbol key, do not use the same label both as a name for one person and as a category unless you clearly distinguish the two uses.

Discrete Claims: One Fixed Subject

A discrete claim concerns one fixed person, thing, event, activity, or group treated as one subject. It uses no quantity operator.

Suppose:

m = Moses

	cp = confronted
Pharaoh
mcp

The formula concerns the one person designated by m. The letters cp remain together as one predicate meaning “confronted Pharaoh.”

A fixed subject need not be known by name. Consider “The masked intruder entered through the window.” The speaker may not know who the intruder was, but the claim still concerns one specific intruder—the one involved in that event. A description can therefore pick out a fixed subject when the context makes the intended subject clear.

Compare that with: “At least one intruder entered through the window.” That claim does not pick out one fixed person. It says only that someone fitting the description did so. It therefore requires the indeterminate form introduced later.

A Group as One Subject

A group may sometimes be treated as one organized subject:

“The jury reached a verdict.”
“The council approved the permit.”
“The team won the championship.”

These claims do not directly say that every juror reached a separate verdict, every council member approved the permit, or every player individually won a championship. The group is treated as one subject acting as a group.

Looking Under the Hood

The compact discrete formula np has a more detailed expansion:

You do not need to memorize this expansion yet. It simply makes three ordinary commitments visible:

$n: the subject tracked by x′ is
the one designated by n
@: that subject is present in the
situation being considered
p: that subject has the predicate p

The symbol x′ is a handle. It is a bookkeeping symbol that lets the formula keep referring to the same subject while several things are said about it.

Suppose n means “the current mayor” and p means “signed the ordinance.” np is false if no one fills that role in the situation being discussed. It is also false if a mayor is present but did not sign the ordinance.

Indeterminate Claims: At Least One

An indeterminate claim says that at least one subject fits a description, without identifying which subject.

An operator is a symbol that governs a formula or part of one and tells us how to interpret it. The symbol + is the indeterminate operator. To be more precise, it is a quantifier—it indicates how to understand the subject-quantity. It means “there is at least one” or “there exists at least one.” A complete claim headed by + is therefore called an indeterminate claim.

The name indeterminate does not mean that the claim’s truth is uncertain. The claim is determinately true or false. What remains indeterminate is which subject satisfies it and how many subjects satisfy it beyond the required minimum of at least one. In standard first-order logic, the same form is commonly described as existentially quantified: it asserts that at least one subject satisfies the description. Philalethic Logic uses indeterminate as the primary name of the claim-form, while retaining existential when discussing its at-least-one commitment and the way + keeps one handle referring to the same required subject throughout its scope.

Suppose:

+cp means at least one subject is both a customer and someone who paid the invoice. The claim does not say which customer paid. It also does not say how many paid beyond at least one. Its fuller handled formula is:

Read the brackets as a package or grouping of qualities assigned to one subject. The + reaches over that entire package. Therefore, one and the same subject must be both c and p. We say that x′ is existentially bound by +: the formula requires at least one subject, and every occurrence of x′ within the scope of that + refers to that same required subject.

If we add another quality:

we might mean that at least one subject is a customer, paid the invoice, and signed the agreement. All three qualities belong to the same subject tracked by x′.

What + Does—and Does Not—Say

+ht says that at least one subject present in the situation under consideration has both h and t. It does not say:

• exactly one subject has both qualities;
• only a few have them;
• most have them;
• every h has t;
• h causes t (or t causes h);
• h necessarily requires t.

The ordinary word some often means at least one and possibly all. In that weak sense, “Some students passed” is still true if every student passed.

In conversation, however, some sometimes suggests some but not all. If a parent says, “Some of the cookies are gone,” listeners may assume that some remain. The + operator does not include that suggestion. The % form expresses “some but not all” directly.

When + Is False

+ht is false when no subject present in the situation under consideration has both h and t. That can happen in two ways:

• no present subject has h;
• some subjects have h, but none of them has t.

For example:

+dp is false if no delivery driver appears in the relevant situation. It is also false if drivers are present but all arrived after noon.

Handles: Keeping Track of the Same Subject

A handle is an internal symbol used to keep track of one subject while several qualities are assigned to it. Every handle is written as a primed lowercase letter. The letters x′, y′, and z′ are common conventions, but any lowercase letter may be used.

A handle works somewhat like a variable in algebra, except that its job is reference rather than calculation. It says, in effect, “Keep talking about this same subject.”

Compare:

and:

The first formula requires one subject that has both a and b. The second requires at least one a and at least one b. The a-subject and b-subject may be the same, but the formula does not require that they be the same.

Consider two ordinary claims: “At least one applicant submitted the form and paid the fee,” and “At least one applicant submitted the form, and at least one applicant paid the fee.” The first could be symbolized as:

This requires one applicant who did both.

The second could be symbolized as:

Here the submitting applicant and the paying applicant may be different people.

A Handled Assignment Does Not Introduce a Subject

The expression x′[a] says only that the subject already being tracked by x′ has a. It does not introduce that subject or by itself assert that the tracked subject is present in the situation.

The + in +x′[a] supplies the “there is at least one” commitment. In a discrete claim, the designator fixes the handle to one identified subject and the discrete form includes situation-relative presence.

The prime identifies the letter as a handle; it does not mean that the tracked subject is fresh, present, designated, or existentially introduced. Those facts are shown by the surrounding formula, assignments, and rule. A handle is normally followed by brackets containing the assigned qualities. When only one simple quality is assigned, the brackets may be omitted: x′p means x′[p].

What Counts as Present?

Quantity ranges over the subjects present in the situation under consideration. Presence is the preferred technical term here. In an ordinary claim directed at Reality, this presence commitment ordinarily carries existential significance; in a story, hypothesis, or merely possible situation, it records presence within that represented situation without asserting Reality-level existence.

In an ordinary claim intended to describe Reality like “At least one customer is waiting in the lobby” (+cl), + indicates the presence of one or more people in the relevant lobby situation in Reality.

In a story, “At least one traveler was wounded” (+tw), the + indicates the presence of at least one relevant subject in the story’s universe.

In a hypothesis, “Suppose at least one support beam is cracked” (+sc), the + indicates the presence of at least one subject in the hypothetical situation.

The technical name for the collection of subjects present in a situation is its subject-domain. The term is useful later, but the basic idea is simple: quantity applies only to the subjects counted as present in the situation under consideration.

The symbol @ records this kind of presence when it must be shown explicitly. Because + already means that at least one present subject fits the description, we normally do not write @ inside an ordinary indeterminate formula.

Portional Claims: Some but Not All

A portional claim says that a category is divided with respect to a predicate. At least one member has the predicate, and at least one member lacks it. The symbol % means “some but not all” in this precise qualitative sense.

Suppose:

%sp means: At least one student passed, and at least one student did not pass. Its fuller handled formula is:

The first half requires a student who passed. The second half requires a student who lacked the quality of passing.

When an indeterminate or portional claim requires an otherwise unspecified present subject, later truth-tree analysis tracks that subject as an instance. The handle is the symbol used to keep track of that instance. Chapter 12 explains how instances are introduced and managed on a tree.

Why Two Handles?

The first half uses x′ and the second uses y′ because the positive and negative instances must be allowed to be different subjects.

When p and _p contradict one another at the same time and in the same respect, one subject cannot satisfy both halves. A true %sp therefore requires at least two students: one who passed and another who did not. For example, “Some witnesses identified the driver, and some did not,” requires at least one witness who identified the driver and at least one different witness who did not.

The Percent Sign Is Not a Percentage

The % symbol does not tell us what percentage passed. It does not state a fraction, majority, or comparison. %sp is compatible with all of these:

• one student passed and ninety-nine failed;
• ninety-nine passed and one failed;
• half passed;
• most passed;
• only a few passed.

It says only that there is at least one student on each side of the division.

Words such as few, many, most, nearly all, and scarcely any add numerical suggestions that % does not preserve. “Most council members approved” may support the weaker conclusion that some approved and some did not, but % alone does not preserve the word most.

When a Portional Claim Is False

%sp is false when either half fails:

• no student passed;
• no student failed to pass.

It is also false if there are no students in the situation.

A one-member category cannot satisfy a genuine portional claim when p and _p contradict each other. Its sole member cannot both have and lack p at the same time and in the same respect.

In Summary

A discrete claim concerns one fixed subject. In sp, s designates the subject and p is the predicate. The fuller expansion x′[$s & @ & p] shows designation, presence in the situation, and predication.

An indeterminate claim uses +. +ht means that at least one subject present in the situation has both h and t. It means at least one and possibly all.

Handles such as x′ and y′ keep track of whether several qualities belong to the same subject. +x′[h & t] requires one subject with both qualities. +x′[h] & +y′[t] allows different subjects.

A handled assignment does not introduce a subject. The quantity operator + introduces an unspecified present subject; a discrete designator fixes the handle to one identified subject.

A portional claim uses %. %ht means that at least one h has t and at least one h lacks t. It normally requires two different instances and says nothing about percentages or majorities.

In discrete sp, s designates one subject. In quantified +ht and %ht, both h and t represent qualities or categories.

A claim’s quantity ranges over subjects present in the situation as it is being considered—whether the claim concerns Reality, a story, or a hypothesis.

Chapter 4 completes the basic forms by introducing extensional and relational universals, neither of which by itself asserts that a member of its subject category is present.

Discussion Questions

1. What question does subject scope answer?
2. What does it mean for a subject to be present in a fictional or hypothetical situation without existing in Reality?
3. Why do generic discrete and quantified formulas use different letter pairs, as in sp and +ht?
4. How can a description designate one fixed subject even when the person’s name is unknown?
5. Why does +ht mean “at least one and possibly all”?
6. What is the difference between +x′[a & b] and +x′[a] & +y′[b]?
7. Why does x′[a] alone neither introduce a subject nor assert that the tracked subject is present?
8. What exactly does %ht assert?
9. Why does a true portional claim normally require two different subjects?
10. What numerical information does %ht deliberately leave unsettled?

Informal Fallacy: Hasty Generalization

Consider this argument. Two inspectors from one company miss defects in two houses. A dissatisfied buyer concludes, “Home inspectors never find anything important.” The experiences may justify concern about those inspections or that company, but they do not yet justify a universal claim about an entire profession.

A hasty generalization draws a broad conclusion from evidence that is too limited, unrepresentative, or poorly selected. It often begins with one case or a few cases and quietly ends with a claim about all, most, or the typical member of a category.

The quantifier shift is the central warning sign. “At least one contractor arrived late” establishes an indeterminate claim. “Some contractors arrive late and some do not” establishes a portional division. Neither claim by itself establishes “All contractors arrive late” or even “Contractors usually arrive late.”

Generalization is not always fallacious. We often reason responsibly from samples, repeated observations, and well-designed studies. The question is whether the evidence is sufficiently broad and representative for the strength of the conclusion. The stronger the quantifier, the stronger the evidential burden.

Ask two questions. What quantity does the evidence actually establish? What justifies moving from that quantity to the conclusion’s broader one?

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Exercise: Identify the Form

Classify each claim as discrete, indeterminate, or portional. Explain briefly.

1. “Ruth remained with Naomi.”
2. “At least one customer paid in cash.”
3. “Some houses on the street have basements, and some do not.”
4. “The committee rejected the proposal.”
5. “Some witnesses remembered the license plate.”
6. “Some disciples understood the parable, and some did not.”
7. “The masked intruder entered through the window.”
8. “At least one appliance in the kitchen is unplugged.”
9. “Some council members approved the permit, and some did not.”

Exercise: Build a Symbol Key

Choose clear labels and symbolize each claim. State whether the first label is a designator or a category predicate.

1. “Daniel prayed.”
2. “At least one nurse arrived before midnight.”
3. “Some voters approved the measure, and some did not.”
4. “The jury reached a verdict.”
5. “At least one apostle was imprisoned.”
6. “Some students completed the assignment, and some did not.”
7. “The current mayor signed the ordinance.”
8. “At least one applicant submitted the form and paid the fee.”

Exercise: Same Subject or Possibly Different Subjects?

For each formula, explain whether one subject must have all the listed qualities or whether different subjects may be involved.

1. +x′[a & b]
2. +x′[a] & +y′[b]
3. +x′[c & p & s]
4. +x′[c & p] & +y′[c & s]
5. %ht
6. +x′[a & b] & +y′[a & _b]

Exercise: What Does the Formula Leave Open?

For each formula, state what it asserts and give two things it does not settle.

1. +ht
2. +x′[a & b & c]
3. %ht
4. sp
5. +x′[a] & +y′[b]

Exercise: Presence and Universe

For each claim, identify the relevant universe and situation. Explain what sort of presence is asserted and whether Reality-level existence is asserted.

1. “At least one customer is waiting in the lobby.”
2. “In the parable, at least one traveler was wounded.”
3. “Suppose at least one support beam is cracked.”
4. “Some dragons guarded the mountain, and some did not.”
5. “The masked intruder entered through the window.”

Exercise: Write Your Own Examples

Write and symbolize:

• one discrete claim about a person, group, event, or activity;
• one indeterminate claim in which a single subject must have three qualities;
• one pair of indeterminate claims in which the two instances may differ;
• one portional claim;
• one misleading ordinary use of “some” whose intended meaning must be clarified.